tag:blogger.com,1999:blog-35685556662811777192024-03-28T00:20:19.584-07:00Excel Master Series BlogLearn How To Do Advanced Statistical Techniques and Solver Optimization in Excel. Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.comBlogger199125tag:blogger.com,1999:blog-3568555666281177719.post-9572665768938481802015-05-04T11:02:00.001-07:002015-05-05T11:06:16.663-07:00How To Create a Completely Automated Shapiro-Wilk Normality Test in Excel in 8 Steps<h1>How To Create a <br /> <br />Completely Automated <br /> <br />Shapiro-Wilk Normality <br /> <br />Test in Excel in 8 Steps</h1> <p>The purpose of this blog article is to show how to create a completely automated Shapiro-Wilk test for normality in Excel. The test is completely automated because the user has only to enter the raw, unsorted data sample to be tested for normality along with the alpha level desired for the test. Upon insertion of the sample data and alpha, the entire test will be automatically run and the test’s output will be immediately returned. </p> <p>As with any hypothesis test, the output states whether to reject or not reject the Null Hypothesis and the specified alpha level. In this case, the Null Hypothesis states that the data sample came from a normally distributed population. This Null Hypothesis is rejected only if Test Statistic W is less than the critical value of W for a given alpha level and sample size.</p> <p align="center"> </p> <h2>Overview of the Shapiro-Wilk Test For Normality</h2> <p>The Shapiro-Wilk Test is a hypothesis test that is widely used to determine whether a data sample is normally distributed. A test statistic W is calculated. If this test statistic is less than a critical value of W for a given level of significance (alpha) and sample size, the Null Hypothesis which states that the sample comes from a normally distributed population is rejected. Keep in mind that passing a hypothesis test for normality only allows one to state that no significant departure from normality was found.</p> <p>The Shapiro-Wilk Test is a robust normality test and is widely-used because of its slightly superior performance against other normality tests, especially with small sample sizes. Superior performance means that it correctly rejects the Null Hypothesis that the data are not normally distributed a slightly higher percentage of times than most other normality tests, particularly at small sample sizes.</p> <p>The Shapiro-Wilk normality test is generally regarded as being slightly more powerful than the Anderson-Darling normality test, which in turn is regarded as being slightly more powerful than the Kolmogorov-Smirnov normality test. The Shapiro-Wilk normality test is affected by tied data values but less so than the Anderson-Darling normality test.</p> <p>An abbreviated description of the steps of the Shapiro-Wilk normality test is as follows:</p> <ul> <li> <p>Sort the raw data in an ascending sort</p> </li> <li> <p>Arrange the sorted data into pairs of successive highest and lowest data values</p> </li> <li> <p>Calculate the difference between the high and low value of each pair </p> </li> <li> <p>Multiply each difference by an “a Value” from a lookup table</p> </li> <li> <p>Calculate test statistic W</p> </li> <li> <p>Compare test statistic W with a W Critical Value obtained from a lookup table</p> </li> <li> <p>Reject the Null Hypothesis stating normality only if W is smaller than W Critical</p> </li> </ul> <p>The complete Shapiro-Wilk test of normality appears in Excel as follows:</p> <p><a href="http://lh3.googleusercontent.com/-mFeKaD_nI7s/VUex7m_ulqI/AAAAAAAA63s/zNS5EVJKP7M/s1600-h/Shapiro_1_Complete_Test_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Complete Shapiro-Wilk Test in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Complete Shapiro-Wilk Test in Excel" src="http://lh3.googleusercontent.com/-7TWDhRUY_hk/VUex-SCMs3I/AAAAAAAA630/ac4Ofl28f8o/Shapiro_1_Complete_Test_600_thumb2.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 1 – Sort Data in an Ascending Sort</h2> <p align="center"> </p> <h3>Enter the Raw Data and Alpha Level</h3> <p>First, paste in the raw, unsorted data and set the alpha level in the yellow cells as follows:</p> <p><a href="http://lh3.googleusercontent.com/-ZPvE5J3CBlM/VUeyBnKSnkI/AAAAAAAA638/J__auY4K4vE/s1600-h/Shapiro_2_Raw_Data_and_Alpha_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Inserting Raw Data and Alpha" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Inserting Raw Data and Alpha" src="http://lh3.googleusercontent.com/-r02PXMTJ7fs/VUeyFCagbeI/AAAAAAAA64E/gvFVXsSn6WE/Shapiro_2_Raw_Data_and_Alpha_600_thu.jpg?imgmax=800" width="404" height="380" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>This Shapiro-Wilk normality test in Excel has the capability to handle up to 20 data points. This data sample contains only 15 data points but the Excel worksheet below indicates the capability for an additional 5 data points in the empty yellow cells. Expanding the capability of this Excel test to handle a larger number of data points would be a straightforward matter of making adjustments to the Excel formulas. </p> <p>The alpha level should be set at 0.01, 0.05, or 0.10 because critical values of W are provided here only for those common alpha levels. If alpha is set at a different level, the following warning appears as a result of the If-Then-Else statement seen in the previous image:</p> <p><a href="http://lh3.googleusercontent.com/-Hs4xNbuR_bI/VUeyHZbYarI/AAAAAAAA64M/iG7xzDpjWDg/s1600-h/Shapiro_3_Alpha_MIstake_5004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Incorrect Alpha" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Incorrect Alpha" src="http://lh3.googleusercontent.com/-rKaHjAM2Pig/VUeyIxfcvmI/AAAAAAAA64U/3EP2c__LsCU/Shapiro_3_Alpha_MIstake_500_thumb2.jpg?imgmax=800" width="404" height="75" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h3>Sort the Raw Data Using Formulas, Not the Excel Sorting Tool</h3> <p>The most efficient way to sort data is with the use of formulas as shown. The Excel sorting tool must be manually re-run whenever the raw data is changed in any way. The following formula will automatically resort the data if any data are changed or added. The following formulas produce an ascending sort. To create a descending sort, simply swap the word <i>LARGE</i> for <i>SMALL</i> within the formula.</p> <p><a href="http://lh3.googleusercontent.com/-RYYnTDjCheQ/VUeyLe-db-I/AAAAAAAA64c/D9OIrGNnTcU/s1600-h/Shapiro_4_Sorting_Data_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Sorting Data With Formulas" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Sorting Data With Formulas" src="http://lh3.googleusercontent.com/-1MBBuALHnFs/VUeyMuUNXFI/AAAAAAAA64k/X2bqbgMTz0U/Shapiro_4_Sorting_Data_600_thumb2.jpg?imgmax=800" width="404" height="384" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 2 – Obtain “a Values” From the “a Value Lookup Table”</h2> <p align="center"> </p> <h3>The Table of a Values</h3> <p>The a Values are constants that are calculated from the means, variances, and covariances of the order statistics of a sample of size n from a normal distribution. The a Value table is shown as follows for data samples varying in size from n = 2 up to n = 50. </p> <p><a href="http://lh3.googleusercontent.com/-CN5wS09zGsc/VUeyN4bMFII/AAAAAAAA64s/ED3I0_ldZYM/s1600-h/Shapiro_7_a_table_1_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - a Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - a Table" src="http://lh3.googleusercontent.com/-wIAb4lPpWr0/VUeyPA60URI/AAAAAAAA640/csKM8GcFmKQ/Shapiro_7_a_table_1_600_thumb2.jpg?imgmax=800" width="404" height="269" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.googleusercontent.com/-vferf7o465w/VUeyQ0D4coI/AAAAAAAA648/EX0JDwLIw1A/s1600-h/Shapiro_8_a_table_2_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - a Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - a Table0" src="http://lh3.googleusercontent.com/-F7rtB3jn7KQ/VUeySHbedXI/AAAAAAAA65E/K_tdn0GobxU/Shapiro_8_a_table_2_600_thumb2.jpg?imgmax=800" width="404" height="269" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.googleusercontent.com/-ATbedeBvC0Q/VUeyTRWfnyI/AAAAAAAA65M/CzKQi7aNZtM/s1600-h/Shapiro_9_a_table_3_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - a Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - a Table" src="http://lh3.googleusercontent.com/-soQCgPqDXAk/VUeyUcA3plI/AAAAAAAA65U/ZZCRC2Twpww/Shapiro_9_a_table_3_600_thumb2.jpg?imgmax=800" width="404" height="270" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.googleusercontent.com/-rQbiAHcYAsk/VUeyVn9ixnI/AAAAAAAA65Y/Np7qY1mKx6U/s1600-h/Shapiro_10_a_table4_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - a Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - a Table" src="http://lh3.googleusercontent.com/-HnVF5xkwhwY/VUeyWj2ERoI/AAAAAAAA65g/AKr1W8XpGpA/Shapiro_10_a_table4_600_thumb2.jpg?imgmax=800" width="404" height="267" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.googleusercontent.com/--h2GBy567DQ/VUeyaJYmEMI/AAAAAAAA65o/3bkuR8w_8MI/s1600-h/Shapiro_11_a_table_5_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - a Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - a Table" src="http://lh3.googleusercontent.com/-S31fpgyXod4/VUeybGoU55I/AAAAAAAA65w/vEvmrRGjotc/Shapiro_11_a_table_5_600_thumb2.jpg?imgmax=800" width="404" height="265" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h3>Create Labels for the a Values</h3> <p>The first step in obtaining the correct a Values is to create the proper number of labels for the a Values. </p> <p>The number of a Values needed depends on the sample size n. If n is an even number, the number of a Values equals n / 2. If n is an odd number, the number of a Values equals (n-1) / 2. This can be implemented with the following formulas:</p> <p><a href="http://lh3.googleusercontent.com/-o7cEIg0u1K8/VUeycV-gQpI/AAAAAAAA654/Pz6cu5uGlxk/s1600-h/Shapiro_5_Numbered_as_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Numbered As" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Numbered As" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_TnEuzDbZAXSCDvgUjVtI2zc0JB-lhoVtFNlWd6B5Nhon8ZF1PSuNWKvp5PIBHAafUHp6qVNhLhRAfaT_XlXv0rzLFRtNZi0SgpiKDH4R4NvulQQsGQ-yU5wke6Qn7BH-YFmGQBGLJ7g2/?imgmax=800" width="404" height="281" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h3>Lookup a Values on the a Value Lookup Table</h3> <p>The a Values can now be looked up on the a Value table previously shown. An Index formula is a straightforward way implementing this. The Excel Index function is its simplest form has the following format:</p> <p>Index(array, row number, column number) and returns the data value located in the specified row and column of the given array. The row and column number are relative to the cell in the upper-left corner of the array. Note that the array containing the a Values starts in cell H58 at the upper left corner and finishes in cell BE82 in the lower right corner.</p> <p><a href="http://lh3.googleusercontent.com/-IwCPARsJgAc/VUeyfd5pF2I/AAAAAAAA66I/mvKYFRiarJQ/s1600-h/Shapiro_6_a_Values_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - A Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - A Values" src="http://lh3.googleusercontent.com/-4VfxM6LzoHk/VUeygMyFVJI/AAAAAAAA66Q/SjgUJFYbQJA/Shapiro_6_a_Values_600_thumb2.jpg?imgmax=800" width="404" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 3 – Pair Up Successive Highest-Lowest Data Values</h2> <p align="center"> </p> <h3>Create X Labels for Data Sample Values</h3> <p>Each sorted data sample value will have an X Label. This is implemented with the following formulas:</p> <p><a href="http://lh3.googleusercontent.com/-p33SXpbv7n0/VUeyhY71JkI/AAAAAAAA66Y/8efe-c7rux0/s1600-h/Shapiro_12_numbered_xs_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Numbered Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Numbered Xs" src="http://lh3.googleusercontent.com/-oFrpO_IaIqk/VUeyifLfbKI/AAAAAAAA66g/RFOABlUbUKk/Shapiro_12_numbered_xs_600_thumb2.jpg?imgmax=800" width="404" height="274" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The following is a close-up of the formulas to create the labels for the X Values:</p> <p><a href="http://lh3.googleusercontent.com/-ZP4Ue0ji4Bc/VUeyjrn58JI/AAAAAAAA66o/1BV_Osz2Tjw/s1600-h/Shapiro_12a_Closeup_Numbered_Xs_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Numbered Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Numbered Xs" src="http://lh3.googleusercontent.com/--zQDd1rTihE/VUeykmOm8jI/AAAAAAAA66w/ap1ZYBSBF_c/Shapiro_12a_Closeup_Numbered_Xs_600_.jpg?imgmax=800" width="404" height="413" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The X Values are then obtained using the following formulas:</p> <p><a href="http://lh3.googleusercontent.com/-VgdKGd5m7z8/VUeylq0iSXI/AAAAAAAA664/UZAobEghNw0/s1600-h/Shapiro_13_X_Values_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - X Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - X Values" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2xjkKj3Plq4S3HFvCxhLmBqWh2xHtNvljcyNgoX3kGi6Gsj5vceAk9OsONsIQjpM3rb2VkeP4duJ_fdvmpzB5NibRxd2hyphenhyphenPa49NHktP06p2P52Y8juF-c8H8o-Hn-99XH9N-2E2coLjuv/?imgmax=800" width="404" height="385" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <strong></strong> <p align="center"> </p> <h3>Create the Upper Value of Each Data Pair</h3> <p>The labels for the upper X Values of each data pair are created as follows:</p> <p><a href="http://lh3.googleusercontent.com/-To8E-fbvFuk/VUeypF9upnI/AAAAAAAA67I/hd7fs7-DUHk/s1600-h/Shapiro_14_Upper_Xs_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Upper Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Upper Xs" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQm-MoCCSY5kkTwAouLlPOOz2k3Ke1YQP2f6e5w91xDiQqpQ3W2PAERT2i2ELltOMYSgBH3YYun6_TBDYEs04TqmXQS_UqNT4ILZa-ljoWJ3zvubbi8f7ryLGqoX0ou4SdQhasyZtNbA7m/?imgmax=800" width="404" height="209" /></a> The following is a close-up of the formulas to create the labels for the upper X Values: </p> <p><a href="http://lh3.googleusercontent.com/-Ex_izTCIuZ8/VUeyxkrMc1I/AAAAAAAA67o/XEfB8TWUqEg/s1600-h/Shapiro_15a_Closeup_Upper_X_Values_6%25255B3%25255D.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Upper Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Upper Xs" src="http://lh3.googleusercontent.com/-b7gOJbXrd4E/VUey07V0l9I/AAAAAAAA670/P_N4N3T8s20/Shapiro_15a_Closeup_Upper_X_Values_6.jpg?imgmax=800" width="404" height="243" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The upper X Values of each data pair are the obtained using the following set of formulas:</p> <p><a href="http://lh3.googleusercontent.com/-1qqaYkX_yg8/VUey2Cxab_I/AAAAAAAA678/ECwYjcz2HK8/s1600-h/Shapiro_15_Upper_X_Values_6009.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Upper X Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Upper X Values" src="http://lh3.googleusercontent.com/-hKrgJrBiTeU/VUey3YAG-pI/AAAAAAAA68E/nEZIEl6cGfc/Shapiro_15_Upper_X_Values_600_thumb5.jpg?imgmax=800" width="404" height="237" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of those formulas:</p> <p><a href="http://lh3.googleusercontent.com/-JPo7fImP7fg/VUey66qNnQI/AAAAAAAA68M/uu2FK8y5Tak/s1600-h/Shapiro_15a_Closeup_Upper_X_Values_6%25255B2%25255D.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Upper X Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Upper X Values" src="http://lh3.googleusercontent.com/-9xr29TmlQOc/VUey7xYRPdI/AAAAAAAA68U/R-0a3yljMb4/Shapiro_15a_Closeup_Upper_X_Values_6%25255B1%25255D.jpg?imgmax=800" width="404" height="243" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p> </p> <h3>Create the Lower Value of Each Data Pair</h3> <p>The labels for the lower X Values of each data pair are created as follows:</p> <p><a href="http://lh3.googleusercontent.com/-iMBNsGHGX7I/VUey-8CDXQI/AAAAAAAA68c/sSo_ZoIVw48/s1600-h/Shapiro_16_Lowers_Xs_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Lower Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Lower Xs" src="http://lh3.googleusercontent.com/-rt-mERy6sAk/VUezAMNhIAI/AAAAAAAA68k/Ku3wsqdTdCc/Shapiro_16_Lowers_Xs_600_thumb2.jpg?imgmax=800" width="404" height="201" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The following is a close-up of the formulas to create the labels for the lower X Values:</p> <p><a href="http://lh3.googleusercontent.com/-DG8f9tCjXCc/VUezCZZl_4I/AAAAAAAA68s/rQDF2P0MtpE/s1600-h/Shapiro_16a_Closeup_Lowers_Xs_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Lower Xs" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Lower Xs" src="http://lh3.googleusercontent.com/-hQXz4IE1Qq8/VUezFD5XYfI/AAAAAAAA680/ta1D-vWWoa0/Shapiro_16a_Closeup_Lowers_Xs_600_th.jpg?imgmax=800" width="404" height="208" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The lower X Values of each data pair are the obtained using the following set of formulas:</p> <p><a href="http://lh3.googleusercontent.com/-NnSZpB3nceo/VUezIUJraaI/AAAAAAAA688/gJx3Ld9ZRe0/s1600-h/Shapiro_17_Lower_X_Values_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Lower X Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Lower X Values" src="http://lh3.googleusercontent.com/-C4eMV1rA6N0/VUezMVUsNUI/AAAAAAAA69E/-c5IiB7EQY8/Shapiro_17_Lower_X_Values_600_thumb2.jpg?imgmax=800" width="404" height="219" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of those formulas:</p> <p><a href="http://lh3.googleusercontent.com/-CVW--FjeEYk/VUezOxtQ27I/AAAAAAAA69M/jinGVGil7Jk/s1600-h/Shapiro_17a_Closeup_Lower_X_Values_6%25255B2%25255D.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Lower X Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Lower X Values" src="http://lh3.googleusercontent.com/-qo-qxJJrWWc/VUezQJ8isxI/AAAAAAAA69U/B6tb6ELu5B8/Shapiro_17a_Closeup_Lower_X_Values_6.jpg?imgmax=800" width="404" height="277" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 4 – Calculate the Difference Within Each Pair</h2> <p><a href="http://lh3.googleusercontent.com/-Xyme17uNk3g/VUezS9Wc64I/AAAAAAAA69c/BuabAlewz9I/s1600-h/Shapiro_18_Differences_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Pair Differences" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Pair Differences" src="http://lh3.googleusercontent.com/-7Uvf_DfFSjo/VUezUHtsI1I/AAAAAAAA69k/3jBkieOhyQs/Shapiro_18_Differences_600_thumb2.jpg?imgmax=800" width="404" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of the difference formulas:</p> <p><a href="http://lh3.googleusercontent.com/-uhCbhFXvsCY/VUezVy7F8HI/AAAAAAAA69s/eSght0LZNX8/s1600-h/Shapiro_18a_Closeup_Differences_4504.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Pair Differences" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Pair Differences" src="http://lh3.googleusercontent.com/-keeuaklOXPM/VUezXhju44I/AAAAAAAA690/nzTpLB9_faY/Shapiro_18a_Closeup_Differences_450_.jpg?imgmax=800" width="404" height="419" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 5 – Calculate a * Difference</h2> <p><a href="http://lh3.googleusercontent.com/-S-VxFP5sx8Y/VUezaYczRNI/AAAAAAAA698/dvoZVj4Wkx4/s1600-h/Shapiro_19_A_Differences_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - A Times Differences" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - A Times Differences" src="http://lh3.googleusercontent.com/-Zk1RLPtOtAo/VUezb8F6f6I/AAAAAAAA6-E/cv3nkOVYx5g/Shapiro_19_A_Differences_600_thumb2.jpg?imgmax=800" width="404" height="223" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of those formulas:</p> <p><a href="http://lh3.googleusercontent.com/-7DyNemXMSd0/VUezdcNnlxI/AAAAAAAA6-M/hL0x-JQqmbk/s1600-h/Shapiro_19a_Closeup_A_Differences_45%25255B2%25255D.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup A Times Differences" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup A Times Differences" src="http://lh3.googleusercontent.com/-hsGWXLIff5Y/VUezfaR8vyI/AAAAAAAA6-U/dvJNhfxjZpk/Shapiro_19a_Closeup_A_Differences_45.jpg?imgmax=800" width="404" height="402" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 6 – Calculate b, SS, and Test Statistic W</h2> <p>b equals the sums of the product of a * pair difference.</p> <p>SS is the sum of the squared deviations of x from the mean x. Excel formula DEVSQ() performs this.</p> <p>Test Statistic W equals b<sup>2</sup>/SS.</p> <p><a href="http://lh3.googleusercontent.com/-ef3EodZ4ryg/VUezh8elyRI/AAAAAAAA6-c/SLkxyHT4nyU/s1600-h/Shapiro_20_b_SS_W_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - b, SS, Test Statistic W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - b, SS, Test Statistic W" src="http://lh3.googleusercontent.com/-dgqureiaLEM/VUezj6XePfI/AAAAAAAA6-k/9qxh3PI96Zw/Shapiro_20_b_SS_W_600_thumb2.jpg?imgmax=800" width="404" height="190" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of these formulas:</p> <p><a href="http://lh3.googleusercontent.com/-NcrFFnttDA8/VUezlxi_fPI/AAAAAAAA6-s/xRvA9XBYelY/s1600-h/Shapiro_20a_Closeup_b_SS_W_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup b, SS, Test Statistic W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup b, SS, Test Statistic W" src="http://lh3.googleusercontent.com/-cKqnEgfzlYY/VUezm9hWgqI/AAAAAAAA6-0/7jFLDDinS2c/Shapiro_20a_Closeup_b_SS_W_600_thumb.jpg?imgmax=800" width="404" height="200" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 7 – Lookup W Critical Values</h2> <p>Each unique combination of sample size and alpha level has its own critical W value. The following is a table of the respective critical W value for each combination of n and alpha up to a sample size of 50 and 3 common levels of alpha.</p> <p><a href="http://lh3.googleusercontent.com/--U52BAMnzqE/VUezphHsqqI/AAAAAAAA6-8/v1r0ajQoHzI/s1600-h/Shapiro_23_Critical_W_1_3754.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Critical W Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Critical W Table" src="http://lh3.googleusercontent.com/-sV1iTMl7yPM/VUeztJzlSmI/AAAAAAAA6_E/oXEtSJ2voys/Shapiro_23_Critical_W_1_375_thumb2.jpg?imgmax=800" width="253" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_MCrFs3Xg6eB2rveoQWH8fgVZ_MVVGg319u2__Xf5il_V9T1Nya2ZS6e_D94VmE20PzZDzunyESz8uZRhUg2yCCFSZVbRNB_9Y2sxXsJeXg3eL4z-Pl9fLjEG0oPCORtVz2FLEGpjB6TE/s1600-h/Shapiro_24_Critical_W_2_3754.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Critical W Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Critical W Table" src="http://lh3.googleusercontent.com/-Ol6_4UxrCPY/VUez5UGIXUI/AAAAAAAA6_U/BZd1DBthGf0/Shapiro_24_Critical_W_2_375_thumb2.jpg?imgmax=800" width="245" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjWSEuj8emnB0qI7Z-WV13RMfR-IfJ-aeot4zkz3yl9UGILM-pL0ug5OFDQq76aeQiEKTgDgqFTkkfbzc60Cbm-c8HV55Tt7UybtLCgmWTui9eKCJL40VwLNapo3hxD4XtVFz-K62ANsLF/s1600-h/Shapiro_21W_Critical_Lookup_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Loookup Critical W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Loookup Critical W" src="http://lh3.googleusercontent.com/-Vmr53Xa5LMM/VUez753p9FI/AAAAAAAA6_k/wB-mlZjus3E/Shapiro_21W_Critical_Lookup_600_thum.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of the formulas. The If-Then-Else statement that looks up the critical W value is the following:</p> <p>=IF(OR(B1=0.01,B1=0.05,B1=0.1),INDEX(BL18:BN67,U14,IF(U12=0.01,1,IF(U12=0.05,2,IF(U12=0.1,3)))),”alpha must be set to 0.01, 0.05, or 0.10”)</p> <p>The INDEX() function looks up the critical W values in the array starting in cell BL18 in the upper left corner and ending in cell BN67 in the lower right corner.</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsHVg3kShZXCA_Uai6JlR_qSe5lx48hycAkfz0eTNqwmUPxVYT1_QOo0UHSOTkKZ4McLLkudqfRh2fNcxzq4pTXZaWSyuzDs81NFEGXz2LiNl3Mtlz9OMYmxcGFD4QoKTVPnof3b0EoW56/s1600-h/Shapiro_21a_Closeup-W_Critical_Lookup_600.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Loookup Critical W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Loookup Critical W" src="http://lh3.googleusercontent.com/-h5eCho6LnPw/VUez9TZnzyI/AAAAAAAA7A4/uLPe8ElHVL0/Shapiro_21a_Closeup-W_Critical_Lookup_600_thumb.jpg?imgmax=800" width="404" height="60" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Step 8 – Determine Whether or Not To Reject Null Hypothesis By Comparing W to W Critical</h2> <p>The Null Hypothesis is rejected only if Test Statistic W is smaller the critical W value for the given sample size and alpha level. The Null Hypothesis states that the sample came from a normally distributed population. The Null Hypothesis is never accepted but only rejected or not rejected. Not rejecting the Null Hypothesis does not automatically imply that the population from which the sample was taken is normally distributed. This merely indicates that there is not even evidence to state that the population is likely not normally distributed for the given alpha level. As with all hypothesis tests, alpha = 1 - level of certainty. If a 95-percent level of certainty is required to reject the Null Hypothesis, the alpha level will be 0.05. </p> <p><a href="http://lh3.googleusercontent.com/-R25iIJ_XjCI/VUez_BfBecI/AAAAAAAA6_8/rbJaLs_b5ng/s1600-h/Shapiro_22_W_Comparison_6004.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Comparison W and Critical W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Comparison W and Critical W" src="http://lh3.googleusercontent.com/-9mqF0csuc0g/VUe0A087oJI/AAAAAAAA7AE/kz-tbq9s1mY/Shapiro_22_W_Comparison_600_thumb2.jpg?imgmax=800" width="404" height="174" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is a close-up of the formulas to compare W with W Critical:</p> <p><a href="http://lh3.googleusercontent.com/-1-DzMfSwcAY/VUe0CqwZ5tI/AAAAAAAA7AM/0LEhbgDZHwA/s1600-h/Shapiro_22a_Closeup_W_Comparison_600.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Closeup Comparison W and Critical W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Closeup Comparison W and Critical W" src="http://lh3.googleusercontent.com/-9HbC080zBgE/VUe0D5I22fI/AAAAAAAA7AU/mwDJeLXcqqA/Shapiro_22a_Closeup_W_Comparison_600%25255B1%25255D.jpg?imgmax=800" width="404" height="101" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The complete Shapiro-Wilk test of normality appears in Excel as follows:</p> <p><a href="http://lh3.googleusercontent.com/-B5eVgeTUuGc/VUe0F3MHVMI/AAAAAAAA7Ac/mYiSWeJGXUY/s1600-h/Shapiro_1_Complete_Test_6009.jpg"><img title="Shapiro-Wilk Normality Test in Excel - Complete Shaprio-Wilk Normality Test in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Shapiro-Wilk Normality Test in Excel - Complete Shaprio-Wilk Normality Test in Excel" src="http://lh3.googleusercontent.com/-mVe-L42X7Go/VUe0IkxuMUI/AAAAAAAA7Ak/8HJ5yzADwak/Shapiro_1_Complete_Test_600_thumb5.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com455tag:blogger.com,1999:blog-3568555666281177719.post-78382426039108740082015-04-13T11:27:00.001-07:002015-04-13T11:38:48.555-07:00Simplifying Excel Ranking Functions: RANK(), RANK.AVG(), RANK.EQ(), PERCENTILE(), PERCENTILE.INC(), PERCENTILE.EXC(), QUARTILE(), QUARTILE.INC(), QUARTILE.EXC()<h1>Simplifying Excel Ranking <br /> <br />Functions: RANK(), <br /> <br />RANK.AVG(), RANK.EQ(), <br /> <br />PERCENTILE(), <br /> <br />PERCENTILE.INC(), <br /> <br />PERCENTILE.EXC(), <br /> <br />QUARTILE(), QUARTILE.INC <br /> <br />(), QUARTILE.EXC()</h1> <p align="center"> </p> <h2>Simplifying RANK(), RANK.EQ(), and RANK.AVG() </h2> <p>Ranking functions RANK(), RANK.EQ(), and RANK.AVG() return the rank of a data value within a data set. Prior to Excel 2010 RANK() was the sole ranking function. In Excel 2010 and beyond RANK() will still work but its functionality has been duplicated by a new ranking function RANK.EQ(). In addition, another useful ranking function RANK.AVG() has been added.</p> <p>The format of each of these functions is as follows:</p> <p>RANK(value to be ranked, data range, [order])</p> <p>RANK.EQ(value to be ranked, data range, [order])</p> <p>RANK.AVG(value to be ranked, data range, [order])</p> <p>The value to be ranked is a specific number or value is a data set that will be ranked. The data range represents the set of values that will provide the basis for the ranking. The order is an optional value. Setting the order to 0 or leaving it blank will rank data in descending order, i.e., the highest value in the data range will be assigned a rank = 1. Setting the order to 1 will rank data in ascending order, i.e., the lowest value in the data range will be assigned a rank = 1.</p> <p>The difference between RANK.EQ() and RANK.AVG() is how tied data values within the data set are ranked. RANK.EQ() assigns the lowest rank of the tied values to all of the tied values. RANK.AVG() assigns the average rank of the tied values to all of the tied values.</p> <p>RANK.AVG() is useful when performing certain nonparametric tests in Excel. Several nonparametric tests require that data be ranked with tied data values being assigned the average rank of the tied values as is done by RANK.AVG(). These nonparametric tests include the Mann-Whitney U Test, the Wilcoxon Signed-Rank Test, and the Friedman Test.</p> <p>The functionality of the RANK() function is shown as follows. The raw data that is being ranked has already been sorted in order to more easily convey the functionality of the RANK() function.</p> <p><a href="http://lh3.ggpht.com/-wsOEu3QqdWU/VSwKVV_dXII/AAAAAAAA6rw/KG7SxFAsUFI/s1600-h/Ranking_Rank_600%25255B4%25255D.jpg"><img title="Ranking Functions in Excel - RANK() Example" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - RANK() Example" src="http://lh6.ggpht.com/-PD0M7f1xya0/VSwKV_NGYjI/AAAAAAAA6r4/CzfoEUHitiw/Ranking_Rank_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="165" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The functionality of the RANK.EQ() function is shown in the following diagram. RANK.EQ() is equivalent to RANK() and should now be used in its place. Note that tied data values are all assigned the lowest rank of the tied values. The raw data that is being ranked has already been sorted in order to more easily convey the functionality of the RANK.EQ() function.</p> <p><a href="http://lh6.ggpht.com/-wrrmtGYYYRQ/VSwKWXMIcLI/AAAAAAAA6sA/AHN3hh-7FFA/s1600-h/Ranking_Rank_Eq_600%25255B5%25255D.jpg"><img title="Ranking Functions in Excel - RANK.EQ() Example" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - RANK.EQ() Example" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGcTlI0t8I-eLHtOhBgsVZ9724TQ4oOcO15qJ6xrdKRcCLUoPBhkpgy3CMOn0LlJcUpcT1hV7G-8fW5ULsCWdFiCdTGtb-zz1Ly5VN4aW5t2YhyX-MazEDpfApbb5hdPlhxMe3_YYWz5O7/?imgmax=800" width="404" height="137" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The functionality of the RANK.AVG() function is shown in the following diagram. Note that tied data values are all assigned the average rank of the tied values. There is no equivalent to this function in versions of Excel prior to 2010. The raw data that is being ranked has already been sorted in order to more easily convey the functionality of the RANK.AVG() function.</p> <p><a href="http://lh3.ggpht.com/-prSaWmykJq8/VSwKXeONOWI/AAAAAAAA6sQ/QsXro6zYiWg/s1600-h/Ranking_Rank_Avg_600%25255B4%25255D.jpg"><img title="Ranking Functions in Excel - RANK.AVG() Example" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - RANK.AVG() Example" src="http://lh5.ggpht.com/-ID6so5fv04o/VSwKX-OF0xI/AAAAAAAA6sY/Sny6yenJIDs/Ranking_Rank_Avg_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="143" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h2>Simplifying PERCENTILE(), PERCENTILE.INC(), and PERCENTILE.EXC()</h2> <p>Percentile functions PERCENTILE(), PERCENTILE.INC(), and PERCENTILE.EXC() return the value at a given percentile in relation to a given data set. For example, given the data set {1, 3, 5, 13, 15} in cells A2:A6, PERCENTILE(A2:A6,0.3) = 3.4. This means that the value 3.4 occupies the 30<sup>th</sup> percentile of the given data set.</p> <p>An example of the use of the percentile functions would be to determine which test scores within a group of test score are at or above the 95<sup>th</sup> percentile of the group. The percentile functions would return that value that would occupy that 95<sup>th</sup> percentile of the group.</p> <p>The percentile functions are typically used to establish a threshold of acceptance or failure.</p> <p>Prior to Excel 2010 PERCENTILE() was the sole percentile function. In Excel 2010 and beyond PERCENTILE() will still work but its functionality has been duplicated by a new percentile function PERCENTILE.INC(). In addition, another percentile function PERCENTILE.EXC() has been added.</p> <p>The format of each of these functions is as follows:</p> <p>PERCENTILE(data range, percentile)</p> <p>PERCENTILE.INC(data range, percentile)</p> <p>PERCENTILE.EXC(data range, percentile)</p> <p>The percentile can be any number between 0 and 1.</p> <p>The value to be ranked is a specific number or value is a data set that will be ranked. The data range represents the set of values that will provide the basis for the ranking. The order is an optional value. Setting the order to 0 or leaving it blank will rank data in descending order, i.e., the highest value in the data range will be assigned a rank = 1. Setting the order to 1 will rank data in ascending order, i.e., the lowest value in the data range will be assigned a rank = 1.</p> <p>Both PERCENTILE.EXC() and PERCENTILE.INC() and its equivalent PERCENTILE() first rank the values in the given data set from 1 (assigned to the lowest value in the data set) to n (assigned to the highest value in the data set). The rank K is then calculated. The main difference between is the method of calculating K as shown in the following Excel algorithms for these functions. If K is not an integer value, linear interpolation between the nearest values in the data set to determine the value that would occupy the given percentile.</p> <p>Another difference between the percentile functions is that PERCENTILE.EXC only works if k is between 1/n and 1-1/n, where n is the number of elements in array. PERCENTILE.INC works for any value of k between 0 and 1.</p> <p>The Excel algorithm for equivalent functions PERCENTILE() and the equivalent PERCENTILE.INC() is as follows:</p> <p><a href="http://lh3.ggpht.com/-9WGHXmL5oV0/VSwKYWQZfUI/AAAAAAAA6sg/J9jFYRrlSEE/s1600-h/Ranking_Percentile_600%25255B4%25255D.jpg"><img title="Ranking Functions in Excel - Excel Algorithm for PRECENTILE() and PERCENTILE.INC()" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - Excel Algorithm for PRECENTILE() and PERCENTILE.INC()" src="http://lh5.ggpht.com/-jSn_H0yk5ps/VSwKZKTccwI/AAAAAAAA6sk/Nr4YYlImyhk/Ranking_Percentile_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="211" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Excel algorithm for PERCENTILE.EXC() is as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDo1PDi5IyUNJykChoOrfX3ph6OwAijzbByIVjmP57kEHO2ZKmEFIowm7pj2hcYaDwVnfW1x4LtTurvPP7pESiGLW78fqI-EyHKnCoSFVLCuaY3iPNMFausEz2UCML6yAR77pvVJPN8Iut/s1600-h/Ranking_Percentile_exc_600%25255B5%25255D.jpg"><img title="Ranking Functions in Excel - Excel Algorithm for PERCENTILE.EXC()" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - Excel Algorithm for PERCENTILE.EXC()" src="http://lh6.ggpht.com/-uh_HCJUl0hw/VSwKZ7aKi5I/AAAAAAAA6s0/wtPT3jrK4aY/Ranking_Percentile_exc_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="191" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h2>Simplifying PERCENTRANK(), PERCENTRANK.INC(), and PERCENTRANK.EXC()</h2> <p>PERCENTRANK(), PERCENTRANK.INC(), and PERCENTRANK.EXC() are the inverse functions of PERCENTILE(), PERCENTILE.INC(), and PERCENTILE.EXC().</p> <p>PERCENTRANK(), PERCENTRANK.INC(), and PERCENTRANK.EXC() return the percentile that a given data value has in relation to set of data values.</p> <p>PERCENTILE(), PERCENTILE.INC(), and PERCENTILE.EXC() return the data value that would occur at a given percentile in relation to a set of data values.</p> <p>The following diagram shows the relationship of the PERCENTRANK functions to the PERCENTILE functions. In each of these cases the percentile is the 30th percentile.</p> <p><a href="http://lh4.ggpht.com/-XaO45y1IUU8/VSwKaQja9bI/AAAAAAAA6tA/YR_SiDOTNmo/s1600-h/Ranking_Percentrank_600%25255B4%25255D.jpg"><img title="Ranking Functions in Excel - Inverse Functions PERCENTRANK() and PERCENTILE()" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking_Percentrank_600" src="http://lh3.ggpht.com/-mxD7yWlGWIk/VSwKa9OvPLI/AAAAAAAA6tI/hyNRESgt6cs/Ranking_Percentrank_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="241" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h2>Simplifying QUARTILE(), QUARTILE,INC(), and QUARTILE.EXC()</h2> <p>The quartile functions are special cases of the percentile functions. The equivalent functions are as follows:</p> <p>QUARTILE(Data range, 0) = PERCENTILE(Data range, 0) = MIN(Data range)</p> <p>QUARTILE(Data range, 1) = PERCENTILE(Data range, .25)</p> <p>QUARTILE(Data range, 2) = PERCENTILE(Data range, .5) = MEDIAN(Data range)</p> <p>QUARTILE(Data range, 3) = PERCENTILE(Data range, .75) </p> <p>QUARTILE(Data range, 4) = PERCENTILE(Data range, 1) = MAX(Data range)</p> <p align="center"> </p> <p>QUARTILE.INC(Data range, 0) = PERCENTILE.INC(Data range, 0) </p> <p>QUARTILE.INC(Data range, 1) = PERCENTILE.INC(Data range, .25)</p> <p>QUARTILE.INC(Data range, 2) = PERCENTILE.INC(Data range, .5) </p> <p>QUARTILE.INC(Data range, 3) = PERCENTILE.INC(Data range, .75) </p> <p>QUARTILE.INC(Data range, 4) = PERCENTILE.INC(Data range, 1) </p> <p align="center"> </p> <p>QUARTILE.EXC(Data range, 0) = PERCENTILE.EXC(Data range, 0) </p> <p>QUARTILE.EXC(Data range, 1) = PERCENTILE.EXC(Data range, .25)</p> <p>QUARTILE.EXC(Data range, 2) = PERCENTILE.EXC(Data range, .5) </p> <p>QUARTILE.EXC(Data range, 3) = PERCENTILE.EXC(Data range, .75) </p> <p>QUARTILE.EXC(Data range, 4) = PERCENTILE.EXC(Data range, 1)</p> <p>The Excel algorithm for equivalent functions QUARTILE() and the equivalent QUARTILE.INC() is as follows:</p> <p><a href="http://lh3.ggpht.com/-rp9ujNrpmW8/VSwKbncYcPI/AAAAAAAA6tQ/pKgscHTPs0U/s1600-h/Ranking_Quartile_600%25255B4%25255D.jpg"><img title="Ranking Functions in Excel - Excel Algorithm for QUARTILE() and QUARTILE.INC()" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - Excel Algorithm for QUARTILE() and QUARTILE.INC()" src="http://lh3.ggpht.com/-4LqsaYOc59Q/VSwKb0g39XI/AAAAAAAA6tY/Gt7i0b3jPfE/Ranking_Quartile_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="210" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Excel algorithm for QUARTILE.EXC() is as follows:</p> <p><a href="http://lh4.ggpht.com/-spWpr2PrGIU/VSwKcrQ9iRI/AAAAAAAA6tg/3Bk3d-vKpB0/s1600-h/Ranking_Quartile_exc_600%25255B7%25255D.jpg"><img title="Ranking Functions in Excel - Excel Algorithm for QUARTILE.EXC()" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Ranking Functions in Excel - Excel Algorithm for QUARTILE.EXC()" src="http://lh6.ggpht.com/-g4yzVUkqSwU/VSwKc50-bpI/AAAAAAAA6to/qL8aiTdltho/Ranking_Quartile_exc_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="201" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com68tag:blogger.com,1999:blog-3568555666281177719.post-50888885793019715662015-04-12T13:02:00.001-07:002015-04-13T09:23:30.191-07:00Automated Data Column Sorting in Excel<h1>Automated Data Column <br /> <br />Sorting in Excel</h1> <h2>Single-Column Sorting Automated With Formulas in Excel</h2> <p>A single column of numeric data can be sorted by using formulas or with the Data Sorting Tool. Using formulas is a much better solution because data sorted by formulas will be automatically resorted if the data is changed or new data is added. The Data Sorting Tool, just like all of the other Data and Data Analysis ToolPak Tools, must be re-run manually when a new sort is required. Unlike Data Tools, formulas automatically recalculate their output when their inputs are changed.</p> <p align="center"> </p> <h3>Descending Sort Using Formulas in Excel</h3> <p>The following image shows the implementation of a descending sort using formulas in Excel. </p> <p>The formula can be typed into the first cell and then copied all the way down, in this case, to cell D200.</p> <p>The LARGE() formula is explained as follows:</p> <p>LARGE(<i>data range</i>, k) returns the kth largest value in the data range. The position count, starting at 1 at the top, is implemented by the following:</p> <p>ROW()-ROW($D$3)</p> <p>ROW() is the number of the current row. If this formula is in cell D<font color="#ff0000"><b>4</b></font>, ROW() = 4 since cell D<font color="#ff0000"><b>4</b></font> is in the <font color="#ff0000"><b>4</b></font><sup>th</sup> row.</p> <p>ROW(D3) = 3 since cell D3 is in the 3<sup>rd</sup> row.</p> <p>In this case, ROW()-ROW($D$3) = 4 – 3 = 1</p> <p>If the formula is in cell D5, ROW()-ROW($D$3) = 5 – 3 = 2</p> <p>Note that the data range in the formula extends from cell B4 all the way down to cell B200. Empty cells in this range are ignored by the sort because of the If-Then-Else statement. </p> <p><a href="http://lh3.ggpht.com/-AyMVVJSfse4/VSrPC9fFixI/AAAAAAAA6pU/woeXX67QaE4/s1600-h/Sorting_Descending_Sort_600%25255B4%25255D.jpg"><img title="Automated Data Column Sorting in Excel - Automated Descending Sort in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Automated Data Column Sorting in Excel - Automated Descending Sort in Excel" src="http://lh5.ggpht.com/-iBeyWqsEQRs/VSrPENyePuI/AAAAAAAA6pc/qkCNPKdSbcQ/Sorting_Descending_Sort_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="266" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h3>Ascending Sort Using Formulas in Excel</h3> <p>A descending sort can be converted into an ascending sort by substituting the word <i>SMALL</i> for <i>LARGE</i> in the formula as follows:</p> <p><a href="http://lh6.ggpht.com/-005I8jcQsMg/VSrPFt6dstI/AAAAAAAA6pk/OmNgDEnv3Ew/s1600-h/Sorting_Ascending_Sort_600%25255B7%25255D.jpg"><img title="Automated Data Column Sorting in Excel - Automated Ascending Sort in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Automated Data Column Sorting in Excel - Automated Ascending Sort in Excel" src="http://lh4.ggpht.com/-vR0LbFyyjiA/VSrPHfeon3I/AAAAAAAA6ps/h6tKvNuaxD0/Sorting_Ascending_Sort_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="277" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h3>New or Changing Data</h3> <p>The advantage of sorting with formulas instead of the Data Sorting Tool is the formulas will automatically resort the data if any data is changed or additional data is added. The following changes were made to the data:</p> <p>-6 was changed to -7 (colored orange)</p> <p>New data points 11, 13, and 14 were added (colored light blue)</p> <p>The data is automatically resorted after these changes and additions. The Data Analysis Sorting Tool would have to be re-run manually to resort the data.</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2hWmAn4NNEkQgZgCMeIIyx21jDSY5gW77zQkaqtwBjKfhyphenhyphenO8u3kQxhwQNS07MvGre0UHElqqVja6lnppET5EZutZC2cQFP-I1PJG1ct4qQ1ahpae4fiyk9dE3q_LO4Nat6uwVXiwjCMZ3/s1600-h/Sorting_Updated_Sort_600%25255B4%25255D.jpg"><img title="Automated Data Column Sorting in Excel - Automated Sorting When Adata Are Added or Changed" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Automated Data Column Sorting in Excel - Automated Sorting When Adata Are Added or Changed" src="http://lh3.ggpht.com/-9iS146WnkUY/VSrPK_jFTXI/AAAAAAAA6p8/9-e9buLTM6o/Sorting_Updated_Sort_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="283" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h2>Multi-Column Sorting With the Data Sorting Tool in Excel</h2> <p>Multiple columns of data that require secondary or tertiary sorts require the use of the Data Sorting Tool. Alphabetic data can also be sorted using this tool. Below are 3 columns of data that will be sorted using a primary sort of Column 3, then a secondary sort of Column 2, and finally a tertiary sort of Column 1. </p> <p><a href="http://lh6.ggpht.com/-3ZdAnjspw_M/VSrPMc8Wd4I/AAAAAAAA6qE/IS0gZRqMuL8/s1600-h/Sorting_Multicolumn_1_Raw_Data_350%25255B4%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Raw Data" src="http://lh4.ggpht.com/-yOM-dfE0IpA/VSrPN0QyPbI/AAAAAAAA6qM/u4cCcQyZIes/Sorting_Multicolumn_1_Raw_Data_350_thumb%25255B2%25255D.jpg?imgmax=800" width="326" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The first step is to highlight the data including the column headers if they are there as follows:</p> <p><a href="http://lh4.ggpht.com/-DYftxJ7VoQI/VSrPPbyUROI/AAAAAAAA6qU/PpFM52Kfvtg/s1600-h/Sorting_Multicolumn_2_Select_Data_350%25255B4%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Selecting Data To Be Sorted AlongWith Column Headers" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Selecting Data To Be Sorted AlongWith Column Headers" src="http://lh3.ggpht.com/-zjkdGiJbXqo/VSrPQty2thI/AAAAAAAA6qc/fsFyf-DQr28/Sorting_Multicolumn_2_Select_Data_350_thumb%25255B2%25255D.jpg?imgmax=800" width="354" height="508" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Data Sorting Tool is found under the Data tab as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLi7zg8wdTsrjSYiclhbe1bcXQ0xUMcwKdGnRDItoQFfyiKIOI3_hff0Pl10tIql4q2xFkXf0OqzshjxF6TaSshcg1owVFQdEMKekGN7dtqIbkfZbFJEwqdS1-blRQdQnIbRgh0OA0JOEq/s1600-h/Sorting_Multicolumn_3_Data_Sort_300%25255B4%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Data Sorting Tool Location Under Data Tab" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Data Sorting Tool Location Under Data Tab" src="http://lh4.ggpht.com/-drsLkY7tHeo/VSrPSLT9fcI/AAAAAAAA6qs/pqb12MzP3wQ/Sorting_Multicolumn_3_Data_Sort_300_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="453" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Clicking on Sort brings up the following dialogue box. The column headers show up because they were highlighted with the original data in Step 1.</p> <p><a href="http://lh3.ggpht.com/-AGK0-xFg6tM/VSrPS_K16eI/AAAAAAAA6q0/om1hrVysa6A/s1600-h/Sorting_Multicolumn_4_Dialogue_Box_600%25255B4%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Sorting Tool Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Sorting Tool Dialogue Box" src="http://lh5.ggpht.com/-fVMsHV93XUY/VSrPTdpBQLI/AAAAAAAA6q8/5AG6kH9kk4E/Sorting_Multicolumn_4_Dialogue_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="187" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The following completed dialogue is set up to perform a primary sort of Column 3, then a secondary sort of Column 2, and finally a tertiary sort of Column 1. </p> <p><a href="http://lh5.ggpht.com/-ycyk2Z3xr8s/VSrPUEp6jnI/AAAAAAAA6rE/siAgTXCBgQQ/s1600-h/Sorting_Multicolumn_5_Dialogue_Box_Completed_600%25255B7%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Sorting Tool Dialogue Box Completed" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Sorting Tool Dialogue Box Completed" src="http://lh4.ggpht.com/-wZbbt2hMfT8/VSrPUwgeqEI/AAAAAAAA6rM/Bl0E-e3AynY/Sorting_Multicolumn_5_Dialogue_Box_Completed_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="189" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Clicking OK performs that sort as follows:</p> <p><a href="http://lh5.ggpht.com/-ehaTNE_CGzw/VSrPViOcxgI/AAAAAAAA6rU/0WZmnHkHL1E/s1600-h/Sorting_Multicolumn_6_Sorted_Data_350%25255B4%25255D.jpg"><img title="Sorting Multicolumn Data in Excel - Sorted Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Sorting Multicolumn Data in Excel - Sorted Data" src="http://lh4.ggpht.com/-VHqvqIQKe5M/VSrPWdZMN8I/AAAAAAAA6rc/2oZz6jKcdkE/Sorting_Multicolumn_6_Sorted_Data_350_thumb%25255B2%25255D.jpg?imgmax=800" width="328" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com113tag:blogger.com,1999:blog-3568555666281177719.post-37128428605605055452015-04-06T12:03:00.001-07:002015-04-11T15:24:29.602-07:00Measures of Variation in Excel<h1>Measures of Variation in <br /> <br />Excel</h1> <h2>Measures of Variation Overview</h2> <p>Measures of variation describe the degree of spread in a random variable. Measures of variation are sometimes referred to as measures of dispersion. Measures of variation are properties describing probability distributions, populations, or samples taken from a population. Quite often the same measures of variation are calculated differently for populations and samples taken from those population.</p> <p>The following measures of variation can be calculated in Excel and will be discussed here:</p> <p><font color="#0000ff"><b>σ</b><sup><b>2</b></sup><b> = Population variance</b></font> – A measure of the degree of dispersion of values of all of the points contained within a population. σ<sup>2</sup> equals the sum of the squared differences between the sample mean and each data value divided by N. Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units.</p> <p><font color="#0000ff"><b>s</b><sup><b>2</b></sup><b> = Sample variance</b></font> – A measure of the degree of dispersion of values of all of the points contained within a sample. s<sup>2</sup> equals the sum of the squared differences between the sample mean and each data value divided by (n – 1).</p> <p><b><font color="#0000ff">σ = Population standard deviation</font></b> – The square root of population variance. Standard deviations are not additive but are expressed in the same units as the data values being described.</p> <p><b><font color="#0000ff">s = Sample standard deviation</font></b> – The square root of sample variance.</p> <p><b><font color="#0000ff">SS = Sum of squares</font></b> - Because variances are additive, the sum of the squares of differences between a mean and data values is used to describe the total variance within specific categories. ANOVA and linear regression both uses the sum of the squares in their calculations. </p> <p><b><font color="#0000ff">MAD – Mean absolute deviation and Median absolute value</font></b> – Mean absolute deviation equals the average distance (absolute distance) between data values and the mean. Median absolute value, which is the average distance between data values and the median, is more robust (less sensitive to outliers) than mean absolute deviation.</p> <p><b><font color="#0000ff">R = Range</font></b> – Range describes the difference between the largest and smallest values of a data set. Range is sensitive to outliers. Range is used in SPC (Statistical Process Control) to describe the spread within each sample when sample size is small (typically 2 – 9). Standard deviation is used in SPC to describe spread within each sample when sample size become larger.</p> <p><b><font color="#0000ff">Interquartile range</font></b> – Interquartile range is the spread or width of the middle two quartiles of the four quartiles that all data of a data set have been divided into. Quartiles divide rank-ordered data into four equal parts, i.e., quartiles Q1, Q2, Q3, and Q4 which contains the highest values. The box of a boxplot contains the middle two quartiles of data while the whiskers contain the outer quartiles of data. Interquartile range is very helpful in finding outliers.</p> <p>Excel also has excellent tools to visually display the degree of variation existing in a data set. The Excel tools that will be described in this blog article that visually display variation of data are the following:</p> <ol> <li> <p style="orphans: 0; widows: 0"><font color="#0000ff"><strong>Histogram</strong></font></p> </li> <li> <p style="orphans: 0; widows: 0"><strong><font color="#0000ff">Box plot</font></strong></p> </li> <li> <p style="orphans: 0; widows: 0"><strong><font color="#0000ff">Scatter plot</font></strong></p> </li> </ol> <h2>Population Variance</h2> <p>Population variance is a measure of the degree of dispersion of values of all of the points contained within a population. σ<sup>2</sup> (sigma squared) equals the sum of the squared differences between the sample mean and each data value divided by N, the total number of data values in a population. µ (mu) is the population mean. The formula for calculating population variance is the following:</p> <p><a href="http://lh4.ggpht.com/-Bw0aOKZg0ds/VSLYBPylmaI/AAAAAAAA6hI/ZJArXe4CnK4/s1600-h/Variance_population%25255B4%25255D.gif"><img title="Variation in Excel - Population Variance" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Population Variance" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHoqeCmr0nwGEmcMEn7qnHgjTKcerE_vG1owi8wuSFxXsYIRpP45nJlznfkjEJLLt3WOe7h8-hhStXihWRe3HHRN8r6OUcRld1AXHKBLYbxs5NNOy0QFpM4wwZjMA7K8myZvjESf_WxvF7/?imgmax=800" width="404" height="47" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Population variance is calculated in Excel 2010 and later with the following formula: </p> <p>σ<sup>2</sup> = VAR.P(<i>data range</i>)</p> <p>Excel 2007 and earlier used the following formula, which also works in later Excel versions:</p> <p>σ<sup>2</sup> = VARP(<i>data range</i>) </p> <p>Note that Greek letters are often used to denote population parameters while letters from the English alphabet are often used to denote sample statistics.</p> <p>Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units. The square root of the variance is the standard deviation. Standard deviations are not additive but they are conveniently expressed in the same units as the original data. </p> <p>It is important to note that population variance is calculated by dividing the sum of the squared by N, the total number of data points in the population. Sample variance, on the other hand, is calculated by dividing the sum of the squares by (n – 1) instead of n, the number of data points in the sample. This will be discussed in detail shortly. </p> <p align="center"> </p> <h2>Sample Variance</h2> <p>Sample variance is a measure of the degree of dispersion of values of all of the points contained within a sample. s<sup>2</sup> equals the sum of the squared differences between the sample mean and each data value divided by (n-1), the total number of data values in sample minus 1. X<sub>avg</sub> is the sample mean. The formula for calculating population variance is the following:</p> <p><a href="http://lh4.ggpht.com/-dW-tBcLOarI/VSLYCOcN3TI/AAAAAAAA6hY/1CbftyQzr6Q/s1600-h/Standard_deviation_sample%25255B7%25255D.gif"><img title="Variation in Excel - Sample Standard Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Sample Standard Deviation" src="http://lh4.ggpht.com/-q1IujlgED7w/VSLYCiA_HYI/AAAAAAAA6hg/3RuEfDXdYGE/Standard_deviation_sample_thumb%25255B3%25255D.gif?imgmax=800" width="404" height="46" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Sample variance is calculated in Excel 2010 and later with the following formula: </p> <p>s<sup>2</sup> = VAR.S(<i>data range</i>)</p> <p>Excel 2007 and earlier used the following formula, which also works in later Excel versions:</p> <p>s<sup>2</sup> = VAR(<i>data range</i>) </p> <p align="center"> </p> <h3>Unbiased Sample Variance</h3> <p>Dividing the sum of the squares from sample data by n instead of (n-1) would produce a biased estimate of the sample variation. The term <i>n-1</i>, known as Bessel’s correction, replaces <i>n</i> as the divisor to remove the bias.</p> <p>An intuitive explanation for the use of Bessel’s correction when calculating sample variance or sample standard deviation is as follows:</p> <p>It is usually the case that when estimating the population standard deviation, σ, from a sample standard deviation, s, the population mean, µ, is also unknown. The sample standard deviation is therefore calculated using the sample mean, x<sub>avg</sub>, in place of the population mean, µ. This substitution creates a slight bias because the sample’s data points are likely to be closer to x<sub>avg</sub> than to µ. The sum of the squares calculated from (xi – x<sub>avg</sub>)<sup>2</sup> will be smaller than (x<sub>i</sub> - µ)<sup>2</sup>. This is compensated for by using Bessel’s correction (n-1) as the divisor instead of n when calculating sample standard deviation or sample variance.</p> <p>Note that if the population mean, µ, is known and used instead of sample mean, x<sub>avg</sub>, during the calculation of sample standard deviation or sample variance, then Bessel’s correction should not be used; the divisor should be n instead of n-1. This is usually not the case however.</p> <p>Bessel’s correction is a consistent estimator because it converges in probability to the population value as the sample size goes to infinity. As sample size increases, the need for the use of Bessel’s correction decreases. When n exceeds 75, the difference between the biased sample standard deviation (using n as the divisor instead of n-1) and the population standard deviation is generally less than 1 percent. The use of uncorrected sample standard deviation for very large samples is generally acceptable.</p> <p align="center"> </p> <h4>Derivation of Bessel’s Correction</h4> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><font face="Verdana, serif"><font color="#000000"><font face="Arial, serif">Bessel’s Correction can be derived in a number of ways. Here is one method that is straightforward and intuitive.</font></font></font></p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-Dio8l6GOF0I/VSLYDJmG06I/AAAAAAAA6ho/7QFDdRSu_No/s1600-h/Standard_deviation_unbiased_derivation_line_1%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh4.ggpht.com/-DN1arecgG6k/VSLYDtzT5VI/AAAAAAAA6hw/QaE95auJYfk/Standard_deviation_unbiased_derivation_line_1_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="41" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-8OZqitNIo60/VSLYEN6mncI/AAAAAAAA6ZE/Kob_TqfDAfY/s1600-h/Standard_deviation_unbiased_derivation_line_3%25255B2%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh3.ggpht.com/-5333a2eAuqo/VSLYEqZWhaI/AAAAAAAA6ZM/dqcVFAqkhu4/Standard_deviation_unbiased_derivation_line_3_thumb.gif?imgmax=800" width="244" height="35" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-JbSGDZjo9J4/VSLYFIduz4I/AAAAAAAA6h4/Irf1YXD0j7Y/s1600-h/Standard_deviation_unbiased_derivation_line_2%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZlzlJNyjtPsm7QlVicV4f4erUp4_Ujzwdd2jgRB3quWLytoAWXVy-LLRwyMgHurjsh_y9M43UHXZOVI4Vrnaj9ohyphenhyphenAA8z-QAcqnF4TLm9wfKNzFX98I888mWALMLxWDatD8a6G98o-8pr/?imgmax=800" width="404" height="50" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh4.ggpht.com/-ysURzaqhZyw/VSLYGLzlA3I/AAAAAAAA6Zk/_-C3Q93o0ag/s1600-h/Standard_deviation_unbiased_derivation_line_1B%25255B2%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh4.ggpht.com/-8kvdBN7NVKQ/VSLYGov_oLI/AAAAAAAA6Zs/TX0o8nx3Jco/Standard_deviation_unbiased_derivation_line_1B_thumb.gif?imgmax=800" width="104" height="30" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh4.ggpht.com/-wVA3x8QtqHo/VSLYHKMNk0I/AAAAAAAA6iI/12uQt8S1F2s/s1600-h/Standard_deviation_unbiased_derivation_line_4%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh6.ggpht.com/-6gyg4pi-Krw/VSLYHUWuEmI/AAAAAAAA6iQ/Q7mrmf0WQSc/Standard_deviation_unbiased_derivation_line_4_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="49" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-MfI2B3aQ9DI/VSLYH4i37gI/AAAAAAAA6iY/L_yxu-T5eZ0/s1600-h/Standard_deviation_unbiased_derivation_line_5%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh4.ggpht.com/-FS_byoaUyWI/VSLYIg3DaeI/AAAAAAAA6ig/O_fowbq2p7c/Standard_deviation_unbiased_derivation_line_5_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="40" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh3.ggpht.com/-wHk_WsHBwIU/VSLYI3WZVVI/AAAAAAAA6io/40qFkDiVzk4/s1600-h/Standard_deviation_unbiased_derivation_line_6%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh6.ggpht.com/-9-YCrTkEL94/VSLYJXE6ioI/AAAAAAAA6iw/2hAJy4hsbJE/Standard_deviation_unbiased_derivation_line_6_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="33" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh3.ggpht.com/-r8tsQ-KNAeA/VSLYJ40dDSI/AAAAAAAA6i4/8vUF4liroBg/s1600-h/Standard_deviation_unbiased_derivation_line_7%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh6.ggpht.com/-SvDOSbR_sdw/VSLYKduyf-I/AAAAAAAA6jA/cx4S0u3Gh3k/Standard_deviation_unbiased_derivation_line_7_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="36" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh3.ggpht.com/-y1wwV6LeIiw/VSLYKzJNQgI/AAAAAAAA6a0/g16f85S6R8c/s1600-h/Standard_deviation_unbiased_derivation_line_8%25255B2%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh4.ggpht.com/-BBfIzti5fzI/VSLYLQiMSDI/AAAAAAAA6a4/tM6a_AYYUHI/Standard_deviation_unbiased_derivation_line_8_thumb.gif?imgmax=800" width="244" height="27" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-xRPvVTsQD3w/VSLYLw-noiI/AAAAAAAA6bE/27Om7FYT4jQ/s1600-h/Standard_deviation_unbiased_derivation_line_9%25255B2%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh3.ggpht.com/-rZAH7-2n8lM/VSLYMdwf2MI/AAAAAAAA6bM/kcCk-xT4hcI/Standard_deviation_unbiased_derivation_line_9_thumb.gif?imgmax=800" width="210" height="53" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh6.ggpht.com/-fPdFHwv3JEc/VSLYMhj9roI/AAAAAAAA6bQ/WsoZ5cl2ofA/s1600-h/Standard_deviation_unbiased_derivation_line_10%25255B2%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh6.ggpht.com/-kSwhT6OhFfU/VSLYNIp3LhI/AAAAAAAA6bc/hKDw0LdBeHc/Standard_deviation_unbiased_derivation_line_10_thumb.gif?imgmax=800" width="244" height="41" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh5.ggpht.com/-7pOVhCmM7ZM/VSLYNjxQavI/AAAAAAAA6bk/GBJwoHgzD_8/s1600-h/Standard_deviation_unbiased_derivation_line_3%25255B5%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh6.ggpht.com/-nsMtULDC6Yk/VSLYOIWjh8I/AAAAAAAA6bo/mlbJNjgMUrA/Standard_deviation_unbiased_derivation_line_3_thumb%25255B1%25255D.gif?imgmax=800" width="244" height="35" /></a> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh4.ggpht.com/-FNaHGRzJu98/VSLYObXA8zI/AAAAAAAA6jI/nUtEqjII9kI/s1600-h/Standard_deviation_unbiased_derivation_line_11%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh3.ggpht.com/-cIDvaW0Gg5w/VSLYOx9N6vI/AAAAAAAA6jQ/9tXg81-vKjc/Standard_deviation_unbiased_derivation_line_11_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="30" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p style="margin-bottom: 0.02in; margin-top: 0.02in"><a href="http://lh4.ggpht.com/-mi2mqiZnpVc/VSLYPQ3kWSI/AAAAAAAA6jY/NvggbqOY32Q/s1600-h/Variance_sample%25255B4%25255D.gif"><img title="Variation in Excel - Bessel's Correction Derivation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Bessel's Correction Derivation" src="http://lh4.ggpht.com/-aukuiM4mtJE/VSLYP5pSRvI/AAAAAAAA6jg/8jc1rumd7Hs/Variance_sample_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="42" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h2>Population Standard Deviation</h2> <p>Population standard deviation, σ (sigma), is the square root of population variance, σ<sup>2</sup>, and is calculated using the following formula:</p> <p><a href="http://lh3.ggpht.com/-gCy_N2pzjDI/VSLYQcOb7mI/AAAAAAAA6jo/jOoazdW1YYo/s1600-h/Standard_deviation_population%25255B4%25255D.gif"><img title="Variation in Excel - Population Standard Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Population Standard Deviation" src="http://lh5.ggpht.com/-kbruwCD3pW8/VSLYQ2njRLI/AAAAAAAA6jw/WN1u8ck1_Y0/Standard_deviation_population_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="49" /></a> </p> <p>µ (mu) = Population mean</p> <p>N = total number of data points in the population</p> <p>Population standard deviation is calculated in Excel 2010 and later with the following formula: </p> <p>σ = STDEV.P(<i>data range</i>)</p> <p>Excel 2007 and earlier used the following formula, which also works in later Excel versions:</p> <p>σ = STDEVP(<i>data range</i>) </p> <p>Note that Greek letters are often used to denote population parameters while letters from the English alphabet are often used to denote sample statistics.</p> <p>Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units. The square root of the variance is the standard deviation. Standard deviations are not additive but they are conveniently expressed in the same units as the original data. </p> <p>It is important to note that population standard deviation is calculated by dividing the sum of the squared by N, the total number of data points in the population. Sample standard deviation, on the other hand, is calculated by dividing the sum of the squares by (n – 1) instead of n, the number of data points in the sample. </p> <p align="center"> </p> <h2>Sample Standard Deviation</h2> <p>Sample standard deviation, s, is the square root of sample variance, s<sup>2</sup>, and is calculated using the following formula:</p> <p><a href="http://lh3.ggpht.com/-P2CfvQhE-AU/VSLYRYdHgZI/AAAAAAAA6j4/6HZON3I6olc/s1600-h/Standard_deviation_sample%25255B9%25255D.gif"><img title="Variation in Excel - Sample Standard Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Sample Standard Deviation" src="http://lh5.ggpht.com/-yk3Lk6_NYEU/VSLYRlyA2SI/AAAAAAAA6kA/eyqDx33U6hQ/Standard_deviation_sample_thumb%25255B5%25255D.gif?imgmax=800" width="404" height="46" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>x<sub>avg</sub> = Population mean</p> <p>n = total number of data points in the sample</p> <p>Sample standard deviation is calculated in Excel 2010 and later with the following formula: </p> <p>s = STDEV.S(<i>data range</i>)</p> <p>Excel 2007 and earlier used the following formula, which also works in later Excel versions:</p> <p>s = STDEV(<i>data range</i>) </p> <p align="center"> </p> <h2>Descriptive Statistics in Excel</h2> <p>Excel provides a tool in the Data Analysis toolpak called Descriptive Statistics that calculates a data column’s sample standard deviation and sample variance along with sample statistics. The Data Analysis toolpak is an Excel add-in that is available in most versions of Excel. This add-in is initially inactive and must be activated by the user before the first use. After this add-in has been activated, Descriptive Statistics can be accessed as follows:</p> <p><b>Data tab / Data Analysis / Descriptive Statistics</b></p> <p>The Descriptive Statistics dialogue box then appears. The data to which Descriptive Statistics will be applied needs to be arranged in a single column. An example of a column of sample data and the completed Descriptive Statistics dialogue box is shown as follows:</p> <p><a href="http://lh4.ggpht.com/-7PSaYvbYVOg/VSLYSBklMjI/AAAAAAAA6kI/bNREd8mW53Y/s1600-h/Variance_Descriptive_Statistics_Diag_Box_600%25255B4%25255D.jpg"><img title="Variation in Excel - Descriptive Statistics - Variance" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Descriptive Statistics - Variance" src="http://lh3.ggpht.com/-LsWuP5wiThE/VSLYSqCmkbI/AAAAAAAA6kQ/0aIA---S7O0/Variance_Descriptive_Statistics_Diag_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="287" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Clicking OK will produce the following output. The Excel formulas needed to calculate the same results are shown as well.</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje4-WYMFC-f6ibnkClR-U8Ouo93CRjdnJwb6Un9ZF795edPjNuar9j0CYmRfQycIfXKAjxKPWSVx23ihFHHTyBA7WWm4E2Rrk0BjWB-N0FekNfRZ-LqaQjHgTmjx06ULb5rAm-8RxhpMs8/s1600-h/Variance_Descriptive_Statistics_Result_600%25255B4%25255D.jpg"><img title="Variation in Excel - Descriptive Statistics - Variance" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Descriptive Statistics - Variance" src="http://lh6.ggpht.com/-NLjNd_iSCjs/VSLYThA1xpI/AAAAAAAA6kg/qrJAQNfNq5E/Variance_Descriptive_Statistics_Result_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="203" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Note that the calculations of standard deviation and variance are done for samples using Bessel’s correction n-1 as the divisor of the sum of squares and not n. Descriptive Statistics is intended to provide sample statistics and not population parameters.</p> <h2>Sum of Squares</h2> <p><b>SS = Sum of squares</b> - Because variances are additive, the sum of the squares of differences between a mean and data values is used to describe the total variance within specific categories. The Excel formula to calculate the sum of the squares of deviations of sample points from the sample mean is as follows:</p> <p>DEVSQ(<i>data range</i>)</p> <p>This formula is the same for all versions of Excel.</p> <p>SS = DEVSQ(<i>data range</i>) = ∑ (x<sub>i</sub> – x<sub>avg</sub>)</p> <p>ANOVA uses the sum of the squares to calculate the within-sample variance, SS<sub>within-groups</sub>, and the between-sample variance, SS<sub>between-groups</sub>. The F Value for each F test in ANOVA is calculated as follows:</p> <p>F Value = [ SS<sub>between-groups</sub> / df<sub>between-groups</sub> ] / [SS<sub>within-groups</sub> / df<sub>within-groups</sub> ]</p> <p>The p Value for each F test in ANOVA is calculated in Excel as follows:</p> <p>p Value = F.DIST.RT( F Value, df<sub>between-groups</sub> , df<sub>within-groups</sub> )</p> <p>If the calculated p Value is smaller than the specified alpha, the factor that is being evaluated in the F test is deemed to be significant, i.e., has an effect on the data values.</p> <p>Linear regression uses the sum of the squares to calculate total variation (SST), explained variation (SSR), and unexplained variation (SSE). R Square of a linear regression is calculated as follows:</p> <p>R Square = (Explained variation) / (Total variation) = SSR / SST</p> <p align="center"> </p> <h2>MAD</h2> <p>MAD can refer to either Mean Absolute Deviation (sometimes called Average Absolute Deviation) or Median Absolute Deviation. It is necessary to clarify which of the two is being calculated when deriving MAD for a set of data.</p> <p align="center"> </p> <h3>Mean Absolute Deviation</h3> <p>Mean Absolute Deviation is calculated by the following formula:</p> <p>Mean Absolute Deviation = 1/n * ∑ (x<sub>i</sub> – x<sub>avg</sub>)</p> <p>Mean Absolute Deviation can be calculated in all versions of Excel with the following Excel formula:</p> <p>AVEDEV(<i>data range</i>)</p> <p>Following is an example of the calculation of Mean Absolute Deviation using both the formula and a more detailed showing each step of the calculation:</p> <p><a href="http://lh5.ggpht.com/-84XmSrjHVcM/VSLYUOoa9FI/AAAAAAAA6ko/rTLlY2PEvQ0/s1600-h/Variance_Mean_Absolute_Deviation_500%25255B4%25255D.jpg"><img title="Variation in Excel - Mean Absolute Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Mean Absolute Deviation" src="http://lh4.ggpht.com/-J8aA-aI-iuc/VSLYUvPrpgI/AAAAAAAA6kw/sK51U4oP8ck/Variance_Mean_Absolute_Deviation_500_thumb%25255B2%25255D.jpg?imgmax=800" width="356" height="484" /></a> </p> <p align="center"> </p> <h3>Median Absolute Deviation</h3> <p>Median Absolute Deviation relies on the median and is therefore more robust than Mean Absolute Deviation. Outliers have a much smaller effect on the median than on the mean.</p> <p>Median Absolute Deviation is calculated by the following formula:</p> <p>Mean Absolute Deviation = 1/n * ∑ (x<sub>i</sub> – Median)</p> <p>Excel does not have a single formula to calculate Median Absolute Deviation as there is for Mean Absolute Deviation. Each individual step of the calculation must be performed as follows:</p> <p><a href="http://lh5.ggpht.com/-z4kHWQyokkk/VSLYVKryAdI/AAAAAAAA6k4/uie0KEQSAaA/s1600-h/Variance_Median_Absolute_Deviation_600%25255B4%25255D.jpg"><img title="Variation in Excel - Median Absolute Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Median Absolute Deviation" src="http://lh5.ggpht.com/-tjogqnO7buE/VSLYVgEQKSI/AAAAAAAA6lA/F7U7TjDl6PI/Variance_Median_Absolute_Deviation_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="463" /></a> </p> <h2>Standard Deviation of a Continuous Distribution</h2> <p>This article has so far discussed the calculation of variation of discrete sets of data. The standard deviation of a population of data distributed according to probability function p(x) can be stated in general terms as follows:</p> <p><a href="http://lh5.ggpht.com/-sONRkSI8KwU/VSLYWMgClxI/AAAAAAAA6lI/hFqWWYnkqa0/s1600-h/Standard_deviation_continuous_variable%25255B4%25255D.gif"><img title="Variation in Excel - Standard Deviation of a Continuous Variable" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Standard Deviation of a Continuous Variable" src="http://lh3.ggpht.com/-HvWm-rkU7ko/VSLYWajT6KI/AAAAAAAA6lQ/yZR-bYo21P8/Standard_deviation_continuous_variable_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="60" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Many of the individual statistical distributions use specific, unique formulas to calculate standard deviation. For example, the variance of data that are distributed according to the binomial distribution is calculated by the formula <i>np(1-p)</i>.</p> <p align="center"> </p> <h2>Effect of Changing Units</h2> <p>Unit can be changed by either adding a constant to each data measurement or by multiplying each data measurement by a constant. These two methods of altering the units of measuring have different effects on the measured variation for the data set as follows:</p> <h3>Adding a Constant to Each Data Value</h3> <p>Adding a constant to each data value does not change the distance between data values. All measures of variability therefore remain unchanged.</p> <h3>Multiplying Each Data Value By a Constant</h3> <p>Multiplying each data value by a constant multiplies range and standard deviation by the value of the constant. The variance is multiplied by the square of the constant.</p> <p align="center"> </p> <h2>Range</h2> <p>Range of a data set simply equals the highest data value minus the lowest data value. Excel does not have a formula that specifically calculates the range of a data set. This must be implemented in Excel as follows:</p> <p>Range = MAX(<i>data range</i>) – MIN(<i>data range</i>)</p> <p>Range is sensitive only to changes in the outermost data values but not changes to any other data values.</p> <p align="center"> </p> <h2>Interquartile Range</h2> <p>Quartiles divide rank-ordered data into four equal parts, i.e., quartiles Q1, Q2, Q3, and Q4 which contains the highest values. Each quartile represent a range into which one quarter of the data points fall. The upper and lower endpoints of each quartile are not data points of the data set but are the borders of the interval that contains one quarter of the data points. </p> <p>The interquartile range is the width of the middle two quartiles combined. This is the distance between the upper boundary of Q2 and the lower boundary of Q3.</p> <p>The boundaries between the quartiles are calculated in Excel using the QUARTILE.EXC() as shown in the following image. The highest and lowest data points within each quartile are calculated in Excel using MIN(), MAX(), INDEX, and MATCH() as shown in the following:</p> <p><a href="http://lh5.ggpht.com/-7IfT35LfWz8/VSLYXIbSdLI/AAAAAAAA6lY/hN0I5HgDHds/s1600-h/Variance_Quartiles_1_600%25255B4%25255D.jpg"><img title="Variation in Excel - Quartiles" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Quartiles" src="http://lh6.ggpht.com/-0-gfkzwpSwM/VSLYXpjR6sI/AAAAAAAA6lg/xvbn1C7o4vE/Variance_Quartiles_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="201" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Here is are close-up images of these Excel commands.</p> <p><a href="http://lh4.ggpht.com/-GYWGQRucqn0/VSLYX0iq-sI/AAAAAAAA6lo/mDli7o_jThU/s1600-h/Variance_Quartiles_closeup_1_600%25255B4%25255D.jpg"><img title="Variation in Excel - Quartiles Close-up" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Quartiles Close-up" src="http://lh6.ggpht.com/-QReSw0b5DPk/VSLYYjdlGoI/AAAAAAAA6lw/5x1uSUIeuYk/Variance_Quartiles_closeup_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="315" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh5.ggpht.com/-s-5yGMlKf2g/VSLYZCIaGtI/AAAAAAAA6l4/LjIgcoCSDaI/s1600-h/Variance_Quartiles_closeup_2_300%25255B4%25255D.jpg"><img title="Variation in Excel - Quartiles Close-up" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Quartiles Close-up" src="http://lh5.ggpht.com/-9dKhrqnQVHc/VSLYZieozII/AAAAAAAA6mA/mFTjdRynUII/Variance_Quartiles_closeup_2_300_thumb%25255B2%25255D.jpg?imgmax=800" width="309" height="484" /></a> </p> <p>The Interquartile range is calculated in Excel 2010 and later with the following:</p> <p>Interquartile range = QUARTILE.EXC(<i>data range</i>,3) – QUARTILE.EXC(<i>data range</i>,1)</p> <p>In Excel 2007 and prior, the Interquartile range is calculated using QUARTILE(). In Excel 2010 QUARTILE() is replaced by QUARTILE.INC(). This produces a slightly different result than the other Excel 2010 function QUARTILE.EXC().</p> <p>In Excel 2007 the Interquartile range is calculated as follows:</p> <p>Interquartile range = QUARTILE(<i>data range</i>,3) – QUARTILE(<i>data range</i>,1)</p> <p>It should be noted that the median of a data set represent the boundary between the 2<sup>nd</sup> and 3<sup>rd</sup> quartiles.</p> <p>The Excel algorithms for the 3 quartile functions – QUARTILE(), QUARTILE.INC(), and QUARTILE.EXC() – are shown as follows:</p> <p><a href="http://lh5.ggpht.com/-6INgVxO4Fy8/VSmeH6i84NI/AAAAAAAA6ok/1j6crHjmylw/s1600-h/Variance_Quartile_INC_600%25255B4%25255D.jpg"><img title="QUARTILE(), QUARTILE.INC() algorithms" style="border-top: 0px; border-right: 0px; border-bottom: 0px; border-left: 0px; display: inline" border="0" alt="QUARTILE(), QUARTILE.INC() algorithms" src="http://lh3.ggpht.com/-podCCsCqiKo/VSmeIYxMYsI/AAAAAAAA6os/e93QJzRMDQQ/Variance_Quartile_INC_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="210" /></a> </p> <p><a href="http://lh6.ggpht.com/-YfxSYLEEbfo/VSmeI7TpcvI/AAAAAAAA6o0/PzOVWyBEybo/s1600-h/Variance_Quartile_EXC_600%25255B7%25255D.jpg"><img title="QUARTILE.EXC() algorithm" style="border-top: 0px; border-right: 0px; border-bottom: 0px; border-left: 0px; display: inline" border="0" alt="QUARTILE.EXC() algorithm" src="http://lh6.ggpht.com/-0sfe0upr3QE/VSmeJTEzyhI/AAAAAAAA6o8/1ZbrPTsnPYM/Variance_Quartile_EXC_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="201" /></a> </p> <p> </p> <h2>Visual Presentation of Variation in Excel</h2> <p>Two tools in Excel that provide a visual representation of the spread of one-dimensional data set are the histogram and the box plot. The spread of two-dimensional data (X-Y data) can be visually presented using a scatterplot graph in Excel.</p> <p align="center"> </p> <h3>Histogram in Excel</h3> <p>A histogram is a bar chart with each bar showing the frequency of occurrence of data values within a specified range. The ranges are called bins. Creating a histogram in Excel requires specifying the upper limits of each bin as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsYLUR9ZEt6SkcrQOof30z-WOIpHnnS1U1wIwuJqSWvr4W9ATsFv7pZ9RuTBVLvbpdijbImW54QTAgLCRzRSQfW2RNFHF967b8gjJstxbUSn4KFW-QVwfciiVUF4jtuby8eqU4_XjKIpcb/s1600-h/Variance_Histogram_Raw_Data_600%25255B4%25255D.jpg"><img title="Variation in Excel - Histogram Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Histogram Raw Data" src="http://lh4.ggpht.com/-z8PFrJy9tx4/VSLYavBrv0I/AAAAAAAA6mQ/KDv7_D265ac/Variance_Histogram_Raw_Data_600_thumb%25255B2%25255D.jpg?imgmax=800" width="296" height="484" /></a> </p> <p>The Excel histogram is one of the tools of the Data Analysis toolpak. The Analysis toolpak is an Excel add-in that is initially inactive in Excel. The user must activate this add-in prior to its first use. The Excel histogram tool can be accessed as follows:</p> <p><b>Data tab / Data Analysis / Histogram</b></p> <p>Doing so brings up the Histogram dialogue box which should be completed as follows: </p> <p><a href="http://lh4.ggpht.com/-kPVB5syvN5E/VSLYa8SSfgI/AAAAAAAA6mY/7vKtitNozDk/s1600-h/Variance_Histogram_Dialogue_Box_600%25255B4%25255D.jpg"><img title="Variation in Excel - Histogram Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Histogram Dialogue Box" src="http://lh5.ggpht.com/-Iw8Th4r0HsI/VSLYbjkYdhI/AAAAAAAA6mc/skFc6_PKsv4/Variance_Histogram_Dialogue_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="287" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Clicking OK produces the following Excel histogram.</p> <p><a href="http://lh4.ggpht.com/-b-Fd9UTntaA/VSLYcHs1B5I/AAAAAAAA6mo/TxlUQSL21I4/s1600-h/Variance_Histogram_600%25255B11%25255D.jpg"><img title="Variation in Excel - Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Histogram" src="http://lh3.ggpht.com/-0z7PRFToXR8/VSLYcoNqoEI/AAAAAAAA6mw/t8Y9fBei450/Variance_Histogram_600_thumb%25255B7%25255D.jpg?imgmax=800" width="404" height="185" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The following two articles in this blog provide detailed descriptions on the following histogram topics:</p> <ol> <li> <p style="orphans: 0; widows: 0">Creating a histogram in Excel using the histogram tool:</p> </li> </ol> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-create-histogram-in-excel-2010_27.html">http://blog.excelmasterseries.com/2014/05/how-to- create-histogram-in-excel-2010_27.html</a></p> <ol> <li> <p style="orphans: 0; widows: 0">Creating an automatically updating histogram in 7 steps in Excel using formulas and a bar chart:</p> </li> </ol> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-create-histogram-in-excel-2010.html">http://blog.excelmasterseries.com/2014/05/how-to-create- histogram-in-excel-2010.html</a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <h3>Frequency Tables</h3> <p>The Excel Histogram tool produces a frequency table that is part of its output. The frequency table displays a count of the number of data points that fall into each bin. This is the basis of the histogram chart shown as follows:</p> <p><a href="http://lh5.ggpht.com/-umgUsadWkWw/VSLYdDbGOxI/AAAAAAAA6m4/LwouvlccImM/s1600-h/Variance_Histogram_600%25255B13%25255D.jpg"><img title="Variation in Excel - Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Histogram" src="http://lh6.ggpht.com/-Lh92vhS2HVM/VSLYddz-4zI/AAAAAAAA6nA/YRWcTjARnko/Variance_Histogram_600_thumb%25255B9%25255D.jpg?imgmax=800" width="404" height="185" /></a> </p> <p>A stand-alone frequency table can also be produced in Excel using the FREQUENCY() formula. The FREQUENCY() formula is an array formula that is implemented as follows.</p> <p align="center"> </p> <p><b>Step 1 – Highlight the Cells That Will Contain the Output</b></p> <p>The raw data should be sorted in a column and the upper limits of each bin should also be listed in a column as shown in the following image.</p> <p>The FREQUENCY() formula produces a frequency table in a separate data column that contains 1 more cell than the number of bins. The upper limits of six bins are listed in this example so there are six total bins. The FREQUENCY() formula will therefore require a column of seven cells to hold its output. </p> <p>The FREQUENCY() is an array formula. The first step in implementing an array formula in Excel is to highlight (select) the cells that will hold the output. Click and drag the mouse to highlight seven empty cells in a column where the output should go. </p> <p><a href="http://lh4.ggpht.com/-fHSFrj0Aik4/VSLYdzcGuCI/AAAAAAAA6nI/D8IcNecHVhI/s1600-h/Variance_Frequency_1_600%25255B4%25255D.jpg"><img title="Variation in Excel - " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variance_Frequency_1_600" src="http://lh4.ggpht.com/-YgWSBq49B5c/VSLYefxEbKI/AAAAAAAA6nQ/FavNVyQJwpg/Variance_Frequency_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="429" /></a> </p> <p align="center"> </p> <p><b>Step 2 – Type in the Formula</b></p> <p>While the seven cells remain highlighted, type in the FREQUENCY() formula as shown.</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyHc6Qw7Gwl1hGKvcm-huII5wEAH1oxMNFFmo-pWJXwX0y7k0Ucks0lESDfJ6NVCQrfz_G3cRTwwWqPNiigh0ppaN28x2VHrheru18fv4wJD0ozUmKWTMcA8LN5VQdI7g1V75jykE8QESF/s1600-h/Variance_Frequency_2_600%25255B4%25255D.jpg"><img title="Variation in Excel - Frequency" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Frequency" src="http://lh4.ggpht.com/-kcjWOqO5JeM/VSLYfcroPeI/AAAAAAAA6ng/RguRNZPAiUU/Variance_Frequency_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="190" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <p><b>Step 3 – Type CTRL – SHIFT – ENTER at the Same Time</b></p> <p>Doing so will produce the frequency table in cells H4:H10 that have been highlighted.</p> <p><a href="http://lh5.ggpht.com/-CYOHsBWE0Gk/VSLYft8o-iI/AAAAAAAA6no/-eEx5kxM2jM/s1600-h/Variance_Frequency_3_600%25255B4%25255D.jpg"><img title="Variation in Excel - Frequency" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Frequency" src="http://lh3.ggpht.com/-1Z_0jCMi3Xc/VSLYgXg7bzI/AAAAAAAA6nw/nTFAnECeHq4/Variance_Frequency_3_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="182" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h3>Box Plots</h3> <p>Box plots visually display the spread of data in each of the four quartiles. The interquartile range, Q2 and Q3, are contained the two boxes. The outer quartiles, Q1 and Q4, are displayed as whiskers above and below the boxes. Q4 is displayed as the upper whiskers and Q1 is displayed as the lower whisker. For this reason boxplots are sometimes called Box-and-Whisker plots. </p> <p>The boundary between the upper box (Q3) and the lower box (Q2) is the median. Box plot diagrams can also be constructed in Excel to display the data’s mean as well. An example of a box plot in Excel and the data from which it was derived is shown as follows:</p> <p></p> <p><a href="http://lh3.ggpht.com/-hTrQy0wxXqA/VSLYgrACDTI/AAAAAAAA6n4/o3aFyAlk128/s1600-h/Variance_Box_Plot_600%25255B4%25255D.jpg"><img title="Variation in Excel - Box Plot" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Box Plot" src="http://lh5.ggpht.com/-2yfRg6uM_S4/VSLYhFjuDII/AAAAAAAA6oA/RbE6cp8rjZI/Variance_Box_Plot_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="295" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The following article in this blog provides step-by-step instructions on how to produce the preceding box plot in 8 steps in Excel:</p> <p><a href="http://blog.excelmasterseries.com/2015/02/box-plots-in-8-steps-in-excel.html">http://blog.excelmasterseries.com/2015/02/box-plots-in-8-steps- in-excel.html</a></p> <p align="center"> </p> <h3>Scatter Plot in Excel</h3> <p>A scatter plot is a diagram using Cartesian coordinates to display the X-Y values of a two-dimensional data set. The Y variable displayed on the vertical axis is often the dependent or response variable while the X variable displayed on the horizontal axis is related to that response variable. The scatter plot is also referred to also referred to as a scatter graph, scatter diagram, or scatter chart.</p> <p>The following Excel scatter plot displays data points which contain a dependent variable (number of parts produced) as the X value and the residual associated with each dependent variable as the Y value. This scatter plot indicates that the residuals have slightly increasing variance as the dependent variable increases in value. This is indicated by the slight fanning-out of the Y values (the residual values) as the X variable (number of parts produced) increases. This scatter plot was used as part of a single-variable linear regression to determine whether one of the required conditions of linear regression, namely that the residual have constant variance, is met. The slight increase in variance of residuals as the dependent variable increases is not significant enough to invalidate the regression that was performed.</p> <p><a href="http://lh5.ggpht.com/-Qbx_YCE00Go/VSLYhmUw4-I/AAAAAAAA6oI/moloLcHSZVs/s1600-h/Variance_Scatter_Plot_600%25255B4%25255D.jpg"><img title="Variation in Excel - Scatter Plot" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Variation in Excel - Scatter Plot" src="http://lh5.ggpht.com/-yH9nXJMFmNw/VSLYiJcKVPI/AAAAAAAA6oQ/5LYwvi127LM/Variance_Scatter_Plot_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="276" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p align="center"> </p> <h2>Variation in Statistical Process Control</h2> <p>Every measurable process contains variation. This variation can be classified in the following two categories:</p> <p><font face="Times New Roman, serif"><font style="font-size: 13pt" size="4"><b>Controlled Variation</b></font></font> – This type of variation is consistent and stable over time and is the result of random statistical variation that is always present at least to some degree. </p> <p><font face="Times New Roman, serif"><font style="font-size: 13pt" size="4"><b>Uncontrolled Variation</b></font></font> – This type of variation changes over time and is attributed to assignable causes. It is the job of management to determine the assignable causes of uncontrolled variation and reduce them as much as possible. </p> <p>Variation in a measurable process can be displayed on control charts. Control charts in SPC (statistical process control) provide information about samples taken from a process. Individual samples are called subgroups. A subgroup is specific number of measurements periodically taken at a single point in a process. </p> <p>An SPC control chart plots the location and dispersion of each subgroup. An SPC control chart consists of a pair of graphs; an upper graph charts the location of each successive subgroup and a lower graph charts the dispersion of each successive subgroup. Two common SPC control charts are the following:</p> <p><font face="Times New Roman, serif"><font style="font-size: 13pt" size="4"><b>X-Bar / R Charts</b></font></font> are running records of subgroup average value (X-Bar) and subgroup range (R). Subgroups usually consist of 3 to 5 data measurements. </p> <p><font face="Times New Roman, serif"><font style="font-size: 13pt" size="4"><b>X-Bar / S Charts</b></font></font> are running records of subgroup average value (X-Bar) and subgroup standard deviation (S). Subgroups usually consist of 8 to 10 data measurements. </p> <p>A process can be deemed to be OOC (out of control) if the location or dispersion values of subgroups fall outside of defined upper or lower control limits or show patterns. The upper and lower control limits are approximately three sigma from the mean value being measured. One sigma is equal to the length of one standard deviation.</p> <p>The interval between subgroups should be set so that changes between subgroups will be maximized. Subgroups should be taken over an interval that is long enough for potential variation to show up. The sampling interval between individual samples within the subgroups should be set so that variation within subgroups is minimized. Subgroup size should be held constant. Typically at least 20 subgroups are needed to judge statistical control and provide a reasonable estimate of parameters of mean and variation.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com71tag:blogger.com,1999:blog-3568555666281177719.post-64568791102619176102015-04-04T13:51:00.001-07:002015-04-08T10:50:57.821-07:00Probability in Excel<h1>Probability in Excel</h1> <h2>Probability Overview</h2> <p>The probability of an event occurring is expressed on a linear scale between 0 and 1. A probability of 0 indicates that there is 0-percent chance of the event occurring and a probability of 1 indicates that there is a 100-percent chance of the event occurring. </p> <p align="center"> </p> <h3>Experiment or Trial</h3> <p>A process that produces a single outcome that is uncertain. An example of an experiment would be the single roll of one dice.</p> <p align="center"> </p> <h3>Outcome</h3> <p>Any possible result of a single trial or experiment. An outcome of a random selection of a single card from a deck of cards has 52 possible outcomes. The sum of the probabilities of all possible outcomes equals 1. The sum of the probability of all possible outcomes can be expressed as follows:</p> <p><a href="http://lh3.ggpht.com/-IE-Ehkr4gaQ/VSBOjyOT5RI/AAAAAAAA6Ts/gSkSjEuaeL8/s1600-h/Probability_Cumulative_All_Possible_Outcomes%25255B8%25255D.gif"><img title="Probability in Excel - Cumulative Probability of All Possible Outcomes" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Cumulative Probability of All Possible Outcomes" src="http://lh3.ggpht.com/-6gdTBQiLtrY/VSBOkZKV1-I/AAAAAAAA6T0/V5VYp88bXE8/Probability_Cumulative_All_Possible_Outcomes_thumb%25255B4%25255D.gif?imgmax=800" width="132" height="68" /></a> </p> <p>If each possible outcome has the same probability of occurring as any other outcome, the probability of each and every outcome, P(E<sub>i</sub>), can be calculated as follows:</p> <p>P(E<sub>i</sub>) = 1 / (number of all possible outcomes)</p> <p>The probability of randomly selecting any specific card from a 52-card deck is equal to 1/52 = 0.0192. This means that there is a 1.92-percent chance of randomly drawing a specific card such as a Queen of Hearts.</p> <p>P(E<sub>i</sub>) = 1 / (number of all possible outcomes)</p> <p>P(Queen of Hearts) = 1/52 = 0.0192 = 1.92%</p> <p>A generalization of concept occurs when objects are distributed according to the universal distribution. All of the possible outcomes contain the same number of objects if a group of objects is universally distributed. Suits of cards in a regular deck are universally distributed because a deck contains 13 diamonds, 13 hearts, 13 clovers, and 13 spades.</p> <p>P(E<sub>i</sub> for universally distributed objects) = (number of objects in each outcome)/(total number of objects)</p> <p>For example, the probability of randomly selecting a card of any specific suit, say queens, would be the following:</p> <p>Pr(Queen) = (number of cards in each suit)/(total number of cards) = 13/52 = 0.25 = 25%</p> <p align="center"> </p> <h3>Sample Space</h3> <p>The set of all possible outcomes of an experiment is called the sample space. The set of all possible outcomes of an experiment are sometimes referred to as the sample space outcomes. This Excel-generated diagram illustrates the sample space of the possible outcomes of the genders of two successive children from a single set of parents:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzha0shLYBv-r4vA4XOH6Vm_oiJ3pAdMQ_5a1dPsCm5L_UEEkZArQ6kSD1N3Hxu8rEJ_ZEA0gcN8JCY_elRQ2eBQdAtIJZRXZp_TzaA8Y8bdmzxeFzuA1KNdQrBWzZky9TnWAHEsMspKx9/s1600-h/Prob_Sample-space_600%25255B4%25255D.jpg"><img title="Probability in Excel - Probability Sample Space" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Probability Sample Space" src="http://lh3.ggpht.com/-b74ZVsH7Obk/VSBOmEVaqiI/AAAAAAAA6UE/dcGs4Yi_T3w/Prob_Sample-space_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="295" /></a> </p> <p>The preceding diagram is called a <u><b>Tree Diagram</b></u>. If the first trial has n<sub>1</sub> possible outcomes and each of those outcomes has the same n<sub>2</sub> possible outcomes, then final total number of possible outcomes for the 2nd trial = n<sub>1</sub> * n<sub>2</sub>. In this case n<sub>1</sub> = n<sub>2</sub> = 2. There are 4 possible outcomes in the sample space after the 2<sup>nd</sup> trial. </p> <p>The number of possible outcomes after the kth trial equals n<sub>1</sub> * n<sub>2</sub> * n<sub>3</sub> * … n<sub>k</sub>. </p> <p>Another example would be a man with 2 jackets and 4 shirts. This man has 8 possible shirt-jacket combinations he can choose from.</p> <p align="center"> </p> <h3>Event</h3> <p>An event is the combination of one or more outcomes that have a single defining characteristic. An event is the set of one or more sample space outcomes that have a defining characteristic. An event derived from the preceding diagram would be the event of at least one of the two children being a girl. This event would be described by the following sample set:</p> <p>Event A = At least one child is a girl = {BG, GB, GG}</p> <p>Pr(Event A) = Pr(at least one child is a girl) = Pr(BG) + Pr(GB) + Pr(GG) = 0.25 + 0.25+ 0.25 = 0.75</p> <p>Another example of an event is the drawing of a red queen from a deck of playing cards. This event is defined the by following set of possible outcomes:</p> <p>Event A = Red Queen = {Queen of Hearts, Queen of Diamonds}</p> <p>P(Queen of Hearts) = 1/52 = 0.0192 = 1.92%</p> <p>P(Queen of Hearts) = P(Queens of Diamonds) = 0.0192</p> <p>Event A = Drawing a Red Queen = Drawing a Queen of Hearts or Queen of Diamonds</p> <p>Probability of Event A = Sum of probabilities of outcomes that are described by the event</p> <p>P(A) = P(Queen of Hearts) + P(Queen of Diamonds)</p> <p>P(A) = 0.0192 + 0.0192 = 0.0385 = 3.85%</p> <p>When calculating the probability of a specific event, it is important to determine whether the event is defining by only one single, possible outcomes or if the event is defined by more than two or more unique outcomes of a single trial or experiment. </p> <p>The probability of Event A is denoted as P(A).</p> <p>The term <i>event</i> is often used interchangeably with the term <i>outcome</i>, which is used to describe one unique possible outcome of an experiment. Although this is not often done, it is a good idea to specify whether the term <i>event</i> is being used to describe a single, unique outcome or is being used to describe a combination of single, unique outcomes. Note that the formula define the probability of an event use the term E<sub>i</sub> to denote outcome i as follows:</p> <p><a href="http://lh4.ggpht.com/-j5HrF_QETSM/VSBOmfwyoAI/AAAAAAAA6PE/kShfw8MFXL4/s1600-h/Probability_Cumulative_All_Possible_Outcomes%25255B5%25255D.gif"><img title="Probability in Excel - Cumulative Probability of All Possible Outcomes" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Cumulative Probability of All Possible Outcomes" src="http://lh3.ggpht.com/-ak3xH4akrtE/VSBOm5nDq2I/AAAAAAAA6PQ/9UXGz67Wv3E/Probability_Cumulative_All_Possible_Outcomes_thumb%25255B1%25255D.gif?imgmax=800" width="132" height="68" /></a> </p> <p align="center"> </p> <h3>Complimentary Event</h3> <p>The complimentary event of Event A is the event of an experiment not producing any of the outcomes which define Event A. This is the event of Event A not occurring. The complementary event of event A is denoted as Event <a href="http://lh6.ggpht.com/-jpRNIBvNu8E/VSBOnE5p-CI/AAAAAAAA6PY/3yhTYOAk7fw/s1600-h/Compliment_of_Event_A%25255B2%25255D.gif"><img title="Compliment of Event A" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Compliment of Event A" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWErZ7LkZmBnwb3gDwgdXIQ0sbHTkNMhOas0j_BolJuQmJddhidiUoygBiiReOgYcw6HNVLus-wSvImlKnECq06-GqnaCWuKbRPJw1oseBeCgd66M0Myypy4b5sJtB7nhnZVtV0IXj2xxt/?imgmax=800" width="18" height="22" /></a> . The complimentary event of Event A contains every possible outcome that is not part of the set of outcomes that are associated with Event A. This Excel-generated Venn Diagram illustrates an event and its complementary event:</p> <p><a href="http://lh4.ggpht.com/-zWpjZWnhOAE/VSBOoJA_NOI/AAAAAAAA6UM/PrtKVH_Y_4c/s1600-h/Prob_Venn_A_Compliment_of_A_600%25255B4%25255D.jpg"><img title="Probability in Excel - Venn Diagram Compliment of A" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram Compliment of A" src="http://lh6.ggpht.com/-d131geAGIN0/VSBOomlDOGI/AAAAAAAA6UU/GRqvIfEGmXw/Prob_Venn_A_Compliment_of_A_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="209" /></a> </p> <p><a href="http://lh4.ggpht.com/-yslBwasYpmQ/VSBOpNhPcfI/AAAAAAAA6Uc/29scdjb7GRA/s1600-h/Probability_Compliment_of_Event_A%25255B4%25255D.gif"><img title="Probability in Excel - Probability of Compliment of Event A" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Probability of Compliment of Event A" src="http://lh4.ggpht.com/-Mlj3XV-Vd7A/VSBOpuO1vfI/AAAAAAAA6Uk/ARRXmCVCIRA/Probability_Compliment_of_Event_A_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="21" /></a> </p> <p>Event A = Drawing a Red Queen</p> <p>P(A) = 0.0385</p> <p>Compliment of Event A = <a href="http://lh5.ggpht.com/-HHxQjHodZgY/VSBOqEXkTiI/AAAAAAAA6Us/sGb9ynT-dWE/s1600-h/Compliment_of_Event_A%25255B6%25255D.gif"><img title="Probability in Excel - Compliment of Event A" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Compliment of Event A" src="http://lh6.ggpht.com/-DQ3B8LCbr_s/VSBOqrbDI8I/AAAAAAAA6U0/V2YWw-X66vE/Compliment_of_Event_A_thumb%25255B2%25255D.gif?imgmax=800" width="18" height="22" /></a> = Not drawing a Red Queen</p> <p>Sometimes the Compliment of Event A is denoted as A' for ease of presentation because many word processing programs, such as MS Word, do not make it simple to create an overline, i.e., a line over a character.</p> <p>P(A') = 1 – P(A) = 1 – 0.0385 = 0.9615 = 96.15%</p> <p>There is a 96.15-percent chance of the compliment of event A not occurring, i.e., that a single card randomly selected from a 52-card deck is not a red queen. Correspondingly, there is a 3.85% chance of event A occurring, i.e., that a single card randomly selected from a 52-card deck is a red queen.</p> <p align="center"> </p> <h3>Independent Events</h3> <p>Events are independent of each other is the occurrence of one of the events does not affect the probability of the any of the other events occurring during any successive trials. Events described by the drawing of specific cards from a deck are independent if the drawn card is replaced back into the deck immediately after each draw. If the drawn card is not replaced back into the deck, the events are not independent because of the reduced set of possible outcomes after each draw. Successive flips of a fair coin are independent events.</p> <p align="center"> </p> <h3>Short-Term Relative Frequency</h3> <p>The short-term relative frequency = (number of actual occurrences of event)/(actual number of trials)</p> <p>If 10 flips of a fair coin have produced 2 heads, the short-term relative frequency of heads is 2/10 = .2 = 20%.</p> <p align="center"> </p> <h3>Long-Term Relative Frequency</h3> <p>P(E), the calculated expected probability that event E will occur, should always be considered to be a long-term relative frequency. The short-term relative frequency will approach the long-term relative frequency as the number of trials increase. Random variance can cause the short-term frequency to differ significantly from the long-term relative frequency (P(E) – the expected probability). </p> <p align="center"> </p> <h3>Mutually Exclusive Events</h3> <p>Events are mutually exclusive of each other if the occurrence of one event precludes the occurrence of any of the other events during a single trial. Events are mutually exclusive if the sample space outcomes that define each event do not contain any of the same individual outcomes that can occur in a single trial. In other words each of the possible outcomes of a single trial is included in the sample set of only one of the events. The event of drawing a red card is mutually exclusive of the event of drawing a black card. </p> <p>This Excel-generated Venn Diagram illustrates two mutually exclusive events:</p> <p><a href="http://lh6.ggpht.com/-pyx8j7RkBAM/VSBOq4F2wUI/AAAAAAAA6U8/x8PNWvtz9Gk/s1600-h/Prob_Venn_Mutually_Exclusive_600%25255B7%25255D.jpg"><img title="Probability in Excel - Venn Diagram Mutual Exclusivity" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram Mutual Exclusivity" src="http://lh4.ggpht.com/-yh3O0r-nlac/VSBOrhwVACI/AAAAAAAA6VE/sW4xoRhdK58/Prob_Venn_Mutually_Exclusive_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="207" /></a> </p> <p>This Excel-generated Venn Diagram illustrates two non-mutually exclusive events:</p> <p><a href="http://lh3.ggpht.com/-C98B1fXfCSc/VSBOsJF6ajI/AAAAAAAA6VM/De9U9sep5is/s1600-h/Prob_Venn_A_OR_B_600%25255B7%25255D.jpg"><img title="Probability in Excel - Venn Diagram A OR B" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram A OR B" src="http://lh4.ggpht.com/-C0tJKYLOsm0/VSBOs0J4FUI/AAAAAAAA6VU/Dd7fOoroJLs/Prob_Venn_A_OR_B_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="209" /></a> </p> <p align="center"> </p> <h3>Independence of Events Does Not Mean Mutual Exclusivity of Events, and Vice Versa</h3> <p>Events can be independent and not mutually exclusive. If event A is defined a drawing a Red Queen and Event B is defined as drawing any red card, these event are independent of each other if the drawn card is replaced back in the deck before the next draw. Randomly drawing a Red Queen on the first draw would not change the probability of randomly drawing a red card on the next draw if the first drawn card is replaced back into the deck before the second draw. These events are not mutually exclusive because the events of drawing a red card contains the same set of outcomes that can occur if a Red Queen is drawn.</p> <p>The events of drawing a red card and drawing a black card are mutually exclusive because none of the outcomes in the sample space of the event of drawing a red card are the same as any of the outcomes in the sample space of the event of drawing a black card. If the first drawn card is not replaced, then the events are not independent of each. The removal of a single red or black card from a deck changes the probability of drawing either a black card or red card on successive draws.</p> <p align="center"> </p> <h4>Sampling With Replacement Ensures That Successive Trials Are Independent</h4> <p>When the sample taken is immediately placed randomly back into the population, the population remains unchanged. The probability of any specific outcome remains unchanged in successive trials when the population is returned to its original state as a result of sampling with replacement. The binomial distribution is often used to calculate the probability of a positive binary event occurring when samples are replaced. An example is as follows:</p> <p>Given a regular deck of 52 playing cards, calculate the probability that 7 out of 10 cards randomly sampled will be red if each sample is immediately replaced before the next one is taken. This probability is calculated using the following Excel formula:</p> <p>Pr(X ≤ k) = BINOM.DIST(k, n, p, TRUE)</p> <p>X = the actual number of times a positive binary occurrence (a red card) occurs during the sampling</p> <p>k = the specified number of positive occurrences (red cards) = 7</p> <p>n = the sample size = the total number of trials (cards sampled) = 10</p> <p>p = probability of a positive occurrence = 0.50 = 50% chance of a red card in a regular deck</p> <p>TRUE indicates the cumulative distribution function, i.e., <i><u>UP TO</u></i> 7 red cards in the sample </p> <p>Pr(X ≤ 7) = BINOM.DIST(7,10,0.50,TRUE) = 0.945313 = 94.53 percent chance of up to 7 red cards in 10 sample drawn from a regular deck if sample are replaced.</p> <p align="center"> </p> <h4>Sampling Without Replacement Ensures That Successive Trials Are Not Independent</h4> <p>When the samples replaced back into the population, the population changes after every sample. The current state of the population is dependent on the outcomes of all of the previous samples. Sampling without replacement ensures that successive trials are not independent of each other. The hypergeometric distribution is often used to calculate the probability of a positive binary event occurring when samples are not replaced. An example is as follows:</p> <p>Given a regular deck of 52 playing cards, calculate the probability that 7 out of 10 cards randomly sampled will be red if samples are not replaced. The probability is calculated using the following Excel formula:</p> <p>Pr(X ≤ k) = HYPGEOM.DIST(k, n, K, N, TRUE)</p> <p>X = the actual number of times a positive binary occurrence (a red card) occurs during the sampling</p> <p>k = the specified number of positive occurrences (red cards) = 7</p> <p>n = the sample size = the total number of trials (cards sampled) = 10</p> <p>K = the number of positive occurrences in the population at the start of sampling = 26</p> <p>N = the population size at the start of the sampling = 52</p> <p>TRUE indicates the cumulative distribution function, i.e., <i><u>UP TO</u></i> 7 red cards in the sample </p> <p>Pr(X ≤ 7) = HYPGEOM.DIST(7,10,26,52,TRUE) = 0.9624324 = 96.24 percent chance of up to 7 red cards in 10 sample drawn from a regular deck if samples are not replaced.</p> <p align="center"> </p> <h2>The Union of Two Events = A <i><u>OR</u></i> B = A U B</h2> <p>The union of two events A and B is the probability either A or B will occur. The union or addition of two events is denoted as follows:</p> <p><a href="http://lh3.ggpht.com/-5kHqCwSWc7I/VSBOtPDOFOI/AAAAAAAA6Vc/CfbAy7SBZ0M/s1600-h/Probability_OR%25255B3%25255D.gif"><img title="Probability in Excel - OR" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - OR" src="http://lh6.ggpht.com/-8yxGIRVnVcM/VSBOtsSrwZI/AAAAAAAA6Vk/c7GGoaeR6Pk/Probability_OR_thumb%25255B1%25255D.gif?imgmax=800" width="404" height="38" /></a> </p> <p> <br /> <br /></p> <h3>Addition of N Mutually Exclusive Events</h3> <p>The addition of mutually exclusive events is calculated by the following formula:</p> <p><a href="http://lh4.ggpht.com/-RFDdEm4VWKs/VSBOuOvHlxI/AAAAAAAA6Vs/Pg-osEIrtWA/s1600-h/Probability_Addition_Rule_Mutually_Exclusive_Events%25255B3%25255D.gif"><img title="Probability in Excel - Addition Rule For Mutually Exclusive Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Addition Rule For Mutually Exclusive Events" src="http://lh5.ggpht.com/-LY1hCU8QKFk/VSBOvPZMU_I/AAAAAAAA6V0/d9_wUMr8lxk/Probability_Addition_Rule_Mutually_Exclusive_Events_thumb%25255B1%25255D.gif?imgmax=800" width="404" height="22" /></a> </p> <p>The following Excel-generated Venn Diagram illustrates the addition of two mutually exclusive events.</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrkjHN_miI9Ld4bMVInKM059vTQB-_jfEsY43QefwJqEOQ8Em65Dinr4ycA0942qRX1K8LT7Yaoj8gt1-6GmgiUdKCHPu8iGnhsndCMgQLFGlY3ZsndQoGt8v4YG2ua7Z1LnfdCNNL-eyw/s1600-h/Prob_Venn_Mutually_Exclusive_600%25255B9%25255D.jpg"><img title="Probability in Excel - Venn Diagram For Mutually Exclusive Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram For Mutually Exclusive Events" src="http://lh5.ggpht.com/-4JCBYf_WcQk/VSBOv35lbNI/AAAAAAAA6WE/YFwjT2j0ptc/Prob_Venn_Mutually_Exclusive_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="207" /></a> </p> <p align="center"> </p> <h2>The Intersection of Two Events = A <i><u>AND</u></i> B = A ∩ B</h2> <p>The intersection of two events A and B is the probability of both A and B occurring. The intersection or multiplication of two events is denoted as follows:</p> <p><a href="http://lh4.ggpht.com/-s_uIzwiJAoI/VSBOwJmrqjI/AAAAAAAA6WM/D13B6NC6gtg/s1600-h/Probablity_AND%25255B4%25255D.gif"><img title="Probability in Excel - AND" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - AND" src="http://lh3.ggpht.com/-0l4mST_yFtM/VSBOwod4VDI/AAAAAAAA6WU/09KM-0FX5Jo/Probablity_AND_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="36" /></a> </p> <p>P(A<sub>1</sub> ∩ A<sub>2</sub>) equals the probability of an object belonging to sets A<sub>1</sub> and A<sub>2</sub>.</p> <p>The following Excel-generated Venn Diagram illustrates the intersection of two events.</p> <p><a href="http://lh4.ggpht.com/-Mtxx1Ke-1Lc/VSBOxIXzDFI/AAAAAAAA6Wc/w_kBAjQb-x0/s1600-h/Prob_Venn_A_AND_B_600%25255B4%25255D.jpg"><img title="Probability in Excel - Venn Diagram A AND B" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram A AND B" src="http://lh4.ggpht.com/-TuHYRsZ_oIo/VSBOxpjd7QI/AAAAAAAA6Wk/s3JO94JB7xQ/Prob_Venn_A_AND_B_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="209" /></a> </p> <p align="center"> </p> <h3>Addition of 2 Non-Mutually Exclusive Events</h3> <p>The addition of non-mutually exclusive events is calculated by the following formula:</p> <p><a href="http://lh3.ggpht.com/-zUobiZk112o/VSBOx-gbYnI/AAAAAAAA6Ws/2utzVaJwxNg/s1600-h/Probability_Addition_Rule_Non-Mutually_Exclusive_Events%25255B9%25255D.gif"><img title="Probability in Excel - Addition Rule For Nonmutually Exclusive Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Addition Rule For Nonmutually Exclusive Events" src="http://lh5.ggpht.com/-yr89ICHiHeA/VSBOyZxvWeI/AAAAAAAA6W0/7blfQrIBv_A/Probability_Addition_Rule_Non-Mutually_Exclusive_Events_thumb%25255B3%25255D.gif?imgmax=800" width="404" height="26" /></a> </p> <p>A<sub>1</sub> ∩ A<sub>2</sub> = A<sub>1</sub> AND A<sub>2</sub> = Intersection between A<sub>1</sub> and A<sub>2</sub> as shown in the following Venn Diagram</p> <p>The following Excel-generated Venn Diagram illustrates the addition of two non-mutually exclusive events.</p> <p><a href="http://lh4.ggpht.com/-v5bnq_IRev0/VSBOytvNKEI/AAAAAAAA6W8/F5vWP0ODQgs/s1600-h/Prob_Venn_A_OR_B_600%25255B8%25255D.jpg"><img title="Probability in Excel - Venn Diagram A OR B" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Venn Diagram A OR B" src="http://lh5.ggpht.com/-xs3_mKJi0uE/VSBOzBZkbwI/AAAAAAAA6XE/v0Rvz2THiqk/Prob_Venn_A_OR_B_600_thumb%25255B4%25255D.jpg?imgmax=800" width="404" height="209" /></a> </p> <p>For example, in a company of 10,000 employees, it is known that 2,000 employees have at least one green car, 3,000 employees have at least one blue car, and 500 employee have both a green car and a blue car. Calculate the probability of a randomly selected employee owning a green car or a blue car.</p> <p>Pr(Own a green car) = Pr(Green) = 2,000/10,000 = 0.2</p> <p>Pr(Own a blue car) = Pr(Blue) = 3,000/10,000) = 0.3</p> <p>Pr(Own a blue car and own a green car) = Pr(Green ∩ Blue) = 500/10,000 = 0.05</p> <p><a href="http://lh4.ggpht.com/-Aztc6gTgNbk/VSBOzsoZlZI/AAAAAAAA6XM/J4t4-vvX4x0/s1600-h/Probability_Addition_Rule_Non-Mutually_Exclusive_Events%25255B10%25255D.gif"><img title="Probability in Excel - Addition Rule For Nonmutually Exclusive Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Addition Rule For Nonmutually Exclusive Events" src="http://lh3.ggpht.com/-A4VvdA_kglg/VSBOzx4vGhI/AAAAAAAA6XU/bAOfiZ4BZ5w/Probability_Addition_Rule_Non-Mutually_Exclusive_Events_thumb%25255B4%25255D.gif?imgmax=800" width="404" height="26" /></a> </p> <p>Pr(Green Or Blue) = Pr(Green) + Pr(Blue) – Pr(Green AND Blue)</p> <p>Pr(Green U Blue) = Pr(Green) + Pr(Blue) – Pr(Green ∩ Blue) = 0.2 + 0.3 – 0.05 = 0.45 = 45%</p> <p>There is a 45-percent probability that a randomly selected employee owns a green or blue car.</p> <p align="center"> </p> <h3>The Intersection of Two Non-Events = A' ∩ B' = (Not A) AND (Not B) </h3> <p>A' ∩ B' = P [ (A' ∩ B' )' ] = 1 – P(A U B)</p> <p align="center"> </p> <h2>Conditional Probability</h2> <p>If the occurrence of one event depends upon the occurrence of another event, a conditional probability is created. If the occurrence of event A depends on the occurrence of event B, the conditional probability of event A is denoted as P(A|B), which is stated as the probability of A given B. The formula for P(A|B) is as follows:</p> <p><a href="http://lh3.ggpht.com/-BBdCJ0T5xb0/VSBO0fFJytI/AAAAAAAA6Xc/EUscRYD0iho/s1600-h/Probability_Conditional%25255B3%25255D.gif"><img title="Probability in Excel - Conditional Probability" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Conditional Probability" src="http://lh5.ggpht.com/-q_2zxAydMyk/VSBO0-l1WwI/AAAAAAAA6Xk/e5JkIiyRwUg/Probability_Conditional_thumb%25255B1%25255D.gif?imgmax=800" width="404" height="49" /></a> </p> <p>For example, 30 people out of a group of 100 exercise regularly. 10 of these exercisers wear Nike shoes. Calculate of a person wearing Nike shoes given that the person exercises regularly.</p> <p>Event A = A person wears Nikes</p> <p>Event B = A person exercises regularly</p> <p>Pr(A) = Pr( Person in the group of 100 wears Nikes) = Not given</p> <p>P(B) = Pr( Person in the group of 100 exercises regularly) = 0.30</p> <p>P(A ∩ B) = Pr( Person in the group of 100 wears Nikes AND exercise regularly) = 0.10</p> <p>P(A|B) = Pr(A ∩ B) / Pr(B) = (0.10) / (0.30) = 1/3 = 0.33 = 33%</p> <p>33-percent of those in the group of 100 who exercise regularly wear Nikes.</p> <p align="center"> </p> <h3>The Multiplication of N Independent Events</h3> <p>Events are either <u><b>independent</b></u> or <u><b>not independent</b></u> of each other. Events are independent if the occurrence of any event does not affect the probability of the any other events occurring. </p> <p>Events A and B are independent if the following are true:</p> <p>P(A|B) = P(A)</p> <p>or equivalently</p> <p>P(B|A) = P(B)</p> <p>The following is the formula for the intersection (AND) of multiple independent events. Note that the probability of the intersection of multiple independent event equals the product of the individual probabilities of each of the events. </p> <p><a href="http://lh6.ggpht.com/-c4ftB2hCC-o/VSBO1GG5-QI/AAAAAAAA6Xs/gpDqUA9WqcU/s1600-h/Probability_Multiplication_Rule_Independent_Events%25255B3%25255D.gif"><img title="Probability in Excel - Multiplication Rule For Independent Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Multiplication Rule For Independent Events" src="http://lh6.ggpht.com/-qZAkfx9_RQA/VSBO1jjv2bI/AAAAAAAA6X0/3b_4cr2-vKM/Probability_Multiplication_Rule_Independent_Events_thumb%25255B1%25255D.gif?imgmax=800" width="404" height="25" /></a> </p> <p>For example, what is the probability of a single roll of two dice producing “snake eyes?” The outcome of “snake eyes” occurs when both of the dice are showing a single dot after the roll.</p> <p>The outcome of separate rolls of dice are independent form each other. The probability of a single dice rolling a 1 is 1/6 = 0.1667 = 16.67%</p> <p>Event A<sub>1</sub> = Dice 1 rolling a 1</p> <p>Event A<sub>2</sub> = Dice 2 rolling a 1</p> <p>Pr(A<sub>1</sub>) = Pr(A<sub>2</sub>) = 0.1667</p> <p>Pr(Dice 1 rolling a 1 AND Dice 2 rolling a 1) = Pr(A<sub>1</sub> AND A<sub>2</sub>) = Pr(A<sub>1</sub> ∩ A<sub>2</sub>) = Pr(A<sub>1</sub>) Pr(A<sub>2</sub>)</p> <p>Pr(A<sub>1</sub> ∩ A<sub>2</sub>) = Pr(A<sub>1</sub>) Pr(A<sub>2</sub>) = (0.1667)(0.1667) = 0.0278 = 2.78%</p> <p>There is a 2.78-percent chance that a single roll of 2 dice will produce “snake eyes.”</p> <p align="center"> </p> <h3>The Multiplication of Non-Independent Events</h3> <p>The probability of the intersection of two non-independent events requires the knowledge of the conditional probability of one of the events as follows:</p> <p><a href="http://lh3.ggpht.com/-qExuigoIgqU/VSBO2LqSShI/AAAAAAAA6X8/sB1fwFpQyEU/s1600-h/Probability_Multiplication_Non-Independent_Events%25255B6%25255D.gif"><img title="Probability in Excel - Multiplication Rule For Non-Independent Events" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Probability in Excel - Multiplication Rule For Non-Independent Events" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpR2vAei45qbnasDSG08Y9-wk_N9BXc35pDdZKKfBfmpGnwzNTnI04HvMoSy9phGg0mBfM-_03iuVvVtKFjJPQ1vpm_HOo3X_mgoOCHTi9BFLlr2_uwx_eCiyfJFPL7GqjbaGxY1hlP4Pp/?imgmax=800" width="404" height="25" /></a> </p> <p>For example, </p> <p>P(A|B) = Pr(A ∩ B) / Pr(B) = (0.10) / (0.30) = 1/3 = 0.33 = 33%</p> <p>33-percent of those in the group of 100 wear Nike shoes given they exercise regularly. 30-percent of that group of 100 exercise regularly. What percentage of that group wear Nikes and exercise regularly?</p> <p>Event A = A person wears Nikes</p> <p>Event B = A person exercises regularly</p> <p>Pr(A|B) = Pr( A person wears Nikes given that the person exercise regularly) = 33% = 0.33</p> <p>Pr(B) = Pr( A person exercises regularly) = 30% = 0.3</p> <p>Pr(A ∩ B) = Pr(B) * Pr(A|B) = (0.3)*(0.33) = 0.1 = 10%</p> <p>10-percent of the group of 100 wear Nikes and exercise regularly.</p> <p>Note that a comparison between the conditional probability calculated previously, Pr(A|B) = 33%, the probability of intersection calculated here, Pr(A ∩ B) = 10%, make a strong implication that Nike is a preferred brand by those who exercise. The larger that group is and the more representative that group is of the general population, the more validity that implication has.</p> <p align="center"> </p> <h2>Law of Total Probability</h2> <p>Assume the Events A<sub>1</sub>, A<sub>2</sub>, .., A<sub>n</sub> are mutually exclusive events whose intersection is sample space S. This means that sample space S is entirely composed of these mutually exclusive (non-overlapping) events and nothing else. E represents any of these events. The Law of Total Probability states that the probability of any of these events (Events A<sub>1</sub>, A<sub>2</sub>, .., A<sub>n</sub>) occurring is the following:</p> <p>P(E) = P(A<sub>1</sub>)P(E|A<sub>1</sub>) + P(A<sub>2</sub>)P(E|A<sub>2</sub>) + P(A<sub>3</sub>)P(E|A<sub>3</sub>) + … + P(A<sub>n</sub>)P(E|A<sub>n</sub>)</p> <p>For example, a factory has 4 machines that produce similar items.</p> <p>The event of any of these machines producing a defect is denoted as Event D. </p> <p>Machine A<sub>1</sub> produces 40% of all items of which 5% are typically defective.</p> <p>Machine A<sub>2</sub> produces 30% of all items of which 6% are typically defective.</p> <p>Machine A<sub>3</sub> produces 20% of all items of which 7% are typically defective.</p> <p>Machine A<sub>4</sub> produces 10% of all items of which 8% are typically defective.</p> <p>From this we can state the following:</p> <p>Probability that an item was made by machine A<sub>1</sub> = 40% so P(A<sub>1</sub>) = 0.4</p> <p>Probability that an item was made by machine A<sub>2</sub> = 30% so P(A<sub>2</sub>) = 0.3</p> <p>Probability that an item was made by machine A<sub>3</sub> = 20% so P(A<sub>3</sub>) = 0.2</p> <p>Probability that an item was made by machine A<sub>4</sub> = 10% so P(A<sub>4</sub>) = 0.1</p> <p>We can also state the following:</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>1</sub> = P(D|A<sub>1</sub>) = 0.05</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>2</sub> = P(D|A<sub>2</sub>) = 0.06</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>3</sub> = P(D|A<sub>3</sub>) = 0.07</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>1</sub> = P(D|A<sub>4</sub>) = 0.08</p> <p>P(D), the probability that a defect will occur on any of the machines is calculated by the Law of Total Probability as follows:</p> <p>P(E) = P(A<sub>1</sub>)P(E|A<sub>1</sub>) + P(A<sub>2</sub>)P(E|A<sub>2</sub>) + P(A<sub>3</sub>)P(E|A<sub>3</sub>) + … + P(A<sub>n</sub>)P(E|A<sub>n</sub>)</p> <p>P(D) = P(A<sub>1</sub>)P(D|A<sub>1</sub>) + P(A<sub>2</sub>)P(D|A<sub>2</sub>) + P(A<sub>3</sub>)P(D|A<sub>3</sub>) + P(A<sub>4</sub>)P(E|A<sub>n</sub>)</p> <p>P(D) = (0.4)(0.05) + (0.3)(0.06) + (0.2)(0.07) + (0.1)(0.8) = 0.06</p> <p>There is a 6 percent chance that an item produced by any of the 4 machines will be defective.</p> <p align="center"> </p> <h2>Bayes’ Theorem</h2> <p>Closely related to the Law of Total Probability is Bayes’ Theorem. Bayes’ Theorem, named after English mathematician Thomas Bayes (1702 – 1761), is sometimes called the <i>theorem on the probability of causes</i> because it calculates the probability of each of the possible causes of an event given that the event occurred.</p> <p>Assume the Events A<sub>1</sub>, A<sub>2</sub>, .., A<sub>n</sub> are mutually exclusive events whose intersection is sample space S. This means that sample space S is entirely composed of these mutually exclusive (non-overlapping) events and nothing else. E represents any of these events. Bayes’ Theorem calculates P(A<sub>k</sub>|E), the probability of a specific event (Ak) occurring given that one of the events did occur, as follows:</p> <p>P(A<sub>k</sub>|E) = P(A<sub>k</sub>) * P(E|A<sub>k</sub>) / P(E)</p> <p>Using data from the previous example, calculating the probability that a defect came from a specific machine given that a defect did occur is done as follows:</p> <p>P(A<sub>k</sub>|E) = P(A<sub>k</sub>) * P(E|A<sub>k</sub>) / P(E)</p> <p>P(A<sub>k</sub>|D) = P(A<sub>k</sub>) * P(D|A<sub>k</sub>) / P(D)</p> <p>Recall the following from the previous example:</p> <p>P(D) = Probability of a defect occurring = 0.06</p> <p>Probability that an item was made by machine A<sub>1</sub> = 40% so P(A<sub>1</sub>) = 0.4</p> <p>Probability that an item was made by machine A<sub>2</sub> = 30% so P(A<sub>2</sub>) = 0.3</p> <p>Probability that an item was made by machine A<sub>3</sub> = 20% so P(A<sub>3</sub>) = 0.2</p> <p>Probability that an item was made by machine A<sub>4</sub> = 10% so P(A<sub>4</sub>) = 0.1</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>1</sub> = P(D|A<sub>1</sub>) = 0.05</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>2</sub> = P(D|A<sub>2</sub>) = 0.06</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>3</sub> = P(D|A<sub>3</sub>) = 0.07</p> <p>Probability that a defect occurred given that the item came from Machine A<sub>1</sub> = P(D|A<sub>4</sub>) = 0.08</p> <p>Using the Bayes’ Theorem formula, P(A<sub>k</sub>|D) = P(A<sub>k</sub>) * P(D|A<sub>k</sub>) / P(D), to calculate the probability that a defect was from a specific machine given the a defect occurred, P(A<sub>k</sub>|D), is done as follows:</p> <p>P(A<sub>k</sub>|D) = P(A<sub>k</sub>) * P(D|A<sub>k</sub>) / P(D)</p> <p>*****************</p> <p>P(A<sub>1</sub>|D) = Probability that machine A<sub>1</sub> produced the defect given that a defect occurred is as follows:</p> <p>P(A<sub>1</sub>|D) = P(A<sub>1</sub>) * P(D|A<sub>1</sub>) / P(D) = (0.4) * (0.05) / (0.06) = 0.33 = 33 percent</p> <p>*****************</p> <p>P(A<sub>2</sub>|D) = Probability that machine A<sub>2</sub> produced the defect given that a defect occurred is as follows:</p> <p>P(A<sub>2</sub>|D) = P(A<sub>2</sub>) * P(D|A<sub>2</sub>) / P(D) = (0.3) * (0.06) / (0.06) = 0.3 = 30 percent</p> <p>*****************</p> <p>P(A<sub>3</sub>|D) = Probability that machine A<sub>3</sub> produced the defect given that a defect occurred is as follows:</p> <p>P(A<sub>3</sub>|D) = P(A<sub>3</sub>) * P(D|A<sub>3</sub>) / P(D) = (0.2) * (0.07) / (0.06) = 0.23 = 23 percent</p> <p>*****************</p> <p>P(A<sub>4</sub>|D) = Probability that machine A<sub>4</sub> produced the defect given that a defect occurred is as follows:</p> <p>P(A<sub>4</sub>|D) = P(A<sub>4</sub>) * P(D|A<sub>4</sub>) / P(D) = (0.1) * (0.07) / (0.06) = 0.12 = 12 percent</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com58tag:blogger.com,1999:blog-3568555666281177719.post-41391010529268887562015-03-24T09:00:00.001-07:002015-05-05T11:10:05.854-07:00Blog Directory of Statistics Topics and Articles<p><strong>Excel Master Series Blog Directory</strong></p> <p>Statistical Topics and Articles In Each Topic</p> <ul> <li><span>Histograms in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/how-to-create-histogram-in-excel-2010_27.html" target="_blank">Creating a Histogram With the Histogram Data Analysis Tool in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/how-to-create-histogram-in-excel-2010.html" target="_blank">Creating an Automatically Updating Histogram in 7 Steps in Excel With Formulas and a Bar Chart</a></span> </li> </ul> </li> <li><span>Bar Charts in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/creating-bar-chart-in-excel-2010-and.html" target="_blank">Creating a Bar Chart in 7 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Box Plots in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/02/box-plots-in-8-steps-in-excel.html" target="_blank">Creating a Box Plot in 8 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Probability in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/04/probability-in-excel.html" target="_blank">Probability in Excel</a></span> </li> </ul> </li> <li><span>Combinations & Permutations in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/combinations-in-excel-2010-and-excel.html" target="_blank">Combinations in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/permutations-in-excel-2010-and-excel_27.html" target="_blank">Permutations in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Time Series Analysis in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2010/11/excel-marketing-forecasting-technique-3.html" target="_blank">Forecasting With Exponential Smoothing in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2010/11/weighted-moving-average-accurate-simple.html" target="_blank">Forecasting With the Weighted Moving Average in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2010/10/excels-most-forecasting-tool-simple.html" target="_blank">Forecasting With the Simple Moving Average in Excel</a></span> </li> </ul> </li> <li><span>Measures of Central Tendency and Variation in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/03/measures-of-central-tendency-in-excel.html" target="_blank">Measures of Central Tendency in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/04/measures-of-variation-in-excel.html" target="_blank">Measures of Variation in Excel</a></span> </li> </ul> </li> <li><span>Simplifying Useful Excel Functions and Tools</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/03/simplifying-sumif-sumifs-countif.html" target="_blank">Simplifying Excel Functions: SUMIF, SUMIFS, COUNTIF, COUNTIFS, AVERAGEIF, and AVERAGEIFS</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/03/simplifying-excel-form-controls-check.html" target="_blank">Simplifying Excel Form Controls: Check Box, Option Button, Spin Button, and Scroll Bar</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/02/simplifying-excel-lookup-functions.html" target="_blank">Simplifying Excel Lookup Functions: VLOOKUP, HLOOKUP, INDEX, MATCH, CHOOSE, and OFFSET</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/03/scenario-analysis-in-excel-with-option.html" target="_blank">Scenario Analysis in Excel With Option Buttons and the What-If Scenario Manager</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/03/simplifying-goal-seek-in-excel.html" target="_blank">Simplifying Goal Seek in Excel</a></span> </li> </ul> </li> <li><span>Sorting and Ranking in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/04/automated-column-sorting-in-excel.html" target="_blank">Automated Data Column Sorting in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/04/simplifying-excel-ranking-functions.html" target="_blank">Simplifying Excel Ranking Functions: RANK(), RANK.AVG(), RANK.EQ(), PERCENTILE(), PERCENTILE.INC(), PERCENTILE.EXC(), QUARTILE(), QUARTILE.INC(), QUARTILE.EXC()</a></span> </li> </ul> </li> <li><span>Normal Distribution in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution_3.html" target="_blank">Overview of the Normal Distribution</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-pdf-in-excel-2010.html" target="_blank">Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-cdf-in-excel-2010.html" target="_blank">Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/solving-normal-distribution-problems-in.html" target="_blank">Solving Normal Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/standard-normal-distribution-in-excel.html" target="_blank">Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/an-important-difference-between-t-and.html" target="_blank">An Important Difference Between the t and Normal Distribution Graphs</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/how-to-empirical-rule-and-chebyshevs.html" target="_blank">The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/how-to-demonstrate-central-limit.html" target="_blank">Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way</a></span> </li> </ul> </li> <li><span>t-Distribution in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/overview-of-t-distribution.html" target="_blank">Overview of the t- Distribution</a></span> </li> </ul> </li> <li><span>Binomial Distribution in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/overview-of-binomial-distribution-in.html" target="_blank">Overview of the Binomial Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/solving-problems-with-binomial.html" target="_blank">Solving Problems With the Binomial Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/normal-approximation-of-binomial.html" target="_blank">Normal Approximation of the Binomial Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/distributions-related-to-binomial.html" target="_blank">Distributions Related to the Binomial Distribution</a></span> </li> </ul> </li> <li><span>z-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution.html" target="_blank">Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/1-sample-z-test-in-excel-2010-and-excel.html" target="_blank">One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-unpooled-z-test-in-excel-2010.html" target="_blank">2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/paired-z-test-in-excel-2010-and-excel.html" target="_blank">Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>t-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/overview-of-t-tests-t-distribution.html" target="_blank">Overview of t-Tests: Hypothesis Tests that Use the t-Distribution</a></span> </li> <li><span>1-Sample t-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/1-sample-t-test-in-excel-2010-and-excel.html" target="_blank">1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-normality-testing-for-1-sample-t.html" target="_blank">Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/1-sample-t-test-effect-size-in-excel.html" target="_blank">1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/1-sample-t-test-power-with-gpower.html" target="_blank">1-Sample t-Test Power With G*Power Utility</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/wilcoxon-signed-rank-test-as-1-sample-t.html" target="_blank">Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/sign-test-as-1-sample-t-test.html" target="_blank">Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>2-Independent-Sample Pooled t-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-pooled-t-test-in-excel-2010.html" target="_blank">2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-variance-tests-levenes-brown.html" target="_blank">Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-normality-tests-kolmogorov_30.html" target="_blank">Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/two-sample-t-test-all-excel-calculations.html" target="_blank">Two-Independent-Sample Pooled t-Test - All Excel Calculations</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-pooled-t-test-effect-size-in.html" target="_blank">2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-pooled-t-test-power-with.html" target="_blank">2-Sample Pooled t-Test Power With G*Power Utility</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/mann-whitney-u-test-as-2-sample-pooled.html" target="_blank">Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-pooled-t-test-single-factor.html" target="_blank">2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups</a></span> </li> </ul> </li> <li><span>2-Independent-Sample Unpooled t-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-unpooled-t-test-in-excel-2010.html" target="_blank">2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/variance-tests-levenes-test-brown.html" target="_blank">Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test </a></span></li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-normality-tests-kolmogorov.html" target="_blank">Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test </a></span></li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-unpooled-t-test-excel.html" target="_blank">2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-unpooled-t-test-effect-size-in.html" target="_blank">Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-unpooled-t-test-test-power.html" target="_blank">Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility</a></span> </li> </ul> </li> <li><span>Paired (2-Sample Dependent) t-Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/paired-t-test-in-excel-2010-and-excel_28.html" target="_blank">Paired t-Test in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-normality-testing-of-paired-t.html" target="_blank">Excel Normality Testing of Paired t-Test Data</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/paired-t-test-excel-calculations.html" target="_blank">Paired t-Test Excel Calculations, Formulas, and Tools</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/paired-t-test-effect-size-in-excel-2010.html" target="_blank">Paired t-Test – Effect Size in Excel 2010, and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/paired-t-test-test-power-with-g-power.html" target="_blank">Paired t-Test – Test Power With G-Power Utility</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/wilcoxon-signed-rank-test-as-paired-t.html" target="_blank">Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/sign-test-in-excel-as-paired-t-test.html" target="_blank">Sign Test in Excel As A Paired t-Test Alternative</a></span> </li> </ul> </li> </ul> </li> <li><span>Hypothesis Tests of Proportion in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/hypothesis-tests-of-proportion-overview.html" target="_blank">Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/1-sample-hypothesis-test-of-proportion.html" target="_blank">1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-sample-pooled-hypothesis-test-of.html" target="_blank">2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2010/03/how-to-duplicate-google-website.html" target="_blank">How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer</a></span> </li> </ul> </li> <li><span>Chi-Square Independence Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/chi-square-independence-test-in-excel.html" target="_blank">Chi-Square Independence Test in 7 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Chi-Square Goodness-Of-Fit Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/chi-square-goodness-of-fit-test-overview.html" target="_blank">Overview of the Chi-Square Goodness-of-Fit Test</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/chi-square-goodness-of-fit-in-excel.html" target="_blank">Chi-Square Goodness- of-Fit Test With Pre-Determined Bins Sizes in 7 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/chi-square-goodness-of-fit-normality.html" target="_blank">Chi-Square Goodness-Of-Fit-Normality Test in 9 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>F Tests in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/f-test-in-excel-2010-and-excel-2013.html" target="_blank">F-Test in 6 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/normality-testing-for-f-test-in-excel.html" target="_blank">Normality Testing For F Test In Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/levenes-and-brown-forsythe-tests-in.html" target="_blank">Levene’s and Brown- Forsythe Tests: F-Test Alternatives in Excel</a></span> </li> </ul> </li> <li><span>Correlation in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/overview-of-correlation-in-excel-2010.html" target="_blank">Overview of Correlation In Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Pearson Correlation in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/pearson-correlation-in-excel-2010-and.html" target="_blank">Pearson Correlation in 3 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/pearson-correlation-r-critical-and-p.html" target="_blank">Pearson Correlation – Calculating r Critical and p Value of r in Excel</a></span> </li> </ul> </li> <li><span>Spearman Correlation in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/spearman-correlation-coefficient-in.html" target="_blank">Spearman Correlation in 6 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Confidence Intervals in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/z-based-confidence-interval-of.html" target="_blank">z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/t-based-confidence-interval-of.html" target="_blank">t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/min-sample-size-to-limit-width-of-excel.html" target="_blank">Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/confidence-interval-of-population.html" target="_blank">Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/min-sample-size-of-confidence-interval.html" target="_blank">Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Simple Linear Regression in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/overview-of-simple-linear-regression-in.html" target="_blank">Overview of Simple Linear Regression in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/simple-linear-regression-example-in.html" target="_blank">Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/residual-evaluation-for-simple.html" target="_blank">Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/residual-normality-tests-in-excel.html" target="_blank">Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/evaluation-of-simple-regression-output.html" target="_blank">Evaluation of Simple Regression Output For Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/all-calculations-performed-by-simple.html" target="_blank">All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/prediction-interval-of-simple.html" target="_blank">Prediction Interval of Simple Regression in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Multiple Linear Regression in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/basics-of-multiple-regression-in-excel.html" target="_blank">Basics of Multiple Regression in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/multiple-linear-regression-example-in.html" target="_blank">Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/linear-regressions-required-assumptions.html" target="_blank">Multiple Linear Regression’s Required Residual Assumptions</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/normality-testing-of-residuals-in-excel.html" target="_blank">Normality Testing of Residuals in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/interpret-excel-output-of-multiple.html" target="_blank">Evaluating the Excel Output of Multiple Regression</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/estimating-prediction-interval-of.html" target="_blank">Estimating the Prediction Interval of Multiple Regression in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2010/03/how-to-use-dummy-variable-regression-in.html" target="_blank">Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel</a></span> </li> </ul> </li> <li><span>Logistic Regression in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/logistic-regression-overview.html" target="_blank">Logistic Regression Overview</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/logistic-regression-performed-in-excel.html" target="_blank">Logistic Regression in 6 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/r-square-for-logistic-regression.html" target="_blank">R Square For Logistic Regression Overview</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/excel-r-square-tests-nagelkerke-cox-and.html" target="_blank">Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/likelihood-ratio-is-better-than-wald.html" target="_blank">Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/excel-classification-table-logistic.html" target="_blank">Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/hosmer-lemeshow-test-in-excel-logistic.html" target="_blank">Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Single-Factor ANOVA in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/overview-of-single-factor-anova.html" target="_blank">Overview of Single-Factor ANOVA</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/single-factor-anova-example-in-excel.html" target="_blank">Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/shapiro-wilk-normality-test-in-excel_29.html" target="_blank">Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/kruskal-wallis-test-alternative-for.html" target="_blank">Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/levenes-and-brown-forsythe-tests-in_29.html" target="_blank">Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison </a></span></li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/single-factor-anova-all-excel.html" target="_blank">Single-Factor ANOVA - All Excel Calculations</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/post-hoc-testing-for-single-factor-anova.html" target="_blank">Overview of Post-Hoc Testing For Single-Factor ANOVA</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/tukey-kramer-post-hoc-test-in-excel-for.html" target="_blank">Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/games-howell-post-hoc-test-in-excel-for.html" target="_blank">Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/effect-size-for-single-factor-anova.html" target="_blank">Overview of Effect Size For Single-Factor ANOVA</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/anova-effect-size-calculation-eta.html" target="_blank">ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/anova-effect-size-calculation-psi-rmsse.html" target="_blank">ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/anova-effect-size-calculation-omega.html" target="_blank">ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/single-factor-anova-test-power-with.html" target="_blank">Power of Single-Factor ANOVA Test Using Free Utility G*Power</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/welchs-anova-test-in-excel-substitute.html" target="_blank">Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/brown-forsythe-f-test-in-excel.html" target="_blank">Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar</a></span> </li> </ul> </li> <li><span>Two-Factor ANOVA With Replication in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/two-factor-anova-with-replication-in.html" target="_blank">Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/variance-tests-levenes-and-brown.html" target="_blank">Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/shapiro-wilk-normality-test-in-excel.html" target="_blank">Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-factor-anova-wrep-effect-size-in.html" target="_blank">2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-post-hoc-tukeys-hsd-test-for-2.html" target="_blank">Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/2-factor-anova-wrep-test-power-with-g.html" target="_blank">2-Factor ANOVA With Replication – Test Power With G-Power Utility</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/scheirer-ray-hope-test-alternative-for.html" target="_blank">Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication</a></span> </li> </ul> </li> <li><span>Two-Factor ANOVA Without Replication in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/two-factor-anova-without-replication-in.html" target="_blank">Two-Factor ANOVA Without Replication in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Randomized Block Design ANOVA in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/01/randomized-block-design-anova-in-excel.html" target="_blank">Randomized Block Design ANOVA in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Repeated-Measures ANOVA in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/01/single-factor-repeated-measures-anova.html" target="_blank">Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/01/sphericity-testing-in-9-steps-for.html" target="_blank">Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/01/effect-size-for-repeated-measures-anova.html" target="_blank">Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html" target="_blank">Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>ANCOVA in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/12/ancova-in-excel.html" target="_blank">Single-Factor ANCOVA in 8 Steps in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Normality Testing in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/02/box-plots-in-8-steps-in-excel.html" target="_blank">Creating a Box Plot in 8 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/02/normal-probability-plot-with-adjustable.html" target="_blank">Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/chi-square-goodness-of-fit-normality.html" target="_blank">Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/05/how-to-create-completely-automated_4.html" target="_blank">How To Create a Completely Automated Shapiro-Wilk Normality Test in Excel in 8 Steps</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-normality-tests-kolmogorov.html" target="_blank">Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel</a></span> </li> </ul> </li> <li><span>Nonparametric Testing in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/mann-whitney-u-test-as-2-sample-pooled.html" target="_blank">Mann-Whitney U Test in 12 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/wilcoxon-signed-rank-test-as-1-sample-t.html" target="_blank">Wilcoxon Signed-Rank Test in 8 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/sign-test-as-1-sample-t-test.html" target="_blank">Sign Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html" target="_blank">Friedman Test in 3 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/scheirer-ray-hope-test-alternative-for.html" target="_blank">Scheirer-Ray-Hope Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/welchs-anova-test-in-excel-substitute.html" target="_blank">Welch's ANOVA Test in 8 Steps Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/brown-forsythe-f-test-in-excel.html" target="_blank">Brown-Forsythe F Test in 4 Steps Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-variance-tests-levenes-brown.html" target="_blank">Levene's Test and Brown-Forsythe Variance Tests in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/chi-square-independence-test-in-excel.html" target="_blank">Chi-Square Independence Test in 7 Steps in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/chi-square-goodness-of-fit-test-overview.html" target="_blank">Chi-Square Goodness-of-Fit Tests in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/chi-square-population-variance-test-in.html" target="_blank">Chi-Square Population Variance Test in Excel</a></span> </li> </ul> </li> <li><span>Post Hoc Testing in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/05/excel-post-hoc-tukeys-hsd-test-for-2.html" target="_blank">Tukey's HSD Post Hoc Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/tukey-kramer-post-hoc-test-in-excel-for.html" target="_blank">Tukey-Kramer Post Hoc Test in Excel</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/games-howell-post-hoc-test-in-excel-for.html" target="_blank">Games-Howell Post Hoc Test in Excel</a></span> </li> </ul> </li> <li><span>Creating Interactive Graphs of Statistical Distributions in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/interactive-statistical-distribution.html" target="_blank">Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-normal.html" target="_blank">Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-chi-square.html" target="_blank">Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-t-distribution-in.html" target="_blank">Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/t-distributions-pdf-in-excel-2010-and.html" target="_blank">Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/t-distributions-cdf-in-excel-2010-and.html" target="_blank">Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-binomial.html" target="_blank">Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-exponential.html" target="_blank">Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-beta-distribution.html" target="_blank">Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-gamma-distribution.html" target="_blank">Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-poisson.html" target="_blank">Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Solving Problems With Other Distributions in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/solving-uniform-distribution-problems.html" target="_blank">Solving Uniform Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-problems-with-multinomial.html" target="_blank">Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-exponential-distribution.html" target="_blank">Solving Exponential Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-beta-distribution-problems-in.html" target="_blank">Solving Beta Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-gamma-distribution-problems-in.html" target="_blank">Solving Gamma Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-poisson-distribution-problems.html" target="_blank">Solving Poisson Distribution Problems in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Optimization With Excel Solver</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/maximizing-lead-generation-with-excel.html" target="_blank">Maximizing Lead Generation With Excel Solver</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/minimizing-cutting-stock-waste-with.html" target="_blank">Minimizing Cutting Stock Waste With Excel Solver</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/optimal-investment-selection-with-excel.html" target="_blank">Optimal Investment Selection With Excel Solver</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/minimizing-shipping-costs-with-excel.html" target="_blank">Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/knapsack-loading-problem-in-excel-solver.html" target="_blank">Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment </a></span></li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/optimizing-bond-portfolio-with-excel.html" target="_blank">Optimizing a Bond Portfolio With Excel Solver</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2014/05/solving-traveling-salesman-problem-with.html" target="_blank">Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers</a></span> </li> </ul> </li> <li><span>Chi-Square Population Variance Test in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2014/06/chi-square-population-variance-test-in.html" target="_blank">Overview of the Chi-Square Population Variance Test in Excel 2010 and Excel 2013</a></span> </li> </ul> </li> <li><span>Analyzing Data With Pivot Tables and Pivot Charts</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2011/12/pivot-tables-how-to-set-up-pivot-table.html" target="_blank">Simplifying Excel Pivot Table and Pivot Chart Setup</a></span> </li> </ul> </li> <li><span>SEO Functions in Excel</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2011/01/excel-seo-functions.html" target="_blank">Top 10 Excel SEO Functions - You'll Like These</a></span> </li> </ul> </li> <li><span>VLOOKUP</span> <ul> <li><span><a href="http://blog.excelmasterseries.com/2015/02/simplifying-excel-lookup-functions.html" target="_blank">Simplifying Excel Lookup Functions: VLOOKUP, HLOOKUP, INDEX, MATCH, CHOOSE, and OFFSET</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2012/01/vlookup-just-like-looking-number-up-in.html" target="_blank">VLOOKUP - Just Like Looking Up a Number in a Telephone Book</a></span> </li> <li><span><a href="http://blog.excelmasterseries.com/2012/01/looking-up-quantity-discount-in-distant.html" target="_blank">VLOOKUP To Look Up a Discount in a Distant Database</a></span> </li> </ul> </li> </ul> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com33tag:blogger.com,1999:blog-3568555666281177719.post-31851496243914977452015-03-17T11:27:00.001-07:002015-03-24T09:31:50.435-07:00Measures of Central Tendency in Excel: Mean, Weighted Mean, Median, Mode, Geometric Mean, Harmonic Mean, and Weighted Harmonic Mean<h1>Measures of Central <br /> <br />Tendency in Excel: Mean, <br /> <br />Weighted Mean, Median, <br /> <br />Mode, Geometric Mean, <br /> <br />Harmonic Mean, and <br /> <br />Weighted Harmonic Mean</h1> <h2>Measures of Central Tendency Overview</h2> <p>Measures of central tendency describe the center of a finite set of data or a theoretical distribution such as the normal distribution. Data is said to have a strong or weak central tendency based on measures of its dispersion such as standard deviation.</p> <p>The following measures of central tendency can be calculated in Excel:</p> <p><b>Mean</b> (arithmetic mean) – the sum of all data values divided by the number of data values</p> <p><b>Weighted mean</b> – an arithmetic mean that incorporates weighting to certain data points</p> <p><b>Median</b> – the middle value that separates the upper half of the data set from the lower half of the data set</p> <p><b>Mode</b> – The most frequent value in the data set</p> <p><b>Geometric mean</b> – the <i>n</i>th root of the product of data, where there are <i>n</i> of these.</p> <p><b>Harmonic mean</b> – the reciprocal of the arithmetic mean of the reciprocals of the data values</p> <p><b>Weighted harmonic mean</b></p> <p> </p> <h3>Mean (Arithmetic mean) in Excel</h3> <p>The mean is simply the sum of a collection of data values in a data set divided by the number of data values in the data set. The mean is very sensitive to outliers, i.e., very high or low values. If significant outliers exist the median can be a more robust measure of central tendency than the mean. Many parametric statistical tests rely on the calculation of the mean. Parametric tests are statistical tests which require that the data set be distributed according to a specified distribution such as the normal distribution. Many nonparametric tests rely on the calculation of the median. Nonparametric tests do not require that the data set be distributed according to a specified distribution.</p> <p>A population mean is denoted by µ, the Greek letter <i>mu</i>.</p> <p>A sample mean is denoted by x_bar or x_avg. </p> <p>The formula to calculate the mean is as follows:</p> <p><a href="http://lh4.ggpht.com/-607lI2iGPN4/VQhyEZ-aEwI/AAAAAAAA6Mo/fGbs3d4WU88/s1600-h/Mean%25255B5%25255D.gif"><img title="Measures of Central Tendency in Excel - Mean Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Mean Formula" src="http://lh4.ggpht.com/-ZQfmoGOE2UY/VQhyE1top3I/AAAAAAAA6Mw/It0fpzRD8K4/Mean_thumb%25255B3%25255D.gif?imgmax=800" width="157" height="64" /></a> </p> <p>The Excel formula to calculate the mean is the following:</p> <p>AVERAGE(<em>data range</em>)</p> <p> </p> <h3>Weighted Mean</h3> <p>The weighted arithmetic mean is used if certain individual data values should be given greater weight due to increased importance or due to a larger number of occurrences of specific values. The formula for the weighted mean is as follows:</p> <p><a href="http://lh3.ggpht.com/-g1wlc2etVdw/VQhyFGyONtI/AAAAAAAA6M4/463iJ1s3mIk/s1600-h/Weighted_Mean%25255B4%25255D.gif"><img title="Measures of Central Tendency in Excel - Weighted Mean Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Weighted Mean Formula" src="http://lh6.ggpht.com/-QEzBNrbWeeY/VQhyFjaHTUI/AAAAAAAA6NA/c1d2WvE5Zjk/Weighted_Mean_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="44" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>An instructor teaches two sections of the same course and gives the same test to both sections. The test average from section 1, which has 30 students, is 84. The test average of section 2, which has 20 students is 93. The average score of all students combined could be calculated using the weighted mean as follows: </p> <p>Weighted Mean = (30*84 + 20*93) / (20 + 30) = 87.6</p> <p> </p> <h3>Median in Excel</h3> <p>Half of the values of a data set are larger than the mean and half of the values are smaller than the mean. If there is no single value, then the median is defined as being the average of the two middle values. There is no widely-accepted symbol for the median so any symbol used to denote the median must be explicitly defined. The median is the 2nd quartile, 5<sup>th</sup> decile, and 50<sup>th</sup> percentile. The median and mode are the only measures of central tendency that can be applied to ordinal data. Ordinal data are data that are ranked but whose values have no specific numerical meaning.</p> <p>The Excel formula to calculate the median is the following:</p> <p>MEDIAN(<em>data range</em>)</p> <p> </p> <h3>Mode in Excel</h3> <p>The mode is the most frequently occurring value in a data set. A data set is said to be unimodal if there is only one data value that occurs significantly more often than the other data values. The PDF (probability density function) of a unimodal data set has a global maximum and no other local maxima. The bell-shaped normal distribution is unimodal. A data set is said to be multimodal if there are more than one value that occurs significantly more often than the other values in a data set. The PDF of a multimodal data set has at least two local maxima. If the PDF has only two local maxima, the data set is said to be bimodal. Three local maxima in the PDF designate the data set as being trimodal.</p> <p>The mode is one of the few descriptors that can be applied to a data set that is nominal. Nominal data are merely labels that have no numerical significance or ranking.</p> <p>The shape of a data set’s distribution can be observed by creating a histogram of the data.</p> <p>The Excel formula to calculate the mode is the following:</p> <p>MODE(<em>data range</em>) – This function is compatible with Excel 2007 and earlier</p> <p>MODE.SING(<em>data range</em>) – This function can be used in Excel 2010 or later. It returns the most frequently occurring value in a data set or data range.</p> <p>MODE.MULT(<em>data range</em>) – This function can be used in Excel 2010 or later. It returns a vertical array of the most frequently occurring values in a data set or data range. For a horizontal array, use the following:</p> <p>TRANSPOSE(MODE.MULT(<em>data range</em>))</p> <p>Perhaps the quickest way to determine the modal values is to create an Excel histogram of the data. An example of this is shown as follows:</p> <p><a href="http://lh6.ggpht.com/-pJ-d8ZcrlaA/VQhyGOkVMyI/AAAAAAAA6NI/Uny-efrmSIo/s1600-h/Histogram_Data_600%25255B4%25255D.jpg"><img title="Measures of Central Tendency in Excel - Histogram Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Histogram Data" src="http://lh5.ggpht.com/-jrXlYst6tWo/VQhyGxpJCwI/AAAAAAAA6NQ/tzP5mDorvI0/Histogram_Data_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="269" /></a> </p> <p><a href="http://lh5.ggpht.com/-3Uv1n5a31Sk/VQhyHZQDPlI/AAAAAAAA6NY/sebwSv9qM_s/s1600-h/Histogram_600%25255B4%25255D.jpg"><img title="Measures of Central Tendency in Excel - Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Histogram" src="http://lh4.ggpht.com/-YJ-LB1rtyds/VQhyH5IUqwI/AAAAAAAA6Ng/p6cahkYBayk/Histogram_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="215" /></a> </p> <p>The histogram chart generated in Excel quickly shows that the data is bimodal and the two modes existing at 3 and 8.</p> <p> </p> <h3>Geometric Mean in Excel</h3> <p>The geometric mean is the <i>n</i>th root of the product of <i>n</i> numbers. </p> <p><a href="http://lh3.ggpht.com/-BxBTHmyxX5E/VQhyIAxG33I/AAAAAAAA6Nk/Xlo8Mnw4ym8/s1600-h/Geometric_Mean%25255B4%25255D.gif"><img title="Measures of Central Tendency in Excel - Geometric Mean Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Geometric Mean Formula" src="http://lh6.ggpht.com/-VeoSODzSy7s/VQhyIWzM0sI/AAAAAAAA6Nw/N5n5vXHltYA/Geometric_Mean_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="63" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">The Excel Formula for nth root is POWER(NUMBER,1/n). The geometric mean formula can be generalized in Excel as follows:</font></font></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">Geometric Mean =POWER(PRODUCT(</font></font><font color="#323232"><font style="font-size: 10pt" size="2"><i>data range</i></font></font><font color="#323232"><font style="font-size: 10pt" size="2">),1/(COUNTA(</font></font><font color="#323232"><font style="font-size: 10pt" size="2"><i>data range</i></font></font><font color="#323232"><font style="font-size: 10pt" size="2">))</font></font></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">COUNTA(<em>data range</em>) counts the number of cells in a range that are not empty.</font></font></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">A more direct way to calculate the geometric mean would be to use the following formula:</font></font></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">Geometric Mean = GEOMEAN(</font></font><font color="#323232"><font style="font-size: 10pt" size="2"><i>data range</i></font></font><font color="#323232"><font style="font-size: 10pt" size="2">)</font></font></p> <p> </p> <h4>Combining Differently-Scaled Metrics into a Single Metric With Geometric Mean</h4> <p>The geometric mean provides a method to combine separate measures created on different scales into a single measure that is representative of all of the measures combined. The geometric mean is a way of normalizing the all measurements so that so that no one measure has a disproportionately large effect on the calculation of the geometric mean.</p> <p>For example, products within a certain category might be rated for quality on a scale of 1 to 10 and also be rated for ease-of-use on a scale of 1 to 100. The geometric mean creates a single measure that combines both of these scores without allowing either of the scores to have a disproportionately large effect.</p> <p>Brand A Quality Rating: 6</p> <p>Brand A Ease-of-Use Rating: 70</p> <p>Brand A Geometric mean = GEOMEAN(6,70) = 20.5</p> <p>Brand B Quality Rating: 8</p> <p>Brand B Ease-of-Use Rating: 25</p> <p>Brand B Geometric mean = GEOMEAN(8,25) = 14.1</p> <p><font color="#323232"><font style="font-size: 10pt" size="2">If the criteria of Quality Rating and Ease-of-Use Rating are considered equally important and are the two main criteria for selecting among brands within the category, Brand A would be the preferred brand as determined by its higher geometric mean of those two measures.</font></font></p> <p><font color="#323232"><font style="font-size: 10pt" size="2">One very important characteristic of the geometric mean is the normalization of the scales of the different measures. The same percent change in either measure will produce the same change in the geometric mean.</font></font></p> <p>Brand B Geometric mean = GEOMEAN(8,25) = 14.1</p> <p>If Brand B’s Quality Rating is reduced by 50 percent from 8 to 4, the geometric mean now becomes the following:</p> <p>Brand B Geometric mean = GEOMEAN(4,25) = 10</p> <p>If Brand B’s Ease-of-Use Rating is reduced by 50 percent from 25 to 12.5, the geometric mean will also equal 10 as follows:</p> <p>Brand B Geometric mean = GEOMEAN(8,12.5) = 10</p> <p> </p> <h4>Calculating Combined Growth Rate With Geometric Mean</h4> <p>Another important use of the geometric mean is to calculate a single “average” growth rate over periods of different growth rates that would produce the same overall amount of growth. For example, an investment of $1,000 would grow to $1,615 if it grew by 15 percent in the 1<sup>st</sup> year, 30 percent in the 2<sup>nd</sup> year, and 8 percent in the 3<sup>rd</sup> year. This can be calculated by the following: </p> <p>1<sup>st</sup> year balance = $1,000 * (1.15) = $1,150</p> <p>2<sup>nd</sup> year balance = $1,150 * (1.30) = $1,495</p> <p>3<sup>rd</sup> year balance = $1,495 * (1.08) = $1,615</p> <p>An average growth rate for all three periods would be calculated using the geometric mean as follows:</p> <p>GEOMEAN(1.15,1.30,1.08) = 1.173154</p> <p>The end balance after 3 years at this constant annual growth rate would be calculated as follows:</p> <p>$1,000 * (1.173154)<sup>3</sup> = $1,615 </p> <p> </p> <h3>Harmonic Mean in Excel</h3> <p>The harmonic mean is often used to calculate an average of different rates or ratios. The harmonic mean is also used to calculate an average of a set of numbers that might have significant outliers. The harmonic mean would provide a truer representation of the average than the arithmetic mean because outliers have a disproportionate effect on the arithmetic mean.</p> <p>The harmonic mean is calculated by the following formula:</p> <p><a href="http://lh4.ggpht.com/-96mphYo90r8/VQhyI0LYgtI/AAAAAAAA6N4/U0KjTJjvPKA/s1600-h/Harmonic_Mean%25255B4%25255D.gif"><img title="Measures of Central Tendency in Excel - Harmonic Mean Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Harmonic Mean Formula" src="http://lh4.ggpht.com/-HQtLp1RrtVI/VQhyJcJ-F4I/AAAAAAAA6OA/evyYHxnIMio/Harmonic_Mean_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="57" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Excel formula to calculate the harmonic mean is the following:</p> <p>Harmonic Mean = HARMEAN(<i>data range</i>)</p> <p> </p> <h4>Calculating the Average Rate Using the Harmonic Mean</h4> <p>One pump operates at 5 gallon/minute, a second pump operates at 7 gallons/minute, and a third pump operates at 9 gallons/minute. The average rate for all three pumps if each pump is used sequentially to pump the same amount as the other two pumps is calculated as follows:</p> <p>HARMEAN(5,7,9) = 6.608392</p> <p>If each pump was operated separately to pump out 40 gallons of water and they operated sequentially (one turned on immediately after another finished pumping 40 gallons of water), the average pumping volume over the entire time that it took all three pumps to collectively pump 120 gallons is 6.608392 gallons/min.</p> <p>That can be verified as follows:</p> <p>Total time = 120 gallons / (6.608392 gallons/minute) = 18.16 minutes</p> <p>Total time = 40 gallons / (5 gal/min) + 40 gallons/(7 gal/min) + 40 gallons(9 gal/min) = </p> <p>= 8 minutes + 5.7143 minutes + 4.4444 minutes = 18.16 minutes</p> <p> </p> <h4>Calculating the Average Speed Using the Harmonic Mean</h4> <p>If equal distances were individually travelled at different speeds, the harmonic mean would be used to calculate the average speed for the entire trip. For example Point A is 120 km from Point B. A train goes from Point A to Point B at 40 km/hour and then back to Point A at 60 km/hour, the average speed would be calculated using the harmonic mean as follows:</p> <p>Average speed = HARMEAN(40,60) = 48 km/hour</p> <p>This is verified as follows:</p> <p>The total trip from A to B and then back to A would take the following amount of time travelling at 48 km/hour:</p> <p>Total time = (120 km + 120 km)/(48 km/hour) = 5 hours</p> <p>The total trip going from A to B at 40 km/hour and then from B to A at 60 km/hour would take the following amount of time:</p> <p>Total time = (120 km)/(40 km/hour) + (120 km)/(60 km/hour) =3 hours + 2 hours = 5 hours</p> <p> </p> <h4>Calculating an Average when Significant Outliers Exist By Using the Harmonic Mean</h4> <p>One of the advantages that the harmonic mean has over the arithmetic mean is that outliers in the data set do not unduly skew the harmonic mean to the same degree that occurs with the arithmetic mean. </p> <p>The following data set has a single significant outlier:</p> <p>(3, 4, 6, 5, 7, 85, 6, 3, 5)</p> <p>The harmonic mean of this data set is calculated in Excel as follows:</p> <p>Harmonic mean = HARMEAN(3,4,6,5,7,85,6,3,5) = 4.988</p> <p>The arithmetic mean of this data set is calculated in Excel as follows:</p> <p>Arithmetic mean = AVERAGE(3,4,6,5,7,85,6,3,5) = 13.778</p> <p>If the single outlier were removed, the harmonic and arithmetic means would be calculated as follows:</p> <p>Harmonic mean = HARMEAN(3,4,6,5,7,6,3,5) = 4.462</p> <p>Arithmetic mean = AVERAGE(3,4,6,5,7,6,3,5) = 4.875</p> <p> </p> <h4>The Harmonic Mean is the Preferred Method in Finance For Averaging Multiples</h4> <p>The harmonic mean gives equal weight to each data point and removes excessive influence that large outliers have on the arithmetic mean. Because of this quality, the harmonic mean is the preferred method in finance for averaging multiples such as the price/earnings ratio. </p> <p> </p> <h4>Calculating the Combined Resistance of Parallel Resistors Using the Harmonic Mean</h4> <p>The cumulative resistance of parallel resistor is calculated using the harmonic mean as follows:</p> <p>Resistor 1 = 40Ω</p> <p>Resistor 1 = 60Ω</p> <p>Their combined resistance is place parallel to each other is the following:</p> <p>Combined resistance = HARMEAN(40,60) = 48Ω</p> <p>The harmonic mean is also in the same fashion to calculate the combined capacitance of capacitors arranged in series.</p> <p> </p> <h3>Weighted Harmonic Mean in Excel</h3> <p>The harmonic mean formula above assumes each rate has produced the same output, i.e., the same distance was travelled at each of the different rates or the same amount of liquid was pumped at each of the different rates. The different output occur at different rates, a weighted harmonic mean should be applied. The formula for this is as follows:</p> <p><a href="http://lh3.ggpht.com/-CTxMkDb9SvY/VQhyJ--3FRI/AAAAAAAA6OE/or273zXOWkY/s1600-h/Weighted_Harmonic_Mean%25255B4%25255D.gif"><img title="Measures of Central Tendency in Excel - Weighted Harmonic Mean Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Measures of Central Tendency in Excel - Weighted Harmonic Mean Formula" src="http://lh3.ggpht.com/-KvM-R6KCIKs/VQhyKdKVifI/AAAAAAAA6OQ/0gsjyrM48KY/Weighted_Harmonic_Mean_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="44" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>There is no easy way of automating this formula in Excel. The weighting and data values must be individually entered into the equation as in the following example:</p> <p>A person travels in a train at 40 km/hour for 100 km and at 60 km/hour for 150 km. Calculate the average speed for the entire trip.</p> <p>The weighting for the first part of the trip, w<sub>1</sub>, is set to 2. The weighting for the second part of the trip, w<sub>2</sub>, is set to 3. Any set of number could be used for w<sub>1</sub> and w<sub>2</sub> as long as w<sub>1</sub> / w<sub>2</sub> = 2/3.</p> <p>The weighted harmonic mean would then be calculated as follows:</p> <p>Weighted Harmonic Mean = (2 + 3) / (2/40 + 3/60) = 50 </p> <p>The average speed for the entire trip is 50 km/hour. This can be verified as follows:</p> <p>The total distance travelled was 250 km. At 50 km/hour, the trip would take 5 hours.</p> <p>Traveling at 40 km/hour for 100 km would take 2.5 hours (100km / 40 km/hour = 2.5 hours)</p> <p>Traveling at 60 km/hour for 150 km would take 2.5 hours (150km / 60 km/hour = 2.5 hours)</p> <p>The total trip time is 5 hours.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com39tag:blogger.com,1999:blog-3568555666281177719.post-90316312939472551402015-03-10T10:58:00.001-07:002015-03-24T09:30:36.956-07:00Simplifying Goal Seek in Excel<h1>Simplifying Goal Seek in <br /> <br />Excel</h1> <p>Goal Seek is one of the tools in the What-If Analysis category, which is located under the Data tab in the ribbon. Goal Seek calculates the value of a single input variable of a computation that will produce a specified value in the outcome.</p> <p>An input variable that is changed in order to produce a specified outcome is called a Decision Variable. The output variable is called the Objective. Goal Seek calculates the value of a single Decision Variable that will produce a specified value of the Objective.</p> <p>Goal Seek is a very limited tool for the following reasons:</p> <ol> <li> <p>Goal Seek can only calculate the value of a single Decision Variable. This means that all other inputs to the computation must remain constant while only the single Decision Variable can be changed. No other input can be varied while Goal Seek is performing its calculations.</p> </li> <li> <p>Goal Seek cannot find a solution if any part of the computation contains a discontinuous formula.</p> </li> </ol> <p>The Excel Solver is the preferred tool when attempting to calculate the value of a Decision Variable that will produce a specified result in the Objective for the following reasons:</p> <ol> <li> <p>Excel Solver can calculate the values of numerous Decision Variables simultaneously for a problem that has multiple inputs.</p> </li> <li> <p>Excel Solver has the capability to operate within constraints that are imposed on any of the variables in the computation.</p> </li> <li> <p>Excel Solver can calculate the value of Decision Variables that can maximize or minimize an Objective in addition to merely calculating the value of Decision Variables that produces an exact value in the Objective.</p> </li> <li> <p>Excel Solver can work with many discontinuous formulas within the computation.</p> </li> <li> <p>Excel Solver can be set up just as quickly as Goal Seek to solve any problem simple enough that can be solved with Goal Seek.</p> </li> </ol> <p>The brief example problem in this article will therefore be solved with both Goal Seek and the Excel Solver to compare the basic operation of the two. The problem to be solved is the following:</p> <p><a href="http://lh4.ggpht.com/-J1oVx7HyUNM/VP8yPdbbAgI/AAAAAAAA6KM/5BwqtBpei3Y/s1600-h/Goal_Seek_1_Initial_Data_400%25255B7%25255D.jpg"><img title="Goal Seek in Excel - Initial Problem Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Initial Problem Data" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFSQUb0Bum5G-7YHX9ZmSwU0IKo2EjTlSxY1QUfnm5tvtiWdvUxBmZJtTKu9NEjZ5MEcOcRVT9BHjWYrD7hUNvt74d8N2EZ5gDzQs9J__RCw8QAVeYTHOQ732DsSIi9y8wJ0LULZYfs2ep/?imgmax=800" width="404" height="196" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The purpose of this exercise is to calculate the value on the Decision Variable (cell B2, which currently contains the value of 4) that will cause the Objective (cell B4, which currently contains the value 625) to assume the value of 3,125. The solution is reached when the Decision Variable equals 5 because 5<sup>5</sup> = 3,125. This simple problem will be solved with both Goal Seek and with the Excel Solver.</p> <p> </p> <h2>Solving a Single-Decision-Variable Problem With Goal Seek</h2> <p>Goal Seek is one of the tools in the What-If Analysis category under the Data tab in the ribbon. Selecting Goal Seek brings up the Goal Seek Dialogue box shown as follows:</p> <p><a href="http://lh6.ggpht.com/-ArFdbRZxD68/VP8yQGAWaBI/AAAAAAAA6Kc/KVVTh2hl8kg/s1600-h/Goal_Seek_2_Selecting_Goal_Seek_600%25255B4%25255D.jpg"><img title="Goal Seek in Excel - Selecting Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Selecting Goal Seek" src="http://lh6.ggpht.com/-5FeY8pH8QhY/VP8yQnGIXZI/AAAAAAAA6Kk/PtCBWL9vhTw/Goal_Seek_2_Selecting_Goal_Seek_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="167" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Goal Seek Dialogue is very simply to configure because Goal Seek changes the value of a single Decision Variable that will produce a specific value in the Objective cell without allowing any constraints to be imposed on any variables in the computation. The completed dialogue box for this problem is shown below:</p> <p><a href="http://lh4.ggpht.com/-5D5UjAck2v8/VP8yQ50pl2I/AAAAAAAA6Ks/AFKAk-p4QYg/s1600-h/Goal_Seek_3_Dialogue_Box_600%25255B4%25255D.jpg"><img title="Goal Seek in Excel - Goal Seek Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Goal Seek Dialogue Box" src="http://lh6.ggpht.com/-fmuYr4i2050/VP8yRmxMEhI/AAAAAAAA6K0/pv2Rkp3kvLg/Goal_Seek_3_Dialogue_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="172" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Clicking OK produces the following solution. Goal Seek occasionally requires computations to be rounded off.</p> <p><a href="http://lh3.ggpht.com/-f5GyjgN_0qQ/VP8ySEgfogI/AAAAAAAA6K8/EwZzdcOz668/s1600-h/Goal_Seek_4_Final_Answer_600%25255B4%25255D.jpg"><img title="Goal Seek in Excel - Goal Seek Final Answer" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Goal Seek Final Answer" src="http://lh4.ggpht.com/-g_okjY2XvgI/VP8ySa9nEEI/AAAAAAAA6LE/50WyWEKbCIE/Goal_Seek_4_Final_Answer_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="162" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Solving a Single-Decision-Variable Problem With Excel Solver</h2> <p>The Excel Solver require just about the same amount of work to set up and solve this problem. The Excel Solver is an add-in that must be activated on a one-time basis before it appears in the ribbon and is available for use. Most versions of Excel ship with the Solver as an add-in that needs to be activated, but some versions of Excel do not. </p> <p>Assuming that the Solver has activated, the Solver will appear under the Data tab in the ribbon as follows:</p> <p><a href="http://lh6.ggpht.com/-d1omiaIfGoM/VP8yS1LD_LI/AAAAAAAA6LM/BgmGI5vtGQw/s1600-h/Goal_Seek_5_Select_Solver_600%25255B4%25255D.jpg"><img title="Goal Seek in Excel - Select Solver" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Select Solver" src="http://lh6.ggpht.com/-LkKhF1ztxRo/VP8yTaNVIVI/AAAAAAAA6LU/HCqwLxzkCqM/Goal_Seek_5_Select_Solver_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="66" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Selecting Solver will produce the following empty dialogue box.</p> <p><a href="http://lh4.ggpht.com/-er2ZSK5mfT0/VP8yT4WliQI/AAAAAAAA6Lc/oKADcpPcaHA/s1600-h/Goal_Seek_6A_Empty_Solver_Dailogue_Box_600%25255B7%25255D.jpg"><img title="Goal Seek in Excel - Empty Solver Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Empty Solver Dialogue Box" src="http://lh6.ggpht.com/--Kq-GEAWib0/VP8yUC1aKGI/AAAAAAAA6Lk/FmGjsM4X7Cc/Goal_Seek_6A_Empty_Solver_Dailogue_Box_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="375" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-fVOBJd4UxfU/VP8yUhDVxLI/AAAAAAAA6Ls/jz3p6ThHw64/s1600-h/Goal_Seek_1_Initial_Data_400%25255B9%25255D.jpg"><img title="Goal Seek in Excel - Initial Problem Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Initial Problem Data" src="http://lh5.ggpht.com/-5vxMkUqj0yE/VP8yVHFxKmI/AAAAAAAA6L0/lp6DOfrHBc4/Goal_Seek_1_Initial_Data_400_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="196" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>For the problem shown here, The Solver dialogue box would be completed as follows:</p> <p><a href="http://lh3.ggpht.com/-F1QBtNA1fpM/VP8yVgUdi0I/AAAAAAAA6L8/_ufh0KoE6Zs/s1600-h/Goal_Seek_6_Solver_Dialogue_Box_600%25255B7%25255D.jpg"><img title="Goal Seek in Excel - Solver Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; float: none; margin-left: auto; display: block; border-top-width: 0px; margin-right: auto" border="0" alt="Goal Seek in Excel - Solver Dialogue Box" src="http://lh3.ggpht.com/-s-ZNAtWJiHc/VP8yWL5HOnI/AAAAAAAA6ME/SKYso1kG2y0/Goal_Seek_6_Solver_Dialogue_Box_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="375" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Note that the Excel Solver has the capability to solve for multiple Decision Variables simultaneously and also has the ability to operate within constraints that the user can impose. The Solver method used is the GRG Nonlinear method. This solving method is the default for Solver and can be used to solve linear and nonlinear equations that are continuous. The details of the different Solver methods are discussed in other articles in this blog that focus specifically on that topic.</p> <p>Selecting Solve produces the following output:</p> <p><a href="http://lh5.ggpht.com/-5deTweu5wuw/VP8yWlBjDCI/AAAAAAAA6MM/o47TpPaDB2o/s1600-h/Goal_Seek_7_Solver_Solution_600%25255B4%25255D.jpg"><img title="Goal Seek in Excel - Solver Solution" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Goal Seek in Excel - Solver Solution" src="http://lh4.ggpht.com/-NNEzo0OGLtc/VP8yXGAwj_I/AAAAAAAA6MU/dxEitoOw6bw/Goal_Seek_7_Solver_Solution_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="213" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Solver creates 3 reports related to different aspects of the computation of its solution. </p> <p>To sum up, it is better to use the Excel Solver to perform any calculation that can be performed with Goal Seek. The Solver can be set up just about as quickly as Goal Seek but has so much more capability and available output information. </p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com16tag:blogger.com,1999:blog-3568555666281177719.post-45894033568270702252015-03-09T15:34:00.001-07:002015-03-24T09:34:13.553-07:00Scenario Analysis in Excel With Option Buttons and the What-If Scenario Manager<h1>Scenario Analysis in Excel <br /> <br />With Option Buttons and <br /> <br />the What-If Scenario <br /> <br />Manager </h1> <p>Two excellent tools in Excel to conveniently perform scenario analysis are Option Buttons and the What-If Scenario Manager. Scenario analysis in Excel involves switching different sets of input values into the same set of formulas to compare the differences in outcome. The classic example is the Best Case-Expected Case-Worst Case set of scenarios. This article will demonstrate in detail how to implement this set of scenarios with both the options buttons and with the What-If Scenario Manager. Here is a brief description of each:</p> <p> </p> <p><u><b>Scenario Analysis With Option Buttons</b></u> – Formerly called a Radio Button, the Option Button to choose between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are common methods employed to implement scenario selection. Both of these techniques will be demonstrated in this article. Switching between different scenarios is as simple as selecting a different option button. An example using options buttons to implement scenario selection is shown as follows:</p> <p><a href="http://lh3.ggpht.com/-f43Tz7_35Kk/VP4fXmOKbyI/AAAAAAAA5_c/hC7hzzsv3Xw/s1600-h/Form_Controls_14_Radio_Scenario_2%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button Scenarios" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button Scenarios" src="http://lh4.ggpht.com/--Lccrs5V2cc/VP4fYO4BPSI/AAAAAAAA5_g/SZtao7qpZ6Y/Form_Controls_14_Radio_Scenario_2_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><u><b>Scenario Analysis With the What-If Scenario Manager</b></u> - </p> <p>The Scenario Manager is a tool found under the What-If Analysis section which is located under the Data tab. The Scenarios are created and stored within the Scenario Manager. When an individual scenario is selected within the Scenario Manager, the value in the output cell is changed to the value specified by the scenario as follows:</p> <p><a href="http://lh6.ggpht.com/-fUxD-ejgBYE/VP4fYt-5MpI/AAAAAAAA5_s/fc3sodAkjAM/s1600-h/Scenarios_14_Select_Scenario_2_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Select Scenario" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select Scenario" src="http://lh6.ggpht.com/-CJYo3fLQ5oM/VP4fZOObepI/AAAAAAAA5_0/8yuXCqxqdoo/Scenarios_14_Select_Scenario_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="328" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-5oIRtZgJ7x8/VP4fZvARBLI/AAAAAAAA5_8/icA8tFwrfaQ/s1600-h/Scenarios_12_Scenario_Formulas_600%25255B7%25255D.jpg"><img title="Scenario Analysis in Excel - Select Formulas" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select Formulas" src="http://lh5.ggpht.com/-RAmjDrYb7ZA/VP4faKzO_9I/AAAAAAAA6AE/RD_2K7yFVxY/Scenarios_12_Scenario_Formulas_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="178" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Scenario Manager must be brought up every time that a scenario change is desired. Option buttons are always visible making them a more convenient option to switch between scenarios.</p> <p> </p> <h2>Turning On the Developer Tab in the Excel Ribbon</h2> <p>Form controls are accessed from the Developer tab. By default the Developer tab does not appear in the ribbon. The Developer tab must be configured to be one of the tabs permanently displayed in the ribbon. This can be quickly implemented with the following steps:</p> <p><b>File / Options / Customize Ribbon</b></p> <p>This will produce the following dialogue box. The checkbox next to Developer will initially be unchecked. Simply check that checkbox and the Developer tab will become a permanently visible part of the ribbon as long as that checkbox remains checked. This is shown in the following images.</p> <p><a href="http://lh4.ggpht.com/-TQRDk1xKvJ8/VP4faspT4aI/AAAAAAAA6AM/oVIARPfFhic/s1600-h/Form_Controls_1_Options_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Form Control Options" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Form Control Options" src="http://lh6.ggpht.com/-PrWrnqu5IfU/VP4fbLFr0NI/AAAAAAAA6AQ/NWLK9rCf3u0/Form_Controls_1_Options_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="328" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-bLwy-gwuBVs/VP4fbrAOwyI/AAAAAAAA6Ac/yE8dnFLBi5Y/s1600-h/Form_Controls_2_Developer_Tab_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Developer Tab" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Developer Tab" src="http://lh4.ggpht.com/-Oq_83sGG8X4/VP4fcDGB32I/AAAAAAAA6Ak/t3igBlOsPbY/Form_Controls_2_Developer_Tab_400_thumb%25255B2%25255D.jpg?imgmax=800" width="255" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Developer tab now appears as a permanent part of the ribbon as long as the checkbox next to Developer remains checked. This is shown as follows:</p> <p><a href="http://lh6.ggpht.com/-JaWnCZQeYg0/VP4fcpvps8I/AAAAAAAA6As/BBaJJRxXOX8/s1600-h/FOrm_Controls_3_Developer_3_500%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Developer Form Controls" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Developer Form Controls" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtiQLWcgAFwz8FkxjU3cZXOQAIp07WkBUWD-hlh-v1pY1j4mH6d5Zp6Q6BEZYz8JvgwIEQInbMbOYNVdeOJINo7xo3dx7gG7ftCRWz83WB6J4DMWJaV5NocjnECtqXCtuv1jvn-Ld5a-lK/?imgmax=800" width="404" height="157" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>All form controls are made available clicking Insert in the Developer tab.</p> <p><a href="http://lh5.ggpht.com/-XR-iUgC7ZNU/VP4fdkio7AI/AAAAAAAA6A8/7DmBE-7mf6o/s1600-h/Form_Controls_4_Select_Form_Control_500%25255B6%25255D.jpg"><img title="Scenario Analysis in Excel - Select Form Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select Form Control" src="http://lh5.ggpht.com/-n9APkBLFXZk/VP4feG8yAKI/AAAAAAAA6BE/d7FIalEUFjE/Form_Controls_4_Select_Form_Control_500_thumb%25255B4%25255D.jpg?imgmax=800" width="404" height="346" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Scenario Analysis With Option Buttons</h2> <p>Formerly called a Radio Button, the Option Button is used to choose between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are common methods employed to implement scenario selection. Both of these techniques will be demonstrated in this article.</p> <p>The option button is creating by clicking the option button icon in the Form Controls menu as follows:</p> <p><a href="http://lh3.ggpht.com/-kQvUzsFxSSw/VP4feubGMII/AAAAAAAA6BM/EK7N4LSntdM/s1600-h/Form_Controls_27_Option_Button_500%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button" src="http://lh3.ggpht.com/-mrSk4InaE28/VP4ffBngJzI/AAAAAAAA6BU/C8RlBRaBAvk/Form_Controls_27_Option_Button_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="352" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the option button icon is clicked, the cursor become the tool to place the option button on the Excel worksheet. The outline of the option button is created by clicking and then dragging the cursor. The location and dimensions of the option button can be changed at any time after the initial option button has been created. The initial option button will appear as follows:</p> <p><a href="http://lh4.ggpht.com/-KnUeqpH5F7M/VP4ffhxvjJI/AAAAAAAA6BY/pyFdQE-nbfg/s1600-h/Form_Controls_28_Create_Option_Button_1_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Create Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Create Option Button" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWLBgf5KuKe7vXNaZdX-OVAJiyEQnX6wY2NUWG8ljMoC_iGYMAikWs7V5ZYMN0nspomG9hmZqAtK3aGzD_9TT2x5iUYfH8bu43A0AC1nFNvYA4yLkBYufmWNHyRblpOtKAjXHOMIRs5M0l/?imgmax=800" width="404" height="202" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The option button must now be formatted. This is accomplished by right-clicking anywhere in the option button to bring up the following short-cut menu. Format Control is selected from the short-cut menu. This brings up the Format Control dialogue box. The cell which will hold the output of the option button must now be specified. In this case that cell is A4.</p> <p>Each new option button that is created will be part of the initial group of option buttons. All option buttons in the same group will collectively send their output to the same cell. Only one option button in a group can selected at any time. The output cell, A4 in this case, will contain the number that is associated with and unique to the specific option button that has been selected.</p> <p><a href="http://lh4.ggpht.com/-Gmn0o49c6mI/VP4fhMExgiI/AAAAAAAA6Bs/CQ0Z85oKtnk/s1600-h/Form_Controls_21_Option_Button_3_Format_600%25255B7%25255D.jpg"><img title="Scenario Analysis in Excel - Format Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Format Option Button" src="http://lh5.ggpht.com/-oFWSRVMa4t4/VP4fhqGkgiI/AAAAAAAA6Bw/HLyt0J60CNs/Form_Controls_21_Option_Button_3_Format_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="391" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The first completed options button and output cell appear as follows</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-yNRv_3LlzWUriDEONRk86-GyAR7PIVm1LDrwpbcCrVlzRH8CCVPZhow1YnaUyq0f9EMBVQ5Z1f6fPhAHmiD9bKCCQivPl1RtItLYYxkn7QKotmxZC_pqweW_Sy8J4SrafaC27SWdNIKy/s1600-h/Form_Controls_24_Option_Button_1_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form_Controls_24_Option_Button_1_400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUSLVk0-LD9Qv-bH817mHBI_rOXaMrydfsBlQ777R51TZ7o5pn0t1E1yJY4erbY3fnO9bK6EL_xVtGkyTRtAP2tNjGtk95cIMaeggYRH98C7TXG-nvvO_P54ZwRKDcFzab8L38_-kZETnM/?imgmax=800" width="404" height="203" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>A second options button can now be created. This second options button will be part of the same group that contains the first options button. The Format Control dialogue box that will appear when the second dialogue box is right-clicked as follows. By default the value of the second options buttons will initially be Unchecked and the output will be linked to the same output cell of the first options box, which is A4. </p> <p>Changing the output cell in the Format Control of an options button will change the location of the output cells for all options buttons in the same group.</p> <p><a href="http://lh5.ggpht.com/-O9yKrwN74TE/VP4fkljQLLI/AAAAAAAA6CM/QvTytxSKer8/s1600-h/Form_Controls_23_Option_Button_2_Format_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Format Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Format Option Button" src="http://lh4.ggpht.com/--ECy9vuJIPE/VP4flPAFjDI/AAAAAAAA6CU/7yx2-y_mZm8/Form_Controls_23_Option_Button_2_Format_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="316" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>A third options button can be added to the group as follows:</p> <p><a href="http://lh4.ggpht.com/--3m31zFg6Tg/VP4fltL_hGI/AAAAAAAA6CY/Trh4grGm0GE/s1600-h/Form_Controls_20_Option_Button_3_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="v" src="http://lh4.ggpht.com/-MD3gqJNht6A/VP4fmLHFVyI/AAAAAAAA6Ck/bUWia1GsvnI/Form_Controls_20_Option_Button_3_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="366" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The labels associated with each options button can be renamed by right-clicking on the original label and then typing in the new label. The result might be as follows: </p> <p><a href="http://lh6.ggpht.com/-KBsqzv7eqxM/VP4fmh6ZzYI/AAAAAAAA6Co/cfTCnZn3bBs/s1600-h/Form_Controls_19_Renaming_Buttons_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Renaming Option Buttons" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Renaming Option Buttons" src="http://lh4.ggpht.com/-NuZmRe7w_js/VP4fm9pqviI/AAAAAAAA6C0/ZFFvV--JzEk/Form_Controls_19_Renaming_Buttons_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="320" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>These three options buttons can be assigned to a single group by selecting the Group Box from the list of Format Controls.</p> <p><a href="http://lh6.ggpht.com/-PqYA2_dzcPM/VP4fnVKGK7I/AAAAAAAA6C8/cyGgIA-BQzg/s1600-h/Form_Controls_16_Insert_Group_Box_400%25255B5%25255D.jpg"><img title="Scenario Analysis in Excel - Insert Group Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Insert Group Box" src="http://lh4.ggpht.com/-Q30yWqBGj5g/VP4fn4vW2AI/AAAAAAAA6DE/UmliNSFLk8s/Form_Controls_16_Insert_Group_Box_400_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="549" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the group box icon is clicked, the cursor become the tool to place the group box on the Excel worksheet. The outline of the group box is created by clicking and then dragging the cursor. The location and dimensions of the check box can be changed at any time after the initial check box has been created. Any options buttons that are inside the group box are part of one group. Any options buttons that are outside of the group box are part of a different group. The initial group box will appear as follows:</p> <p><a href="http://lh5.ggpht.com/-MQth4o3S3ls/VP4fockxnWI/AAAAAAAA6DM/hKU1lqr2Few/s1600-h/Form_Controls_18_1st_Group_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 1st Group" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 1st Group" src="http://lh4.ggpht.com/-mnojP5sKTec/VP4fo0LaHrI/AAAAAAAA6DU/HdPnmON7W9w/Form_Controls_18_1st_Group_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="381" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-PDRPQYDJ9jc/VP4fpQPr-GI/AAAAAAAA6DY/_MeFcsvqCWE/s1600-h/Form_Controls_18_New_Group_400%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - New Group" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - New Group" src="http://lh3.ggpht.com/-m_PORGO7aVk/VP4fp9a59gI/AAAAAAAA6Dk/SWjtC5PAAQk/Form_Controls_18_New_Group_400_thumb%25255B2%25255D.jpg?imgmax=800" width="369" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Option Button is designed to create the ability to choose between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are straightforward methods to implement scenario selection. An example of each of the two methods are shown as follows: </p> <p><a href="http://lh3.ggpht.com/-hDlI9rd24lk/VP4fqRyfMcI/AAAAAAAA6Ds/fXQLedZ6qqk/s1600-h/Form_Controls_13_Radio_Scenario_1_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button Scenario 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button Scenario 1" src="http://lh3.ggpht.com/-heHGGwWd9a8/VP4fq4l5yII/AAAAAAAA6Dw/hh3qVPXarfQ/Form_Controls_13_Radio_Scenario_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-cUEp2lFa_Do/VP4frDz4tDI/AAAAAAAA6D4/qxFiBLRM3oo/s1600-h/Form_Controls_13A_Radio_Scenario_1_Closeup_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button Scenario 1 Closeup" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button Scenario 1 Closeup" src="http://lh5.ggpht.com/-wnZHASzoU1E/VP4frh9Y1-I/AAAAAAAA6EA/7d4UONv0Rys/Form_Controls_13A_Radio_Scenario_1_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-YifFWI3DUn4/VP4fsCBq6nI/AAAAAAAA6EM/Bvzo_uiANI4/s1600-h/Form_Controls_14_Radio_Scenario_2_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button Scenario 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button Scenario 2" src="http://lh5.ggpht.com/-nqoB9DK-hu4/VP4fsmT4orI/AAAAAAAA6EQ/7tZKkfspMCU/Form_Controls_14_Radio_Scenario_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Conditional formatting was applied to display negative values in red instead of black as shown as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnpFtcpv6yPW92Y9EPRlSZr8a0-FSDpADU6TWmYpy6EIqouzHHE5zSmZ-xY3OjuZHlrWBE_gMpJ86iKF6ywNthuLAbuR1Kb911ebZ-HfEJrXarTRx0qT0st-UEUcuYuc2H2n3z33xWd-5b/s1600-h/Form_Controls_15_Radio_Scenario_3_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Option Button Scenario 3" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Option Button Scenario 3" src="http://lh3.ggpht.com/-YVFl62J7xDg/VP4ftrfCVuI/AAAAAAAA6Ek/CiLpos03Wmo/Form_Controls_15_Radio_Scenario_3_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Scenario Analysis With the What-If Analysis Scenario Manager</h2> <p>The Scenario Manager is a tool found under the What-If Analysis section which is located under the Data tab. The Scenarios are created and stored within the Scenario Manager. When an individual scenario is selected within the Scenario Manager, the value in the output cell is changed to the value specified by the scenario.</p> <h3 class="western">Creating a Scenario in the What-If Scenario Manager </h3> <p>The Scenario Manager is one of the tools in the drop-down menu under What-If Analysis, which is under the Data tab.</p> <p><a href="http://lh4.ggpht.com/-W-KtgpO6Yz4/VP4fuCiJYgI/AAAAAAAA6Es/yMJh0Vaa8U0/s1600-h/Scenarios_1_Select_What-If_600%25255B7%25255D.jpg"><img title="Scenario Analysis in Excel - Select What-If" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select What-If" src="http://lh5.ggpht.com/-OoslZmr8BFA/VP4funrYRlI/AAAAAAAA6E0/SkpcS_Rot3w/Scenarios_1_Select_What-If_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="163" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The following empty dialogue box appears when Scenario Manager has been selected.</p> <p><a href="http://lh3.ggpht.com/-QFXi5rVsYvQ/VP4fvEM0CJI/AAAAAAAA6E8/IRRflDINxeY/s1600-h/Scenarios_2_Scenario_Mgr_Dialogue_Box_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Scenario Manager Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Scenario Manager Dialogue Box" src="http://lh5.ggpht.com/--bgEhqJIaRs/VP4fvq9wWqI/AAAAAAAA6FE/f9mLbCwDKYs/Scenarios_2_Scenario_Mgr_Dialogue_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="409" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Add a scenario by selecting Add. This brings up the following Edit Scenario dialogue box, which has been filled in with the basic information about the first scenario titled Best Case. The output cells for this scenario is A4. </p> <p><a href="http://lh5.ggpht.com/-9gwPORPlk3I/VP4fwN2aD7I/AAAAAAAA6FM/9cKP1ypi1wQ/s1600-h/Scenarios_3_1st_Scenario_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 1st Scenario" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 1st Scenario" src="http://lh5.ggpht.com/-rkx0Hb-Fsqg/VP4fwwpnvfI/AAAAAAAA6FQ/oqxbyiwu9xw/Scenarios_3_1st_Scenario_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="320" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The value that will appear in this output cell when the scenario is set in the Scenario Values dialogue box, which appears after OK is clicked in the Edit Scenario dialogue box.</p> <p><a href="http://lh5.ggpht.com/-hw4WxHYYEaQ/VP4fxCVGhJI/AAAAAAAA6Fc/Qq4R3NAj3Ok/s1600-h/Scenarios_3_1st_Scenario_Value_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 1st Scenario Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 1st Scenario Value" src="http://lh6.ggpht.com/-KNQQnXz2z7k/VP4fxnqJiZI/AAAAAAAA6Fk/g5Wgw8NPhEs/Scenarios_3_1st_Scenario_Value_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="165" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>A second scenario titled Most Expected Case is created in a very similar fashion. In this case, the output cell, A4, will display a 2 when this scenario is selected.</p> <p><a href="http://lh4.ggpht.com/-6X5UtT3-v78/VP4fyIHR71I/AAAAAAAA6Fs/esRBcwdO-UY/s1600-h/Scenarios_5_2nd_Scenario_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 2nd Scenario" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 2nd Scenario" src="http://lh5.ggpht.com/-5arSrXmP_0M/VP4fyoSETbI/AAAAAAAA6F0/gyceZO8Adn8/Scenarios_5_2nd_Scenario_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="321" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-urBeWU742nw/VP4fzKBFtUI/AAAAAAAA6F4/Vpf9zGSPxho/s1600-h/Scenarios_6_2nd_Scenario_Value_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 2nd Scenario Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 2nd Scenario Value" src="http://lh4.ggpht.com/-i97q7fC_SSk/VP4fzn9nfaI/AAAAAAAA6GE/5anjpzY8W_Y/Scenarios_6_2nd_Scenario_Value_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="165" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Finally a third scenario is added titled Worst Case. This will also be linked to cell A4 and will produce the value of 3.</p> <p><a href="http://lh6.ggpht.com/-Xs0yPdGlBL4/VP4f0OAD2UI/AAAAAAAA6GM/ZLqBDYC0zv8/s1600-h/Scenarios_7_3rd_Scenario_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - 3rd Scenario" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - 3rd Scenario" src="http://lh3.ggpht.com/-4oj-BjfQvqU/VP4f0m46UoI/AAAAAAAA6GQ/KDWcQUxVBXI/Scenarios_7_3rd_Scenario_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="323" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-xgKZyBGYQcQ/VP4f1JN3ijI/AAAAAAAA6Gc/Trc_cyR4ySE/s1600-h/Scenarios_8_3rd_Scenario_Value_600%25255B7%25255D.jpg"><img title="Scenario Analysis in Excel - 3rd Scenario Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; float: none; margin-left: auto; display: block; border-top-width: 0px; margin-right: auto" border="0" alt="Scenario Analysis in Excel - 3rd Scenario Value" src="http://lh3.ggpht.com/-SxEbEYqSQkc/VP4f1VYX9yI/AAAAAAAA6Gg/cg8FIoBf4co/Scenarios_8_3rd_Scenario_Value_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="163" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The output cell A4 can control which set of input values are applied to the analysis through the use of If-Then-Else statements or CHOOSE() commands as follows:</p> <p><a href="http://lh6.ggpht.com/-icv94ZIOmEE/VP4f1yZshnI/AAAAAAAA6Gs/Ptq7yiPUJ7E/s1600-h/Scenarios_12_Scenario_Formulas_600%25255B9%25255D.jpg"><img title="Scenario Analysis in Excel - Scenario Formulas" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Scenario Formulas" src="http://lh3.ggpht.com/-G5yWK0lnOJQ/VP4f2JMkooI/AAAAAAAA6Gw/AP6dvBgGOfk/Scenarios_12_Scenario_Formulas_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="178" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-u7z4Vq4Yw2U/VP4f2t18N9I/AAAAAAAA6G8/UsUc5y-fQiI/s1600-h/Scenarios_13_Scenario_Formulas_Closeup_600%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenarios_13_Scenario_Formulas_Closeup_600" src="http://lh6.ggpht.com/-19puracZS5g/VP4f3AV3vmI/AAAAAAAA6HE/ZV7ch4I868Y/Scenarios_13_Scenario_Formulas_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="205" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The scenario is selected from the What-If Scenario Manager as follows:</p> <p><a href="http://lh5.ggpht.com/-1cOGtOySMAc/VP4f3lgUrzI/AAAAAAAA6HM/RkPdMZf15Sg/s1600-h/Scenarios_1_Select_What-If_600%25255B9%25255D.jpg"><img title="Scenario Analysis in Excel - Select What-If" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select What-If" src="http://lh4.ggpht.com/-THE7aZTtoHw/VP4f363-0ZI/AAAAAAAA6HU/ePBoEKhggbU/Scenarios_1_Select_What-If_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="163" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-ZeYl3yI250c/VP4f4k50XwI/AAAAAAAA6Hc/Wo1aXqpbKeI/s1600-h/Scenarios_10_Show_Scenario_2%25255B4%25255D.jpg"><img title="Scenario Analysis in Excel - Select Scenario 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Scenario Analysis in Excel - Select Scenario 2" src="http://lh6.ggpht.com/-kudyV7NnPEA/VP4f4yd87oI/AAAAAAAA6Hg/Fivq_VmoTTg/Scenarios_10_Show_Scenario_2_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="320" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This sets the value of cell A4 to 2.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com13tag:blogger.com,1999:blog-3568555666281177719.post-49975798594818460732015-03-09T09:55:00.001-07:002015-03-24T09:37:32.954-07:00Simplifying Excel Form Controls: Check Box, Option Button, Spin Button, and Scroll Bar<h1>Simplifying Excel Form <br /> <br />Controls: Check Box, <br /> <br />Option Button, Spin <br /> <br />Button, and Scroll Bar </h1> <p>Excel form controls provide interactivity and convenience to an Excel worksheet. With form controls a user can quickly run through scenarios, add or subtract large pieces of formulation, and quickly scroll through a large range of values within a single cell. The most commonly used form controls are the Check Box, the Option Button, the Spin Button, and the Scroll Bar. Here is a brief description of what each does:</p> <p><u><b>Check Box</b></u> – The Check Box toggles the value in a specified cell between TRUE and FALSE. This enables a user to turn on or turn off large blocks of formulation by simply checking or unchecking a check box. Utilizing If-Then-Else statements linked to the TRUE/FALSE cell is a common way of toggling on or off the inclusion of blocks of formulation or individual items. This will be shown in this article. An example using multiple Excel check boxes is as follows:</p> <p><a href="http://lh6.ggpht.com/-7CLHWa8L9t8/VP3PyMHUYnI/AAAAAAAA50I/wi7IFcJMxZs/s1600-h/Form_Controls_12_3_Checkboxes_2_600%25255B7%25255D.jpg"><img title="Form Controls in Excel - Check Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box" src="http://lh4.ggpht.com/-mqCod2Uy2io/VP3PyR9JoiI/AAAAAAAA50Q/cQkHOWcuMGA/Form_Controls_12_3_Checkboxes_2_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="260" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><u><b>Option Button</b></u> – Formerly called a Radio Button, the Option Button is used to switch between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are common methods employed to implement scenario selection. Both of these techniques will be demonstrated in this article. An example using options buttons to implement scenario selection is shown as follows:</p> <p><a href="http://lh6.ggpht.com/-BPA09ABcpVw/VP3Py4zRHjI/AAAAAAAA50Y/vgs0iN4ldqg/s1600-h/Form_Controls_14_Radio_Scenario_2_600%25255B7%25255D.jpg"><img title="Form Controls in Excel - Option Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Box" src="http://lh3.ggpht.com/-P4gxoWVEhs4/VP3PzWu4aYI/AAAAAAAA50g/4NjHisxGaVY/Form_Controls_14_Radio_Scenario_2_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><u><b>Spin Button</b></u> – Clicking a Spin Button increases or decreases the value in a specified cell by an incremental amount that was defined when the spin button was initially created and formatted. An example of a spin button in use is shown as follows:</p> <p><a href="http://lh5.ggpht.com/-9ZlWGLvfvA0/VP3PzvsZ2DI/AAAAAAAA50o/f_fGjmVPoYg/s1600-h/Form_Controls_34_Spin_Button_Final_600%25255B8%25255D.jpg"><img title="Form Controls in Excel - Spin Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Spin Button" src="http://lh3.ggpht.com/-TaGRl6aM2lc/VP3P0Aet7DI/AAAAAAAA50w/h8Lc6QIzTzg/Form_Controls_34_Spin_Button_Final_600_thumb%25255B4%25255D.jpg?imgmax=800" width="244" height="221" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><u><b>Scroll Bar</b></u> – Scrolling up or down with a scroll bar increases or decreases the value in a specified cell in a convenient and rapid fashion. An example of a scroll bar in use is shown as follows:</p> <p><a href="http://lh6.ggpht.com/-4ZRgmInisuU/VP3P0vn2vtI/AAAAAAAA504/SVi6UGU1OPI/s1600-h/Form_Controls_40_Final_Scroll_Bar_400%25255B6%25255D.jpg"><img title="Form Controls in Excel - Scroll Bar" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Scroll Bar" src="http://lh3.ggpht.com/-yJ-my111CsI/VP3P1Ob9p0I/AAAAAAAA51A/DlaJ5AdlfRY/Form_Controls_40_Final_Scroll_Bar_400_thumb%25255B2%25255D.jpg?imgmax=800" width="244" height="231" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Turning On the Developer Tab in the Excel Ribbon</h2> <p>Form controls are accessed from the Developer tab. By default the Developer tab does not appear in the ribbon. The Developer tab must be configured to be one of the tabs permanently displayed in the ribbon. This can be quickly implemented with the following steps:</p> <p><b>File / Options / Customize Ribbon</b></p> <p>This will produce the following dialogue box. The checkbox next to Developer will initially be unchecked. Simply check that checkbox and the Developer tab will become a permanently visible part of the ribbon as long as that checkbox remains checked. This is shown in the following images.</p> <p><a href="http://lh4.ggpht.com/-R6oDj-jNc_c/VP3P1qJcc0I/AAAAAAAA51I/9B0SI7jZQEI/s1600-h/Form_Controls_1_Options_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Active Developer Tab" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Active Developer Tab" src="http://lh4.ggpht.com/-VU1pk7ATrHc/VP3P2DnKrXI/AAAAAAAA51Q/nafno6RZLpw/Form_Controls_1_Options_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="328" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-r72dVYuQ1WM/VP3P2m9O6uI/AAAAAAAA51Y/MaHSEEOGnHs/s1600-h/Form_Controls_2_Developer_Tab_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Active Developer Tab" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Active Developer Tab" src="http://lh4.ggpht.com/-xG90tnO9XJ4/VP3P3GwClII/AAAAAAAA51g/jC1pXtstnVs/Form_Controls_2_Developer_Tab_400_thumb%25255B2%25255D.jpg?imgmax=800" width="255" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Developer tab now appears as a permanent part of the ribbon as long as the checkbox next to Developer remains checked. This is shown as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQXdk_vOO_EImQ-77OEijEsx2aEnupAoRExcVYs3-yxpM6bxdzy21GwIXh2hOl4CAXsuLAICBb7gy_tVi9ckTXWHVL9D60v5fI8PktIt1q34V787aGk7xkbOAa-9suyPlHhC09Ml949TFy/s1600-h/FOrm_Controls_3_Developer_3_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Developer Tab" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Developer Tab" src="http://lh4.ggpht.com/-w7TTMCalTuU/VP3P4OXGwDI/AAAAAAAA51w/NWXvdKsIclU/FOrm_Controls_3_Developer_3_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="157" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>All form controls are made available clicking Insert in the Developer tab.</p> <p><a href="http://lh4.ggpht.com/-ztsrRtL5MFc/VP3P4gXksxI/AAAAAAAA514/7tyqmhUh7JI/s1600-h/Form_Controls_4_Select_Form_Control_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Select Form Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Select Form Control" src="http://lh4.ggpht.com/-Vp3PIpN_0HU/VP3P5KIyOhI/AAAAAAAA52A/xk6LP-C5qgs/Form_Controls_4_Select_Form_Control_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="346" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Check box</h2> <p>The check box toggles the value in a specified cell between TRUE and FALSE. This enables a user to turn on or turn off large blocks of formulation by simply checking or unchecking a check box. Each check box is independent of the other check boxes. Utilizing If-Then-Else statements linked to the TRUE/FALSE cell is a common way of toggling on or off the inclusion of blocks of formulation or individual items.</p> <p>The checkbox is creating by clicking the check box icon in the Form Controls menu as follows:</p> <p><a href="http://lh5.ggpht.com/-sg1XOkoR2Mw/VP3P5s15pgI/AAAAAAAA52I/MZE7v-voLxw/s1600-h/Form_Controls_4_Check_Box_Form_Control_400%25255B7%25255D.jpg"><img title="Form Controls in Excel - Check Box Form Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box Form Control" src="http://lh6.ggpht.com/-AddEXxVla1w/VP3P6TPbG4I/AAAAAAAA52Q/OBnUZOtv2cg/Form_Controls_4_Check_Box_Form_Control_400_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="260" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the check box icon is clicked, the cursor become the tool to place the check box on the Excel worksheet. The outline of the check box is created by clicking and then dragging the cursor. The location and dimensions of the check box can be changed at any time after the initial check box has been created. The initial check box will appear as follows:</p> <p><a href="http://lh5.ggpht.com/-BfBaWTV9FyA/VP3P6oZB-NI/AAAAAAAA52Y/YUTYpg8DMxA/s1600-h/Form_Control_5_Make_Select_Box_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Create Check Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Create Check Box" src="http://lh3.ggpht.com/-1ptTRbrOLYs/VP3P69tCv1I/AAAAAAAA52g/8mhHJuUNAK0/Form_Control_5_Make_Select_Box_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The check box must now be formatted. This is accomplished by right-clicking anywhere in the check box to bring up the following short-cut menu. Format Control is selected from the short-cut menu as follows: </p> <p><a href="http://lh6.ggpht.com/-hOLKWywZCOU/VP3P7c9DU5I/AAAAAAAA52o/wT4fqV-OOa8/s1600-h/Form_Control_6_Format_Control_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - CHeck Box Format Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - CHeck Box Format Control" src="http://lh6.ggpht.com/-PZ1WenNE_68/VP3P73WRIVI/AAAAAAAA52w/LIv9yWDFdcM/Form_Control_6_Format_Control_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="396" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The following Format Control dialogue box then appears.</p> <p><a href="http://lh3.ggpht.com/-T5trYV5hSZ0/VP3P8MH9FqI/AAAAAAAA524/9Db_CWiGY0k/s1600-h/Form_Controls_6_Diag_Box_Empty_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Check Box Empty Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box Empty Dialogue Box" src="http://lh5.ggpht.com/-cAck_if33mM/VP3P8n5JyiI/AAAAAAAA53A/6TuVaOyns1s/Form_Controls_6_Diag_Box_Empty_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="373" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Type in the cell that will contain the check box’s output, which toggles between TRUE and FALSE. Selecting Unchecked specifies that check box will initially be unchecked on the worksheet as follows:</p> <p><a href="http://lh3.ggpht.com/-pp-AOGZuq6Q/VP3P8ypWM7I/AAAAAAAA53I/zPlKeABWk4o/s1600-h/Form_Controls_7_Control_Charateristics_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Check Box Control Characteristics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box Control Characteristics" src="http://lh3.ggpht.com/-bUXLry1lFp4/VP3P9cmQ7KI/AAAAAAAA53Q/rLEI_YFLn1U/Form_Controls_7_Control_Charateristics_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="373" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-26Lzisb90kk/VP3P9zsFg4I/AAAAAAAA53Y/361VLe8LjZw/s1600-h/Form_Controls_8_Checked_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Form Controls Checked" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Form Controls Checked" src="http://lh5.ggpht.com/-KmhvxrI6a4I/VP3P-NCTOQI/AAAAAAAA53g/Sed2FvldRqA/Form_Controls_8_Checked_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="216" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-75Q7zW2O_K8/VP3P-vIu5CI/AAAAAAAA53o/ol5v-rl9uU8/s1600-h/Form_Controls_9_Unchecked_300%25255B5%25255D.jpg"><img title="Form Controls in Excel - Form Controls Unchecked" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; margin-left: 0px; display: inline; border-top-width: 0px; margin-right: 0px" border="0" alt="Form Controls in Excel - Form Controls Unchecked" src="http://lh3.ggpht.com/-RSMN4FVHRa8/VP3P_HVF6zI/AAAAAAAA53w/qaS7317cjLU/Form_Controls_9_Unchecked_300_thumb%25255B3%25255D.jpg?imgmax=800" width="404" align="left" height="247" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>If desired, the check box can be outlined and colored with the Format Control dialogue box as follows:</p> <p><a href="http://lh5.ggpht.com/-lNA5rlFTY0U/VP3P_gfBfxI/AAAAAAAA534/rCzFR76sXE0/s1600-h/Form_Control_10_Colors_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Setting Color for Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Setting Color for Control" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg89JxlYLSmUwFdHjPrUyoYuVv_cybRvi6GKB6HK_QjSbKHXD8_EFBGt3TPD-Gzd5IrgMw6VSDaNwPCPc-ex8qfaaOjMkvduoIfL_yHfm91BY04Cl_czyca36lClSQs9vCtTO2N8hAkDgku/?imgmax=800" width="404" height="373" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-QE5me1lohp0/VP3QAQ_CmxI/AAAAAAAA54I/7nCHIdy6QxA/s1600-h/Form_Controls_10__First_Checkbox_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - First Check Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - First Check Box" src="http://lh6.ggpht.com/-RfYizIUjZXM/VP3QBrsCvuI/AAAAAAAA54Q/00wPXBhCgEw/Form_Controls_10__First_Checkbox_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="217" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>One of the most common uses of the check box is to toggle between inclusion and exclusion of items within computation. An If-Then-Else statement linked to the TRUE/FALSE cell is a common method to implement this toggle as follows:</p> <p><a href="http://lh3.ggpht.com/-UlSz_4Umgj0/VP3QBxBq-0I/AAAAAAAA54Y/jbkSEAn6DtU/s1600-h/Form_Controls_11_3_Checkboxes_1_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Check Box 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box 1" src="http://lh4.ggpht.com/-r2-zeC8Lppc/VP3QCS_t6KI/AAAAAAAA54g/9LsZiK3I4-w/Form_Controls_11_3_Checkboxes_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="265" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-yJzb-gTb_Ug/VP3QC_91g-I/AAAAAAAA54o/37h9nwR_WN8/s1600-h/Form_Controls_12_3_Checkboxes_2_600%25255B9%25255D.jpg"><img title="Form Controls in Excel - Check Box 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Check Box 2" src="http://lh3.ggpht.com/-m8hDdfHcJtE/VP3QDc0MRZI/AAAAAAAA54w/yA1HGDCIBws/Form_Controls_12_3_Checkboxes_2_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="260" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Option Button</h2> <p>Formerly called a Radio Button, the Option Button is used to switch between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are common methods employed to implement scenario selection. Both of these techniques will be demonstrated in this article.</p> <p>The option button is creating by clicking the option button icon in the Form Controls menu as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj45S_aGIrA0cKjVHAM6IgIiQYDKBmY4LNCTuy-pOPopgZ6yZE4d7ayB2BJZWOTR2B8GIaSKnoMrGX6lPAwLrqr-iqKOuXDrNbzq1resu0q6aZ8ebY-AgkZh8swuzKntxbQWJVDHIPmAlDm/s1600-h/Form_Controls_27_Option_Button_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button" src="http://lh4.ggpht.com/-OX6V7mD1ZsM/VP3QEOfipNI/AAAAAAAA55A/6z_BjLkULVY/Form_Controls_27_Option_Button_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="352" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the option button icon is clicked, the cursor become the tool to place the option button on the Excel worksheet. The outline of the option button is created by clicking and then dragging the cursor. The location and dimensions of the option button can be changed at any time after the initial option button has been created. The initial option button will appear as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWcNYlBOPO1VJ1iywAF5F9CB7IGMTj375CVZrzoV7FHNGdqP1IlYRfKToneWLB5vToXGjAq5sylVgbl3tid4cU0zaGHu_Hth9chdrazzJ0-pwSPpkwCEQcwGkfJl2Uyq1YOB7hi-7TnxeQ/s1600-h/Form_Controls_28_Create_Option_Button_1_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Create Option Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Create Option Button" src="http://lh4.ggpht.com/-NBh6_mx2Pe4/VP3QFKcKN3I/AAAAAAAA55Q/c0yWd7fzQq0/Form_Controls_28_Create_Option_Button_1_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="202" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The option button must now be formatted. This is accomplished by right-clicking anywhere in the option button to bring up the following short-cut menu. Format Control is selected from the short-cut menu. This brings up the Format Control dialogue box. The cell which will hold the output of the option button must now be specified. In this case that cell is A4.</p> <p>Each new option button that is created will be part of the initial group of option buttons. All option buttons in the same group will collectively send their output to the same cell. Only one option button in a group can selected at any time. The output cell, A4 in this case, will contain the number that is associated with and unique to the specific option button that has been selected.</p> <p><a href="http://lh3.ggpht.com/-76quHBRnOqE/VP3QFgOvplI/AAAAAAAA55Y/DQZs1f89-mU/s1600-h/Form_Controls_25_Option_Button_1_Format_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button Format" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button Format" src="http://lh6.ggpht.com/-1RDqI7_gdXs/VP3QGAkUyWI/AAAAAAAA55g/tZOI-Awgx4o/Form_Controls_25_Option_Button_1_Format_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="305" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The first completed options button and output cell appear as follows</p> <p><a href="http://lh6.ggpht.com/-i6cUszjyXPM/VP3QGhOykZI/AAAAAAAA55o/Nfh8Jf6inVU/s1600-h/Form_Controls_24_Option_Button_1_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button 1" src="http://lh4.ggpht.com/-F1FsKxj1-wM/VP3QHO0GPoI/AAAAAAAA55w/6D6baeI6Ndo/Form_Controls_24_Option_Button_1_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="203" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>A second options button can now be created. This second options button will be part of the same group that contains the first options button. The Format Control dialogue box that will appear when the second dialogue box is right-clicked as follows. By default the value of the second options buttons will initially be Unchecked and the output will be linked to the same output cell of the first options box, which is A4. </p> <p>Changing the output cell in the Format Control of an options button will change the location of the output cells for all options buttons in the same group.</p> <p><a href="http://lh6.ggpht.com/-d7kuGjLNbTg/VP3QHozCHsI/AAAAAAAA554/NvUleEkr9VY/s1600-h/Form_Controls_23_Option_Button_2_Format_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button 2" src="http://lh5.ggpht.com/-hBpUdrqZHB8/VP3QH6dUjmI/AAAAAAAA56A/ZtcT3Hk4jq4/Form_Controls_23_Option_Button_2_Format_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="316" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>A third options button can be added to the group as follows:</p> <p><a href="http://lh3.ggpht.com/-WFsIB1deXJk/VP3QIWdEZTI/AAAAAAAA56I/CiL84zx-WeI/s1600-h/Form_Controls_20_Option_Button_3_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button 3" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button 3" src="http://lh3.ggpht.com/-RAVYhJ4LmCE/VP3QI-6evBI/AAAAAAAA56Q/gPkAExv28nE/Form_Controls_20_Option_Button_3_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="366" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The labels associated with each options button can be renamed by right-clicking on the original label and then typing in the new label. The result might be as follows: </p> <p><a href="http://lh4.ggpht.com/-uyEpQE6aTJs/VP3QJRtZX7I/AAAAAAAA56Y/N1U-NqKUbLo/s1600-h/Form_Controls_19_Renaming_Buttons_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Renaming Buttons" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Renaming Buttons" src="http://lh4.ggpht.com/-7mBPML9lUoo/VP3QJjbw_xI/AAAAAAAA56g/YDgaLQEnb1k/Form_Controls_19_Renaming_Buttons_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="320" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>These three options buttons can be assigned to a single group by selecting the Group Box from the list of Format Controls.</p> <p><a href="http://lh3.ggpht.com/-66_xRe7ZC1Y/VP3QKH9tW9I/AAAAAAAA56o/BlNcQA0hzWw/s1600-h/Form_Controls_16_Insert_Group_Box_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Insert Group Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Insert Group Box" src="http://lh6.ggpht.com/-i6kIpUX0zJQ/VP3QKpQmiAI/AAAAAAAA56w/zQYMSMt5D40/Form_Controls_16_Insert_Group_Box_400_thumb%25255B2%25255D.jpg?imgmax=800" width="356" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the group box icon is clicked, the cursor become the tool to place the group box on the Excel worksheet. The outline of the group box is created by clicking and then dragging the cursor. The location and dimensions of the check box can be changed at any time after the initial check box has been created. Any options buttons that are inside the group box are part of one group. Any options buttons that are outside of the group box are part of a different group. The initial group box will appear as follows:</p> <p><a href="http://lh6.ggpht.com/-_XnQNxpEGjU/VP3QLBdUZlI/AAAAAAAA564/HcwppSuSKg8/s1600-h/Form_Controls_18_1st_Group_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - 1st Group" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - 1st Group" src="http://lh6.ggpht.com/-sltkxaJh16k/VP3QLon2iUI/AAAAAAAA57A/UyPS-3O5mb8/Form_Controls_18_1st_Group_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="381" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-tw6DJA3-GU4/VP3QLxc5keI/AAAAAAAA57I/8T5uz1SsLgg/s1600-h/Form_Controls_18_New_Group_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - New Group" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - New Group" src="http://lh5.ggpht.com/-SYRqZjdHLFk/VP3QMZweG8I/AAAAAAAA57Q/_LUeCbXnTAU/Form_Controls_18_New_Group_400_thumb%25255B2%25255D.jpg?imgmax=800" width="369" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Option Button is designed to create the ability to choose between multiple options. This quite commonly used to perform scenario analysis. Linking If-Then-Else statements or the CHOOSE() formula to the options buttons are straightforward methods to implement scenario selection. An example of each of the two methods are shown as follows: </p> <p><a href="http://lh4.ggpht.com/-TcARW705iAY/VP3QMq6kReI/AAAAAAAA57Y/7KvVAIW6dqw/s1600-h/Form_Controls_13_Radio_Scenario_1_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Option Button Scenario 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button Scenario 1" src="http://lh6.ggpht.com/-eZjxI7nU_jA/VP3QNHhIeNI/AAAAAAAA57g/VUOexvEstqg/Form_Controls_13_Radio_Scenario_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-4FWrgkF0PRs/VP3QNhOCspI/AAAAAAAA57o/iZRercRCL38/s1600-h/Form_Controls_13A_Radio_Scenario_1_Closeup_600%25255B5%25255D.jpg"><img title="Form Controls in Excel - Option Button Scenario 1 Closeup" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button Scenario 1 Closeup" src="http://lh3.ggpht.com/-KidlHsCjvI8/VP3QOLndgOI/AAAAAAAA57w/-dmbHVfVK9k/Form_Controls_13A_Radio_Scenario_1_Closeup_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-zGpNjPA8XN0/VP3QOim9g8I/AAAAAAAA574/j6-ydsEA8mc/s1600-h/Form_Controls_14_Radio_Scenario_2_600%25255B10%25255D.jpg"><img title="Form Controls in Excel - Option Button Scenario 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; margin-left: 0px; display: inline; border-top-width: 0px; margin-right: 0px" border="0" alt="Form Controls in Excel - Option Button Scenario 2" src="http://lh6.ggpht.com/-ZCf9GP_Jgm8/VP3QPAOozTI/AAAAAAAA58A/pfIDlnnI8rE/Form_Controls_14_Radio_Scenario_2_600_thumb%25255B6%25255D.jpg?imgmax=800" width="404" align="left" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Conditional formatting was applied to display negative values in red instead of black as shown as follows:</p> <p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizC-5BvutDcacGiSrn9qjIRGiVIgZmDPkIMTE5Hu4pYD2cUdrk_WePAMOcHmGFZnFGtcIEODDykxxdY2vFnDk2IuSHSSftNSAMgE76GOoyrbtTSD-OtfZXMBIaUZs55-kYqQoZU-ResGT8/s1600-h/Form_Controls_15_Radio_Scenario_3_600%25255B5%25255D.jpg"><img title="Form Controls in Excel - Option Button Scenario 3" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Option Button Scenario 3" src="http://lh3.ggpht.com/-0ARebr4Ukvc/VP3QQKAVw1I/AAAAAAAA58Q/cmPSKdaJZ70/Form_Controls_15_Radio_Scenario_3_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="161" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Spin Button</h2> <p>Clicking a Spin Button increases or decreases the value in a specified cell by an incremental amount that was defined when the spin button was initially created and formatted.</p> <p>The spin button is creating by clicking the spin button icon in the Form Controls menu as follows:</p> <p><a href="http://lh6.ggpht.com/-SnuzZYhFj8Y/VP3QQsUeWvI/AAAAAAAA58Y/b1otJKL71F0/s1600-h/Form_Controls_29_Select_Spin_Button_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Select Spin Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Select Spin Button" src="http://lh5.ggpht.com/-pfJo9KD7Xs8/VP3QRKRXLhI/AAAAAAAA58g/Jp0xi2_QKjQ/Form_Controls_29_Select_Spin_Button_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="347" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>When the spin button icon is clicked, the cursor become the tool to place the spin button on the Excel worksheet. The outline of the spin button is created by clicking and then dragging the cursor. The location and dimensions of the spin button can be changed at any time after the initial spin button has been created. The initial spin button will appear as follows:</p> <p><a href="http://lh5.ggpht.com/-UeKrie_4ObQ/VP3QRZzq9UI/AAAAAAAA58o/VWkQ65aTd5M/s1600-h/Form_Controls_30_Insert_Spin_Button_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Insert Spin Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Insert Spin Button" src="http://lh6.ggpht.com/-kw2BJD88KyI/VP3QR1MZQ5I/AAAAAAAA58w/2qLyHQKwANs/Form_Controls_30_Insert_Spin_Button_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="354" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-ceV3rImUvqE/VP3QSbV5hEI/AAAAAAAA584/uhO3RqDdyZc/s1600-h/Form_Controls_31_Format_Control_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Spin Button Format Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Spin Button Format Control" src="http://lh4.ggpht.com/-KEGO0Jepnsk/VP3QSjeUiII/AAAAAAAA59A/YeAcbdbuRqY/Form_Controls_31_Format_Control_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="324" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The spin button must now be formatted. This is accomplished by right-clicking anywhere in the spin button to bring up the following short-cut menu. Format Control is selected from the short-cut menu. This brings up the Format Control dialogue box. The cell which will hold the output of the spin button must now be specified. In this case that cell is A4. The range, incremental change, and initial value of the spin button are declared as follows:</p> <p><a href="http://lh4.ggpht.com/-ZLPCP40F1-w/VP3QTB5S21I/AAAAAAAA59I/xYlJxVvs83s/s1600-h/Form_Controls_32_Format_Diag_Box_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Spin Button Format Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Spin Button Format Dialogue Box" src="http://lh4.ggpht.com/-SP_HLExJOfA/VP3QT1n_fyI/AAAAAAAA59Q/CjQzgJ15shc/Form_Controls_32_Format_Diag_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="375" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The completed spin button and output cell appear as follows:</p> <p><a href="http://lh3.ggpht.com/-YY6Q2T65BQI/VP3QURk0UjI/AAAAAAAA59Y/vwei6CtuBvQ/s1600-h/Form_Controls_33_Spin_Button_Initial_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Initial Spin Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Initial Spin Button" src="http://lh4.ggpht.com/-zX0ZfGvoyEY/VP3QU4EPP_I/AAAAAAAA59g/TN2G2VzC0fE/Form_Controls_33_Spin_Button_Initial_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="372" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Clicking an up or down arrow changes the value in the output cell by one increment if the new value is within the range specified in the Format Control dialogue box. 19 clicks of the up arrow would produce this value in the output cell:</p> <p><a href="http://lh3.ggpht.com/-rN8Dugth_0U/VP3QVMkhVeI/AAAAAAAA59o/z6Kj7hk0HhQ/s1600-h/Form_Controls_34_Spin_Button_Final_600%25255B10%25255D.jpg"><img title="Form Controls in Excel - Final Spin Button" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Final Spin Button" src="http://lh4.ggpht.com/-sSRoNF7wcD8/VP3QVv0gYfI/AAAAAAAA59w/mM3bMKTZb_g/Form_Controls_34_Spin_Button_Final_600_thumb%25255B6%25255D.jpg?imgmax=800" width="404" height="366" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Scroll Bar</h2> <p>Scrolling up or down with a scroll bar increases or decreases the value in a specified cell in a convenient and rapid fashion.</p> <p>The scroll bar is creating by clicking the scroll bar icon in the Form Controls menu as follows:</p> <p><a href="http://lh5.ggpht.com/-KLlOhMOX_YA/VP3QWNhfyzI/AAAAAAAA594/eyc63km6P70/s1600-h/Form_Controls_35_Select_Scroll_Bar_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Select Scroll Bar" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Select Scroll Bar" src="http://lh6.ggpht.com/-fuFj1w2ihfY/VP3QWm1ijyI/AAAAAAAA5-A/n9z_U-PYdeI/Form_Controls_35_Select_Scroll_Bar_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="347" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.ggpht.com/-LgsdW1yvDUQ/VP3QXNgFBSI/AAAAAAAA5-I/wUbXC56YqZ8/s1600-h/Form_Controls_36_Create_Scroll_Bar_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Create Scroll Bar" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Create Scroll Bar" src="http://lh6.ggpht.com/-PHA5Rv87FIk/VP3QXQhfBBI/AAAAAAAA5-Q/WIciKI7pKdQ/Form_Controls_36_Create_Scroll_Bar_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="364" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The scroll bar must now be formatted. This is accomplished by right-clicking anywhere in the scroll bar to bring up the following short-cut menu. Format Control is selected from the short-cut menu. This brings up the Format Control dialogue box. The cell which will hold the output of the scroll bar must now be specified. In this case that cell is A4. The range, incremental change, and initial value of the scroll bar are declared as follows:</p> <p><a href="http://lh6.ggpht.com/-Wul_7flBeng/VP3QYAtEagI/AAAAAAAA5-Y/ppm2Tuboc_I/s1600-h/Form_Controls_37_Format_Control_500%25255B4%25255D.jpg"><img title="Form Controls in Excel - Scroll Bar Format Control" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Scroll Bar Format Control" src="http://lh3.ggpht.com/-3NuqsziFT9U/VP3QYkjDCiI/AAAAAAAA5-g/Z_9UM3eJr08/Form_Controls_37_Format_Control_500_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="314" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-FsghhOFdN4c/VP3QZKh0sCI/AAAAAAAA5-o/6-BknS6hifA/s1600-h/Form_Controls_38_Format_Diag_Box_600%25255B4%25255D.jpg"><img title="Form Controls in Excel - Scroll Bar Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Scroll Bar Dialogue Box" src="http://lh3.ggpht.com/-1_NSBj5LoOQ/VP3QZlt73jI/AAAAAAAA5-w/orOvZ8NdV5U/Form_Controls_38_Format_Diag_Box_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="375" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The completed scroll bar and output cell appear as follows:</p> <p><a href="http://lh6.ggpht.com/-pfkXDnazhtA/VP3QZ5cRjiI/AAAAAAAA5-4/axBC6iPidEA/s1600-h/Form_Controls_39_Initial_Scroll_Bar_400%25255B4%25255D.jpg"><img title="Form Controls in Excel - Initial Scroll Bar" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Initial Scroll Bar" src="http://lh4.ggpht.com/-j_PmZ6wHsNk/VP3QaUz5y7I/AAAAAAAA5_A/uR3uW8RzRug/Form_Controls_39_Initial_Scroll_Bar_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="375" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Scrolling up or down with the scroll bar rapidly changes the output cells value as follows:</p> <p><a href="http://lh6.ggpht.com/-vqS1r0K5aaI/VP3QbJpId2I/AAAAAAAA5_I/8otA8V4sSOc/s1600-h/Form_Controls_40_Final_Scroll_Bar_400%25255B8%25255D.jpg"><img title="Form Controls in Excel - Final Scroll Bar" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Form Controls in Excel - Final Scroll Bar" src="http://lh5.ggpht.com/-ayFgkXDJr9E/VP3QbkiFD5I/AAAAAAAA5_Q/UMAZzsRew1c/Form_Controls_40_Final_Scroll_Bar_400_thumb%25255B4%25255D.jpg?imgmax=800" width="404" height="383" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com4tag:blogger.com,1999:blog-3568555666281177719.post-91234184186834190692015-03-02T19:26:00.001-08:002015-03-24T09:40:22.934-07:00Simplifying SUMIF, SUMIFS, COUNTIF, COUNTIFS, AVERAGEIF, and AVERAGEIFS<h1>Simplifying SUMIF, <br /> <br />SUMIFS, COUNTIF, <br /> <br />COUNTIFS, AVERAGEIF, <br /> <br />and AVERAGEIFS </h1> <p>The format and actual formulas for SUMIF, SUMIFS, COUNTIF, COUNTIFS, AVERAGEIF, and AVERAGEIFS usually have to be looked up by Excel users. The following presents a quick visual reference for the formulas and uses of those six Excel formulas.</p> <h2>SUMIF</h2> <p><a href="http://lh4.ggpht.com/-X6olB6_eS-0/VPUpmRJdj0I/AAAAAAAA5kc/Cq50gzYu8-I/s1600-h/Av_IF_5_SUMIF_Table_600%25255B4%25255D.jpg"><img title="SUMIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="SUMIF in Excel" src="http://lh3.ggpht.com/-LuKdgJn89jM/VPUpmw9uBWI/AAAAAAAA5kg/CMrgsaP_2N0/Av_IF_5_SUMIF_Table_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="219" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-XTKq4zR1tyw/VPUpnSaJmbI/AAAAAAAA5ks/ln6Skvx8P-Q/s1600-h/Av_IF_6_SUMIF_Formulas_600%25255B4%25255D.jpg"><img title="SUMIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="SUMIF in Excel" src="http://lh4.ggpht.com/-IElftbVYmeM/VPUpnz1DlKI/AAAAAAAA5kw/IpFJnKLxScI/Av_IF_6_SUMIF_Formulas_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="233" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>SUMIFS</h2> <p><a href="http://lh6.ggpht.com/-Yoj3TuQKsZQ/VPUpoY3fXCI/AAAAAAAA5k4/dj3aqXvDIms/s1600-h/Av_IF_7_SUMIFS_Table_600%25255B4%25255D.jpg"><img title="SUMIFS in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="SUMIFS in Excel" src="http://lh5.ggpht.com/-DT2CDMluh_s/VPUpoxZYyRI/AAAAAAAA5lA/ZuKOda6w8uI/Av_IF_7_SUMIFS_Table_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="232" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-W9_j_TysUWw/VPUppdGfM6I/AAAAAAAA5lM/hDVQdQAvXZg/s1600-h/AV_IF_8_SUMIFS_Formulas_600%25255B4%25255D.jpg"><img title="SUMIFS in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="SUMIFS in Excel" src="http://lh5.ggpht.com/-fXfqy6Azn_E/VPUppwVIXsI/AAAAAAAA5lQ/0vH1kabBnM4/AV_IF_8_SUMIFS_Formulas_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="194" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>COUNTIF</h2> <p><a href="http://lh4.ggpht.com/-ZSP0aSKOh3I/VPUpqWRNwyI/AAAAAAAA5lY/_b2lM3R42Hc/s1600-h/Av_IF_9_COUNTIF_Table_600%25255B4%25255D.jpg"><img title="COUNTIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIF in Excel" src="http://lh6.ggpht.com/-QlG9an9XgnE/VPUpq01lvFI/AAAAAAAA5lg/PkU5PiSrOHo/Av_IF_9_COUNTIF_Table_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="227" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-rRhUJRiiqFY/VPUprRd8mkI/AAAAAAAA5ls/mdw4w5llxWQ/s1600-h/Av_IF_10_COUNTIF_Formulas_600%25255B4%25255D.jpg"><img title="COUNTIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIF in Excel" src="http://lh3.ggpht.com/-59Xi5pNvF68/VPUprzfywXI/AAAAAAAA5lw/WFsF6lgDRZs/Av_IF_10_COUNTIF_Formulas_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="223" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>COUNTIFS</h2> <p><a href="http://lh4.ggpht.com/-rsVTBecsQtY/VPUpsQQxAFI/AAAAAAAA5n0/edT4o7G9rjM/s1600-h/Av_IF_11_COUNTIFS_Table_600%25255B2%25255D.jpg"><img title="COUNTIFS in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIFS in Excel" src="http://lh4.ggpht.com/-WDCl5IjJxAI/VPUps67jk-I/AAAAAAAA5n8/s1L6PuwPVpM/Av_IF_11_COUNTIFS_Table_600_thumb%25255B1%25255D.jpg?imgmax=800" width="404" height="233" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-6W-IWoK2OvI/VPUptvKQ__I/AAAAAAAA5mM/ek6k6mhQIUg/s1600-h/Av_IF_12_COUNTIFS_Formulas_600%25255B7%25255D.jpg"><img title="COUNTIFS in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIFS in Excel" src="http://lh5.ggpht.com/-bW2j_uNBRAc/VPUpuJkHf_I/AAAAAAAA5mQ/ka8dLKt8ZqA/Av_IF_12_COUNTIFS_Formulas_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="205" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>AVERAGEIF</h2> <p><a href="http://lh4.ggpht.com/-lBAWQo7opxs/VPUpulSsO_I/AAAAAAAA5mc/nQ90-2FDv0Q/s1600-h/Av_IF_1_AVERAGEIF_1_Table_600%25255B9%25255D.jpg"><img title="AVERAGEIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="AVERAGEIF in Excel" src="http://lh6.ggpht.com/-WLzgL40yQWc/VPUpvM6Y98I/AAAAAAAA5mg/S_-Dd8Zi_qU/Av_IF_1_AVERAGEIF_1_Table_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-JJEdaZaIFbw/VPUpv2IVwKI/AAAAAAAA5ms/ordld3rbeNo/s1600-h/Av_IF_2_AVERAGEIF_Formulas_600%25255B16%25255D.jpg"><img title="AVERAGEIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="AVERAGEIF in Excel" src="http://lh6.ggpht.com/-VfYWdC1EO8c/VPUpwIGk6jI/AAAAAAAA5mw/wdSkLrg7u1M/Av_IF_2_AVERAGEIF_Formulas_600_thumb%25255B12%25255D.jpg?imgmax=800" width="404" height="293" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>AVERAGEIFS</h2> <p><a href="http://lh3.ggpht.com/-8upuqnIPWvI/VPUpwnzn9wI/AAAAAAAA5ng/I14REauWRM4/s1600-h/Av_IF_11_COUNTIFS_Table_600%25255B1%25255D.jpg"><img title="COUNTIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIF in Excel" src="http://lh5.ggpht.com/-KVd5pvmqYVM/VPUpxGsbY-I/AAAAAAAA5no/Z04Oa0Zaiis/Av_IF_11_COUNTIFS_Table_600_thumb.jpg?imgmax=800" width="404" height="233" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-iRVIzTfc-7E/VPUpxuChLAI/AAAAAAAA5nM/dsRzlh-DVDY/s1600-h/Av_IF_12_COUNTIFS_Formulas_600%25255B9%25255D.jpg"><img title="COUNTIF in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="COUNTIF in Excel" src="http://lh4.ggpht.com/-CA5Un8j8H0o/VPUpyYNhRJI/AAAAAAAA5nQ/Xyo63k1aNO0/Av_IF_12_COUNTIFS_Formulas_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="205" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com6tag:blogger.com,1999:blog-3568555666281177719.post-87929048107190951962015-02-24T13:47:00.001-08:002015-03-24T18:58:31.270-07:00Simplifying Excel Lookup Functions: VLOOKUP, HLOOKUP, INDEX, MATCH, CHOOSE, and OFFSET<h1>Simplifying Excel Lookup <br /> <br />Functions: VLOOKUP, <br /> <br />HLOOKUP, INDEX, MATCH, <br /> <br />CHOOSE, and OFFSET </h1> <p>The difference between Excel reference functions VLOOKUP, HLOOKUP, INDEX, MATCH, CHOOSE, and OFFSET is often not clear to every Excel user. This article will illustrate the most basic use of each of these lookup function formulas so that the differences and most common use of each is clear and more intuitive. Each one of these lookup functions will be used to look up the same item in the same array in order to illustrate the differences between the formulas.</p> <h2>VLOOKUP</h2> <p>VLOOKUP is used most often to look up an item in a table that has row headers. VLOOKUP looks up the correct row header and then locates the desired item in the specified column of that row. VLOOKUP looks up and down vertically for the correct row header. In its simplest form, VLOOKUP can be implemented with the following formula:</p> <p>VLOOKUP( <font color="#ff0000"><font size="3"><b>Row Header</b></font></font>, Array, <font color="#0000cc"><font size="3"><b>Column Number</b></font></font>)</p> <p><a href="http://lh5.ggpht.com/-jFKQYY2uQ1M/VOzxPHJwk-I/AAAAAAAA5a4/E4HHirOaJ2c/s1600-h/VLOOKUP_1_600%25255B4%25255D.jpg"><img title="VLOOKUP in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="VLOOKUP in Excel - Lookup and Reference Functions in Excel" src="http://lh5.ggpht.com/-JCWrAJcHsms/VOzxPkiOKnI/AAAAAAAA5bA/ltIMabO96E8/VLOOKUP_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="206" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h2>HLOOKUP</h2> <p>HLOOKUP is used most often to look up an item in a table that has column headers. HLOOKUP looks up the correct column header and then locates the desired item in the specified row of that column. HLOOKUP looks back and forth horizontally for the correct column header. In its simplest form, HLOOKUP can be implemented with the following formula:</p> <p>HLOOKUP( <font color="#0000cc"><font size="3"><b>Column Header</b></font></font>, Array, <font color="#ff0000"><font size="3"><b>Row Number</b></font></font>)</p> <p><a href="http://lh4.ggpht.com/-FL10reWOgFQ/VOzxQL8376I/AAAAAAAA5bI/DLczAns4byE/s1600-h/HLOOKUP_1_600%25255B4%25255D.jpg"><img title="HLOOKUP in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="HLOOKUP in Excel - Lookup and Reference Functions in Excel" src="http://lh6.ggpht.com/-jXOGiM0Ghfc/VOzxQh7f_bI/AAAAAAAA5bQ/kfx4sWmKwjg/HLOOKUP_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="213" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h2>INDEX</h2> <p>INDEX is best used to look up an item in a table when the item’s relative position in the table (row number and column number relative to the upper left corner of the table) is known. In its simplest form, INDEX can be implemented with the following formula:</p> <p>INDEX( Array, <font color="#ff0000"><font size="3"><b>Row Number</b></font></font>, <font color="#0000cc"><font size="3"><b>Column Number </b></font></font>)</p> <p><a href="http://lh4.ggpht.com/-IusNLXyxtQo/VOzxROVvaSI/AAAAAAAA5bU/cDFMCiErKOo/s1600-h/INDEX_1_600%25255B17%25255D.jpg"><img title="INDEX in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="INDEX in Excel - Lookup and Reference Functions in Excel" src="http://lh3.ggpht.com/-WfmgEoOkf5k/VOzxRqtoT-I/AAAAAAAA5bc/uIev_Wulu0w/INDEX_1_600_thumb%25255B9%25255D.jpg?imgmax=800" width="404" height="169" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h2>CHOOSE</h2> <p>CHOOSE returns a value from a list that is in the list position specified by the formula. In its simplest form, CHOOSE can be implemented with the following formula:</p> <p>CHOOSE( <font color="#ff0000"><font style="font-size: 11pt" size="2"><b>List Position</b></font></font>, List Item 1 Address, List Item 2 Address, List Item 3 Address, … )</p> <p><a href="http://lh6.ggpht.com/-ZURTu_zcJuQ/VOzxR5fa1II/AAAAAAAA5bk/RHYAsMTWWrY/s1600-h/CHOOSE_1_600%25255B4%25255D.jpg"><img title="CHOOSE in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="CHOOSE in Excel - Lookup and Reference Functions in Excel" src="http://lh5.ggpht.com/-Nhwncp7lIY8/VOzxSYCjF0I/AAAAAAAA5bs/ZyJcgb-u5x4/CHOOSE_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="155" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>It can also be implemented by the following formula:</p> <p>CHOOSE( <font color="#ff0000"><font style="font-size: 11pt" size="2"><b>List Position</b></font></font>, List Item 1, List Item 2, List Item 3, … , List Item n )</p> <p><a href="http://lh6.ggpht.com/-86SfdICwmrw/VOzxSx2iKcI/AAAAAAAA5b0/ubEYD0mnERw/s1600-h/CHOOSE_2_600%25255B4%25255D.jpg"><img title="CHOOSE in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="CHOOSE in Excel - Lookup and Reference Functions in Excel" src="http://lh6.ggpht.com/-pdidrXlH1N4/VOzxTQ-ojkI/AAAAAAAA5b4/5-RF98tl7GY/CHOOSE_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="78" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h2>OFFSET</h2> <p>OFFSET returns in item the cell that is a certain number of rows below and columns to the right of a reference cell. In its simplest form, OFFSET is implemented with the following formula:</p> <p>OFFSET( <font color="#00b050"><font size="3"><b>Reference Cell</b></font></font>, <font color="#0000cc"><font size="3"><b>Rows Down</b></font></font>, <font color="#ff0000"><font size="3"><b>Columns to the Right</b></font></font><font color="#ff0000"><font size="3"> </font></font>)</p> <p><a href="http://lh4.ggpht.com/-278RBSXc9ZM/VOzxTnq51II/AAAAAAAA5cE/SSQzk5tcayk/s1600-h/OFFSET_1_600%25255B4%25255D.jpg"><img title="OFFSET in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="OFFSET in Excel - Lookup and Reference Functions in Excel" src="http://lh5.ggpht.com/-mRsNSW725PY/VOzxUPgMD7I/AAAAAAAA5cM/XvNoO_krPCg/OFFSET_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h2>MATCH</h2> <p>MATCH returns the relative list position of a specified item in a list. In its simplest form, MATCH is implemented with the following formula:</p> <p>MATCH ( <font size="3"><b>Item</b></font>, array, 0) 0 indicates that there must be an exact of the item in the list for a position to be returned.</p> <p><a href="http://lh5.ggpht.com/-Ive0a5etLuo/VOzxU7yJnQI/AAAAAAAA5cU/CFiEn8D3ifE/s1600-h/MATCH_1_600%25255B4%25255D.jpg"><img title="MATCH in Excel - Lookup and Reference Functions in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="MATCH in Excel - Lookup and Reference Functions in Excel" src="http://lh6.ggpht.com/-wjqE52Mx0FQ/VOzxVflA7EI/AAAAAAAA5cc/qqwRDTlZJ_Y/MATCH_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="169" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com3tag:blogger.com,1999:blog-3568555666281177719.post-33448738018015125052015-02-09T21:27:00.001-08:002015-03-24T19:03:46.073-07:00Normal Probability Plot With Adjustable Confidence Bands in 9 Steps in Excel<h1>Normal Probability Plot <br /> <br />With Adjustable <br /> <br />Confidence Interval Bands <br /> <br />in 9 Steps in Excel</h1> <p><a href="http://lh4.ggpht.com/-cHjN05jwhZI/VNmWQCrG2oI/AAAAAAAA47s/Y-V7rByVxrg/s1600-h/Norm_Prob_1_Final_Graph_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel" src="http://lh3.ggpht.com/-ak6Pn_bJE7w/VNmWQ-e-6eI/AAAAAAAA47w/ikvWyvY8MZ4/Norm_Prob_1_Final_Graph_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="257" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Overview</h2> <p>The Normal Probability Plot is a graphical tool used to determine whether sample data is approximately normally distributed. Sample data are plotted against a line that the sample data would fall on if the sample was perfectly normally distributed. Sample data points that do not fall on this line represent deviations from the normal distribution.</p> <p>Confidence interval bands can be added to a Normal Probability Plot to provide a visual representation of how wide a confidence interval would need to be to contain the majority of the sample data points if the data were normally distributed. The wider that the confidence interval must be to contain the data points, the less likely it is that the data are normally distributed. </p> <p>The Normal Probability Plot is based upon the following unique characteristic of the normal distribution:</p> <p>There is a linear relationship between data values and Z Scores of normally-distributed data. A graph of data value vs. data Z Score will be a perfectly straight line if the data are perfectly normally distributed. A Normal Probability Plot graphs the sample data values against the line of perfect normal distribution, i.e., the line that the data values would fall on if the data were perfectly normally distributed. </p> <p>The line of perfect normal distribution for a given data set is fairly straightforward to construct on an Excel scatterplot. Upper and lower confidence interval bands can also be added to that scatterplot. Finally the actual sample data points can be plotted to show their proximity to the line of perfect normally distribution and also to determine whether they fall within the upper and lower bands of the specified confidence interval. </p> <p>A histogram is probably a better tool to apply to make a quick determination of whether a data set might be normally distributed. A histogram is highly dependent on the choice of bin size and location but will directly show the shape of the data set’s distribution. A Normal Probability Plot does provide visual indication of the length of tail and whether the data is skewed to either side but the histogram is the more intuitive tool to evaluate the shape of a data set’s distribution.</p> <p>Placing adjustable confidence interval bands on a Normal Probability Plot makes the Normal Probability Plot more useful. The bands can be adjusted to determine how large of a confidence interval must be to contain the data if they are normally distributed.</p> <p>Here is the 9-step process to construct a Normal Probability Plot with adjustable confidence interval bans in Excel: </p> <h2>Step 1 – Sort Sample Data</h2> <p>The first step in plotting a data sample within a Normal Probability Plot is to sort the data in ascending order. Sorting can be performed automatically by copying the formula in cell D10 down to cell D28 as shown below. Changing the word <i>SMALL</i> to <i>LARGE</i> convert this ascending sort to a descending sort. This method of sorting using formulas is more convenient than using the Excel sorting tool because the formulas will automatically resort the data if any of the raw data changes. Excel’s sorting tool must be re-run manually every time a new sort is needed. The automatic sorting formula in cell D10 to be copied down to cell D28 is the following:</p> <p>=IF($B10=””,””,SMALL($B$10:$B$28,ROW()-ROW($D$9)))</p> <p><a href="http://lh4.ggpht.com/-5ojU-lgufGE/VNmWRB3OQMI/AAAAAAAA478/pLN3GwOr0BI/s1600-h/Norm_Prob_2_Raw%252520Data_700_Tall%25255B5%25255D.jpg"><img title="Normal Probability Plot in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Raw Data" src="http://lh3.ggpht.com/-bVj5k4kxMfU/VNmWRts_91I/AAAAAAAA48A/RzmhMzVQvP8/Norm_Prob_2_Raw%252520Data_700_Tall_thumb%25255B3%25255D.jpg?imgmax=800" width="77" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh4.ggpht.com/-kVURhQG_V2A/VNmWSIVKnhI/AAAAAAAA48M/pMxPHtmsPJE/s1600-h/Norm_Prob_3_Sorted_Sample_Values_600%25255B8%25255D.jpg"><img title="Normal Probability Plot in Excel - Sorted Sample Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Sorted Sample Values" src="http://lh6.ggpht.com/-paZ1PjARSXI/VNmWSqCCqKI/AAAAAAAA48Q/BbN6Z6rWEqI/Norm_Prob_3_Sorted_Sample_Values_600_thumb%25255B6%25255D.jpg?imgmax=800" width="404" height="309" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 2 – Calculate Expected Z <br /> <br />Scores </h2> <p>The actual Z Scores of the sorted sample data must first be calculated. The expected Z Scores will correspond to the actual Z Score except that each expected Z Score will be evenly spaced from each other as they increase from smallest to largest Z Score. Each expected Z Score will have the same incremental increase from the previous Z Score. </p> <p>Z Score is calculated by the following formula:</p> <p>Z Score = (Data Value – Average Data Value) / (Data Sample Standard Deviation)</p> <p>The lowest and highest Z Scores are used to determine the total range in the Z Score values of the data set. The increment of Z Score between each expected Z Score point is calculated by the following formula:</p> <p>Z Score increment = (Total Z Score Range) / (n – 1)</p> <p>n = the number of data points in the data set</p> <p>These calculations in Excel are shown as follows:</p> <p>Data sample count, mean, and sample standard deviation are first calculated in Excel.</p> <p><a href="http://lh6.ggpht.com/-A0zr1GvdAeA/VNmWSzr9VRI/AAAAAAAA48c/JZgC4m0fJXY/s1600-h/Norm_Prob_5_Sample_Mean_Count_StDev_550%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Sample Mean, Count, Standard Deviation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Sample Mean, Count, Standard Deviation" src="http://lh5.ggpht.com/-II-2iQWzflA/VNmWTZG-GcI/AAAAAAAA48k/wNPqWdVTOLI/Norm_Prob_5_Sample_Mean_Count_StDev_550_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="275" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The actual Z Score of each sample data point can now be calculated.</p> <p><a href="http://lh5.ggpht.com/-oE9tGVLV6zI/VNmWT350jmI/AAAAAAAA48s/YvTosllwVvs/s1600-h/Norm_Prob_4_Z_Actual_700_Tall%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Z Score Actual" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Z Score Actual" src="http://lh6.ggpht.com/-GVyAigGLdxw/VNmWUdsPE4I/AAAAAAAA480/Q3lNlK0DlaY/Norm_Prob_4_Z_Actual_700_Tall_thumb%25255B2%25255D.jpg?imgmax=800" width="350" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p></p> <p></p> <p>The total range of actual Z Score values can now be calculated. The expected Z Score incremental increase can be calculated as soon as the total range of Z Score values and the number of data points is known.</p> <p><a href="http://lh5.ggpht.com/-nnlj_341Qyw/VNmWVMZPGFI/AAAAAAAA484/ST1IIx_HmxY/s1600-h/Norm_Prob_6_Z_Range_Increment_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Z Score Range Increment" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Z Score Range Increment" src="http://lh4.ggpht.com/-C2RecpQsrJs/VNmWVdjdZGI/AAAAAAAA49E/_llAx2Ifrbc/Norm_Prob_6_Z_Range_Increment_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="147" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The expected Z Scores for the sample data points if they were normally distributed can now be calculated by starting at the lowest actual Z Score value and adding the incremental increase to each consecutive Z Score until the highest actual Z Score value is reached. </p> <p>It is important to note that the expected Z Scores calculated in this step will be the X-axis values of all points plotted on the Normal Probability Plot. The four sets of data values that will be plotted on the Normal Probability Plot using these expected Z Scores as their x-axis values are the following:</p> <ol> <li> <p align="justify">The normal distribution line</p> </li> <li> <p align="justify">The upper confidence interval band</p> </li> <li> <p align="justify">The lower confidence interval band</p> </li> <li> <p align="justify">The actual sample data points</p> </li> </ol> <p>Expected Z Scores are calculated in Excel as follows:</p> <p><a href="http://lh5.ggpht.com/-H9WisbWLnyo/VNmWV-0wYwI/AAAAAAAA49M/zMMgHLMnYfk/s1600-h/Norm_Prob_7_Z_Score_Exp_700_Tall%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Z Score Expected" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Z Score Expected" src="http://lh3.ggpht.com/-NPeYqoTmmA4/VNmWWcZ4EJI/AAAAAAAA49U/5fBEz839JyQ/Norm_Prob_7_Z_Score_Exp_700_Tall_thumb%25255B2%25255D.jpg?imgmax=800" width="266" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 3 – Calculate Expected Data <br /> <br />Values if Data Are Normally <br /> <br />Distributed</h2> <p>Calculating the data values that would occur if the data were normally distributed is implemented with the following two-step process:</p> <ol> <li> <p>Calculate the CDF (Cumulative Distribution Function or F(Y)) of each data point. This can be done using the Excel formula NORM.S.DIST(Z Score, TRUE). The Z Scores used are the expected Z Scores if the data were normally distributed.</p> </li> <li> <p>Calculate each expected data point’s value with the following Excel formula:</p> </li> </ol> <p>NORM.INV( F(Y), Sample Mean, Sample Standard Deviation)</p> <p>The normal distribution has the unique property that there is a linear relationship between Z Scores and data values if the data are normally distributed. </p> <p><a href="http://lh5.ggpht.com/-eEbTL8mD2sk/VNmWW3oQChI/AAAAAAAA49c/tXfWkR1tHtQ/s1600-h/Norm_Prob_8_F%252528Y%252529_Exp_700_Tall%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - CDF Expected" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - CDF Expected" src="http://lh3.ggpht.com/-XM0JPJ443nM/VNmWXqA3GNI/AAAAAAAA49k/gAuvCc4U5yE/Norm_Prob_8_F%252528Y%252529_Exp_700_Tall_thumb%25255B2%25255D.jpg?imgmax=800" width="322" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh5.ggpht.com/-bvY7RKVCYD8/VNmWYFxniwI/AAAAAAAA49s/yUuc8bHhuNQ/s1600-h/Norm_Prob_9_Y_Exp_700%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Expected Sample Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Expected Sample Values" src="http://lh4.ggpht.com/-Usao-5LvqC8/VNmWYpuwkPI/AAAAAAAA49w/ZEyOH2ZT5GI/Norm_Prob_9_Y_Exp_700_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="397" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 4 – Calculate Half Width of <br /> <br />Confidence Interval </h2> <p>The half-width of the confidence interval of the mean Y value at any X value of a fitted line is shown here. This formula is commonly used to construct the confidence interval about a regression line.</p> <p>It should be noted that the confidence interval bands for the normal probability plot normal distribution line are curved with the narrowest part of the band occurring at the mean X value. Intuitively this occurs because there is less certainty of mean Y values as X values get further and further from the mean X value. The (X<sub>i</sub> – X_bar) term in the formula has its lowest value at the mean X value. The confidence interval bands will be at their narrowest point here at the mean X value as a result.</p> <p><a href="http://lh4.ggpht.com/-qn4cfOIx_eI/VNmWY9IDt-I/AAAAAAAA494/OY9q3uSsIds/s1600-h/CI_half-width_Norm_Prob_Plot_long%25255B4%25255D.gif"><img title="Normal Probability Plot in Excel - Confidence Interval Half-Width" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Confidence Interval Half-Width" src="http://lh5.ggpht.com/-dFJVHC6pHMc/VNmWZRyuWDI/AAAAAAAA4-A/G_Vk61CuuNk/CI_half-width_Norm_Prob_Plot_long_thumb%25255B2%25255D.gif?imgmax=800" width="404" height="44" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The individual pieces of this formula will be calculated in this section. The Alpha (The level of significance) is user-adjustable and is currently set at a = 0.05. Reducing the size of a will cause the confidence interval bands to widen. The smaller alpha is, the greater the specified level of confidence is. The wider the confidence interval, the greater level of certainty exists that the confidence interval will contain the actual mean. For the Normal Probability Plot, the wider the confidence bands needs to be to contain the sample data points surrounding the normal distribution line, the less likely it is that the sample data are normally distributed.</p> <p>The degrees of freedom is set at n – 2 because the values of two parameters, X and Y, must be known before an error term can be calculated. </p> <p><a href="http://lh3.ggpht.com/-aiPQsIQN43Y/VNmWZw16_JI/AAAAAAAA4-I/J5tfcr97w_o/s1600-h/Norm_Prob_11_Alpha_t_df_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Alpha, t-Value, Degrees of Freedom" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Alpha, t-Value, Degrees of Freedom" src="http://lh4.ggpht.com/-JHR9eozTbq4/VNmWaPTGxXI/AAAAAAAA4-U/rwFSFEuI-SE/Norm_Prob_11_Alpha_t_df_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="190" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The t Value above is calculated using the following Excel formula:</p> <p>=T.INV(1 – Alpha/2, n-2)</p> <p>=T.INV(1-T2/2,T4)</p> <p><a href="http://lh4.ggpht.com/-6pEqDJ6dsEE/VNmWavAMQuI/AAAAAAAA4-Y/b0Dx5aWVTDA/s1600-h/Norm_Prob_12_SE_of_Y_Values_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Standard Error of Y Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Standard Error of Y Values" src="http://lh4.ggpht.com/-1ErhjOu4UcM/VNmWbFneMwI/AAAAAAAA4-g/vLF6Gm9UUQU/Norm_Prob_12_SE_of_Y_Values_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="97" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The Standard Error of the Y Values is calculated using the following Excel formula:</p> <p>SE of Y Values = STEYX(Array Sorted Sample Values, Array Z Scores)</p> <p>SE of Y Values = STEYX(D10:D28,I10:I28)</p> <p><a href="http://lh5.ggpht.com/-ZN-CKTPF4Y8/VNmWbXBWujI/AAAAAAAA4-s/9ayP_jNTs0Q/s1600-h/Norm_Prob_13_SSx_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Sum of Squared Deviations From Mean X" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Sum of Squared Deviations From Mean X" src="http://lh5.ggpht.com/-f4sIx75fBGI/VNmWb9LkhAI/AAAAAAAA4-0/TIS8yeWE2rY/Norm_Prob_13_SSx_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="99" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The Sum of Squared X Deviations From the Mean is calculated with the following Excel formula:</p> <p>=DEVSQ(X Array) = DEVSQ(Array Z Scores) = DEVSQ(I10:I28)</p> <p><a href="http://lh3.ggpht.com/-QMj0xLvOZ_4/VNmWcePoIXI/AAAAAAAA4-8/KJ1apKWgl_M/s1600-h/Norm_Prob_9_Xi-X_bar_Squared_700_Tall%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - (Xi - X_bar)Squared" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - (Xi - X_bar)Squared" src="http://lh6.ggpht.com/-JBL5XTSFZOc/VNmWc0vRWgI/AAAAAAAA4_E/S0nKLoVOa5s/Norm_Prob_9_Xi-X_bar_Squared_700_Tall_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="381" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The above piece of the Confidence Interval formula is the reason that the upper and lower confidence interval bands are curved. The confidence interval about the mean estimated Y value (the normal distribution line) is tightest at the mean X value (mean Z Score). The confidence interval about the mean Y value gets wider the further the X value is from the X mean. </p> <p>The individual pieces of the formula can now be combined to calculate the half-width of the confidence interval at each X value. The expected Z Scores are the X Values.</p> <p><a href="http://lh4.ggpht.com/-v5vBfisXT8E/VNmWdYUV0fI/AAAAAAAA4_M/6kjweAey-xM/s1600-h/Norm_Prob_14_CI_Half_Width_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Confidence Interval Half-Width" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Confidence Interval Half-Width" src="http://lh5.ggpht.com/-xikhyQwG7RA/VNmWd4xAZWI/AAAAAAAA4_Q/p1znAZqTZSg/Norm_Prob_14_CI_Half_Width_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="322" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 5 – Calculate Upper and Lower <br /> <br />Bands of the Confidence Interval</h2> <p>The location and width of the confidence interval at each X value (Expected Z Score) can now be calculated.</p> <p><a href="http://lh4.ggpht.com/-uuZ44EMI8K0/VNmWeYEltyI/AAAAAAAA4_c/k5ry6uCckac/s1600-h/Norm_Prob_15_Upper_CI_700_Tall%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Upper Confidence Interval Band" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Upper Confidence Interval Band" src="http://lh5.ggpht.com/-ep_Go7Fxbdc/VNmWe5eQfVI/AAAAAAAA4_k/meyPO3lDQ60/Norm_Prob_15_Upper_CI_700_Tall_thumb%25255B2%25255D.jpg?imgmax=800" width="234" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh6.ggpht.com/-CvCfmfYV5qw/VNmWfTef3oI/AAAAAAAA4_s/KJj7FhlerCY/s1600-h/Norm_Prob_16_Lower_CI_700_Tall%25255B7%25255D.jpg"><img title="Normal Probability Plot in Excel - Lower Confidence Interval Band" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Lower Confidence Interval Band" src="http://lh6.ggpht.com/-CP9_4ilUhxk/VNmWf6wN_QI/AAAAAAAA4_0/S08voK2t9cE/Norm_Prob_16_Lower_CI_700_Tall_thumb%25255B3%25255D.jpg?imgmax=800" width="232" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The following four data sets of Y values will be graphed to form the Normal Probability Plot:</p> <ol> <li> <p align="justify">Expected sample values will be plotted to form the normal distribution line.</p> </li> <li> <p align="justify">Upper confidence interval values will be plotted to form the upper band of the confidence interval.</p> </li> <li> <p align="justify">Lower confidence interval values will be plotted to form the lower band of the confidence interval.</p> </li> <li> <p align="justify">Actual sorted sample values will be plotted individually.</p> </li> </ol> <p>Each of the above data sets of Y values will use the same X values, i.e., the Expected Z Scores.</p> <p>This graphing will be performed as follows:</p> <h2>Step 6 – Graph the Normal <br /> <br />Distribution Line</h2> <p>All four data sets will be graphed using the following Excel scatterplot graph with straight lines and markers. </p> <p><a href="http://lh3.ggpht.com/-Pl8NXVL06YE/VNmWgePMseI/AAAAAAAA4_4/1oF7af_h3jA/s1600-h/Norm_Prob_17_Select_Scatter_Chart_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Select Scatter Chart" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Select Scatter Chart" src="http://lh5.ggpht.com/-MQb6OC-WKyI/VNmWgl_lV7I/AAAAAAAA5AE/QsWoyipZPCs/Norm_Prob_17_Select_Scatter_Chart_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="237" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The first data series to be added the chart is the Normal Distribution Line which consists of Expected Z Scores as the X values and Expected Sample Values as they values. This data series is added as follows:</p> <p><a href="http://lh5.ggpht.com/--ftXWC37F_8/VNmWhbV-P0I/AAAAAAAA5AM/Txpf0f7AYhg/s1600-h/Norm_Prob_18_Series_1_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Plotting Series 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Plotting Series 1" src="http://lh6.ggpht.com/-d3q-Rvbb8-Y/VNmWhkKi-hI/AAAAAAAA5AQ/PrPcPm-NoTM/Norm_Prob_18_Series_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="173" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>This data will appear on the graph as follows. Note that the data graphs as a straight line because of the unique property of the normal distribution that a linear relationship exists between the Z Scores and data values of normally-distributed data.</p> <p><a href="http://lh5.ggpht.com/-HA_2au63pEE/VNmWiEVjdPI/AAAAAAAA5Ac/rrNXjm-AVIU/s1600-h/Norm_Prob_19_Series_1_Initial_Graph_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Plotting Series 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Plotting Series 1" src="http://lh6.ggpht.com/-fN6PurWN2Qc/VNmWin0qOsI/AAAAAAAA5Ag/UrW9qZVE7w8/Norm_Prob_19_Series_1_Initial_Graph_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="247" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a name="_GoBack"></a>By default Excel configures the vertical (Y) axis to cross the horizontal (X) axis at the point X = 0. The vertical axis needs to be moved out of the way to the left side of the graph. This is accomplished by double-clicking the horizontal axis and specifying the vertical axis crosses at X = -2 as is shown when the format Axis dialogue box pops up with the double-click.</p> <p><a href="http://lh3.ggpht.com/-lbaDjuLZhn8/VNmWjE4tYSI/AAAAAAAA5As/hARpGcKyul0/s1600-h/Norm_Prob_20_Move_Vertical_Axis_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Move Vertical Axis to Left" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Move Vertical Axis to Left" src="http://lh3.ggpht.com/-xH_KH0VdrG8/VNmWjbx8f4I/AAAAAAAA5Aw/kBC24uMFbCg/Norm_Prob_20_Move_Vertical_Axis_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="230" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The vertical axis is now moved over. This plotted Normal Distribution Line should be only a line. The markers (round dots in the line) should be removed. Double-click on the line and select Format Data Series. Change the Marker Options from Automatic to None and the markers will disappear, leaving only the line.</p> <p><a href="http://lh4.ggpht.com/-3FnZXNLc1K8/VNmWj3fTVVI/AAAAAAAA5A8/w_sZ5umyYrw/s1600-h/Norm_Prob_20_Move_Vertical_Axis_600%25255B9%25255D.jpg"><img title="Normal Probability Plot in Excel - Move Vertical Axis to Left" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Move Vertical Axis to Left" src="http://lh5.ggpht.com/-udhJjQa1hlk/VNmWkeGO7lI/AAAAAAAA5BE/eJDrDvR9nO0/Norm_Prob_20_Move_Vertical_Axis_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="230" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>There is a lot of empty space in the graph above and below the plotted line. Double-click on the vertical axis and reset the minimum and maximum axis values to 130,000 and 270,000 in the Format Axis dialogue box.</p> <p><a href="http://lh5.ggpht.com/-4zwuU9c7DhQ/VNmWkkFNdMI/AAAAAAAA5BM/_SOKzzKc1-g/s1600-h/Norm_Prob_22_Remove_Marker_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Remove Markers" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Remove Markers" src="http://lh3.ggpht.com/-L7zGGaVHYJo/VNmWlGIpHWI/AAAAAAAA5BU/UElNYLTjfnU/Norm_Prob_22_Remove_Marker_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="179" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh6.ggpht.com/-i8NSktClB8s/VNmWlrb2_II/AAAAAAAA5Bc/iTIDaGfUH5I/s1600-h/Norm_Prob_24_Remove_Marker_2_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Remove Markers" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Remove Markers" src="http://lh3.ggpht.com/-Lt7EdHCHU9I/VNmWmFaZ-CI/AAAAAAAA5Bg/NcFVeqQPI6Y/Norm_Prob_24_Remove_Marker_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="198" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The first data series, the Normal Distribution Line, will now appear as follows:</p> <p><a href="http://lh4.ggpht.com/-RZfnLnVbczQ/VNmWmazRpTI/AAAAAAAA5Bs/0zESFRKowvQ/s1600-h/Norm_Prob_25_Markers_Gone_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Remove Markers" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Remove Markers" src="http://lh4.ggpht.com/-GTNAO0SIG-Y/VNmWnLOGvRI/AAAAAAAA5Bw/ku8lryJffR4/Norm_Prob_25_Markers_Gone_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="247" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 7 – Graph the Upper and <br /> <br />Lower Confidence Interval Bands</h2> <p>The upper and lower confidence interval bands are designated as the next two data series to be included in the graph. Right-click on the graph anywhere and click Select Data. Add the two data series in the following manner. Once again, the X values for both data series will be the Expected Z Scores.</p> <p><a href="http://lh6.ggpht.com/-240e_BjbJWY/VNmWnUyL1OI/AAAAAAAA5B8/v7w1NOZs87Q/s1600-h/Norm_Prob_26_Series_2_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Plotting Series 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Plotting Series 2" src="http://lh5.ggpht.com/-0-uauxvgNmw/VNmWn7IUiQI/AAAAAAAA5CA/iKDxhFvCbng/Norm_Prob_26_Series_2_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="148" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh6.ggpht.com/-zCRfIogcIZo/VNmWoRiXOFI/AAAAAAAA5CI/msIiUeeGnZ8/s1600-h/Norm_Prob_27_Series_3_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Plotting Series 3" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Plotting Series 3" src="http://lh4.ggpht.com/-GHV01AmNkVE/VNmWooUIqGI/AAAAAAAA5CQ/y9mY3lZGxZ8/Norm_Prob_27_Series_3_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="153" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Remove the markers from of the confidence interval series and they should appear graphed as follows. Note that the confidence interval band is at its narrowest point at the mean X value, which is the mean Expected Z Score of 0.</p> <p><a href="http://lh6.ggpht.com/-8vjR3INa3VU/VNmWpHmMPkI/AAAAAAAA5Cc/Tnkd8Yo11p8/s1600-h/Norm_Prob_28_Chart_With_Bands_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel With Confidence Interval Bands " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel With Confidence Interval Bands" src="http://lh3.ggpht.com/-O8HpSmfv1MQ/VNmWprVjSLI/AAAAAAAA5Cg/ZYUutxgmkio/Norm_Prob_28_Chart_With_Bands_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 8 – Graph Sorted Sample Values</h2> <p>The sorted sample values are the last series to be plotted. Once again the X values of this series will be the Expected Z Scores.</p> <p><a href="http://lh5.ggpht.com/-NBZyxMAQ5zI/VNmWqPlnD3I/AAAAAAAA5Cs/5etnPIO3Bs0/s1600-h/Norm_Prob_28_Series_4_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Plotting Series 4" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Plotting Series 4" src="http://lh5.ggpht.com/-Xom2NuAP5gU/VNmWqUeGxPI/AAAAAAAA5Cw/K1C8ULf_PmI/Norm_Prob_28_Series_4_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>In the case of this plotted data series, only the markers should be plotted and not the connecting lines. Double-click on the plotted data series to bring up the Format Data Series dialogue box. Select No Line to remove the connecting lines.</p> <p><a href="http://lh3.ggpht.com/-JvS2mHu1hCM/VNmWq8UeyxI/AAAAAAAA5C8/RVCQhQ_-GyM/s1600-h/Norm_Prob_29_Remove_Series_4_Lines_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Remove Connecting Lines" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Remove Connecting Lines" src="http://lh5.ggpht.com/-eyg3kL6MUcE/VNmWrHHv0MI/AAAAAAAA5DA/lFLTZH2Yn7E/Norm_Prob_29_Remove_Series_4_Lines_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="157" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The markers can also be configured here to have a blue color. The graph will now appear as follows:</p> <p><a href="http://lh4.ggpht.com/-Uy_85KiW97M/VNmWrncoUmI/AAAAAAAA5DM/CCR0Y8-ET4I/s1600-h/Norm_Prob_30_Series_Plotted_600%25255B7%25255D.jpg"><img title="Normal Probability Plot in Excel - Series Plotted" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Series Plotted" src="http://lh4.ggpht.com/-D_zikky9eXg/VNmWsFHq1qI/AAAAAAAA5DQ/v-X8kQAYYm0/Norm_Prob_30_Series_Plotted_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Step 9 – Add Chart Title, Axes <br /> <br />Titles, and Legend</h2> <p>Chart elements such as chart title, legend, and axis title can be added to the chart by clicking on the chart and selecting Design / Add Chart Elements. </p> <p><a href="http://lh5.ggpht.com/-FB36Vc2U_HA/VNmWse3GtLI/AAAAAAAA5Dc/M7uzRflPACQ/s1600-h/Norm_Prob_31_Add_Chart_Elements_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Add Chart Elements" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Add Chart Elements" src="http://lh6.ggpht.com/-XVDujrX1A9o/VNmWsw9tK0I/AAAAAAAA5Dg/Ap5wcEy9Xfo/Norm_Prob_31_Add_Chart_Elements_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="239" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The final chart appears as follows:</p> <p><a href="http://lh5.ggpht.com/-1lio19jb-aQ/VNmWtQTylpI/AAAAAAAA5Ds/W87vT14jIbE/s1600-h/Norm_Prob_32_Final_Graph_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Final Graph" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Final Graph" src="http://lh5.ggpht.com/-QWR3t-d-m2E/VNmWt-qsITI/AAAAAAAA5Dw/A08fqCcX7oE/Norm_Prob_32_Final_Graph_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="257" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The alpha of the chart was initial set to 0.05. Reducing alpha will widen the confidence interval. This the result of the following formula:</p> <p>Level of Confidence = 1 – Level of Significance (alpha)</p> <p>The above chart shows that the 95-percent confidence level does not contain many of the data points.</p> <p>Reducing alpha to 0.0001 causes the confidence interval bands to widen as shown. </p> <p><a href="http://lh6.ggpht.com/-kqhNu8TkrzM/VNmWuAZk7gI/AAAAAAAA5D4/DdCzxHwbCmo/s1600-h/Norm_Prob_33_Alpha_0001_400%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Resetting Alpha to 0.0001" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Resetting Alpha to 0.0001" src="http://lh4.ggpht.com/-1X5o7Jl_QOo/VNmWugJ8EII/AAAAAAAA5EA/lgaSwUUFtVU/Norm_Prob_33_Alpha_0001_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="99" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh6.ggpht.com/-9baudmTvYzw/VNmWu141SBI/AAAAAAAA5EM/4qxJfnjObLE/s1600-h/Norm_Prob_34_Graph_Alpha_0001_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Wider Confidence Interval" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Wider Confidence Interval" src="http://lh3.ggpht.com/-DMMVm9Dzlm4/VNmWvSlUzfI/AAAAAAAA5EQ/8yBru-f8dnc/Norm_Prob_34_Graph_Alpha_0001_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="254" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The Normal Probability Plot is a quick indicator of normality and should be used in combination with other normality-testing techniques. Here is an Excel histogram of this sample data set which shows the general shape of the data’s distribution:</p> <p><a href="http://lh4.ggpht.com/-bZ3q16lCUZY/VNoa4dIQyUI/AAAAAAAA5Ek/VWXkQKKRROg/s1600-h/Norm_Prob_35_Histogram_600%25255B4%25255D.jpg"><img title="Normal Probability Plot in Excel - Histogram of Same Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Normal Probability Plot in Excel - Histogram of Same Data" src="http://lh5.ggpht.com/-rjZHRELxqBc/VNoa400rB0I/AAAAAAAA5Eo/ak0WjPQV8DU/Norm_Prob_35_Histogram_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="327" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com10tag:blogger.com,1999:blog-3568555666281177719.post-4001359476107658182015-02-03T09:54:00.001-08:002015-03-24T19:09:11.404-07:00Box Plots in 8 Steps in Excel<h1>Box Plots in 8 Steps in <br /> <br />Excel</h1> <h2>Overview</h2> <p>A box plot is a simple method of displaying data by splitting the data in quartiles. Each quartile of a data set contains one quarter of the data points. The Q1 quartile contains the quarter of the data set that has the lowest values. Quartile Q4 contains the quarter of the data set that has the highest values. An example of a box plot is the following:</p> <p><a href="http://lh5.ggpht.com/-VMvtzwf7HhA/VNEKhIzTu7I/AAAAAAAA4uU/CsLwZzGkGGM/s1600-h/Box_Plot_40_Final_Graph_600%25255B7%25255D.jpg"><img title="Box Plots in Excel - Final Graph" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Final Graph" src="http://lh6.ggpht.com/-UDAxZm6HWrI/VNEKhQIfl2I/AAAAAAAA4uc/5yBLMbZ5M4w/Box_Plot_40_Final_Graph_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The box plot is sometimes called the box-and-whisker diagram because the highest and lowest quartiles of data are represented by whiskers above and below the boxes that contain quartiles 2 and 3 of the data set. The inner 50-percent of the data are contained in quartiles 2 and 3. The outer 50-percent of the data points are contained in quartiles 1 and 4. The border between the 2nd quartile (purple box) and the 3rd quartile (yellow box) is the data set’s median. One half of all data points are above and below the median. Sometimes box plots also plot the mean is done here.</p> <p>The following data set can be used to create a box plot in Excel:</p> <p><a href="http://lh6.ggpht.com/-IptKBBTY5T4/VNEKiNmgR1I/AAAAAAAA4ug/ICZudRni8qw/s1600-h/Box_Plot_1_Raw_Data_700_Length%25255B11%25255D.jpg"><img title="Box Plots in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Raw Data" src="http://lh5.ggpht.com/-mvGsead7opQ/VNEKiltxkGI/AAAAAAAA4us/N5v2NXlHcIk/Box_Plot_1_Raw_Data_700_Length_thumb%25255B9%25255D.jpg?imgmax=800" width="148" height="772" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>A box plot in Excel of the data would appear as follows:</p> <p><a href="http://lh5.ggpht.com/--YpWuUqK308/VNEKjRGCCAI/AAAAAAAA4u0/s30msoSBTi8/s1600-h/Box_Plot_4_Basic_Box_Plot_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Basic Box Plot" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Basic Box Plot" src="http://lh5.ggpht.com/-ngcVdAqF1HU/VNEKjxgIG2I/AAAAAAAA4u4/EdZBHd7glhc/Box_Plot_4_Basic_Box_Plot_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="403" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The box plot was created with Excel charting tools using the following information:</p> <p><a href="http://lh5.ggpht.com/-ul1gBQ2O4jg/VNEKkdG6tOI/AAAAAAAA4vE/v1VBhv36I2U/s1600-h/Box_Plot_2_Info_Summary_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Information Summary" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Information Summary" src="http://lh6.ggpht.com/-gJQXm7jrhgw/VNEKk7y90uI/AAAAAAAA4vM/6ZkvhpLMZEY/Box_Plot_2_Info_Summary_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="270" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The information is displayed in the box plot as follows:</p> <p><a href="http://lh6.ggpht.com/-zKDGDsjfG-M/VNEKlYCDZbI/AAAAAAAA4vQ/SKurmc4zLJs/s1600-h/Box_Plot_3_Explanation_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Box Plot Explanation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Box Plot Explanation" src="http://lh3.ggpht.com/-jZB3kc9sJPo/VNEKlyCWLVI/AAAAAAAA4vc/1i1DIVWEy5c/Box_Plot_3_Explanation_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="289" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <h2>Information That Box Plots Provide</h2> <p>Box plots provide basic information about the distribution of data. Box plots provide visual information:</p> <ol> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2">The </font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>variation or spread of the data</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"> is indicated by the width of each quartile. The wider a quartile is, the greater the spread or variance in that quartile is likely to be.</font></font></font></font></font></font></p> </li> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>Any skew of the data set</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"> is indicated by the lengths of the upper and lower whiskers relative to each other or by how close the median and mean are to each other. A data set is skewed in a direction if a histogram of the data set would show a long tail in that direction. The skew is in the direction of the long data tail. A longer upper whisker would indicate a skew in the positive direction while a longer lower whisker would indicate that the data is skewed in the negative direction. If the mean is greater than the median, the data likely has a longer tail in the positive direction and is therefore likely to be positively skewed. When the mean is less than the median, a negative skew is indicated.</font></font></font></font></font></font></p> </li> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>The range of data</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"> is indicated by the difference between the end points of the upper and lower whiskers, which are the min and max data values.</font></font></font></font></font></font></p> </li> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>The interquartile range</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"> (IQR) is the middle half of all data points. 50-percent of all data points lie in the 2</font></font></font><font color="#000000"><sup><font face="Arial, serif"><font size="2">nd</font></font></sup></font><font color="#000000"><font face="Arial, serif"><font size="2"> and 3</font></font></font><font color="#000000"><sup><font face="Arial, serif"><font size="2">rd</font></font></sup></font><font color="#000000"><font face="Arial, serif"><font size="2"> quartiles that are the boxes shown in the Q2 and Q3 boxes </font></font></font></font></font></font></p> </li> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2">Box plots provide the following </font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>measures of central tendency</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2">: the mean and the median.</font></font></font></font></font></font></p> </li> <li> <p style="margin-bottom: 0.11in; line-height: 107%"><font color="#00000a"><font face="Calibri, serif"><font style="font-size: 11pt" size="2"><font color="#000000"><font face="Arial, serif"><font size="2">Box plot provide </font></font></font><font color="#000000"><font face="Arial, serif"><font size="2"><u><b>visual indication of whether a factor is significant</b></u></font></font></font><font color="#000000"><font face="Arial, serif"><font size="2">. A factor that is significant will display a visible difference in central tendency and dispersion in a box plot of sample groups of different levels of that factor. This is demonstrated as follows:</font></font></font></font></font></font></p> </li> </ol> <p>The following two images show box plots of three samples that have the same means in both images. The difference is the degree of dispersion. The small amount of within-group variance makes the differences between the sample groups readily apparent in a box plot. </p> <p><a href="http://lh4.ggpht.com/-BTlMztYLeMA/VNEKmRoRlSI/AAAAAAAA4vk/aCzoU9IJ5DM/s1600-h/Box_Plot_4A_Box_Plot_Small_Wintin-Group_Variance_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Small Within-Group Variance" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Small Within-Group Variance" src="http://lh4.ggpht.com/-7nRogpZtQ_k/VNEKm8y3oPI/AAAAAAAA4vs/KEbEhIhbeMQ/Box_Plot_4A_Box_Plot_Small_Wintin-Group_Variance_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="445" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The relatively larger amount of within-group variance in the sample groups in the second image that follows obscures the difference between the sample groups. ANOVA (Analysis of Variance) combines the differences in sample means and sample variances to determine whether at least one sample group is significantly different than the other sample groups and therefore likely to have come from a different population than the other sample groups. </p> <p><a href="http://lh3.ggpht.com/-v7m21Vcl01U/VNEKnXpM63I/AAAAAAAA4v0/w3tco6-hiGA/s1600-h/Box_Plot_4B_Large_Within-Group_Variance_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Large Within-Group Variance" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Large Within-Group Variance" src="http://lh4.ggpht.com/-Aan-IUX5TDY/VNEKn98P8KI/AAAAAAAA4v8/Lo9_qoFKr3Y/Box_Plot_4B_Large_Within-Group_Variance_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="443" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Box plot are basic analysis tools and do not provide information such as the shape of the distribution of a data set. A histogram is a good tool to create a visual representation of that shape. </p> <p>Below are the steps to create a box plot in Excel:</p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Sort the Sample Groups in Descending Order </font></b></font></font></font></h2> <p>This is not absolutely necessary but simply makes the data a bit more intuitive to work with. The Excel Sort tool can easily be used for this.</p> <p><a href="http://lh5.ggpht.com/-6bZjJ6IpCD8/VNEKomxRZOI/AAAAAAAA4wE/mgTK8LZRgo0/s1600-h/Box_Plot_5_Raw_Data_Sorted_700_Length%25255B4%25255D.jpg"><img title="Box Plots in Excel - Sorted Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Sorted Raw Data" src="http://lh3.ggpht.com/-ddXi9-6axKU/VNEKozUqbcI/AAAAAAAA4wM/btAxyyKCBS8/Box_Plot_5_Raw_Data_Sorted_700_Length_thumb%25255B2%25255D.jpg?imgmax=800" width="396" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Calculate Measures of Central Tendency and Dispersion for Each Sample Group</font></b></font></font></font></h2> <p><a href="http://lh4.ggpht.com/-_iZyxQ0uVdQ/VNEKpmq4kQI/AAAAAAAA4wU/SgroVVFDMzY/s1600-h/Box_Plot_6_Data_Table_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Data Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Data Table" src="http://lh3.ggpht.com/-t0lnEvDbm2M/VNEKp7rqBsI/AAAAAAAA4wY/gwh8MnuvmlM/Box_Plot_6_Data_Table_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="177" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Create the Bottom Sections of a Stacked Column Chart</font></b></font></font></font></h2> <p>The box plots for the sample are created with a stacked column chart in Excel. The chart’s first data series becomes the bottom section of the stacked columns. The second data series that is established for the chart becomes the second section of the columns. The third data series that is established for the chart becomes the third section of the columns and so on.</p> <p>The bottom section, which is created in this step, goes from 0 to Q1, the top of the first quartile. This bottom section will be configured to have no fill or outline and will therefore be invisible.</p> <p>The next higher section will be the Q2 quartile. The section of the stacked column will be configured with purple fill and black outline.</p> <p>The next higher section will be the Q3 quartile. This section of the stacked column will be configured with yellow fill and black outline.</p> <p>The upper whisker is then created by attaching a positive error bar that has the length of the Q4 quartile to the Q3 column section.</p> <p>The lower whisker is then created by attaching a negative error bar that has the length of the Q1 quartile to the bottom, invisible column section.</p> <p>Create a stacked chart by selecting the Insert tab and then selecting the following 2-D Column, Stacked Column chart option: </p> <p><a href="http://lh3.ggpht.com/-LzLQyA1kP2A/VNEKqSHg0EI/AAAAAAAA4wk/yqSVGR6922o/s1600-h/Box_Plot_7_Select_Stacked_Column_Chart_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Stacked Column Chart" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Stacked Column Chart" src="http://lh3.ggpht.com/-CyVZTzvcptE/VNEKrM-h5PI/AAAAAAAA4ws/8wk55JdkOtA/Box_Plot_7_Select_Stacked_Column_Chart_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="298" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>This creates a blank chart. To add the first data series, which will become the bottom section of the columns, right-click on the blank chart and then click Select Data in the pop-up menu that appears with the right-click on the chart as shown below: </p> <p><a href="http://lh6.ggpht.com/-FSWGgJ5hNtU/VNEKrtz-ZGI/AAAAAAAA4w0/Lcn_pjdgULg/s1600-h/Box_Plot_8_Select_Data_600%25255B10%25255D.jpg"><img title="Box Plots in Excel - Select Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Select Data" src="http://lh5.ggpht.com/-AdrqrklxGrE/VNEKsAZnFYI/AAAAAAAA4w4/IiKO2ZPPyvw/Box_Plot_8_Select_Data_600_thumb%25255B4%25255D.jpg?imgmax=800" width="404" height="193" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The Select Data dialogue box then appears. Add the first data series by clicking Add. This will bring up the Edit Series dialogue box as shown below. In this dialogue box input the location of the series name and the series data values. This will be a blank series and the name is designated by the blank cell F10. The series values are in cells G10:I10, which contains a copy of the Q1 values in row 6. </p> <p><a href="http://lh4.ggpht.com/-92AKWo8oZ-Y/VNEKsSRpjRI/AAAAAAAA4xA/j8PRdFEBtuk/s1600-h/Box_Plot_9A_1st_Data_Series-Blank_Closeup_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - 1st Data Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 1st Data Series" src="http://lh5.ggpht.com/-U2Id4t-ejVs/VNEKs_8kHfI/AAAAAAAA4xI/FyjGhdc97fI/Box_Plot_9A_1st_Data_Series-Blank_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="137" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-EnU0mzKxJ68/VNEKtUNf32I/AAAAAAAA4xQ/Por_wrmIelI/s1600-h/Box_Plot_9_1st_Data_Series-Blank_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - 1st Data Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 1st Data Series" src="http://lh5.ggpht.com/-58zt8hDN9CM/VNEKt89sLNI/AAAAAAAA4xc/3QgxykJq42E/Box_Plot_9_1st_Data_Series-Blank_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="166" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This produces the following graph, which contains the bottom section of the stacked columns. The default color for this section is blue.</p> <p><a href="http://lh3.ggpht.com/-mmCBl3Uegb8/VNEKuUtQwAI/AAAAAAAA4xg/392lpk6Q7as/s1600-h/Box_Plot_10_Blank_Series_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - 1st Data Seriesl" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 1st Data Series" src="http://lh4.ggpht.com/-vey8mYv2zOY/VNEKuvFAenI/AAAAAAAA4xo/ngcGabCKtEo/Box_Plot_10_Blank_Series_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="144" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>To remove the blue color, right click on any of the blue columns and then select the No Fill option of the pop-up menu that appears with the right-click. It should also be noted that this data series has also been designated having no name by selecting cell F10, which is a blank cell, to provide the name for the data series.</p> <p><a href="http://lh6.ggpht.com/-U9hMnni6jvo/VNEKvHbsi4I/AAAAAAAA4x0/_EuGA3TF-ts/s1600-h/Box_Plot_11_No_FIll_Blank_Data_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - 1st Data Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 1st Data Series" src="http://lh5.ggpht.com/-uJeEQJbWJJk/VNEKvl814nI/AAAAAAAA4x4/lPpl1eH1kKE/Box_Plot_11_No_FIll_Blank_Data_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="156" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>After the blue color is removed, the bottom sections of the stacked columns are still there, but they are now invisible as shown in the following diagram:</p> <p><a href="http://lh4.ggpht.com/-bNm3HqnyyhY/VNEKwLhuteI/AAAAAAAA4yA/vDMLU0QRl6I/s1600-h/Box_Plot_12_Blank_Chart_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Blank Chart" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Blank Chart" src="http://lh3.ggpht.com/-xth63XDlQgY/VNEKwbBM5hI/AAAAAAAA4yM/4oTWxwsP7YQ/Box_Plot_12_Blank_Chart_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="254" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Create the Next Highest Section of the Stacked Columns</font></b></font></font></font></h2> <p>The next section of each stacked column will be the purple box representing the Q2 data. This section will sit on top of the invisible bottom column section that extends to the end of the Q1 quartile (which is the beginning of the Q2 quartile that is now being constructed).</p> <p>This column section is created by right-clicking the chart and then clicking the Select Data from the pop-up menu that appears with the right-click.</p> <p><a href="http://lh3.ggpht.com/-BQoBMdsHJGk/VNEKwzMyjfI/AAAAAAAA4yU/Xph9evsLmJE/s1600-h/Box_Plot_8_Select_Data_600%25255B12%25255D.jpg"><img title="Box Plots in Excel - Select Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Select Data" src="http://lh4.ggpht.com/-OU_rLXuUsIM/VNEKxb1Ba8I/AAAAAAAA4yc/XhF7oZBkyJc/Box_Plot_8_Select_Data_600_thumb%25255B6%25255D.jpg?imgmax=800" width="404" height="193" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The Select Data dialogue box then appears. Add the second data series by clicking Add. This will bring up the Edit Series dialogue box as shown below. In this dialogue box input the location of the series name and the series data values. This will be Q2 quartile series and the name is designated as 2Q Box from that value in cell the blank cell F11. The series values are in cells G11:I11, which contains a copy of the Q2 values in row 7.</p> <p><a href="http://lh6.ggpht.com/-Gmb8l02Cecc/VNEKxyvNHXI/AAAAAAAA4yk/kLSxK7KyDNw/s1600-h/Box_Plot_13A_2Q_Box_Series_Closeup_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - 2Q Box Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 2Q Box Series" src="http://lh3.ggpht.com/-edTY_UTBl5o/VNEKyQys46I/AAAAAAAA4ys/83Epdx8dAuA/Box_Plot_13A_2Q_Box_Series_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="96" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.ggpht.com/-YheBVW-ZBqQ/VNEKzDZ9ekI/AAAAAAAA4y0/wMbNDjSOe40/s1600-h/Box_Plot_13_2Q_Box_Series_600%25255B7%25255D.jpg"><img title="Box Plots in Excel - 2Q Box Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - 2Q Box Series" src="http://lh6.ggpht.com/-Z3H_HBZrK28/VNEKzWwj1RI/AAAAAAAA4y8/oafmV5wfJmU/Box_Plot_13_2Q_Box_Series_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="211" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The default color for these column sections is orange. This change be changed by right-clicking any one of the orange boxes on the chart and then selecting purple fill and a solid line outline in the dialogue box that appears with the right-click as shown below:</p> <p><a href="http://lh4.ggpht.com/-TQkdgyBVrz0/VNEK0H7aAbI/AAAAAAAA4zA/dZh8pDNZecE/s1600-h/Box_Plot_14_Series_Fill_Outline_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Series Fill Options" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Series Fill Options" src="http://lh6.ggpht.com/-Gy-8tJivjGU/VNEK0QqHZMI/AAAAAAAA4zM/wIKWggCRwK8/Box_Plot_14_Series_Fill_Outline_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="280" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The current state of the graph is now shown as follows. The yellow boxes representing the Q3 quartile data will be created in the next step.</p> <p><a href="http://lh4.ggpht.com/-Z6Y4V-sg7Pc/VNEK0_q_brI/AAAAAAAA4zU/XGTfB0It6zQ/s1600-h/Box_Plot_15_Series_Filled_and_Outlined_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Series Filled and Outlined" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Series Filled and Outlined" src="http://lh6.ggpht.com/-u6M3WYcKQNE/VNEK1ZA7rGI/AAAAAAAA4zc/8ZXtSIxPzCY/Box_Plot_15_Series_Filled_and_Outlined_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="267" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 5) Create the Next Highest Section of the Stacked Columns</font></b></font></font></font></h2> <p>The next section of each stacked column will be the yellow box representing the Q3 data. This section will sit on top of the purple column section that extends to the end of the Q2 quartile.</p> <p>This column section is created by right-clicking the chart and then clicking the Select Data from the pop-up menu that appears with the right-click.</p> <p><a href="http://lh6.ggpht.com/-iZje6MRcQJw/VNEK1-HpHxI/AAAAAAAA4zg/cJY20BgObOg/s1600-h/Box_Plot_8_Select_Data_600%25255B14%25255D.jpg"><img title="Box Plots in Excel - Select Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Select Data" src="http://lh6.ggpht.com/-JxA7eRgT7Tc/VNEK2ZHv0uI/AAAAAAAA4zs/moInKjX01OE/Box_Plot_8_Select_Data_600_thumb%25255B8%25255D.jpg?imgmax=800" width="404" height="193" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Select Data dialogue box then appears. Add the second data series by clicking Add. This will bring up the Edit Series dialogue box as shown below. In this dialogue box input the location of the series name and the series data values. This will be Q3 quartile series and the name is designated as 3Q Box from that value in cell the blank cell F12. The series values are in cells G12:I12, which contains a copy of the Q3 values in row 8.</p> <p><a href="http://lh5.ggpht.com/-6U9iuw42pf0/VNEK2-S8itI/AAAAAAAA4z0/F2iVFahej6M/s1600-h/Box_Plot_16_Q3_Series_Add_Closeup_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Q3 Box Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Q3 Box Series" src="http://lh5.ggpht.com/-rn4BIhILKv8/VNEK3KxDRhI/AAAAAAAA4z8/yJgwY0G0MKY/Box_Plot_16_Q3_Series_Add_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="93" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-w04gGbz69Qs/VNEK33SbAAI/AAAAAAAA40A/nSrT57-rHis/s1600-h/Box_Plot_16_Q3_Series_Add_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Add Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Add Series" src="http://lh4.ggpht.com/-B5UEgn_7Qvk/VNEK4XvJGuI/AAAAAAAA40M/Q8zmyyiYA7A/Box_Plot_16_Q3_Series_Add_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The chart now appears as follows. The default color of the third section of the stacked column is grey.</p> <p><a href="http://lh4.ggpht.com/-cKxmN1zAvHE/VNEK42Iny2I/AAAAAAAA40Q/_jFGqWHhe10/s1600-h/Box_Plot_17_New_Q3_Box_Series_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Q3 Box Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Q3 Box Series" src="http://lh6.ggpht.com/-ucPZl_r-Bww/VNEK5cP7vPI/AAAAAAAA40c/cPcsIgu5X3Q/Box_Plot_17_New_Q3_Box_Series_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="251" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The upper box can now be converted from grey to yellow with a block outline by right-clicking on any of the grey boxes and select yellow fill and black outline from the dialogue box that appears with the right-click.</p> <p><a href="http://lh4.ggpht.com/-1KVHsFZUlw0/VNEK5_YfPjI/AAAAAAAA40k/9H2TZGQPboc/s1600-h/Box_Plot_18_3Q_Box_Series_Filled_Outlined_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Filled Outline" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Filled Outline" src="http://lh4.ggpht.com/-LrGYYoV4B6Y/VNEK6ZnlMqI/AAAAAAAA40s/ixrOGvQigQ0/Box_Plot_18_3Q_Box_Series_Filled_Outlined_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="251" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The stacked data columns only have the generic labels 1, 2, and 3 on the horizontal axis. Horizontal axis labels can be added by right-clicking the chart once again. This will bring up the Select Data dialogue box. Click the Edit button for the Horizontal Axis Labels as shown below: </p> <p><a href="http://lh5.ggpht.com/-_-4d_JfVf6k/VNEK6yZUdkI/AAAAAAAA40w/5q74SWB2cRs/s1600-h/Box_Plot_19_Horizontal_Labels_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Horizontal Labels" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Horizontal Labels" src="http://lh5.ggpht.com/-Rdn0D9xIaeI/VNEK7QAeEDI/AAAAAAAA408/jFcRKInWY9E/Box_Plot_19_Horizontal_Labels_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="215" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Designate the horizontal axis labels to be the contents of cells G2:I2, which are Sample 1, Sample 2, and Sample 3. </p> <p><a href="http://lh3.ggpht.com/-t-5cAHC5-U8/VNEK7wsCUVI/AAAAAAAA41E/YpsXlmqT1xo/s1600-h/Box_Plot_20_Selecting_Labels_Closeup_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Selecting Labels" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Selecting Labels" src="http://lh5.ggpht.com/-lz2ME_3h2gE/VNEK8qApYzI/AAAAAAAA41I/n0FNBJCQ2MM/Box_Plot_20_Selecting_Labels_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="99" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-tWhJ2mj7z9Q/VNEK9IJkfEI/AAAAAAAA41U/vQGcsocR6AE/s1600-h/Box_Plot_20_Selecting_Labels_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Selecting Labels" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Selecting Labels" src="http://lh3.ggpht.com/-huFnGZw6mOk/VNEK9qVTkWI/AAAAAAAA41c/iNHIk8t4ng8/Box_Plot_20_Selecting_Labels_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="177" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The resulting chart is show as follows and is now ready to have the upper and lower whiskers installed.</p> <p><a href="http://lh5.ggpht.com/-9aF_ghTD97w/VNEK-BmHeoI/AAAAAAAA41k/noB9yvOUIBI/s1600-h/Box_Plot_21_Resulting_Labels_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Horizontal Labels" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Horizontal Labels" src="http://lh5.ggpht.com/-O0QWhynuSf8/VNEK-qV1RiI/AAAAAAAA41s/o8SDwSIf5so/Box_Plot_21_Resulting_Labels_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 6) Create the Upper Whiskers</font></b></font></font></font></h2> <p>The whiskers will extend either up or down from the top of the data box that they are attached to. The upper whiskers, which represent the Q4 quartile, will extend from the top of the Q3 data box. Do that by first selecting any yellow Q3 box. This will highlight all three Q3 boxes along with the data row that they represent.</p> <p><a href="http://lh4.ggpht.com/-OS8PaDVMHtU/VNEK_JaOwvI/AAAAAAAA410/BRXaXCISwag/s1600-h/Box_Plot_22_Select_Q3_Series_600%25255B7%25255D.jpg"><img title="Box Plots in Excel - Q3 Box Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Q3 Box Series" src="http://lh3.ggpht.com/-ca0J8aDzcyg/VNEK_lKDUcI/AAAAAAAA414/4UgTSnHyoUM/Box_Plot_22_Select_Q3_Series_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="155" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Add the Error bar by clicking <b>Design / Add Chart Element / Error Bars / More Error Bar Options</b> as shown in the following diagram:</p> <p><a href="http://lh4.ggpht.com/-LPITH7Bhm08/VNELAOCCLhI/AAAAAAAA42E/Mr49SJUCC5w/s1600-h/Box_Plot_23_Q3_Error_Bars_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Error Bars" src="http://lh4.ggpht.com/-I42nyPVreJU/VNELArPgRUI/AAAAAAAA42M/lHvapI3N94k/Box_Plot_23_Q3_Error_Bars_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="322" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Select Custom Error Bars from the dialogue box and then select the Whisker+ data in cells G14:I14 as show in the following diagram:</p> <p><a href="http://lh4.ggpht.com/-MkXdCx9R1y4/VNELBfYtfdI/AAAAAAAA42U/_56Dn_V4Jz8/s1600-h/Box_Plot_25A_Error_Bar_Closeup_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Error Bars" src="http://lh5.ggpht.com/-nLJHyfLyKIk/VNELByZID2I/AAAAAAAA42c/dZrIT30M4_Q/Box_Plot_25A_Error_Bar_Closeup_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="136" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-sd-KsQZ_Z4g/VNELCtMzugI/AAAAAAAA42k/_ms5k1k0xv4/s1600-h/Box_Plot_25_Error_Bar_Values_Closeup%25255B4%25255D.jpg"><img title="Box Plots in Excel - Error Bar Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Error Bar Values" src="http://lh6.ggpht.com/-J4NjsH9sCok/VNELDIskSTI/AAAAAAAA42s/fITsqGwnLNQ/Box_Plot_25_Error_Bar_Values_Closeup_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="136" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh4.ggpht.com/-F3DMFkOKi0A/VNELDliMfvI/AAAAAAAA420/di00UqCe-5U/s1600-h/Box_Plot_25_Error_Bar_Values_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Error Bar Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Error Bar Values" src="http://lh6.ggpht.com/-f3v7KNzZQNo/VNELEaHHNgI/AAAAAAAA428/DG38jCEykJo/Box_Plot_25_Error_Bar_Values_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="141" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The final step is to format the Error Bars. The Format Error Bars dialogue box is on the right of the page as shown in the previous image. The upper Error Bar should be configured in the Plus direction, with a Cap, and using the values from the Whisker+ row as the values for the length. </p> <p><a href="http://lh3.ggpht.com/-UJ8xZQuLMTs/VNELEqKzgOI/AAAAAAAA43E/HYcYFX0RQII/s1600-h/Box_Plot_24_Error_Bar_Choices_1_400%25255B4%25255D.jpg"><img title="Box Plots in Excel - Error Bar Choices" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Error Bar Choices" src="http://lh3.ggpht.com/-lJszBLp_C7o/VNELFIDVMII/AAAAAAAA43M/_7s8dpNFZMk/Box_Plot_24_Error_Bar_Choices_1_400_thumb%25255B2%25255D.jpg?imgmax=800" width="322" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The upper whiskers will now appear as follows:</p> <p><a href="http://lh5.ggpht.com/-H1SHUVFpZ70/VNELFmwhnFI/AAAAAAAA43U/FwnKnj5_TPo/s1600-h/Box_Plot_26_Upper_Whiskers%25255B5%25255D.jpg"><img title="Box Plots in Excel - Upper Whiskers" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Upper Whiskers" src="http://lh5.ggpht.com/-sY-wq4CqB58/VNELGGOLMpI/AAAAAAAA43Y/LeOXYUEtfGI/Box_Plot_26_Upper_Whiskers_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="251" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p> </p> <h2><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 7) Create the Lower Whiskers</font></b></font></font></font></h2> <p>The whiskers will extend either up or down from the top of the data box that they are attached to. The lower whiskers, which represent the Q1 quartile, will extend from the top of the invisible Q1 data section of the stacked column. Do that by first selecting any transparent Q1 section. This will highlight all three transparent Q1 sections along with the data row that they represent.</p> <p><a href="http://lh6.ggpht.com/-WDErxKoPyb0/VNELGwIxqQI/AAAAAAAA43k/a5g1PQ_XLeg/s1600-h/Box_Plot_27_Select_Blank_Series_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Select Blank Series" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Select Blank Series" src="http://lh3.ggpht.com/-J0hJD-grOLM/VNELHP9ZLVI/AAAAAAAA43o/0CRMGWeLIeI/Box_Plot_27_Select_Blank_Series_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="173" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Add the Error bar by clicking <b>Design / Add Chart Element / Error Bars / More Error Bar Options</b> as shown in the following diagram:</p> <p><a href="http://lh3.ggpht.com/-22hKvW9wxAg/VNELH_ayQiI/AAAAAAAA430/HT38KEBapIk/s1600-h/Box_Plot_28_New_Error_Bars_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - New Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - New Error Bars" src="http://lh3.ggpht.com/-cvuRmd2xe8o/VNELIaJUPkI/AAAAAAAA434/5NQJms--has/Box_Plot_28_New_Error_Bars_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="238" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Set the Error Bar direction to Minus, add a Cap, and designate the Whisker- row data to be the length of these whiskers.</p> <p><a href="http://lh3.ggpht.com/-z5aEDzC6Mxk/VNELIiQT4WI/AAAAAAAA44E/QQj9YaVWbQw/s1600-h/Box_Plot_29_Error_Bars_Configure_1_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Configure Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Configure Error Bars" src="http://lh6.ggpht.com/-PmXk-3dJteI/VNELJPLUQHI/AAAAAAAA44I/mVC8Lp2O-Dc/Box_Plot_29_Error_Bars_Configure_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="278" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-bq15eUJv-Zo/VNELJv00J3I/AAAAAAAA44Q/7vbcE1STBsA/s1600-h/Box_Plot_30_Error_Values_Closeup_400%25255B4%25255D.jpg"><img title="Box Plots in Excel - Configure Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Configure Error Bars" src="http://lh6.ggpht.com/-rGacWqhDovQ/VNELKCLMBsI/AAAAAAAA44c/kAp652vQMPI/Box_Plot_30_Error_Values_Closeup_400_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="283" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-2ut06AgjdjY/VNELLJZJyaI/AAAAAAAA44k/RXssjOMFo5Q/s1600-h/Box_Plot_30_Error_Values_Closeup%25255B4%25255D.jpg"><img title="Box Plots in Excel - Configure Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Configure Error Bars" src="http://lh3.ggpht.com/-7HnOxsdEiYo/VNELLq1jWJI/AAAAAAAA44o/C2_XZKOOIwU/Box_Plot_30_Error_Values_Closeup_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="140" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Here is the resulting box plot graph with the upper and lower whiskers attached to the Q2/Q3 boxes.</p> <p><a href="http://lh5.ggpht.com/-PdnqXWIccZE/VNELMFdf2RI/AAAAAAAA440/lOVdTSi5tkg/s1600-h/Box_Plot_31_Lower_Error_Bars_600%25255B4%25255D.jpg"><img title="Box Plots in Excel - Lower Error Bars" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Box Plots in Excel - Lower Error Bars" src="http://lh4.ggpht.com/-RCRzsSWo7iY/VNELMtmkV-I/AAAAAAAA448/dV8Ldg9sxEM/Box_Plot_31_Lower_Error_Bars_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="251" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com1tag:blogger.com,1999:blog-3568555666281177719.post-85420366101440826462015-01-20T15:35:00.001-08:002015-03-24T19:12:15.724-07:00Randomized Block Design ANOVA in Excel<h1>ANOVA Using Randomized <br /> <br />Block Design in Excel</h1> <h2>Randomized Block Design ANOVA Overview</h2> <p>Randomized block design is a method used to perform single-factor ANOVA while partially removing the effects of another variable, sometimes called a confounding variable. A confounding variable is an additional variable that might be affecting all of the data values in unpredictable ways and possibly obscuring the intended result of the single-factor ANOVA test, which is to determine whether the original factor has a significant effect on data values.</p> <p>Randomized block design is equivalent to two-factor ANOVA without replication. In Excel, randomized block design is implemented with the following Data Analysis tool: Two-Factor ANOVA Without Replication. </p> <p>Data most suitable for analysis with randomized block design have much of the overall variance explained by two relatively unrelated factors. The data are placed into unique cells each having a unique combination of levels of the two factors. Randomized block design requires that each unique cell contains only one data point so that each unique combination of levels of the two factor is represented by a single data point.</p> <p><a href="http://lh6.ggpht.com/-htpew6GeFR0/VL7l7N2D_rI/AAAAAAAA4oY/aw6AfZ3m28c/s1600-h/Ran_Bl_ANOVA_1_Raw_Data_600%25255B14%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh5.ggpht.com/-_0Jnwkh3Yfg/VL7l7tEAc5I/AAAAAAAA4og/y7qw_v8HIv4/Ran_Bl_ANOVA_1_Raw_Data_600_thumb%25255B6%25255D.jpg?imgmax=800" width="404" height="119" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The levels of the confounding factor are referred to as blocks. Each block contains a single random data point from each the levels of the main factor being isolated and tested. Hence the name <i>randomized block design</i>. Following is an example of data correctly arranged for ANOVA testing using randomized block design. The main factor has its levels divided across columns with each level being referred to in this case as a treatment. The confounding factor has its levels divided among rows with each level being referred to as a block. Each unique combination of treatment/block is represented by a single data point as follows:</p> <p></p> <p>The main purpose of performing ANOVA testing using randomized block design is to isolate the effect of the main factor (whose levels are called treatments) from the effect of the confounding factor (whose levels are called blocks) so that the effect of the main factor can be analyzed with greater clarity.</p> <p>One simple example of randomized block design might be to test the effects of four different types of fertilizer on crop yields. Each of the treatments of the main factor represents the use of one of the unique type of the four available fertilizers. Each measurement taken represents the total amount of the same crop harvested from the same-sized plot of farm land treated with the respective type of fertilizer.</p> <p>This same experiment was performed on four different farms. Each farm has its own unique set of conditions that affect crop growth such as different rainfall and soil nutrients. The confounding factor is the farm upon which each fertilizer experiment took place. Each farm is described as being its own block or level of the confounding factor. The fertilizer experiment was replicated exactly on each of the four different farm but individual differences inherent to each farm have an unpredictable effect on crop growth and are considered to be a confounding factor. ANOVA using randomized block design attempts to separate the effects of the confounding factor from the effects of the main factor so that the main factor can be analyzed with greater clarity.</p> <p>ANOVA using randomized block design has the same overall purpose as single-factor ANOVA, i.e., determining whether the main factor has a significant effect on data values. A significant effect is defined as the occurrence of at least one of the sample groups of the main factor (one of the treatment groups) having a significantly different mean than any of the other treatment groups.</p> <p>ANOVA using randomized block design has more power than single-factor ANOVA when applied to a data set. Test power in ANOVA is the ability to detect a significant difference between sample group means if such a difference exists. The greater power of randomized block design is evidenced by the lower p Value that this test calculates than single-factor ANOVA does when applied to the same data set. </p> <p>ANOVA tests are hypothesis tests. The Null Hypotheses of all ANOVA tests state all sample groups analyzed with the same F test came from the same population. The lower the p Value calculated by a hypothesis test, the greater is the probability that at least one of the sample groups came from a different population than the other samples groups that were analyzed in the same F test. ANOVA using randomized block design will calculate a lower p Value than single-factor ANOVA when analyzing the significance of the effect of the main factor. </p> <p>The reason is that ANOVA using randomized block design attempts to separate the effects of the confounding factor from the main factor while single-factor ANOVA does not. ANOVA using randomized block design is still not the most desirable (powerful) test to determine when a factor significantly affect data values when a confounding factor also exists. The small sample group sizes that are mandated by the structure of randomized block design generally cause this test to have relatively low power. This is discussed as follows.</p> <h2>ANOVA With Randomized Block</h2> <h2>Design Usually Has Very Low Power</h2> <p>Randomized block design specifies that only one data value be collected for each unique combination of main-factor level/confounding-factor level. This nearly always ensures that the test will be of extremely low power because of the small total number of data points collected for the entire test. The power of a test is its ability to detect a significant difference if one exists. A test with low power has a high probability of making a type 2 error, i.e., a false negative – failing to detect a difference that actually exists).</p> <p>Test power is nearly always increased when more random, representative sample data are collected and added to the correct sample groups. If the overall intention of the randomized block design test is isolate the effects of the main factor from the effects of the confounding variable, a more effective (powerful) alternative is simply to collect more data and analyze the larger sample groups with two-factor ANOVA with replication using the following one-step Excel Data Analysis tool: Two-Way ANOVA With Replication. Implementation of this technique will be shown in detail later in this blog article.</p> <p>Statistical tests become more and more powerful as the total number of data points within sample groups increases. This is accomplished by replicating levels of the confounding variable. Each additional row of data consists of data</p> <h2>Null and Alternative Hypotheses for</h2> <h2>ANOVA With Randomized Block</h2> <h2>Design</h2> <p>The Null Hypothesis for ANOVA with randomized block design is exactly like that of single-factor ANOVA and states that the sample groups of the main factor <i>ALL</i> come from the same population. An equivalent of that would be to state that the respective populations from the sample groups were drawn all have the same population mean. This would be written as follows:</p> <p>Null Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>1</b></sub> = µ<sub><b>2</b></sub> = … = µ<sub><b>k</b></sub> (k equals the number of sample groups)</p> <p>Note that Null Hypothesis is not referring to the sample means, x_bar<sub><b>1</b></sub> , x_bar<sub><b>2</b></sub> , … , x_bar<sub><b>k</b></sub>, but to the population means, µ<sub><b>1</b></sub> , µ<sub><b>2</b></sub> , … , µ<sub><b>k</b></sub>.</p> <p>The Alternative Hypothesis for ANOVA with randomized block design states that <i>at least one</i> sample group is likely to have come from a different population. Like single-Factor ANOVA, ANOVA with randomized block design is an omnibus test meaning that the test does not clarify which groups are different or how large any of the differences between the groups are. This Alternative Hypothesis only states whether <i>at least one</i> sample group is likely to have come from a different population. An equivalent of that would be to state that the population that at least one sample group was drawn from has a different population mean that the populations that the other sample groups were drawn from. This would be written as follows:</p> <p>Alternative Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>i</b></sub> ≠ µ<sub><b>j</b></sub> for some i and j</p> <h2>ANOVA With Randomized Block</h2> <h2>Design Example in Excel</h2> P>The article will continue using the example of comparing the effects of four types of fertilizer (the four treatments, which are the levels of the main factor) on plots of farm land that exist on four different farms (the four blocks, which are the levels of the confounding factor). <p>Each farm has four identical plots of land. Each of the four plots will be fertilized with one of the four types of fertilizers. Growing conditions are identical in all plots within the same farm. Each farm has its own unique set of conditions that affect crop growth such as different rainfall and soil nutrients. The confounding factor is the farm upon which each fertilizer experiment took place. Each farm is described as being its own block or level of the confounding factor. The fertilizer experiment was replicated exactly on each of the four different farm but individual differences inherent to each farm have an unpredictable effect on crop growth and are considered to be a confounding factor.</p> <p>The same type of crop will be planted in all 16 plots (each of the four farms has four plots). The weight of the total crop harvested from each of the 16 plots is recorded as follows:</p> <p><a href="http://lh6.ggpht.com/-6E2sVEXzje4/VL7l8Jer_pI/AAAAAAAA4oo/SOMldCPMAbg/s1600-h/Ran_Bl_ANOVA_1A_ANOVA_Raw_Data_600%25255B7%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh4.ggpht.com/-dpsfwHLb1V4/VL7l8g0HWlI/AAAAAAAA4os/3nPmSCllV-Q/Ran_Bl_ANOVA_1A_ANOVA_Raw_Data_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="106" /></a> </p> <p>Converting the above into a generic randomized block design would be done as follows:</p> <p><a href="http://lh5.ggpht.com/-nQ9-NbgxU7g/VL7l9NqCFDI/AAAAAAAA4o4/0BOYadflTe0/s1600-h/Ran_Bl_ANOVA_1_Raw_Data_600%25255B16%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh3.ggpht.com/-8lzQrLP7Y3Q/VL7l9qyCE4I/AAAAAAAA4pA/P5oD2R_MlTI/Ran_Bl_ANOVA_1_Raw_Data_600_thumb%25255B8%25255D.jpg?imgmax=800" width="404" height="119" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>It is best to arrange the raw data as shown above with blocks on separate rows and treatments on separate columns. The reason is that ANOVA using randomized block design usually has very low power due to the small amount of sample data. Test power is defined as the ability to detect a difference from sample data when a difference actually exists between the populations from which the samples were drawn.</p> <p>This test can be made more powerful by collecting more data. Each block of data would have to be replicated the sample number of times. For example if the entire experiment were replicated three times on each farm, the data would appear similar to the following:</p> <p><a href="http://lh4.ggpht.com/-aK6WiC0cBJ4/VL7l-H-pj7I/AAAAAAAA4pI/HWIJUirMDRI/s1600-h/Ran_Bl_ANOVA_6_More_Data_600%25255B7%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Larger Amount of Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Larger Amount of Raw Data" src="http://lh6.ggpht.com/-cMbgCDATEAQ/VL7l-vPxVII/AAAAAAAA4pQ/y8o4jv6eLz0/Ran_Bl_ANOVA_6_More_Data_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="234" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p></p> <p>This data matrix would now be analyzed with the Excel Data Analysis tool Two-Factor ANOVA With Replication. This would no longer be randomized block design because randomized block design specifies that only one data point for each unique combination of block and treatment, which is shown once again in the original raw data matrix.</p> <p><a href="http://lh3.ggpht.com/-5xA0Me7VYco/VL7l_FhsSwI/AAAAAAAA4pU/ar3x5dTRpow/s1600-h/Ran_Bl_ANOVA_1_Raw_Data_600%25255B18%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh4.ggpht.com/-_Yh9ocFW0fc/VL7l_oqHlwI/AAAAAAAA4pg/jTOTi2YjWXM/Ran_Bl_ANOVA_1_Raw_Data_600_thumb%25255B10%25255D.jpg?imgmax=800" width="404" height="119" /></a> </p> <p>ANOVA using randomized block design can be performed in Excel in as follows:</p> <h3 class="western" align="justify">Required Assumptions For ANOVA Using Randomized Block Design</h3> <h4 class="western" align="left">ANOVA Using Randomized Block Design Has Assumptions That Are Very Similar To Single-Factor ANOVA</h4> <p>ANOVA using randomized block design has the following seven required assumptions whose validity should be confirmed before this test is applied. The seven required assumptions are the following:</p> <p><font color="#0000cc"><font face="Arial, serif"><b>1) Independence of Sample Group Data</b></font></font> Sample groups must be differentiated in such a way that there can be no cross-over of data between sample groups. No data observation in any sample group could have been legitimately placed in another sample group. No data observation affects the value of another data observation in the same group or in a different group. This is verified by an examination of the test procedure. In this case each of the 16 data points (the crop yields from the 16 plots of farm land) are independent because no crop yield affects the size of any other crop yield.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>2) Sample Data Are Continuous</b></font></font> Sample group data (the dependent variable’s measured value) can be ratio or interval data, which are the two major types of continuous data. Sample group data cannot be nominal or ordinal data, which are the two major types of categorical data. The measurement taken is the weight of the yield of each of the plots. Weight is a continuous variable that is classified as ratio because the zero point indicates an absence of the variable and not an arbitrary point on a scale.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>3) Both Independent Variables (Factors) Are Categorical</b></font></font> The main factor and the confounding factor are both categorical variables. The levels of the main factor are often called treatments and the levels of the confounding factor are called blocks. The main factor is the type of fertilizer being used. The confounding factor is the choice of farm.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>4) Extreme Outliers Removed If Necessary</b></font></font> ANOVA is a parametric test that relies upon calculation of the means of sample groups. Extreme outliers can skew the calculation of the mean. Outliers should be identified and evaluated for removal in all sample groups. Occasional outliers are to be expected in normally-distributed data but all outliers should be evaluated to determine whether their inclusion will produce a less representative result of the overall data than their exclusion. There were no values of crop yields that could be considered extremely outliers.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>5) Normally-Distributed Data In All Sample Groups</b></font></font> ANOVA is a parametric test having the required assumption the dependent-variable data from each sample group come from a normally-distributed population. Each sample group’s dependent-variable data should be tested for normality. Normality testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that normality tests will provide accurate results. Like single-factor ANOVA, ANOVA using randomized block design is relatively robust to minor deviation from sample group normality. Verifying normality of treatment groups (sample groups that represent levels of the main variables) is much more important than verifying normality of blocks (sample groups that represent levels of the confounding variable) because the objective of this ANOVA is to determine whether the main factor has a significant effect on data values. The only important F test is the F test that will be performed on the treatment groups (the sample groups that represent levels of the main factor). The sample groups that are part of this F test should be normally distributed. </p> <p>Normality testing cannot reliably performed here because the main factor sample groups (treatment groups) have only four data points. That is too few. </p> <p><font color="#0000cc"><font face="Arial, serif"><b>6) Relatively Similar Variances In Sample Groups of the Same F Test</b></font></font> ANCOA requires that sample groups that are analyzed in the same F test have similar variances. Similar sample variances indicate that the populations from which the samples were taken have similar population variances. The sample variances do not have to be exactly equal but do have to be similar enough so the variance testing of the sample groups will not detect significant differences. The variances of sample groups are considered similar if no sample group variance is more than twice as large as the variance of another sample group. Variance testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that variance tests will provide accurate results. Verifying similarity of variances of treatment groups (sample groups that represent levels of the main variables) is much more important than verifying similarity of variances of blocks (sample groups that represent levels of the confounding variable) because the objective of this ANOVA is to determine whether the main factor has a significant effect on data values. The only important F test is the F test that will be performed on the treatment groups (the sample groups that represent levels of the main factor). The sample groups that are part of this F test should have similar sample variances.</p> <p>Variance comparison cannot reliably performed here because the main factor sample groups (treatment groups) have only four data points. That is too few. </p> <p><font color="#0000cc"><font face="Arial, serif"><b>7) Only One Data Point Is Collected For Each Unique Combination of Block and Treatment</b></font></font></p> <p>This structure is what defines the randomized block design. Because there is no additional data collection for any levels of any factors, this test is performed with two-factor ANOVA without replication (which is the equivalent of ANOVA with randomized block design). If additional data were collected, e.g., two data points collected for each unique combination of treatment/block, then two-factor ANOVA with replication would be used instead of randomized block design.</p> <p>All of the required assumptions of ANOVA using randomized block design have been validated except for sample group normality and sample group homogeneity of variance for the main factor, this test can now be performed in Excel as follows:</p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Run Two-Way ANOVA Without Replication on the Raw Data Matrix </font></b></font></font></font></p> <p>Two-Factor ANOVA Without Replication is one of the Excel Data Analysis tools.</p> <p><a href="http://lh3.ggpht.com/-Kfqo7pvB8W8/VL7mAOYgrsI/AAAAAAAA4po/4aW_XN_38ag/s1600-h/Ran_Bl_ANOVA_1_Raw_Data_600%25255B20%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh6.ggpht.com/-7PT_J4D8Czo/VL7mARzi1vI/AAAAAAAA4pw/qdrH3yJVHCg/Ran_Bl_ANOVA_1_Raw_Data_600_thumb%25255B12%25255D.jpg?imgmax=800" width="404" height="119" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The dialogue box for the Data Analysis tool Two-Factor ANOVA Without Replication is completed as follows:</p> <p><a href="http://lh3.ggpht.com/-BW063LoC6MI/VL7mBLk0SSI/AAAAAAAA4p4/LRBozciWANY/s1600-h/Ran_Bl_ANOVA_4_2-way_ANOVA_wo_rep_Diag_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" src="http://lh3.ggpht.com/-bsmGE46T3ug/VL7mBpjmdJI/AAAAAAAA4p8/k6JkDWPqJ-s/Ran_Bl_ANOVA_4_2-way_ANOVA_wo_rep_Diag_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="253" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The following output is produced:</p> <p><a href="http://lh5.ggpht.com/-D5RiEIBQLCc/VL7mCHQG8HI/AAAAAAAA4qI/MZOrCC0Z_KQ/s1600-h/Ran_Bl_ANOVA_5_2-way_ANOVA_wo_Rep_Output_600%25255B7%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Output" src="http://lh4.ggpht.com/-euLcYR0HrB0/VL7mCYEQg-I/AAAAAAAA4qM/gy1lQ4s50uQ/Ran_Bl_ANOVA_5_2-way_ANOVA_wo_Rep_Output_600_thumb%25255B3%25255D.jpg?imgmax=800" width="404" height="277" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>In this case the ANOVA test using randomized block design did not detect a difference for an alpha of 0.05 as evidenced by the p value of 0.109581 for the Column factor (treatments – types of fertilizer). </p> <p>Although ANOVA using randomized block design is a relatively weak test (as will soon be demonstrated), it still has more power than single-factor ANOVA when applied to the same raw data set. Applying the Excel Data Analysis tool Single-Factor ANOVA, here is the completed dialogue box:</p> <p><a href="http://lh4.ggpht.com/-BxhlXaImuVw/VL7mC2a-a6I/AAAAAAAA4qY/2xgLPUhK-RA/s1600-h/Ran_Bl_ANOVA_2_1-way_ANOVA_Diag_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" src="http://lh4.ggpht.com/--X83k9vYDgE/VL7mDRMQeNI/AAAAAAAA4qc/sroOckDdAzY/Ran_Bl_ANOVA_2_1-way_ANOVA_Diag_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="297" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>This produces the resulting output:</p> <p><a href="http://lh5.ggpht.com/-iMYrBWmGMbk/VL7mD0N78BI/AAAAAAAA4qk/9lPjWIBcN9Q/s1600-h/Ran_Bl_ANOVA_3_1-way_ANOVA_Output_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Output" src="http://lh5.ggpht.com/-0H2HelUwIeI/VL7mETIdk0I/AAAAAAAA4qw/hpLdXE9PmLU/Ran_Bl_ANOVA_3_1-way_ANOVA_Output_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="213" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Single-factor ANOVA calculated a significantly larger p Value of 0.312711 for the same raw data set. Even so, ANOVA using randomized block design has relatively low power because of the small number of total data points. Test power can be quickly obtained using the well-known online utility G*Power as follows: </p> <h3>Power Analysis of Two-Factor ANOVA Without Replication</h3> <p>The accuracy of a statistical test is very dependent upon the sample size. The larger the sample size, the more reliable will be the test’s results. The accuracy of a statistical test is specified as the Power of the test. A statistical test’s Power is the probability that the test will detect an effect of a given size at a given level of significance (alpha). The relationships are as follows:</p> <p>α (“alpha”) = Level of Significance = 1 – Level of Confidence</p> <p>α = probability of a type 1 error (a false positive)</p> <p>α = probability of detecting an effect where there is none</p> <p>Β (“beta”) = probability of a type 2 error (a false negative)</p> <p>Β = probability of not detecting a real effect</p> <p>1 - Β = probability of detecting a real effect</p> <p>Power = 1 - Β</p> <p>Power needs to be clarified further. Power is the probability of detecting a real effect <i>of a given size at a given Level of Significance (alpha)</i> at a given total sample size and number of groups. </p> <p>The term Power can be described as the accuracy of a statistical test. The Power of a statistical test is related with alpha, sample size, and effect size in the following ways:</p> <p>1) The larger the sample size, the larger is a test’s Power because a larger sample size increases a statistical test’s accuracy.</p> <p>2) The larger alpha is, the larger is a test’s Power because a larger alpha reduces the amount of confidence needed to validate a statistical test’s result. Alpha = 1 – Level of Confidence. The lower the Level of Confidence needed, the more likely a statistical test will detect an effect.</p> <p>3) The larger the specified effect size, the larger is a test’s Power because a larger effect size is more likely to be detected by a statistical test.</p> <p>If any three of the four related factors (Power, alpha, sample size, and effect size) are known, the fourth factor can be calculated. These calculations can be very tedious. Fortunately there are a number of free utilities available online that can calculate a test’s Power or the sample size needed to achieve a specified Power. One very convenient and easy-to-use downloadable Power calculator called G-Power is available at the following link at the time of this writing:</p> <p><a href="http://www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3/">http://www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3/</a></p> <p>Power calculations are generally used in two ways: </p> <p>1) <i><b>A priori</b></i><i> </i>- Calculation of the minimum sample size needed to achieve a specified Power to detect an effect of a given size at a given alpha. This is the most common use of Power analysis and is normally conducted <i>a priori</i> (before the test is conducted) when designing the test. A Power level of 80 percent for a given alpha and effect size is a common target. Sample size is increased until the desired Power level can be achieved. Since Power equals 1 – Β, the resulting Β of the targeted Power level represents the highest acceptable level of a type 2 error (a false negative – failing to detect a real effect). Calculation of the sample size necessary to achieve a specified Power requires three input variables: </p> <p>a) <b>Power level </b>– This is often set at .8 meaning that the test has an 80 percent to detect an effect of a given size.</p> <p>b) <b>Effect size</b> - Effect sizes are specified by the variable f. Effect size f is calculated from a different measure of effect size called η<sup><b>2</b></sup> (eta square). η<sup><b>2</b></sup> = SS<sub><b>Between_Groups</b></sub> / SS<sub><b>Total</b></sub><sub> </sub>These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.</p> <p>The relationship between effect size f and effect size η<sup><b>2</b></sup> is as follows:</p> <p><a href="http://lh3.ggpht.com/-BBmSLx-RsBU/VL7mEhyqqZI/AAAAAAAA4q4/c-oEP_9So90/s1600-h/Size_Effect_f%25255B2%25255D.png"><img title="Randomized Block Design ANOVA in Excel - Size Effect f" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Size Effect f" src="http://lh4.ggpht.com/-6lRm1V5CD9Y/VL7mFNHVFHI/AAAAAAAA4rA/J4Xa4jmjukY/Size_Effect_f_thumb.png?imgmax=800" width="128" height="70" /></a> </p> <p></p> <p>Jacob Cohen in his landmark 1998 book <i>Statistical Analysis for the Behavior Sciences</i> proposed that effect sizes could be generalized as follows:</p> <p>η<sup><b>2</b></sup> = 0.01 for a small effect. A small effect is one that not easily observable.</p> <p>η<sup><b>2</b></sup> = 0.05 for a medium effect. A medium effect is more easily detected than a small effect but less easily detected than a large effect.</p> <p>η<sup><b>2</b></sup> = 0.14 for a small effect. A large effect is one that is readily detected with the current measuring equipment.</p> <p>The above values of η<sup><b>2</b></sup> produce the following values of effect size f:</p> <p>f = 0.1 for a small effect</p> <p>f = 0.25 for a medium effect</p> <p>f = 0.4 for a large effect</p> <p>c)<b> Alpha</b> – This is commonly set at 0.05.</p> <h4>Performing <i>a priori</i> Power Analysis for the Main Effect of Factor 1</h4> <p>The G*Power utility will be used in an <i>a priori</i> manner to demonstrate how incredibly low the Power of two-factor ANOVA without replication is. The example used in this chapter will be analyzed. The data set and the Excel output of this example are shown as follows:</p> <p><a href="http://lh4.ggpht.com/-JRzO1fACzLA/VL7mFuAj9_I/AAAAAAAA4rI/Rk3N5wvkq4M/s1600-h/Ran_Bl_ANOVA_1A_ANOVA_Raw_Data_600%25255B9%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Raw Data" src="http://lh3.ggpht.com/-JdZfKKQztcg/VL7mGPMA4zI/AAAAAAAA4rQ/5EOO1Xqh2lg/Ran_Bl_ANOVA_1A_ANOVA_Raw_Data_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="106" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh3.ggpht.com/-CbKq_ipmdaE/VL7mG8zdvAI/AAAAAAAA4rY/pHgymPtppqg/s1600-h/Ran_Bl_ANOVA_5_2-way_ANOVA_wo_Rep_Output_600%25255B9%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Output" src="http://lh4.ggpht.com/-_n90lRnjylk/VL7mHSaAtuI/AAAAAAAA4rc/inclxVSe4o4/Ran_Bl_ANOVA_5_2-way_ANOVA_wo_Rep_Output_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="277" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>Two-Factor ANOVA without replication has two factors. There is no factor to account for the effect of interaction between these two factors. Each factor has its own unique Power that must be calculated. The Power for each factor is the probability that the ANOVA test will detect an effect of a given size caused by that factor. A separate Power calculation can be calculated for each of the two factors in this example. </p> <p>Power analysis performed a priori calculates how large the total sample size must be to achieve a specific Power level to detect an effect of a specified size at a given alpha level. <i>A priori</i> Power analysis of the main effect of factor 1 of this example is done as follows:</p> <p>The following parameters must be entered into the G*Power for <i>a priori</i> analysis for the general ANOVA dialogue box on the left side of the dialogue box shown below:</p> <p><u><b>Power (1 – Β)</b></u>: 0.8 – This is commonly used Power target. A test that achieves a Power level of 0.8 has an 80 percent chance of detecting the specified effect.</p> <p><u><b>Effect size</b></u>: 0.55 – This is a <i>very large</i> effect. This analysis will calculate the sample size needed to achieve an 80 percent probability of detecting an effect of this size. The effect size was set at such a large number in order to keep the required number of data points to a manageable size for further analysis, which will be done as part of this article. The larger the effect to be detected by the test, the smaller is the total number of samples needed by the test.</p> <p><u><b>α (alpha)</b></u>: 0.05 </p> <p><u><b>Numerator df</b></u>: 3 – The degrees of freedom specified for a test of a main effect of a factor equals the number of factor levels – 1. Factor 1 (Fertilizer Type) has 4 levels or treatments. This numerator df therefore equals 4 – 1 = 3. Note that this is the same df that is specified in the Excel ANOVA output for factor 1. </p> <p><u><b>Number of groups</b></u>: 16 – The number of groups equals (number of levels in factor 1) x (number f levels in factor 2). This equals 4 x 6 = 16. The number of groups is equal to the total number of unique treatment cells. Each unique treatment cell exists for each unique combination of levels between the factors.</p> <p><a href="http://lh3.ggpht.com/-_8PWDuiSTak/VL7mHy21ILI/AAAAAAAA4rk/jdh6w_Za0Og/s1600-h/G-Power_2_way_ANOVA_w_Rep_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Test Power - G Power" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Test Power - G Power" src="http://lh5.ggpht.com/-KXSkv4da2NI/VL7mIBNiJcI/AAAAAAAA4rw/CTYwaqkNxX0/G-Power_2_way_ANOVA_w_Rep_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="175" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>The calculated output, which appears on the right side of the dialogue box after it is run, specifies that 42 data points would be required to achieve a power level of 0.8007730 to detect an effect of size 0.55 for an alpha = 0.05 if the main factor has 4 levels (treatments) and the 2<sup>nd</sup> factor (the confounding factor) has 4 levels (blocks).</p> <p>G*Power also produces the following graph which indicates that the current test with 16 data points has a test power of less than 0.20. This means that the probability of this test indicating a difference in the populations from which the samples were taken if a significant difference actually exists is less than 20 percent.</p> <p><a href="http://lh3.ggpht.com/-wDnbvKdZ14E/VL7mIzbtmnI/AAAAAAAA4r4/sxNY5cXOQ4Y/s1600-h/Ran_Bl_ANOVA_0_Sample_Size_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Test Power - G Power" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Test Power - G Power" src="http://lh5.ggpht.com/-hW-_wz1rXzI/VL7mJbQt00I/AAAAAAAA4sA/CrLQ3RaBLxg/Ran_Bl_ANOVA_0_Sample_Size_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="330" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The solution to increase test power is clearly to increase the amount of data being tested. G*Power calculations and the previous G*Power graph indicate that at least 43 data points must be analyzed to bring this test up to a power level of 80 percent.</p> <p>This would be accomplished by simply collecting more data. Collecting more data would involve replicating all blocks an equal number of times. Each unique combination of treatment-block on the raw data matrix is called a treatment cell. All treatment cells for two-factor ANOVA must have the same number of data points. This is known has having <i><b>balanced</b></i> data.</p> <p>The initial test analyzed 16 data points. G*Power indicates that at least total 43 data points must be analyzed to raise the test power to at least 80 percent, which is generally considered to be an acceptable power level. </p> <h2 class="western" align="left">Analyzing the Data With Two-</h2> <h2 class="western" align="left">Factor ANOVA With Replication in </h2> <h2 class="western" align="left">Excel</h2> <p>If each block were replicated three times, the total number of data points would be 48 (16 * 3 = 48). Practical implementation of that would require that each farm conduct the experiment three times and record each result. The raw data matrix from such an effort would look similar to the following:</p> <p><a href="http://lh5.ggpht.com/-kExT757CJCw/VL7mJ6HjZiI/AAAAAAAA4sI/gjXZQmarZdQ/s1600-h/Ran_Bl_ANOVA_6_More_Data_600%25255B9%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - More Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - More Raw Data" src="http://lh4.ggpht.com/-S30FqMw_6ck/VL7mKnxn-sI/AAAAAAAA4sQ/D0MQOzGmHZE/Ran_Bl_ANOVA_6_More_Data_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="234" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p></p> <p>Note that each treatment cell has three data points. The data is therefore balanced as required by two-factor ANOVA. Single-factor ANOVA does not require that sample groups be the same size.</p> <p>Data arranged in this fashion can be evaluated using two-factor ANOVA with replication. There is an Excel Data Analysis tool exactly for this. Its dialogue box can be brought up and completed as follows:</p> <p><a href="http://lh4.ggpht.com/-X_NYY_Rqlv8/VL7mLBIp33I/AAAAAAAA4sY/pra6DkUyYBI/s1600-h/Ran_Bl_ANOVA_7_2-way_ANOVA_w_rep_Diag_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Dialogue Box" src="http://lh3.ggpht.com/-5frCPgEwNL0/VL7mLur21JI/AAAAAAAA4sg/BZrN0LBFRH8/Ran_Bl_ANOVA_7_2-way_ANOVA_w_rep_Diag_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="263" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>This would produce the following output:</p> <p><a href="http://lh6.ggpht.com/-k47sJZfKVV0/VL7mMfg6Y0I/AAAAAAAA4so/oEXh4slzWAE/s1600-h/Ran_Bl_ANOVA_8_2-Way_ANOVA_w_rep_Output_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Output" src="http://lh3.ggpht.com/-VL1hjuwDtdY/VL7mM1TdIMI/AAAAAAAA4ss/rveXTkAGVOM/Ran_Bl_ANOVA_8_2-Way_ANOVA_w_rep_Output_600_thumb%25255B2%25255D.jpg?imgmax=800" width="361" height="484" /></a> </p> <p></p> <p>This test displays more power as evidenced by the lower p Value of 0.061243 for the column factor (the choice of fertilizer treatments). Larger amounts of appropriate data nearly always increase the power of a statistical test.</p> <p>An additional benefit of more data is the ability to verify the two required assumptions that remained unverified for the two-factor ANOVA without replication: normality and variance similarity in all groups of the main factor (the treatment groups). </p> <h3>Normality Testing of Treatment Groups</h3> <p>A histogram for each treatment group can be quickly constructed using the Excel Data Analysis histogram tool. The results are shown in the following diagrams. Normality is not perfect but the histograms don’t deviate from normality too significantly given that the modal values (most frequently occurring) are generally near the center of each histogram. </p> <p><a href="http://lh5.ggpht.com/-9YTjxna6S44/VL7mNWBhxPI/AAAAAAAA4s4/g4iROWonXxU/s1600-h/Ran_Bl_ANOVA_9_Hist_1_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" src="http://lh6.ggpht.com/-R1xWhMUdvCU/VL7mNq01XDI/AAAAAAAA4s8/3Jiw2tghPgU/Ran_Bl_ANOVA_9_Hist_1_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="261" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh6.ggpht.com/-_oLKQvg5sY4/VL7mOMVLf6I/AAAAAAAA4tI/OKaHH7-NUcc/s1600-h/Ran_Bl_ANOVA_10_Hist_2_600%25255B7%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" src="http://lh5.ggpht.com/-4z4uwFJBZls/VL7mOt98mTI/AAAAAAAA4tM/QiGQqdlujz4/Ran_Bl_ANOVA_10_Hist_2_600_thumb%25255B5%25255D.jpg?imgmax=800" width="404" height="266" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh4.ggpht.com/-xrNlDi1p6Y0/VL7mPF-tv0I/AAAAAAAA4tY/T-RjhwlzpiE/s1600-h/Ran_Bl_ANOVA_11_Hist_3_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" src="http://lh5.ggpht.com/-d9lMasJDoyc/VL7mPYaZMTI/AAAAAAAA4tg/bhIVXg4d_Po/Ran_Bl_ANOVA_11_Hist_3_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="253" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p><a href="http://lh5.ggpht.com/-2OI3_p0vHqM/VL7mP1gT_FI/AAAAAAAA4to/oyxNS6osRZ0/s1600-h/Ran_BL_ANOVA_12_Hist4_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - Normailty Testing Histogram" src="http://lh6.ggpht.com/-wBKA5_bAg3A/VL7mQaTmbMI/AAAAAAAA4ts/UwDHYggJJjw/Ran_BL_ANOVA_12_Hist4_600_thumb%25255B2%25255D.jpg?imgmax=800" width="404" height="264" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <h3>Variance Comparison of Treatment Groups</h3> <p>Sample groups display homoscedasticity (similarity of variances) if the variance of any sample group is not more than 4 times that variance of any other sample group. The following ANOVA output, which was previously calculated, shows that the variances of each of the treatment sample groups are similar to each other. </p> <p><a href="http://lh3.ggpht.com/-BTi4Sb5888U/VL7mRM_szRI/AAAAAAAA4t4/hGenuJ-t0pY/s1600-h/Ran_Bl_ANOVA_13_Variances_600%25255B4%25255D.jpg"><img title="Randomized Block Design ANOVA in Excel - ANOVA Sample Group Variance Comparison" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Randomized Block Design ANOVA in Excel - ANOVA Sample Group Variance Comparison" src="http://lh4.ggpht.com/-juywdsyo-Mk/VL7mRtnPfrI/AAAAAAAA4t8/At56LXeTPEs/Ran_Bl_ANOVA_13_Variances_600_thumb%25255B2%25255D.jpg?imgmax=800" width="372" height="484" /></a> <br /><i>(Click On Image To See a Larger Version)</i> </p> <p>It can now be stated that all of the required assumptions for two-factor ANOVA have been verified.</p> <h3>Conclusion</h3> <p><i><b><span style="background: #ffff00">ANOVA using randomized block design (two-factor ANOVA without replication) nearly always tests too little data to be considered reliable.</span></b></i> The small group sizes that occur with two-way ANOVA without replication reduce the test’s Power to an unacceptable level. Small group size also prevents validation of ANOVA’s required assumptions of data normality within groups and similar variances of all groups within each factor. </p> <p>The correct solution is to collect more data and perform two-factor ANOVA with replication.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com4tag:blogger.com,1999:blog-3568555666281177719.post-62702735597559268282015-01-11T11:35:00.001-08:002015-03-24T19:17:07.328-07:00Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel<h1>Single-Factor Repeated <br /> <br />Measures ANOVA in 4 <br /> <br />Steps in Excel</h1> <p>This is one of the following four articles on Repeated-Measures ANOVA</p> <p><a href="http://blog.excelmasterseries.com/2015/01/single-factor-repeated-measures-anova.html">Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/sphericity-testing-in-9-steps-for.html">Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/effect-size-for-repeated-measures-anova.html">Effect Size For Repeated-Measures ANOVA in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html">Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel</a></p> <h2>Overview</h2> <p>Repeated-measures ANOVA is very similar to single-factor ANOVA except that the sample groups consists of measures taken on the same group of subjects at different time periods or under different conditions. Single-factor ANOVA requires that data sample points in any sample group be completely independent of data points of any other sample points. Sample groups in a repeated-measure ANOVA test are related to each other because each sample group consists of measures taken on the same group of subjects as all other sample groups.</p> <p>An easy way conceptualize repeated-measure ANOVA is to view it as an extension of the paired T-test. The most common use of the paired T-test is to determine whether a significant difference exists between before and after-measurements taken on a group of subjects. If an additional measurement was taken on each subject at an intermediate time period, three sample groups (before, middle, after) of data measurements taken on the same subjects would be produced. </p> <p>The main uses of repeated-measure ANOVA are the following:</p> <p>1) Taking the same measurement on subjects of the same group at different time periods to determine if there is any significant difference in that measure at different points in time.</p> <p>2) Taking the same measurement on subjects of the same group in different conditions to determine if there is a significant difference in that measure in different conditions. Conditions are sometimes referred to as <i>treatments</i>.</p> <p>Repeated-measures ANOVA removes most or all of the variance between the subjects leaving only the between-group variance and error (unexplained) variance. To illustrate that point, imagine that repeated-measures ANOVA was performed on twenty subjects who were undergoing a training program. All of the subjects took the same test at three points in time: at the beginning of the training, in the middle of the training, and at the end of the training. The objective of the repeated-measures ANOVA test is to determine whether the average test score had changed at any point in the training. </p> <p>ANOVA is an omnibus test, meaning that it only indicates that one sample group comes from a different population than the other sample groups. ANOVA by itself does not indicate which specific sample groups are different. <i>Post hoc</i> testing is used to determine where sample differences are significant.</p> <p>The differences in abilities of the individual subjects will likely generate a significant amount of variation in the test scores for each of the three sample groups. Each sample group contains the test scores of all tests taken at one of the three points in time. Variation resulting from ability differences of each individual needs to be removed in order to determine whether there are any real differences in average test scores in any of the three time periods. Repeated-measures ANOVA removes variation attributed to the difference among subjects leaving only the between-group variance and error (unexplained) variance.</p> <p>SS<sub>within</sub> = Variation within groups</p> <p>SS<sub>error</sub> = Unexplained variation</p> <p>SS<sub>subjects</sub> = Variation attributed to individual differences between test subjects</p> <p><u><b>Single-Factor ANOVA</b></u> (requires all data points in a sample groups to be totally independent of each other)</p> <p>SS<sub>error</sub> = SS<sub>within</sub></p> <p>MS<sub>between</sub> = SS<sub>between</sub> / df<sub>between</sub></p> <p>MS<sub>within</sub> = SS<sub>within</sub> / df<sub>within</sub></p> <p>F Value = MS<sub>between</sub> / MS<sub>within</sub></p> <p>p Value = F.DIST.RT(F Value, df<sub>between</sub>, df<sub>within</sub>)</p> <p><u><b>Repeated-measures ANOVA</b></u> (data points in different sample group are all taken from the same group of subjects and are therefore not independent of each other)</p> <p>SS<sub>error</sub> = SS<sub>within</sub> - SS<sub>subjects</sub></p> <p>SS stand for “sum of squares,” which is how variance is calculated.</p> <p>Note that the variance attributed to error (SS<sub>error</sub>) is now smaller as a result of removing variance associated with differences among individual subjects (SS<sub>subjects</sub>).</p> <p>MS<sub>between</sub> = SS<sub>between</sub> / df<sub>between</sub></p> <p>MS<sub>error</sub> = SS<sub>error</sub> / df<sub>error</sub></p> <p>F Value = MS<sub>between</sub> / MS<sub>error</sub></p> <p>p Value = F.DIST.RT(F Value, df<sub>between</sub>, df<sub>error</sub>)</p> <p>The F Value for repeated-measures ANOVA will be significantly larger than the F Value of the test if it were performed as single-factor ANOVA. This causes the p Value to be smaller for repeated-measures ANOVA thus making repeated-measures ANOVA the more powerful than single-factor ANOVA would be if applied to the same data (which it should not be because the data in all sample groups are taken from the same subjects are therefore not independent of each other).</p> <p>df<sub>error</sub>, which is calculated by (n-1)(k-1,) will be slightly less than df<sub>within</sub> but the F value of repeated-measures ANOVA is increased significantly more. This ultimately reduces the p Value of repeated-measures ANOVA so that it is less than the p Value of the comparable single-factor ANOVA thus making repeated-measures ANOVA the more sensitive (powerful) test.</p> <h2>Null and Alternative Hypotheses for Repeated-Measures ANOVA</h2> <p>The Null Hypothesis for repeated-measures ANOVA is exactly like that of single-factor ANOVA and states that the sample groups <i>ALL</i> come from the same population. This would be written as follows:</p> <p>Null Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>1</b></sub> = µ<sub><b>2</b></sub> = … = µ<sub><b>k</b></sub> (k equals the number of sample groups)</p> <p>Note that Null Hypothesis is not referring to the sample means, s<sub><b>1</b></sub> , s<sub><b>2</b></sub> , … , s<sub><b>k</b></sub>, but to the population means, µ<sub><b>1</b></sub> , µ<sub><b>2</b></sub> , … , µ<sub><b>k</b></sub>.</p> <p>The Alternative Hypothesis for ANCOVA states that <i>at least one</i> sample group is likely to have come from a different population. Like single-factor ANOVA, repeated-measure ANOVA is an omnibus test that does not clarify which groups are different or how large any of the differences between the groups are. This Alternative Hypothesis only states whether <i>at least one</i> sample group is likely to have come from a different population.</p> <p>Alternative Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>i</b></sub> ? µ<sub><b>j</b></sub> for some i and j</p> <h2>Single-Factor Repeated-Measures ANOVA (Within-Subjects) Example in Excel</h2> <p>A company implemented a four-week training program to reduce clerical errors. Five employees underwent this training program. The number of clerical errors that each trainee committed during each week as the training progressed was recorded. These data are shown as follows:</p> <p><a href="http://lh4.ggpht.com/-62TqqFD0bTY/VLLThnx7W1I/AAAAAAAA4dg/soZj_SITd0A/s1600-h/Rep_ANOVA_1_Raw_Data_72_600_RGB%25255B4%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Raw Data" src="http://lh4.ggpht.com/-_s9Om_6xTkI/VLLTiUN80bI/AAAAAAAA4do/Gbpm38XZmFo/Rep_ANOVA_1_Raw_Data_72_600_RGB_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Single-factor repeated-measures ANOVA (within subjects) will be performed on this data to determine whether the average number clerical errors changed during any week of the training after removing the variation in clerical errors due to individual differences between trainees (subjects).</p> <p>Each of the subjects who underwent the training can be described by the following two variables used in repeated-measures ANOVA:</p> <p>Independent Variable – This is the categorical variable of Training Method type. The overall objective of the repeated-measures ANOVA is to determine if levels of the factor of training program type produced significantly different sales increases.</p> <p>Dependent Variable – This continuous variable is the monthly sales increase for each salesperson who underwent either of the two training programs. Note that values of this variable are colored black.</p> <p>Repeated-measures ANOVA can be performed in Excel in four steps. Before performing these steps, repeated-measures ANOVA’s required assumptions will be listed below as follows:</p> <h3>Repeated-Measures ANOVA’s Required Assumptions</h3> <h4>Repeated-Measures ANOVA Has the Following Same Required Assumptions as Single-Factor ANOVA</h4> <p>Like single-factor ANOVA, repeated-measures ANOVA has the following required assumptions whose validity should be confirmed before this test is applied:</p> <p><font color="#0000cc"><font face="Arial, serif"><b>1) Sample Data Are Continuous</b></font></font> Sample group data (the dependent variable’s measured value) can be ratio or interval data, which are the two major types of continuous data. Sample group data cannot be nominal or ordinal data, which are the two major types of categorical data.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>2) Independent Variable is Categorical</b></font></font> The determinant of which group each data observation belongs to is a categorical, independent variable. Repeated-measures ANOVA uses a single categorical variable that has at least two levels. All data observations associated with each variable level represent a unique data group and will occupy a separate column on the Excel worksheet.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>3) Extreme Outliers Removed If Necessary</b></font></font> Repeated-measures ANOVA is a parametric test that relies upon calculation of the means of sample groups. Extreme outliers can skew the calculation of the mean. Outliers should be identified and evaluated for removal in all sample groups. Occasional outliers are to be expected in normally-distributed data but all outliers should be evaluated to determine whether their inclusion will produce a less representative result of the overall data than their exclusion.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>4) Normally-Distributed Data In All Sample Groups</b></font></font> Repeated-measures ANOVA is a parametric test having the required assumption the dependent-variable data from each sample group come from a normally-distributed population. Each sample group’s dependent-variable data should be tested for normality. Normality testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that normality tests will provide accurate results. Like single-factor ANOVA, repeated-measures ANOVA is relatively robust to minor deviation from sample group normality.</p> <h4>Repeated-measures ANOVA Has the Following Additional Require Assumption</h4> <p><a name="_GoBack"></a><font color="#0000cc"><font face="Arial, serif"><b>5) Sphericity In All Sample Groups</b></font></font> Single-factor ANOVA requires that sample groups are obtained from populations that have similar variances. Repeated-measures ANOVA has a similar but more rigorous requirement called Sphericity. Sphericity exists when the variance of the differences of all combinations of groups is equal. Sample groups represent measurements taken from the same set of subjects under different conditions or at different times. The differences referred to by Sphericity are the differences between data valuess in different sample groups that are taken from the same subject.</p> <p>Violation of Sphericity makes the test more likely to perform a Type 1 error, i.e., a false positive. If the requirement of Sphericity is violated, a correction can be applied to the df<sub>between</sub> and df<sub>error</sub> which will increase the test’s p Value making the test less likely to report a false positive (making the test more conservative). Sphericity testing and any necessary df adjustment will be explained and demonstrated in the blog article following this one.</p> <p>Variance testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that variance tests will provide accurate results.</p> <h3><u>Single-Factor Repeated-Measures ANOVA in Excel - The 4 Steps </u></h3> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Run Single-Factor ANOVA on the Sample Groups</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Calculate Subject Means, Group Means, and the Grand Mean</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Calculate SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">subjects</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">, SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">error</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">, and df</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">error</font></sub></b></font></font></font></p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Create an Updated ANOVA Table for Repeated-Measure ANOVA</font></b></font></font></font></p> <p>Here is a detailed description of the performance of each step:</p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Run Single-Factor ANOVA on the Sample Groups</font></b></font></font></font> </p> <p>After the data is correctly arranged on the Excel worksheet, use the Excel data analysis tool <b>ANOVA: Single-Factor</b> to perform single-factor ANOVA. This dialogue box will appear and should be filled as follows:</p> <p><a href="http://lh5.ggpht.com/-diIToZKAyjM/VLLTi2trp4I/AAAAAAAA4dw/vVnVo5fqWnY/s1600-h/Rep_ANOVA_2_Single-Factor_ANOVA_72_600%25255B4%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - Single-Factor ANOVA" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Single-Factor ANOVA" src="http://lh6.ggpht.com/-F45_trYYG3I/VLLTjnDJU4I/AAAAAAAA4d4/WU8V0ebRZlc/Rep_ANOVA_2_Single-Factor_ANOVA_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="153" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Running the Single-Factor ANOVA tool in Excel on the above data produces the following output:</p> <p><a href="http://lh6.ggpht.com/-MA3SdxsWIxs/VLLTkDNHUOI/AAAAAAAA4eA/PzUmAUzA8nI/s1600-h/Rep_ANOVA_3_Single-Factor_ANOVA_Output_72_600%25255B4%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - ANOVA Ouput" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - ANOVA Ouput" src="http://lh5.ggpht.com/-BJKAgBFm5cs/VLLTkyIfxDI/AAAAAAAA4eI/d4iTCNHFTGY/Rep_ANOVA_3_Single-Factor_ANOVA_Output_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="245" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The single-factor ANOVA output is shown above. The p Value of 0.0767 indicates that single-factor ANOVA did not detect a significant difference between the group means if a 95-percent confidence level is required (alpha is set at 0.05). Keep in mind that Single-Factor ANOVA by itself is not an appropriate test to run on this data because the data in the different sample groups are not independent as required by single-factor ANOVA because the data in the different sample groups are all taken from the same set of subjects.</p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Calculate Subject Means, Group Means, and the Grand Mean</font></b></font></font></font> </p> <p><a href="http://lh5.ggpht.com/-pjm-bqJDfWQ/VLLTlUCZ26I/AAAAAAAA4eQ/piFG3ZkEWTg/s1600-h/Rep_ANOVA_4_Calculate_Means_72_600%25255B4%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - Calculating Means" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Calculating Means" src="http://lh5.ggpht.com/-j4WvYGu8fBc/VLLTmLni0SI/AAAAAAAA4eY/ga-8buwvcYM/Rep_ANOVA_4_Calculate_Means_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="169" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Calculate SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">subjects</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">, SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">error</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">, and df</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">error</font></sub></b></font></font></font></p> <p></p> <p></p> <p><a href="http://lh4.ggpht.com/-BfxCU4zpYs4/VLLTm9qSG8I/AAAAAAAA4eg/cMmCMM3noIc/s1600-h/Rep_ANOVA_5_Calcuating_SS_1_72_600%25255B7%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - Calculating SS" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Calculating SS" src="http://lh3.ggpht.com/-Yzq67vKrsuk/VLLTnlE-kcI/AAAAAAAA4eo/BR2DGw7wPrY/Rep_ANOVA_5_Calcuating_SS_1_72_600_thumb%25255B5%25255D.jpg?imgmax=800" width="400" height="237" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-wkS0nAOkSQA/VLLToVAeYmI/AAAAAAAA4ew/rpU9eZmXFYg/s1600-h/Rep_ANOVA_6_Calculating_SS_2_72_600%25255B4%25255D.jpg"><img title="V" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Calculating SS" src="http://lh4.ggpht.com/-UFRdHnNOYEM/VLLTpOXdLSI/AAAAAAAA4e4/pCoioVBYpPc/Rep_ANOVA_6_Calculating_SS_2_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="412" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>SS<sub>subjects</sub> = 13,358.70</p> <p>SS<sub>error</sub> = 7,809.30</p> <p>df<sub>error</sub> = 12</p> <p>When the variance attributed to differences between the test subjects (SS<sub>subjects</sub>) is removed from the within-group variance (SS<sub>within</sub>), only the much smaller unexplained variance (SS<sub>error</sub>) is left. This makes the test more powerful by dramatically increasing the F Value, which reduces the p Value.</p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Create an Updated ANOVA Table for Repeated-Measure ANOVA</font></b></font></font></font></p> <p></p> <p><a href="http://lh5.ggpht.com/-KfjhG-CVobM/VLLTpz-bNtI/AAAAAAAA4fA/dMPC2IlXhWk/s1600-h/Rep_ANOVA_7_Updated_ANOVA_Table_72_600%25255B4%25255D.jpg"><img title="Repeated_Measures ANOVA in Excel - Updated ANOVA Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated_Measures ANOVA in Excel - Updated ANOVA Table" src="http://lh4.ggpht.com/-UBjJe3L1I8o/VLLTqkHXZ6I/AAAAAAAA4fI/JiOERAp5UKE/Rep_ANOVA_7_Updated_ANOVA_Table_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="348" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Note that the p Value has dropped from 0.0767 single-factor ANOVA to 0.0123 in repeated-measure ANOVA. Repeated-measures ANOVA detected a significant difference between the sample group averages that single-factor ANOVA did not.. </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Sphericity Testing</font></b></font></font></font></p> <p>Sphericity testing should now be conducted. Sphericity exists when the variances of the differences between data pairs from the same subjects are the same across all possible combinations of sample groups. Two hypothesis tests are used to determine if Sphericity exists. The weaker but more popular of the two Sphericity hypothesis tests is Mauchly’s test. The more powerful but less frequently employed Sphericity test is called John, Nagao, and Sugiura’s Test of Sphericity. Both of these tests will be demonstrated in detail on the data used in this example in the next blog article. </p> <p>If the Null Hypothesis of either Mauchly’s Sphericity Test or John, Nagao, and Sugiura’s Test of Sphericity can be rejected, then a correction should be applied to df<sub>between</sub> and df<sub>error</sub> that will ultimately make the test more conservative and less powerful by increasing the final p Value of the repeated-measures ANOVA.</p> <p>If the Sphericity requirement has been violated, then the degree to which Sphericity is violated needs to be calculated. The statistic that describes how much Sphericity is violated is called Epsilon (<font face="Calibri, serif">?</font>). Epsilon is a number between 1 and 0. The further from 1 that Epsilon is, the greater is the violation of Sphericity. </p> <p>Sphericity can only be estimated because the available data are sample data and not population data. There are two methods commonly used to estimate Epsilon: the Geisser-Greenhouse procedure and the Huynd-Feldt procedure. The estimate of Sphericity (Epsilon) that each of these procedures calculates is used to correct df<sub>between</sub> and df<sub>error</sub> in a way that makes test less powerful by increasing the final p value.</p> <p>The blog article following this one will provided detailed instructions on how to perform the Geisser-Greenhouse procedure and the Huynd-Feldt procedure in Excel on the data used in this example and make then corrections to the degrees of freedom.</p> <h2>Calculating Effect Size in Repeated-Measures ANOVA</h2> <p>Effect size is a way of describing how effectively the method of data grouping allows those groups to be differentiated. A blog article following this one will provide detailed instructions on how to calculated effect size for repeated-measures ANOVA with the data used in this example.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com0tag:blogger.com,1999:blog-3568555666281177719.post-61771112337699350882015-01-11T09:10:00.001-08:002015-03-24T19:27:58.133-07:00Sphericity Testing in 9 Steps For Repeated-Measures ANOVA in Excel<h1>Sphericity Testing in <br /> <br />9 Steps For Repeated <br /> <br />Measures ANOVA in Excel</h1> <p>This is one of the following four articles on Repeated-Measures ANOVA</p> <p><a href="http://blog.excelmasterseries.com/2015/01/single-factor-repeated-measures-anova.html">Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/sphericity-testing-in-9-steps-for.html">Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/effect-size-for-repeated-measures-anova.html">Effect Size For Repeated-Measures ANOVA in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html">Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel</a></p> <h2>Overview</h2> <p>Single-factor ANOVA has a requirement that the variances of all sample groups be similar. The rule of thumb for single-factor ANOVA is that no sample group’s variance should more than four time the variance of any other sample group. </p> <p>Repeated-measures ANOVA has a similar but more rigorous requirement called sphericity. Sphericity exists when the variances of the differences between data pairs from the same subjects are the same across all possible combinations of sample groups. </p> <p>Violation of sphericity makes repeated-measures ANOVA more likely to produce Type 1 Errors, i.e., detecting a significant difference when one does not exist – a false positive. When sphericity is shown to be violated, a correction is applied to the test that makes the test more conservative (less likely to detect a significant difference) by increasing the test result’s p Value. Tests of sphericity and the correction calculation will be demonstrated in Excel in this blog article.</p> <h3>Two Sphericity Tests</h3> <p>Two hypothesis tests are used to determine if Sphericity exists. The weaker but more popular of the two sphericity hypothesis tests is <u><b>Mauchly’s Test of Sphericity</b></u>. The more powerful but less frequently employed sphericity test is called <u><b>John, Nagao, and Sugiura’s Test of Sphericity</b></u>. Both of these tests will be demonstrated in detail on the data used in this example in the next blog article. </p> <p>If the Null Hypothesis of either Mauchly’s Sphericity Test or John, Nagao, and Sugiura’s Test of Sphericity can be rejected, then a correction should be applied to df<sub>between</sub> and df<sub>error</sub> that will ultimately make the test more conservative and less powerful by increasing the final p Value of the repeated-measures ANOVA.</p> <h3>Epsilon Estimation Corrections</h3> <p>If the sphericity requirement has been violated, then the degree to which sphericity is violated needs to be calculated. The statistic that describes how much sphericity is violated is called Epsilon (<font face="Calibri, serif">?</font>). Epsilon is a number between 1 and 0. The further from 1 that Epsilon is, the greater is the violation of sphericity. </p> <p>Sphericity can only be estimated because the available data are sample data and not population data. There are two methods commonly used to estimate Epsilon: the <u><b>Geisser-Greenhouse procedure</b></u> and the <u><b>Huynd-Feldt procedure</b></u>. The estimate of Sphericity (Epsilon) that each of these procedures calculates is used to correct df<sub>between</sub> and df<sub>error</sub> in a way that makes test less powerful by increasing the final p value.</p> <p>The blog article following this one will provided detailed instructions on how to perform the Geisser-Greenhouse procedure and the Huynd-Feldt procedure in Excel on the data used in this example and make then corrections to the degrees of freedom. </p> <h2>Sphericity Testing For Repeated-Measures ANOVA Example in Excel</h2> <p>The example used in this article to demonstrate sphericity testing for repeated-measures ANOVA in Excel is the same example as in the previous blog article about how to perform repeated-measure ANOVA in Excel. A company implemented a four-week training program to reduce clerical errors. Five employees underwent this training program. The number of clerical errors that each trainee committed during each week as the training progressed was recorded. These data are shown as follows:</p> <p><a href="http://lh4.ggpht.com/-CA9GULpwi48/VLKts4HmQXI/AAAAAAAA4Tw/lN4qoVqhekU/s1600-h/Rep_ANOVA_1_Raw_Data_72_600_RGB%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - " src="http://lh3.ggpht.com/-EKkDsKcwD7c/VLKtt08JDgI/AAAAAAAA4T4/8vWhzg9nNaI/Rep_ANOVA_1_Raw_Data_72_600_RGB_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="199" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The data were analyzed with both single-factor ANOVA and with repeated-measures ANOVA with the following results:</p> <p><a href="http://lh5.ggpht.com/-byBST50_kn4/VLKtwO7ScNI/AAAAAAAA4UA/Neoe4s4W8eU/s1600-h/Rep_ANOVA_7_Updated_ANOVA_Table_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Updated ANOVA Table" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Updated ANOVA Table" src="http://lh4.ggpht.com/-28v079K7kMQ/VLKtwUnQ8gI/AAAAAAAA4UI/ejx8eiIo6Go/Rep_ANOVA_7_Updated_ANOVA_Table_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="348" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Repeated-measures ANOVA detects a significant difference between sample groups (p Value = 0.0123) while single-factor ANOVA does not (p Value = 0.0767) if a 95-percent level of confidence (alpha = 0.05) is required.</p> <p>Single-factor repeated-measures ANOVA has the following five required assumptions:</p> <p>1) Sample data are continuous.</p> <p>2) The independent variable is categorical</p> <p>3) Extreme outliers have been removed in necessary</p> <p>4) Sample groups are normally distributed</p> <p>5) Sphericity</p> <p>Assuming that first four required assumptions for repeated-measures ANOVA have been met, sphericity should now be evaluated.</p> <p>Sphericity exists when the variances of the differences between data pairs from the same subjects are the same across all possible combinations of sample groups. Remember that all sample groups for a repeated-measures ANOVA test consist of measurements taken from the same set of subjects at different time intervals or in different conditions. Violation of sphericity makes a repeated-measures ANOVA test more likely to produce a false positive (a Type 1 error). If sphericity is shown to exist, a correction should be applied to both degrees of freedom which will increase the final p Value of the repeated-measures ANOVA test. Increasing the p Value reduces the power of the test (makes it less likely that the test will detect a difference) in order to compensate for the test’s increased tendency to produce a false positive result due to the data’s violation of sphericity.</p> <p>Following are the 9 steps of sphericity testing and Epsilon estimation correction:</p> <h3><u>Sphericity Testing and Estimated Epsilon Correction for Single-Factor Repeated-Measures ANOVA in Excel - The 9 Steps </u></h3> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Create the Covariance Matrix From the Sample Groups</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Calculate the Eigenvalues of the Covariance Matrix</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Conduct Mauchly’s Test of Sphericity</font></b></font></font></font></p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Conduct John, Nagao, and Sugiura’s Test of Sphericity</font></b></font></font></font></p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 5) Determine Whether Sphericity Exists</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 6) Calculate the Greenhouse-Geisser Correction</font></b></font></font></font> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 7) Calculate the Huyhn-Feldt Correction</font></b></font></font></font></p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 8) Apply the Correction to the Degrees of Freedom and Recalculate p Value</font></b></font></font></font></p> <p>Here is a detailed description of the performance of each step:</p> <h2>Step 1) Create the Covariance Matrix From the Sample Groups</h2> <p>Both sphericity tests require the use of Eigenvalues that are calculated from the covariance matrix of the raw data sample matrix. Eigenvalues are the characteristic roots of a set of linear equations that a matrix represents. </p> <p>The first step in determining the Eigenvalues of the covariance matrix is to create the covariance matrix from the raw data sample matrix. Creating the covariance matrix from the raw data matrix is performed in Excel as follows. The Excel formulas are shown for the cells in column BE29:BE32.</p> <p><a href="http://lh3.ggpht.com/-2t0fuavhnT0/VLKtyC_89mI/AAAAAAAA4UQ/ZTvkZQNdv30/s1600-h/Rep_ANOVA_7b_Covariance_Matrix_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Covariance Matrix" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Covariance Matrix" src="http://lh6.ggpht.com/-D02Eh6IuX0Q/VLKt0H2fB4I/AAAAAAAA4UY/93w-OXu0SO8/Rep_ANOVA_7b_Covariance_Matrix_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="274" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The diagonals of the covariance matrix should agree with the results of the single-factor ANOVA previously performed on the raw data. Variance, σ<sup>2</sup>, is a special case of covariance, σ<sub>xy</sub>, when the two variables are the same.</p> <p>σ(X,X) = σ<sup>2</sup>(X)</p> <p><a href="http://lh6.ggpht.com/-kE11hidnqC4/VLKt12bICUI/AAAAAAAA4Ug/vLtj34NSqvs/s1600-h/Rep_ANOVA_7c_ANOVA_Output_72_500%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - ANOVA Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - ANOVA Output" src="http://lh3.ggpht.com/-EPmXO2Xoc5Q/VLKt3ozOZcI/AAAAAAAA4Uo/86EJC8I2iuM/Rep_ANOVA_7c_ANOVA_Output_72_500_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="214" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <h2>Step 2) Calculate the Eigenvalues of the Covariance Matrix</h2> <h3>Method 1 – Use an Online Tool</h3> <p>The easiest way to find all Eigenvalues for any symmetric matrix is to use an automated online tool such as the following:</p> <p><a href="http://www.bluebit.gr/matrix-calculator/">http://www.bluebit.gr/matrix-calculator/</a></p> <p>Input the Covariance matrix data as follows:</p> <p><a href="http://lh3.ggpht.com/-szplVMirp9M/VLKt53gZBWI/AAAAAAAA4Uw/JQ9ae8EW8_Y/s1600-h/Rep_ANOVA_8_Eigenvalue_Online_Data_Entry_72_600%25255B5%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Eigenvalue Online Data Entry" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Eigenvalue Online Data Entry" src="http://lh4.ggpht.com/-0RwXWLSxdJA/VLKt7JoNKPI/AAAAAAAA4U4/PONmh0BZbVg/Rep_ANOVA_8_Eigenvalue_Online_Data_Entry_72_600_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="295" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Hitting the Calculate button will produce the following output:</p> <p><a href="http://lh3.ggpht.com/-9J-KxF0a2Z8/VLKt8n4WmFI/AAAAAAAA4VA/EZzuUUAXy00/s1600-h/Rep_ANOVA_9_Eigenvector_Online_Output_72_600%25255B6%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Eigenvector Online Output" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Eigenvector Online Output" src="http://lh3.ggpht.com/-fUQVZDAiIJc/VLKt-IL06BI/AAAAAAAA4VI/LbiWv6RJSnU/Rep_ANOVA_9_Eigenvector_Online_Output_72_600_thumb%25255B4%25255D.jpg?imgmax=800" width="400" height="257" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>We now know that the Covariance matrix has the following 4 Eigenvalues: </p> <p>λ<sub>4</sub> = 5073.607973</p> <p>λ<sub>3</sub> = 211.9174057</p> <p>λ<sub>2</sub> = 7.37740705</p> <p>λ<sub>1</sub> = 0.087529249</p> <h3>Method 2 – Calculate the Eigenvalues in Excel</h3> <p>This method requires a little more work and a bit of guessing. The first step is to create the Identity matrix of the Covariance matrix as follows:</p> <p><a href="http://lh3.ggpht.com/-qE32fVtJWpg/VLKt__75fCI/AAAAAAAA4VQ/GGhr-QOn0XY/s1600-h/Rep_ANOVA_10_Identity_Matrix_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Identity Matrix" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Identity Matrix" src="http://lh5.ggpht.com/-ie9yrs6Sx5A/VLKuBrY8GhI/AAAAAAAA4VY/5GWw3OI7AZ4/Rep_ANOVA_10_Identity_Matrix_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="254" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The next step is to create the [A-cI] matrix. </p> <p>[A – cI] = Covariance Matrix – Covariance Matrix Eigenvalue * Identity Matrix </p> <p>The Covariance Matrix, A, is contained in cells BZ23:CC26</p> <p>The Identity Matrix is contained in cells BZ29:CC32</p> <p>The Covariance Matrix Eigenvalue is contained in cell BZ47. It has been initially set to a value of 10000.</p> <p>In Excel, to create the [A-cI] matrix in cells BZ39:CC42, perform the following actions:</p> <p>1) Highlight cells BZ39:CC42</p> <p>2) While these cells are still highlighted (the active cells), type =BZ23:CC26-BZ47*BZ29:CC32</p> <p>3) Simultaneously hit CTRL/SHIFT/ENTER and the [A-cI] matrix will be created in cells BZ39:CC42</p> <p><a href="http://lh6.ggpht.com/-95c1YgzUwB4/VLKuDB5n_iI/AAAAAAAA4Vg/OV2qHiGEWcM/s1600-h/Rep_ANOVA_11_Calculating_A-cI_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Calculating A-cI" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Calculating A-cI" src="http://lh6.ggpht.com/-3yyEBWgT4L8/VLKuDknpAEI/AAAAAAAA4Vs/MAy3uXwxvpM/Rep_ANOVA_11_Calculating_A-cI_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="321" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Once again, the [A-cI] matrix is contained in cells BZ39:CC42. </p> <p>[A – cI] = Covariance Matrix – Covariance Matrix Eigenvalue * Identity Matrix</p> <p>The Determinant of the [A – cI] is contained in cell BZ50 and is calculated by the following formula:</p> <p>MDETERM(BZ39:CC42)</p> <p>The Eigenvalues that are needed for both sphericity tests are the Eigenvalues of the Covariance matrix.</p> <p>An Eigenvalue of matrix A is a number that causes the determinant of matrix [A – cI] to be 0. In this case matrix A is the Covariance matrix.</p> <p>The Excel tool Goal Seek can be used to find the values of the Eigenvalue in cell BZ47 that causes Det [A – cI] to assume a value of 0. The Goal Seek tool is part of the What-If Analysis found under the Data tab in Excel.</p> <p>Goal Seek is a relatively straight-forward tool that calculates the value in a specific cell (the Eigenvalue in cell BZ47) that will cause another cell (Det [A – cI] in cell BZ50) to assume a specified value (0).</p> <p>A 4X4 matrix should have 4 Eigenvalues. The quickest way to find those 4 Eigenvalues is to first find the highest and lowest values and then use a bit of guesswork to find the middle 2 values.</p> <p>Find the highest Eigenvalue by setting the value in cell BZ47 to 10000 and running Goal Seek to find the value of BZ47 that will make Det[A – cI] (cell BZ50) equal 0. This Goal Seek operation is shown as follows:</p> <p><a href="http://lh3.ggpht.com/-cZPPIlo5SJA/VLKuEAAY90I/AAAAAAAA4V0/J6Cgc89zNfc/s1600-h/Rep_ANOVA_12_Goal_Seek_1_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh3.ggpht.com/-x5Ou2bgHkhU/VLKuEtfWyII/AAAAAAAA4V8/DyFkMjRvKp4/Rep_ANOVA_12_Goal_Seek_1_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="278" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Clicking OK will calculate the number in cell BZ47 that is closest to 10000 that will cause cell BZ50 to assume a value of 0. This value is cell BZ47 is the highest of the 4 Eigenvalues and has the following value:</p> <p>λ<sub>4</sub> = 5073.607973</p> <p>Note that this value agrees with the largest Eigenvalue found by the online tool.</p> <p><a href="http://lh5.ggpht.com/-dyaBZQMfBi0/VLKuFG1GWyI/AAAAAAAA4WE/lVHRnxG8ANM/s1600-h/Rep_ANOVA_13_Goal_Seek_2_72_600%25255B7%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh6.ggpht.com/-pE1eAB3f2pU/VLKuGpftdvI/AAAAAAAA4WM/ISyhNx4PbHM/Rep_ANOVA_13_Goal_Seek_2_72_600_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="166" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The next step is to find the lowest Eigenvalue. Do this by setting cell BZ47 (the Eigenvalue of the Covariance matrix) to a very low value. In this case cell BZ47 was set to -10000 and Goal Seek was run again.</p> <p><a href="http://lh5.ggpht.com/-6NmYOVLk3Ws/VLKuITsFNBI/AAAAAAAA4WU/hG8_tziuJ0w/s1600-h/Rep_ANOVA_14_Goal_Seek_3_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh5.ggpht.com/-8ALC71RR-aI/VLKuJofchyI/AAAAAAAA4Wc/TldYn60NqEw/Rep_ANOVA_14_Goal_Seek_3_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="280" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Clicking OK will calculate the number in cell BZ47 that is closest to -10000 that will cause cell BZ50 to assume a value of 0. This new value in cell BZ47 is the lowest of the 4 Eigenvalues and has the following value:</p> <p>λ<sub>1</sub> = 0.087529249</p> <p>Note that this value agrees with the lowest Eigenvalue found by the online tool.</p> <p><a href="http://lh3.ggpht.com/-yha0tfSkABo/VLKuLNzAyEI/AAAAAAAA4Wk/BLCrRsjszqg/s1600-h/Rep_ANOVA_15_Goal_Seek_4_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh6.ggpht.com/-7j7YJtW2lbs/VLKuMgapt5I/AAAAAAAA4Ws/fLulceHzplA/Rep_ANOVA_15_Goal_Seek_4_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="135" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Finding the middle 2 Eigenvalues involves a bit of guess work. One of the those 2 Eigenvalues values can be found by setting the Eigenvalue in cell BZ47 to the halfway point between the lowest Eigenvalue (0.087529249) and the highest Eigenvalue (5073.607973). This middle value is 2536. Running Goal Seek with Cell BZ47 set at 2536 is shown as follows:</p> <p><a href="http://lh5.ggpht.com/-SVa7NSgNI5o/VLKuNp08-eI/AAAAAAAA4W0/02uhRte7K_0/s1600-h/Rep_ANOVA_16_Goal_Seek_5_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh6.ggpht.com/-hQ5NKKMOegg/VLKuOwHItjI/AAAAAAAA4W8/3mfUZyBmZ84/Rep_ANOVA_16_Goal_Seek_5_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="241" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>This newly calculated Eigenvalue of 211.9174057 shown below is the 2<sup>nd</sup> highest of the 4 Eigenvalues.</p> <p><a href="http://lh5.ggpht.com/-3HYzlQMZr_g/VLKuP9mrsCI/AAAAAAAA4XE/I4Z_86OgkNY/s1600-h/Rep_ANOVA_17_Goal_Seek_6_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh4.ggpht.com/-6CLWodR4YBw/VLKuRI0dFJI/AAAAAAAA4XM/KGUrzkia9OE/Rep_ANOVA_17_Goal_Seek_6_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="136" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Running Goal Seek a final time with cell BZ47 set at the halfway point between the 2<sup>nd</sup> highest Eigenvalue (211.9174057) and the lowest Eigenvalue (0.087529249) will produce the 3<sup>rd</sup> highest Eigenvalue. The halfway point is 105 and that Goal Seek operation is shown as follows:</p> <p><a href="http://lh6.ggpht.com/-24FbYoLbIpU/VLKuRj2dwvI/AAAAAAAA4XU/2bH0Xei-nwI/s1600-h/Rep_ANOVA_18_Goal_Seek_7_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh6.ggpht.com/-GgsYgFhnQ_g/VLKuSLl3D3I/AAAAAAAA4Xc/a2ejhoZDoas/Rep_ANOVA_18_Goal_Seek_7_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="247" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>This final Goal Seek operation produces the 3<sup>rd</sup> highest Eigenvalue which is the following:</p> <p>λ<sub>2</sub> = 7.37740705</p> <p><a href="http://lh6.ggpht.com/-oF0vMcbNuHA/VLKuSgUJCbI/AAAAAAAA4Xk/eL8X4dtLWaE/s1600-h/Rep_ANOVA_19_Goal_Seek_8_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh4.ggpht.com/-cuTe4fl2P-4/VLKuTLYpFVI/AAAAAAAA4Xs/pJAPuQu8PQo/Rep_ANOVA_19_Goal_Seek_8_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="139" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>We now know that the Covariance matrix has the following 4 Eigenvalues: </p> <p>λ<sub>4</sub> = 5073.607973</p> <p>λ<sub>3</sub> = 211.9174057</p> <p>λ<sub>2</sub> = 7.37740705</p> <p>λ<sub>1</sub> = 0.087529249</p> <p>These Eigenvalues of the Covariance Matrix agree with the Eigenvalues calculated by the online tool.</p> <h2>Step 3) Conduct Mauchly’s Test of Sphericity in Excel</h2> <p>Mauchly’s Test of Sphericity is the most widely used hypothesis test to evaluate whether sphericity exists in the raw data. Mauchly’s test is, however not the most accurate test because of its small-sample tendency toward Type 2 errors (a false negative, i.e., failing to detect a difference when there is one) and its large-sample tendency toward Type 1 errors (a false positive, i.e., detecting a difference when none exists). A more powerful test is the John, Nagao, and Sugiura’s Test of Sphericity which will be discussed in the next step.</p> <p>Mauchly’s test, which is a hypothesis test, can be summed up as follows:</p> <p>The Null Hypothesis states that sphericity exist, i.e., the variances of the differences between data pairs from the same subjects are the same across all possible combinations of sample groups. This is another way of stating that the covariances (the off-diagonal elements of the covariance matrix) are equal.</p> <p>Test Statistic W is calculated from k (the number of sample groups), n (the number of data points in each sample group), and the 4 Eigenvalues that were just calculated.</p> <p>Critical values of Test Statistic W are available but W can be quickly transformed into X<sub>w</sub><sup>2</sup></p> <p>The distribution of X<sub>w</sub><sup>2 </sup>can be approximated by the Chi-Square distribution with degrees of freedom df = k/2*(k-1). </p> <p>This hypothesis test’s p Value can then be calculated in Excel as follows:</p> <p>p Value = CHISQ.DIST.RT(X<sub>w</sub><sup>2</sup>,df)</p> <p>A p Value of less than alpha (usually set at 0.05) indicates that the Null Hypothesis stating that sphericity exists can be rejected.</p> <p>All of these calculations in Excel are shown as follows:</p> <p>The first step is to calculate k (the number of sample groups), n (the number of data points in each sample group), and degrees of freedom, df<sub>W</sub>. The degrees of freedom for Mauchly’s test is found by this formula:</p> <p><a href="http://lh3.ggpht.com/-OcfikmzWkzw/VLKuUMqoZnI/AAAAAAAA4Hw/Jqj1zcWCZFw/s1600-h/Mauchlys_df%25255B2%25255D.png"><img title="Mauchlys_df" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Mauchlys_df" src="http://lh5.ggpht.com/-qI2nDBd3a6c/VLKuVc-FCuI/AAAAAAAA4H4/HG1wIPPDp1E/Mauchlys_df_thumb.png?imgmax=800" width="161" height="45" /></a> </p> <p></p> <p>Calculation of k, n, and df<sub>W</sub> are shown as follows:</p> <p><a href="http://lh3.ggpht.com/-dmmN48OHkVs/VLKuWt6aoPI/AAAAAAAA4X0/h1e54a6GOKU/s1600-h/Rep_ANOVA_20_Mauchly_1_72_300%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel" src="http://lh6.ggpht.com/-vtiBXaN7y9o/VLKuYES_DkI/AAAAAAAA4X8/6Rbx-cGJ7xU/Rep_ANOVA_20_Mauchly_1_72_300_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="379" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The next step is to calculate f from k and n with this formula:</p> <p><a href="http://lh3.ggpht.com/-JP5WVRq4GTI/VLKuZDWwmSI/AAAAAAAA4IY/JD1OwR9wqaA/s1600-h/Mauchlys_f%25255B2%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Mauchlys F" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys F" src="http://lh6.ggpht.com/-2MLkUFu8rEo/VLKuaDuj8II/AAAAAAAA4Io/jg4aIEObuQI/Mauchlys_f_thumb.png?imgmax=800" width="225" height="55" /></a> </p> <p><a href="http://lh3.ggpht.com/-u4-3tzqFras/VLKubfmL_6I/AAAAAAAA4YE/PyVGc93R42Y/s1600-h/Rep_ANOVA_21_Mauchly_2_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Mauchlys F Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys F Calculation" src="http://lh3.ggpht.com/-sk2Fhq2pEk8/VLKuc5HpeQI/AAAAAAAA4YM/ANc9XDdTEZw/Rep_ANOVA_21_Mauchly_2_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="121" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Test Statistic W is then calculated using k and the 4 Eigenvalues of the Covariance matrix using the following formula:</p> <p><a href="http://lh3.ggpht.com/-8JWPjIuL1m4/VLKudpHG2iI/AAAAAAAA4JE/FfqE2kdGiYY/s1600-h/Mauchlys_W%25255B2%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Mauchlys W" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys W" src="http://lh5.ggpht.com/-qLnXu951yfI/VLKueni8FqI/AAAAAAAA4JM/Y4bMMar2CKU/Mauchlys_W_thumb.png?imgmax=800" width="197" height="55" /></a> </p> <p></p> <p>These calculations in Excel are shown as follows:</p> <p><a href="http://lh3.ggpht.com/-X6BMx1ysHPs/VLKufnEGwXI/AAAAAAAA4YU/l5veYZZkmNw/s1600-h/Rep_ANOVA_22_Mauchly_3_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Mauchlys W Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys W Calculation" src="http://lh3.ggpht.com/-YyTSaPjR09I/VLKugzzJ9eI/AAAAAAAA4Yc/u2x6Dv3l4W8/Rep_ANOVA_22_Mauchly_3_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="219" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>There are critical values calculated for Test Statistic W but an easier way to perform this hypothesis test is to convert W into X<sub>w</sub><sup>2</sup> using the following formula:</p> <p><a href="http://lh6.ggpht.com/-BjMGsOF1QIw/VLKuiAFrljI/AAAAAAAA4Yk/VWauV8ryGg4/s1600-h/Mauchlys_Chi-Square_Statistic%25255B4%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Mauchlys Chi-Square Statistic" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys Chi-Square Statistic" src="http://lh6.ggpht.com/-p8t_mgMysPg/VLKuibg-sAI/AAAAAAAA4Ys/LzMua_MHAGU/Mauchlys_Chi-Square_Statistic_thumb%25255B2%25255D.png?imgmax=800" width="400" height="19" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>These calculations are shown in Excel as follows:</p> <p><a href="http://lh3.ggpht.com/-vHJVKvzdEDU/VLKui1Q9sXI/AAAAAAAA4Y0/S6r2Qkls82A/s1600-h/Rep_ANOVA_13_Goal_Seek_2_72_600%25255B9%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Goal Seek" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Goal Seek" src="http://lh3.ggpht.com/-zuhu9QKiSmg/VLKujqD6OHI/AAAAAAAA4Y8/ihkX8U1_L-s/Rep_ANOVA_13_Goal_Seek_2_72_600_thumb%25255B5%25255D.jpg?imgmax=800" width="400" height="166" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The distribution of X<sub>w</sub><sup>2</sup> can be approximated by the Chi-Square distribution with df<sub>w</sub> degrees of freedom.</p> <p>A p Value for this hypothesis test can then be found with the following formula:</p> <p><a href="http://lh6.ggpht.com/-3Q1RH9LQWwo/VLKukUVO7EI/AAAAAAAA4KQ/UFJTdvTugKg/s1600-h/Mauchlys_p%25255B2%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Mauchlys p" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys p" src="http://lh3.ggpht.com/-4vbwlsmcfJU/VLKuldWTXRI/AAAAAAAA4KY/qUffS9cUyvQ/Mauchlys_p_thumb.png?imgmax=800" width="240" height="20" /></a> </p> <p></p> <p>These calculations in Excel are shown as follows:</p> <p><a href="http://lh5.ggpht.com/-wXfNJ7cAmcc/VLK0dLdO7cI/AAAAAAAA4ZE/CBItdXY-1us/s1600-h/Rep_ANOVA_24_Mauchly_5_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Mauchlys p Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Mauchlys p Calculation" src="http://lh6.ggpht.com/-S_AY8rdb7-g/VLK0dqdVMcI/AAAAAAAA4ZM/SLmb7Fbes3U/Rep_ANOVA_24_Mauchly_5_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="183" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The small p Value of this Mauchly’s Test of Sphericity indicates departure from sphericity. It is important to note that not using all of the Eigenvalues significantly weakens the power of the test. When Eigenvalues are calculated manually in Excel instead of using an automated tool, occasionally the highest Eigenvalue is not located. </p> <p>The p Value for Mauchly’s Test is calculated using only the 3 lowest Eigenvalues but not the highest one. The p value is significantly larger as a result. This means that the test has lost some of its power.</p> <p></p> <h2>Step 4) Conduct John, Nagao, and Sugiura’s Test of Sphericity in Excel</h2> <p>Mauchly’s Test of Sphericity is the most widely used hypothesis test to evaluate whether sphericity exists in the raw data. Mauchly’s test is, however not the most accurate test because of its small-sample tendency toward Type 2 errors (a false negative, i.e., failing to detect a difference when there is one) and its large-sample tendency toward Type 1 errors (a false positive, i.e., detecting a difference when none exists). A more powerful test is the John, Nagao, and Sugiura’s Test of Sphericity which will be discussed here.</p> <p>John, Nagao, and Sugiura’s test, which is a hypothesis test, can be summed up as follows:</p> <p>The Null Hypothesis states that sphericity exist, i.e., the variances of the differences between data pairs from the same subjects are the same across all possible combinations of sample groups. This is another way of stating that the covariances (the off-diagonal elements of the covariance matrix) are equal.</p> <p>Test Statistic V is calculated from the 4 Eigenvalues that were just calculated.</p> <p>Critical values of Test Statistic V are available but V can be quickly transformed into X<sub>V</sub><sup>2</sup></p> <p>The distribution of X<sub>V</sub><sup>2 </sup>can be approximated by the Chi-Square distribution with degrees of freedom df = k/2*(k-1) - 1. </p> <p>This hypothesis test’s p Value can then be calculated in Excel as follows:</p> <p>p Value = CHISQ.DIST.RT(X<sub>V</sub><sup>2</sup>,df)</p> <p>A p Value of less than alpha (usually set at 0.05) indicates that the Null Hypothesis stating that sphericity exists can be rejected.</p> <p>All of these calculations in Excel are shown as follows:</p> <p>The first step is to calculate k (the number of sample groups), n (the number of data points in each sample group), and degrees of freedom, df<sub>V</sub>. The degrees of freedom for John, Nagao, and Sugiura’s test is found by this formula:</p> <p><a href="http://lh3.ggpht.com/-NlJPbcmdGyE/VLK0eNygfNI/AAAAAAAA4PU/Vs2VG3Rnq1M/s1600-h/JNS_df%25255B2%25255D.png"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s degrees of freedom" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s degrees of freedom" src="http://lh5.ggpht.com/--X2RHhL88OY/VLK0enyYahI/AAAAAAAA4Pg/L0gaDgvptWc/JNS_df_thumb.png?imgmax=800" width="196" height="45" /></a> </p> <p></p> <p>These calculations are performed in Excel as follows:</p> <p><a href="http://lh4.ggpht.com/-vv_VeDk45CE/VLK0fcwgQkI/AAAAAAAA4ZU/7qrPO8rVM7Y/s1600-h/Rep_ANOVA_26_JNS_1_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s degrees of freedom Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s degrees of freedom Calculation" src="http://lh6.ggpht.com/-oTW5g_3u7hc/VLK0fzBMz-I/AAAAAAAA4Zc/d7-Ux8vdpso/Rep_ANOVA_26_JNS_1_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Test Statistic V is then calculated from the 4 Eigenvalues with the following formula:</p> <p><a href="http://lh6.ggpht.com/-nDbqK4ZLBow/VLK0gUH2xkI/AAAAAAAA4P4/Q86KWlUnfTs/s1600-h/JNS_V%25255B5%25255D.png"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Test Statistic V" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Test Statistic V" src="http://lh5.ggpht.com/-pe_3gTnQat8/VLK0h6rnNmI/AAAAAAAA4QA/dMxPyg51x8c/JNS_V_thumb%25255B1%25255D.png?imgmax=800" width="153" height="67" /></a>  </p> <p></p> <p>These calculations are performed in Excel as follows:</p> <p><a href="http://lh5.ggpht.com/-u0WdXx7MkiY/VLK0jIkyR2I/AAAAAAAA4Zk/JOZQIcvJaco/s1600-h/Rep_ANOVA_27_JNS_2_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Test Statistic V Calculations" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Test Statistic V Calculations" src="http://lh4.ggpht.com/-ZiqP4i15XTY/VLK0kH7Z1gI/AAAAAAAA4Zs/_6j5u_Nf2VY/Rep_ANOVA_27_JNS_2_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="273" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>There are critical values calculated for Test Statistic V but an easier way to perform this hypothesis test is to convert V into X<sub>V</sub><sup>2</sup> using the following formula:</p> <p><a href="http://lh5.ggpht.com/-WI-dJnGfXmA/VLK0kiiA2NI/AAAAAAAA4Z0/fAjwOh_BTn8/s1600-h/JNS__Chi-Square_Statistic%25255B4%25255D.png"><img title="Repeated-Measure ANOVA in Excel - " style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="JNS__Chi-Square_Statistic" src="http://lh3.ggpht.com/-84RNDevVvKU/VLK0lMGYJMI/AAAAAAAA4Z8/pe6apRbGu_M/JNS__Chi-Square_Statistic_thumb%25255B2%25255D.png?imgmax=800" width="400" height="34" /></a> </p> <p></p> <p>These calculations are shown in Excel as follows:</p> <p><a href="http://lh4.ggpht.com/-BTI47XJ84mo/VLK0l_weoYI/AAAAAAAA4aE/s2PvLLcqS7Q/s1600-h/Rep_ANOVA_28_JNS_3_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Chi-Square Statistic Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s Chi-Square Statistic Calculation" src="http://lh4.ggpht.com/-vux8ZO76FDY/VLK0mQocWsI/AAAAAAAA4aM/0dX70HHCUho/Rep_ANOVA_28_JNS_3_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="175" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The distribution of X<sub>V</sub><sup>2</sup> can be approximated by the Chi-Square distribution with df<sub>V</sub> degrees of freedom.</p> <p>A p Value for this hypothesis test can then be found with the following formula:</p> <p><a href="http://lh6.ggpht.com/-xjW3MkkR-7M/VLK0m_BYFhI/AAAAAAAA4Q0/HS3Smx5Mgjk/s1600-h/JNS_p%25255B2%25255D.png"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s p Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s p Value" src="http://lh4.ggpht.com/-E7csmhekDjo/VLK0nf9oT1I/AAAAAAAA4RA/CmgQtxChaGw/JNS_p_thumb.png?imgmax=800" width="240" height="21" /></a> </p> <p></p> <p>These calculations in Excel are shown as follows:</p> <p><a href="http://lh5.ggpht.com/-dlXAZKlrkiQ/VLK0n2FfQ1I/AAAAAAAA4aU/flZvr_8wSYY/s1600-h/Rep_ANOVA_29_JNS_4_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s p Value Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - John, Nagao, and Sugiura’s p Value Calculation" src="http://lh3.ggpht.com/-1RAkA6uK3kc/VLK0opli2NI/AAAAAAAA4ac/52e3yFemcyE/Rep_ANOVA_29_JNS_4_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="198" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The small p Value of this John, Nagao, and Sugiura’s Test of Sphericity indicates departure from sphericity. Usually this test is more powerful than Mauchly’s Test but it was not in this case as evidenced by Mauchly’s smaller p Value. </p> <h2>Step 5) Determine Whether Sphericity Exists</h2> <p>Sphericity exists if the Null Hypothesis for both Mauchly's test of Sphericity and John, Nagao, and Sugiura's Test of Sphericity cannot be rejected.</p> <p>The Null Hypotheses of both sphericity tests state that the covariances (the off-diagonal elements of the covariance matrix) are equal. If either test's p Value is less than alpha (usually set at 0.05), then the test's Null Hypothesis that sphericity existis can be rejected. </p> <p>If this Null Hypothesis is rejected, an adjustment should be made reducing the degrees of freedom of both between subjects and for error. The reduction in degrees of freedom will increase the final p value for the Repeated ANOVA thereby reducing the overall power of the repeated ANOVA test.</p> <p>p Value<sub>Mauchly</sub> = 0.00167</p> <p>p Value<sub>John, Nagao, and Sugiura</sub> = 0.00536 </p> <p>Both p Values are less than alpha (0.05).</p> <p>The Null Hypothesis that sphericity exists is rejected.</p> <p>The degrees of freedom for the error term and for between subjects must now be adjusted by one of two methods. The next step is to determine which of the two adjustments to apply to the degrees of freedom used to calculate the final p Value for the repeated-measure ANOVA.</p> <p>If sphericity cannot be rejected, the original p Value calculated for the repeated-measures ANOVA is the final answer. If sphericity is violated, the p value calculated for the repeated-measures ANOVA test will need to be increased (making the test less powerful in order to account for the violation of sphericity) by applying a correction to both degrees of freedom used to calculate that p Value. The two variations of this correction are discussed in the following steps.</p> <h2>Step 6) Calculate the Greenhouse-Geisser Correction in Excel</h2> <p>If the required assumption of sphericity has been violated, the repeated-measures ANOVA test become too liberal, i.e., develops a tendency to produce Type 1 errors (false positives – detecting a difference when none exists). To counteract this tendency an adjustment should be made which will increase the ANOVA test’s final p Value by an amount that corresponds to the degree to which sphericity has been violated. Increasing the p Value makes the test less powerful and therefore less likely to register a false positive.</p> <p>When sphericity is found to be violated, there are two corrections that are commonly applied that will increase the test’s p Value thereby reducing the likelihood of the test registering a false positive (a type 1 error). The two corrections are the Greenhouse-Geisser correction and the Huyhn-Feldt correction. The degree to which sphericity is violated and also the relative cost of a Type 1 versus a Type 2 error determines which of the two corrections should be applied.</p> <h3>Epsilon, Є, Equals the Index of Sphericity Between Populations</h3> <p>The index of sphericity between population data is called Epsilon, <font face="Calibri, serif">?</font>. Epsilon is a number between 0 and 1. A value of 1 indicates perfect sphericity between populations. Epsilon can only be estimated because only samples of the populations are available for analysis. The two methods of estimating the population Epsilon are Є<sub>GG</sub> (the Greenhouse-Geisser estimate of Epsilon) and Є<sub>HF</sub> (the Huyhn-Feldt estimate of Epsilon). One of these estimates of Epsilon is designated as the correction that the degrees of freedom (df<sub>between</sub> and df<sub>error</sub>) are each multiplied by when calculating the revised (increased) p Value for the repeated-measure ANOVA test to account for the test’s increased tendency to register a false positive result due to violation of data sphericity.</p> <p>As mentioned, Epsilon is a number between 0 and 1. The smaller that the estimate of Epsilon becomes, the great will be the revised p Value when the correction is applied by multiplying by degrees of freedom by Epsilon. The repeated-measures ANOVA test’s p Value is calculated in Excel as follows:</p> <p>p Value = CHISQ.DIST.RT(F Value, df<sub>between</sub>, df<sub>error</sub>)</p> <p>The correction does not have any effect on the F value. Decreasing both degrees of freedom will increase the p Value, which makes the test less likely to register a false positive (commit a Type 1 error).</p> <p>Є<sub>GG</sub>, the Greenhouse-Geisser estimate of Epsilon, is viewed as being slightly too low and therefore too conservative. The lower the estimate of Epsilon is, the greater will be the test’s p Value. The greater the test’s p Value, the more conservative the test becomes. Є<sub>GG</sub> is often used when Epsilon is estimated to be less than 0.75. The overall estimate of Epsilon is the average between Є<sub>GG</sub> and Є<sub>HF</sub>.</p> <p>Є<sub>HF</sub>, the Huyhn-Feldt estimate of Epsilon, is viewed as being slightly too high and overestimates Epsilon. Є<sub>HF</sub> is used when Epsilon is estimated to be more than 0.75.</p> <p>Є<sub>GG</sub>, the Greenhouse-Geisser estimate of Epsilon, is calculated using the following formula:</p> <p><a href="http://lh3.ggpht.com/-N3ab72XUY8E/VLK0pdp7yII/AAAAAAAA4ak/qwTzyX6Pqwg/s1600-h/Epsilon_GG%25255B4%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon" src="http://lh3.ggpht.com/-DTDseJta7XE/VLK0qrpmNoI/AAAAAAAA4as/Nq9emQ2nlDc/Epsilon_GG_thumb%25255B2%25255D.png?imgmax=800" width="400" height="29" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>k = Number of groups</p> <p>Mean <sub>Cov_Matrix_Diag</sub> = Mean of Diagonals of Covariance Matrix = Mean of Raw Data Group Variances </p> <p>Mean <sub>Cov_Matrix</sub> = Mean of Covariance Matrix</p> <p>SS <sub>Cov_Matrix</sub> = Sum of Squares of Covariance Matrix</p> <p>SS <sub>Cov_Matrix_Column_Means</sub> = Sum of Squares of Means of Covariance Matrix Rows</p> <p>These calculations are shown in Excel as follows:</p> <p><a href="http://lh3.ggpht.com/-pXbe64WseDI/VLK0sYevhHI/AAAAAAAA4a0/EME7PTano5g/s1600-h/Rep_ANOVA_29a_GG_Covariance_Matrix_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" src="http://lh6.ggpht.com/-tzo3L2W6Xck/VLK0syYY5WI/AAAAAAAA4a8/dVpxN03QBOw/Rep_ANOVA_29a_GG_Covariance_Matrix_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="307" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-dDqJrbCFVX8/VLK0tiqlqSI/AAAAAAAA4bE/0vkRm-u_1F8/s1600-h/Rep_ANOVA_29b_CC_Middle_Calculations_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" src="http://lh3.ggpht.com/-QnWfk12LQVM/VLK0uJAk9QI/AAAAAAAA4bM/xtsqBqReFho/Rep_ANOVA_29b_CC_Middle_Calculations_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="349" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-UTYeBNgDRUw/VLK0uiRmEdI/AAAAAAAA4bU/5aEEMsWa4V4/s1600-h/Rep_ANOVA_29c_GG_Final_Calculation_72_600%25255B7%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser Epsilon Calculations" src="http://lh4.ggpht.com/-fLpO3gXILqM/VLK0vDfVNzI/AAAAAAAA4bc/kVhb-o3VJxU/Rep_ANOVA_29c_GG_Final_Calculation_72_600_thumb%25255B5%25255D.jpg?imgmax=800" width="400" height="180" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Є<sub>GG</sub>, the Greenhouse-Geisser estimate of Epsilon, is calculated to be 0.415</p> <h2>Step 7) Calculate the Huyhn-Feldt Correction in Excel</h2> <p>Є<sub>HF</sub>, the Huyhn-Feldt estimate of Epsilon, is calculated using the following formula:</p> <p><a href="http://lh3.ggpht.com/-E27yzlvojrE/VLK2gfzH1dI/AAAAAAAA4bk/RzaULOIihIA/s1600-h/Epsilon_HF%25255B5%25255D.png"><img title="Repeated-Measure ANOVA in Excel - Huyhn-Feldt Epsilon" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Huyhn-Feldt Epsilon" src="http://lh5.ggpht.com/-47qhoMvIjz4/VLK2g7sHiyI/AAAAAAAA4bs/VQd8WHrpC7I/Epsilon_HF_thumb%25255B3%25255D.png?imgmax=800" width="281" height="43" /></a> </p> <p></p> <p>These calculations are shown in Excel as follows:</p> <p><a href="http://lh4.ggpht.com/-gxq5unsb0HY/VLK2hPVZrmI/AAAAAAAA4b0/JYlsvg_x4ts/s1600-h/Rep_ANOVA_29d_HF_Final_Calculations_72_600%25255B6%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Huyhn-Feldt Epsilon Calculations" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Huyhn-Feldt Epsilon Calculations" src="http://lh4.ggpht.com/-gtYc1pdaB3s/VLK2hqpHEbI/AAAAAAAA4b8/7gfZfwmQTb8/Rep_ANOVA_29d_HF_Final_Calculations_72_600_thumb%25255B4%25255D.jpg?imgmax=800" width="400" height="189" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>Є<sub>HF</sub>, the Huyhn-Feldt estimate of Epsilon, is calculated to be 0.51302.</p> <h2>Step 8) Determine Which Epsilon Correction To Apply When Sphericity Is Violated</h2> <p>The relationship between the repeated-measures ANOVA test’s power, the p Value, Є<sub>HF</sub>, and Є<sub>GG</sub> is shown in the following diagram:</p> <p><a href="http://lh4.ggpht.com/-DNVqlYLkL8o/VLK2iD1Fo2I/AAAAAAAA4cE/TbOEtTaA4rw/s1600-h/Rep_ANOVA_29e_GG_HF_Relationship_Graph_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser - Huyhn-Feldt Comparison" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - Greenhouse-Geisser - Huyhn-Feldt Comparison" src="http://lh6.ggpht.com/-rYSEji5cci4/VLK2ivkFZ-I/AAAAAAAA4cM/TgIso-QQprc/Rep_ANOVA_29e_GG_HF_Relationship_Graph_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="203" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The lower bound of the estimates of Epsilon is calculated by 1/(k-1). Note that the lower the estimate of Epsilon is, the lower becomes the test power and the greater becomes the p Value. The higher the test’s p Value becomes, the less likely it is that the test will detect a difference. The lower that the estimate of Epsilon is, the higher the p Value becomes.</p> <p><font face="Times New Roman, serif"><font style="font-size: 13pt" size="4"><b><u>Determining Which Correction To Apply</u></b></font></font></p> <p>There are two rules that can be applied to determine which of the two corrections to apply when sphericity is found to be violated.</p> <p><b>Rule 1) </b></p> <p>If Type 1 Error (False Positive) is more expensive, use the conservative (lower power) adjustment. This is the Greenhouse-Geisser adjustment, Є<sub>GG</sub>. </p> <p>If Type 2 Error (False Negative) is more expensive, use the adjustment that increases test power. This is the Huynh-Feldt adjustment, Є<sub>HF</sub>. </p> <p></p> <p><b>Rule 2) </b></p> <p>If the Estimated Epsilon (the average of Є<sub>GG</sub> and Є<sub>HF</sub>) is less than 0.75, use Є<sub>GG</sub>. If the Estimated Epsilon is greater than 0.75, use Є<sub>HF</sub>. If the Estimated Epsilon is very low and sample size is relatively large (n equals at least k + 10), MANOVA can be used in placed of Repeated-Measure ANOVA because MANOVA does not require sphericity. </p> <p>In this case, the Estimated Epsilon is approximately 0.46 and there is no mention of the cost difference between Type 1 and Type 2 errors. The Greenhouse-Geisser correction, Є<sub>GG</sub>, will therefore be used to adjust the degrees of freedom used to calculate the final p Value for this Repeated-Measure ANOVA.</p> <h2>Step 9) Apply the Correction to the Degrees of Freedom and Recalculate the p Value in Excel</h2> <p>Below is the repeated-measures ANOVA table before the correction is applied:</p> <p><a href="http://lh3.ggpht.com/-TJ8-v81i5P4/VLK2jFYk23I/AAAAAAAA4cU/zycgJiS-7jM/s1600-h/Rep_ANOVA_29f_ANOVA_Table_Pre-Correction_72_600%25255B4%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - ANOVA Table Before Correction" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - ANOVA Table Before Correction" src="http://lh3.ggpht.com/-cdfWjDYEZkQ/VLK2j4MhdRI/AAAAAAAA4cc/V_K3pYk8bKo/Rep_ANOVA_29f_ANOVA_Table_Pre-Correction_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="179" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The correction to be applied is the following: </p> <p>The p Value will be calculated by multiplying df<sub>between</sub> and df<sub>error</sub> by the Greenhouse-Geisser correction, Є<sub>GG</sub>. Note that this only affects the final p Value but not the F Value. </p> <p>Є<sub>GG</sub> = 0.415770093</p> <p>Below is the Repeated-Measure ANOVA table after the Greenhouse-Geisser Correction is applied to df<sub>between</sub> and df<sub>error</sub>.</p> <p><a href="http://lh3.ggpht.com/-zzAcGgvO3js/VLK2kTI_gyI/AAAAAAAA4ck/YSGgmw_TpQ8/s1600-h/Rep_ANOVA_29g_ANOVA_Table_Post-Correction_72_600%25255B7%25255D.jpg"><img title="Repeated-Measure ANOVA in Excel - ANOVA Table After Correction" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measure ANOVA in Excel - ANOVA Table After Correction" src="http://lh4.ggpht.com/-4YfbZ0LCUl8/VLK2lJL5vbI/AAAAAAAA4cs/IZwLK2pG-gU/Rep_ANOVA_29g_ANOVA_Table_Post-Correction_72_600_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="97" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>The correction used to compensate for the lack of sphericity reduced the power of the test to the point that a difference was not detected, if alpha is set to 0.05.</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com2tag:blogger.com,1999:blog-3568555666281177719.post-50916616868165336482015-01-11T07:50:00.001-08:002015-03-24T19:41:24.651-07:00Effect Size For Repeated-Measures ANOVA in Excel<h1>Effect Size For Repeated- <br /> <br />Measure ANOVA in Excel</h1> <p>This is one of the following four articles on Repeated-Measures ANOVA</p> <p><a href="http://blog.excelmasterseries.com/2015/01/single-factor-repeated-measures-anova.html">Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/sphericity-testing-in-9-steps-for.html">Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/effect-size-for-repeated-measures-anova.html">Effect Size For Repeated-Measures ANOVA in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html">Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel</a></p> <h2>Overview of Effect Size</h2> <p>Effect size is a way of describing how effectively the method of data grouping allows those groups to be differentiated. A simple example of a grouping method that would create easily differentiated groups versus one that does not is the following.</p> <p>Imagine a large random sample of height measurements of adults of the same age from a single country. If those heights were grouped according to gender, the groups would be easy to differentiate because the mean male height would be significantly different than the mean female height. If those heights were instead grouped according to the region where each person lived, the groups would be much harder to differentiate because there would not be significant difference between the means and variances of heights from different regions.</p> <p>Because the various measures of effect size indicate how effectively the grouping method makes the groups easy to differentiate from each other, the magnitude of effect size tells how large of a sample must be taken to achieve statistical significance. A small effect can become significant if a larger enough sample is taken. A large effect might not achieve statistical significance if the sample size is too small.</p> <p><b><u>The Most Common Measure of Effect Size</u></b></p> <p>The most common measure of effect size of single-factor ANOVA is the following:</p> <p>η<sup>2</sup> – eta squared </p> <p>(Greek letter “eta” rhymes with “beta”)</p> <h2>Eta Square (η<sup>2</sup>)</h2> <p>Eta square quantifies the percentage of variance in the dependent variable (the variable that is measured and placed into groups) that is explained by the independent variable (the method of grouping). If eta squared = 0.35, then 35 percent of the variance associated with the dependent variable is attributed to the independent variable (the method of grouping).</p> <p>Eta square provides an overestimate (a positively-biased estimate) of the explained variance of the population from which the sample was drawn because eta squared estimates only the effect size on the sample. The effect size on the sample will be larger than the effect size on the population. This bias grows smaller is the sample size grows larger.</p> <p>Eta square is affected by the number and size of the other effects.</p> <p>η<sup>2</sup> = SS <sub>Between_Groups</sub> / SS<sub>Total</sub> These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.</p> <p>Magnitudes of eta-squared are generally classified exactly as magnitudes of r<sup>2</sup> (the coefficient of determination) are as follows: = 0.01 is considered a small effect. = 0.06 is considered a medium effect. = 0.14 is considered a large effect. Small, medium, and large are relative terms. A large effect is easily discernible but a small effect is not.</p> <p><strong><u>Calculating Eta Squared (η<sup>2</sup>) in Excel</u></strong></p> <p>Eta squared is calculated with the formula</p> <p>η<sup>2</sup> = SS<sub>Between_Groups</sub> / SS<sub>Total</sub> </p> <p>and is implemented in Excel on the data set as follows:</p> <p><a href="http://lh3.ggpht.com/-kqXO3HKTVTg/VLKbqXoLjEI/AAAAAAAA35Q/0qs3Dgh27bQ/s1600-h/Rep_ANOVA_29j_eta-square_Generic_Example_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA - Effect Size Generic Example for Single-Factor ANOVA" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA - Effect Size Generic Example for Single-Factor ANOVA" src="http://lh5.ggpht.com/-rDTf97FL9y0/VLKbrMuSwBI/AAAAAAAA35Y/ON44StDbSUk/Rep_ANOVA_29j_eta-square_Generic_Example_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="299" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>An eta-squared value of 0.104 would be classified as a medium-size effect.</p> <p>Magnitudes of eta-squared are generally classified exactly as magnitudes of r<sup>2</sup> (the coefficient of determination) are as follows: = 0.01 is considered a small effect. = 0.06 is considered a medium effect. = 0.14 is considered a large effect. Small, medium, and large are relative terms. A large effect is easily discernible but a small effect is not.</p> <p>The raw data matrix for the repeated-measures ANOVA example used in these blog articles is the following:</p> <p><a href="http://lh3.ggpht.com/-Hih3-aCAHxs/VLKbrteykNI/AAAAAAAA35g/s0fFVFVSTmE/s1600-h/Rep_ANOVA_31_Friedman_Raw_Data_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA - Raw Data Matrix" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA - Raw Data Matrix" src="http://lh3.ggpht.com/-RL91oHAPsUI/VLKbsb1PVWI/AAAAAAAA35o/kQUSkXk06Kg/Rep_ANOVA_31_Friedman_Raw_Data_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="177" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>If single-factor ANOVA were performed on this data and eta-squared was calculated from this data, the result would be the following:</p> <p><a href="http://lh4.ggpht.com/-8bpv0m2yxOo/VLKbtOgFmcI/AAAAAAAA35w/-u6CROihd4A/s1600-h/Rep_ANOVA_29h_Eta_Square_One-Way_ANOVA_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA - Effect Size Performed on Single-factor ANOVA of That Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA - Effect Size Performed on Single-factor ANOVA of That Data" src="http://lh5.ggpht.com/-BSdO90hYbG0/VLKbtkvQk2I/AAAAAAAA354/FtAVji9IbV0/Rep_ANOVA_29h_Eta_Square_One-Way_ANOVA_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="396" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p>After conducting a repeated-measure ANOVA test and then applying a correction for lack of sphericity, eta-squared is calculated in the same fashion as follows:</p> <p><a href="http://lh3.ggpht.com/-IKZqGHA1LOU/VLKbuob6Z5I/AAAAAAAA36A/SZCqmV7doYs/s1600-h/Rep_ANOVA_29i_Eta_Square_Rep_ANOVA_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA - Effect Size Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA - Effect Size Calculation" src="http://lh4.ggpht.com/-NCsVrXuRcg8/VLKbvJTtQ6I/AAAAAAAA36I/9f6fmUBLpog/Rep_ANOVA_29i_Eta_Square_Rep_ANOVA_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="255" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <p>The variance attributed to the subjects, SS<sub>subjects</sub>, has been entirely removed from the Repeated-Measure ANOVA test and should not added back into the total Variance, SS<sub>total</sub>, to calculate eta square.</p> <p>This represents a very large effect size. This means that the difference between the group means is very noticeable if the variation attributed to the subjects is removed and the only remaining variance is between groups and statistical noise (SS<sub>error</sub>).</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com0tag:blogger.com,1999:blog-3568555666281177719.post-38205543101116959322015-01-10T20:36:00.001-08:002015-03-24T19:44:07.845-07:00Friedman Test For Repeated-Measures ANOVA in 3 Steps in Excel<h1>Friedman Test For <br /> <br />Repeated-Measure ANOVA <br /> <br />in 3 Steps in Excel</h1> <p>This is one of the following four articles on Repeated-Measures ANOVA</p> <p><a href="http://blog.excelmasterseries.com/2015/01/single-factor-repeated-measures-anova.html">Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/sphericity-testing-in-9-steps-for.html">Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/effect-size-for-repeated-measures-anova.html">Effect Size For Repeated-Measures ANOVA in Excel</a></p> <p><a href="http://blog.excelmasterseries.com/2015/01/friedman-test-for-repeated-measures.html">Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel</a></p> <h2>Overview of Friedman Test</h2> <p>All types of ANOVA tests require that the any data sample that will undergo an individual F Test be normally distributed. The reason for the ANOVA requirement of sample group normality is that the F Test is particularly susceptible to deviation from normality. As with many other types of ANOVA, repeated-measure ANOVA has an alternative nonparametric test that can be substituted if sample group normality cannot be confirmed. This is the Friedman Test.</p> <p>The Friedman Test is a nonparametric alternative for single-factor, repeated-measures ANOVA when sample groups are not normally distributed. The Friedman Test is also an alternative for single-factor, repeated-measures ANOVA when the dependent variable is ordinal instead of continuous as required by ANOVA. An ordinal variable is a categorical variable whose value indicates a rank or order among other data points. The Likert scale is an example of an ordinal data scale. The Likert scale is often associated with survey in which a respondent rates something with ratings such as “good,” “very good,” and “excellent.”</p> <p>The Friedman Test somewhat resembles the Kruskal-Wallis test that is used as a nonparametric substitute test for single-factor ANOVA expect that the Friedman Test has fewer required assumption. </p> <p>The Kruskal-Wallis Test requires that sample groups have similar distributions. A histogram of each sample group will quickly show whether that condition has been met. The Friedman Test does not have that required assumption.</p> <h2>Friedman Test Required <br /> <br />Assumptions</h2> <p>The Freidman Test has only the following three assumptions:</p> <p><font color="#0000cc"><font face="Arial, serif"><b>1) Sample Data Are Continuous or Ordinal</b></font></font> Sample group data (the dependent variable’s measured value) can be ratio or interval data, which are the two major types of continuous data. sample data can be ordinal which is a type of categorical data in which the data labels indicate a ranking or order of the data. Sample group data cannot be nominal data, which is a type of categorical data in which order has no meaning because data labeling does not indicate any data order.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>2) Independent Variable is Categorical</b></font></font> The determinant of which group each data observation belongs to is a categorical, independent variable. Repeated-measures ANOVA uses a single categorical variable that has at least two levels. All data observations associated with each variable level represent a unique data group and will occupy a separate column on the Excel worksheet.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>3) Sample Data Are Randomly Sampled From a Population</b></font></font> </p> <p><u><strong>The Friedman test procedure can be summarized as follows:</strong></u></p> <p><font color="#0000cc"><font face="Arial, serif"><b>1) Samples are placed next to each other and ranked.</b></font></font> Repeated-measures ANOVA consists of sample group taken from the same subjects at different times or in different conditions. The data points that were taken from one subject are ranked only against each other. This is very different than the Kruskal-Wallis Test which ranks each data point against all other data points in all samples.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>2) The rankings of all points in each sample group are summed.</b></font></font> Each sum of rankings is then squared. Each of these terms is then added together to produce a sum of the squares of the sum of rankings for each group.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>3) Test Statistic F is calculated from this information.</b></font></font></p> <p><font color="#0000cc"><font face="Arial, serif"><b>4) The significance of Test Statistic F is determined.</b></font></font> If the Friedman test is performed on small groups of samples, Test Statistic F is compared to critical F Values in a table. If samples groups are larger, the distribution of Test Statistic F can be approximated by the Chi-Square distribution with k-1 degrees of freedom. A p Value can be calculated in Excel with the following formula:</p> <p>p Value = CHISQ.DIST.RT(Test Statistic F, k - 1)</p> <p>The Friedman Test result is determined to be significant if the p Value is smaller than the specified alpha, which is often set at 0.05.</p> <p>The Null Hypothesis for the Friedman tests states that the sums of ranks for all sample groups are the same. This is a one-tailed test due to the Chi-Square distribution which states that the sum of ranks of at least one sample group is greater than the sum of ranks of any other sample group.</p> <p>If the test result is found to be significant, the Null Hypothesis can be rejected.</p> <p>Here is a detailed description of each step in Excel:</p> <h2>Step 1 - Rank the Values of Data <br /> <br />Points For Each Subject</h2> <p>Here is the raw data matrix:</p> <p><a href="http://lh6.ggpht.com/-1QulZk88ODU/VLH9lzMRMQI/AAAAAAAA32c/Hd2ozdLy6Ss/s1600-h/Rep_ANOVA_31_Friedman_Raw_Data_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA Friedman Test - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - Raw Data" src="http://lh5.ggpht.com/-fvErO36Z2aI/VLH9msz5EqI/AAAAAAAA32k/fXaW7OuObFk/Rep_ANOVA_31_Friedman_Raw_Data_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="177" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <p>Note that data values are ranked only against other data values taken from the same subject. Repeated-measures ANOVA takes measurements from the same group of subjects at different time intervals or in different conditions. Ranking must only be performed on data values taken from the same subjects.</p> <p>Data values that are the same are assigned the average of the rankings that they occupy. The Excel command RANK.AVG() should be used to create those average rankings as follows:</p> <p><a href="http://lh4.ggpht.com/-VAMLoK6GX90/VLH9ndVlFvI/AAAAAAAA32s/KLFNqYpiXts/s1600-h/Rep_ANOVA_32_Friedman_Ranking_Data_72_600%25255B7%25255D.jpg"><img title="Repeated-Measures ANOVA Friedman Test - Ranking Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - Ranking Data" src="http://lh5.ggpht.com/-HXD1hdsdxN0/VLH9oFzzE9I/AAAAAAAA320/ZL6LH_DZo_4/Rep_ANOVA_32_Friedman_Ranking_Data_72_600_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="105" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <h2>Step 2 - Calculate Σ(R<sup>2</sup>)</h2> <p>The sum of the squares of the sum of each sample group’s rankings is calculated in Excel as shown below:</p> <p><a href="http://lh5.ggpht.com/-bQbhPUjrzSs/VLH9ozi0bDI/AAAAAAAA328/BWpRJ-EMYA8/s1600-h/Rep_ANOVA_33_Friedman_Sum_R_Squared_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA Friedman Test - Sum of R Square" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - Sum of R Square" src="http://lh6.ggpht.com/-YE05FbY6twg/VLH9ppRmPMI/AAAAAAAA33E/83uI-kof1As/Rep_ANOVA_33_Friedman_Sum_R_Squared_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="285" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <h2>Step 3 - Calculate Test Statistic F</h2> <p>The formula for the Friedman Test Statistic F when the Friedman Test is used as a nonparametric alternative to single-factor repeated-measure ANOVA is as follows:</p> <p><a href="http://lh5.ggpht.com/--hjFC-rst_I/VLH9qWm-G8I/AAAAAAAA33M/9lGQbSoyt1M/s1600-h/Friedman_Test_Statistic_Repeated_Measure_ANOVA%25255B4%25255D.png"><img title="Repeated-Measures ANOVA Friedman Test - Test Statistic Formula" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - Test Statistic Formula" src="http://lh5.ggpht.com/-0vkPlU0S1wE/VLH9q0LUWZI/AAAAAAAA33U/sQPorYiitWg/Friedman_Test_Statistic_Repeated_Measure_ANOVA_thumb%25255B2%25255D.png?imgmax=800" width="400" height="74" /></a> </p> <p></p> <p>These calculations are performed in Excel as follows:</p> <p><a href="http://lh6.ggpht.com/-ZR10LG7Q_O8/VLH9rnNq5wI/AAAAAAAA33c/qQRjWERUY2A/s1600-h/Rep_ANOVA_34_Friedman_Calculate_F_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA Friedman Test - F Calculation" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - F Calculation" src="http://lh5.ggpht.com/-vaIlZCDrm8s/VLH9sHZrhYI/AAAAAAAA33k/RegVXj1Jpw4/Rep_ANOVA_34_Friedman_Calculate_F_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="249" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <h2>Step 4 - Determine if the Test <br /> <br />Result Is Significant</h2> <p>The distribution of F can be approximated by the Chi-Square distribution with (k-1) degrees of freedom if either k = 5 OR n > 15. If both of these two conditions is not met, the small-sample procedure is used to determine the test result is significant.</p> <p>The test result is deemed significant if the small-sample procedure indicates that Test Statistic F is greater than the critical F value that will be looked up on a chart.</p> <p>If k = 5 OR n > 15 the large-sample procedure will be used. This procedure calculates a p value. If the p Value is smaller than the designated alpha (usually set at 0.05), the test result is deemed to be significant. </p> <p>A significant test result indicates that the Friedman Test’s Null Hypothesis can be rejected.</p> <p>The Null Hypothesis for the Friedman tests states that the sums of ranks for all sample groups are the same. </p> <h3>Small-Sample Procedure</h3> <p>If either of these two conditions does not occur, Test Statistic F must be compared with the following table of Critical F Values for k = 3 and 4.</p> <p><a href="http://lh6.ggpht.com/-1B8YfkGaQ_I/VLH9s5Rdx-I/AAAAAAAA33s/6pyIJ-lf0L4/s1600-h/Rep_ANOVA_35_Friedman_Critical_Values_72_600%25255B4%25255D.jpg"><img title="Repeated-Measures ANOVA Friedman Test - Small Sample Critical Values" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="Repeated-Measures ANOVA Friedman Test - Small Sample Critical Values" src="http://lh3.ggpht.com/-xhphdpTZzJ4/VLH9t3Zq9SI/AAAAAAAA330/KIPy2NGSbEU/Rep_ANOVA_35_Friedman_Critical_Values_72_600_thumb%25255B2%25255D.jpg?imgmax=800" width="322" height="480" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p></p> <p></p> <p>F = 7.98 </p> <p>n = 5</p> <p>k = 4</p> <p>a = 0.05</p> <p>Critical F = 7.800</p> <p>The test result is determined to be significant if the Test Statistic F is greater than the Critical F value for the respective n, k, and a. </p> <p>The Friedman test therefore does indicate a difference between the average ranks of each group. </p> <p>It should be noted that the sample size is very small. Small sample size greatly reduces the power of the Friedman test.</p> <h3>Large-Sample Procedure</h3> <p>If either k = 5 OR n > 15, then the distribution of test Statistic F can be approximated by the Chi-Square distribution with (k-1) degrees of freedom. </p> <p>If that is the case, the test's p Value can be determined by the following Excel equation: </p> <p>p Value = CHISQ.DIST.RT(Test Statistic F, k - 1) </p> <p>p Value = 0.050331098 =CHISQ.DIST.RT(7.98,4-1)</p> <p>This result is very close to the result found by comparing the Test Statistic to the Critical F Value. Both methods show that the test has just barely achieved significance. This result should not be viewed with great confidence because such a small sample size greatly reduces the power of this already relatively weak nonparametric test. </p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com8tag:blogger.com,1999:blog-3568555666281177719.post-73711075289432240172014-12-12T12:45:00.001-08:002015-03-24T19:46:54.118-07:00Single-Factor ANCOVA in 8 Steps in Excel<h1>Single-Factor ANCOVA <br /> <br />in Excel in 8 Steps</h1> <h2>Overview</h2> <p>Just like single-factor ANOVA, ANCOVA is used to determine if there is a real difference between the means of two or more sample groups of continuous data. Like ANOVA, ANCOVA answers the following question: Is it likely that all sample groups came from the same population? Unlike ANOVA, ANCOVA allows the researcher to remove the affects of an outside, unrelated factor or variable that might cause ANOVA to produce an incorrect result.</p> <p>Suppose that a group of boys were the subjects of an ANOVA test to determine if the factor of height had an effect on size of a boy’s vocabulary. ANOVA would very likely indicate that the factor of height has a significant affect on the size of a boy’s vocabulary. This is particularly true if the boys are sampled over a wide range of ages. Older boys will generally be taller and have a larger vocabulary than younger boys. The causal factor for increased vocabulary would intuitively be age, not height. If all boys were sampled from the same age group, then height should not be a factor in the size of a boy’s vocabulary. The natural corollary to this statement is that removing the effects of age would allow a researcher to much more accurately determine if height has a significant effect on vocabulary size.</p> <p>That is exactly what ANCOVA does. ANCOVA is used in place of single-factor ANOVA if a third “confounding” variable is suspected of affecting the measured values of the data points in each of the ANOVA groups. Single-factor ANOVA is considered to have two variables; the independent variable is a categorical variable that separates data points into different groups and the dependent variable is the continuous scale that is used to measure the data values. ANCOVA removes that affect of a third “confounding” variable on the measured data values of the Dependent variable.</p> <h2>ANCOVA = Analysis of Covariance</h2> <p>ANCOVA, Analysis of Covariance, can be used in place of single-factor ANOVA to remove the effects of an outside factor that might be confounding the results of the ANOVA test. The outside factor is a third variable is called a "Covariate," a “Covariate Variable,” a “confounding variable,” or a “nuisance variable.” ANCOVA is essentially a single-factor ANOVA performed after the variance attributed to the third variable has been statistically removed.</p> <p>Single-factor ANOVA analyzes sample groupings of objects that described by two variables. The variable that designates the group into which the object is assigned is a categorical variable and is referred to in ANOVA as the independent variable. Single-factor ANOVA is used to determine whether the factor described by the variable has a significant effect on data values, specifically whether samples grouped according to different levels of the factor (different values of the categorical variable) have significantly different means.</p> <p>The other variable of single-factor ANOVA is the continuous variable used as the measurement scale for all data points in the sample groups. This variable is known as the dependent variable.</p> <p>If an additional factor is known to affect these measured values of the data points, this factor is designated as the third variable and is the Covariate variable. The Covariate Variable must be a continuous variable and must have a known value for each sample data point. ANCOVA performs the same function and calculations of single-factor ANOVA after removing the variance attributed to the Covariate factor or variable. </p> <h2>Null and Alternative Hypotheses for ANCOVA</h2> <p>The Null Hypothesis for ANCOVA is exactly like that of single-factor ANOVA and states that the sample groups <i>ALL</i> come from the same population. This would be written as follows:</p> <p>Null Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>1</b></sub> = µ<sub><b>2</b></sub> = … = µ<sub><b>k</b></sub> (k equals the number of sample groups)</p> <p>Note that Null Hypothesis is not referring to the sample means, s<sub><b>1</b></sub> , s<sub><b>2</b></sub> , … , s<sub><b>k</b></sub>, but to the population means, µ<sub><b>1</b></sub> , µ<sub><b>2</b></sub> , … , µ<sub><b>k</b></sub>.</p> <p>The Alternative Hypothesis for ANCOVA states that <i>at least one</i> sample group is likely to have come from a different population. Like single-Factor ANOVA, ANCOVA does not clarify which groups are different or how large any of the differences between the groups are. This Alternative Hypothesis only states whether <i>at least one</i> sample group is likely to have come from a different population.</p> <p>Alternative Hypothesis = H<sub><b>0</b></sub>: µ<sub><b>i</b></sub> ? µ<sub><b>j</b></sub> for some i and j</p> <h2>ANCOVA Example in Excel</h2> <p>A company is attempting to determine if either of two sales-training methods produces a significantly better increase in a salesperson’s monthly sales. The company simultaneously sent one group of ten salespeople through one of the training programs and sent another group of ten salespeople through the other training program. The change in average monthly sales for each salesperson after completion of the program was recorded.</p> <p>A single-factor ANOVA of the sales increases for each of the two groups did not evidence any significant different between the effectiveness of the two training programs in producing sales increases. This will be shown within the example.</p> <p>A senior manager of the company noticed that the salespeople who had more years of experience seemed to have had higher sales increases after the training as a whole than salespeople with less experience. The senior manager speculated that a salesperson’s amount of experience had a direct effect on that salesperson’s ability to benefit from additional training program.</p> <p>The senior manager wanted to analyze the post-training sales results to determine if one training program significantly outperformed the other but also wanted to remove the effects of the amount of previous experience which might make a difference the salespeople’s ability to implement the training and increase sales.</p> <p>ANCOVA is the correct statistical tool for this task. This example will demonstrate ANCOVA detecting a difference in sample groups that single-factor ANOVA could not. </p> <p>Below is the sample data:</p> <p>  <a href="http://lh3.ggpht.com/-z87QxU2tbq8/VItuewqczNI/AAAAAAAA3u0/29Lo9S1bi7E/s1600-h/ANCOVA_1_Raw_Data_RGB_72%25255B12%25255D.jpg"><img title="ANCOVA in Excel - Raw Data" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Raw Data" src="http://lh5.ggpht.com/-Pnya6Mrk_zA/VItufkQnqnI/AAAAAAAA3u8/U5nnB8vwWdc/ANCOVA_1_Raw_Data_RGB_72_thumb%25255B6%25255D.jpg?imgmax=800" width="400" height="307" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Each of the subjects who underwent the training can be described by the following three variables used in ANCOVA:</p> <p>Independent Variable – This is the categorical variable of Training Method type. The overall objective of the ANCOVA is to determine if levels of the factor of training program type produced significantly different sales increases.</p> <p>Dependent Variable – This continuous variable is the monthly sales increase for each salesperson who underwent either of the two training programs. Note that values of this variable are colored black.</p> <p>Covariate Variable – This continuous variable is the number of years of prior experience that each salesperson had prior to undergoing the training for either of the two programs. It is speculated that this variable might have significantly affected the salespeople’s abilities to implement the training program and increase sales. One of the objectives of this ANCOVA is to remove effects of this variable so that the results will more accurately determine whether one training program is more effective than the other. Note that values of this variable are colored red.</p> <p>ANCOVA can be performed in Excel in 8 steps. Before performing these steps, ANCOVA’s required assumptions will be listed below as follows: <br /></p> <h3>ANCOVA’s Required Assumptions</h3> <h4>ANCOVA Has the Following Same Required Assumptions as Single-Factor ANOVA</h4> <p>Like single-factor ANOVA, ANCOVA has the following six required assumptions whose validity should be confirmed before this test is applied. The six required assumptions are the following:</p> <p><font color="#0000cc"><font face="Arial, serif"><b>1) Independence of Sample Group Data</b></font></font> Sample groups must be differentiated in such a way that there can be no cross-over of data between sample groups. No data observation in any sample group could have been legitimately placed in another sample group. No data observation affects the value of another data observation in the same group or in a different group. This is verified by an examination of the test procedure.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>2) Sample Data Are Continuous</b></font></font> Sample group data (the dependent variable’s measured value) can be ratio or interval data, which are the two major types of continuous data. Sample group data cannot be nominal or ordinal data, which are the two major types of categorical data.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>3) Independent Variable is Categorical</b></font></font> The determinant of which group each data observation belongs to is a categorical, independent variable. ANCOVA uses a single categorical variable that has at least two levels. All data observations associated with each variable level represent a unique data group and will occupy a separate column on the Excel worksheet.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>4) Extreme Outliers Removed If Necessary</b></font></font> ANCOVA is a parametric test that relies upon calculation of the means of sample groups. Extreme outliers can skew the calculation of the mean. Outliers should be identified and evaluated for removal in all sample groups. Occasional outliers are to be expected in normally-distributed data but all outliers should be evaluated to determine whether their inclusion will produce a less representative result of the overall data than their exclusion.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>5) Normally-Distributed Data In All Sample Groups</b></font></font> ANCOVA is a parametric test having the required assumption the dependent-variable data from each sample group come from a normally-distributed population. Each sample group’s dependent-variable data should be tested for normality. Normality testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that normality tests will provide accurate results. Like single-factor ANOVA, ANCOVA is relatively robust to minor deviation from sample group normality. There are a number of different ways of testing sample groups for normality. The simplest way is to create a histogram of the data in each sample group. Each histogram should be bell-shaped, which would indicate that more points in each data sample were close to the sample's mean than away from it. A normal probability plot for each sample group could also be constructed. A number of other normality hypothesis tests can also be applied to each data sample to determine whether to reject each test's Null Hypothesis that the sample group is normally distributed. The more well-known of these tests include the Kolmogorov-Smirnov test, the Anderson-Darling test, and  the Shapiro-Wilk test. The Shapiro-Wilk test is considered the most powerful of those three is applied most often. The following link leads to step-by-step instructions in this blog on how to perform the Shapiro-Wilk test in Excel on data samples for single-factor ANOVA: <a href="http://blog.excelmasterseries.com/2014/05/shapiro-wilk-normality-test-in-excel_29.html" target="_blank">http://blog.excelmasterseries.com/2014/05/shapiro-wilk-normality-test-in-excel_29.html</a></p> <p><font color="#0000cc"><font face="Arial, serif"><b>6) Relatively Similar Variances In All Sample Groups</b></font></font> ANCOVA requires that sample groups are obtained from populations that have similar variances. This requirement is often worded to state that the populations must have equal variances. The variances do not have to be exactly equal but do have to be similar enough so the variance testing of the sample groups will not detect significant differences. The variances of sample groups are considered similar if no sample group variance is more than twice as large as the variance of another sample group. Variance testing becomes significantly less powerful (accurate) when a group’s size fall below 20. An effort should be made to obtain group sizes that exceed 20 to ensure that variance tests will provide accurate results. Two common hypothesis tests used to determine whether sample variances are similar are Levene's test and the Brown-Forsythe test. Here is a link leading to another article in this blog showing exactly how to perform both tests in Excel on sample groups for single-factor ANOVA:   <a href="http://blog.excelmasterseries.com/2014/05/levenes-and-brown-forsythe-tests-in_29.html" target="_blank">http://blog.excelmasterseries.com/2014/05/levenes-and-brown-forsythe-tests-in_29.html</a></p> <h4>ANCOVA Has the Following Additional Required Assumptions</h4> <p><font color="#0000cc"><font face="Arial, serif"><b>7) Covariate Data Are Continuous</b></font></font> Covariate values at each data point can be ratio or interval data, which are the two major types of continuous data.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>8) Covariate Data Have a Linear Relationship With Dependent Variable Data</b></font></font> This can be quickly observed by creating a scatterplot of Covariate/Dependent data points. Linearity or nonlinearity between the two variables will be quickly observable on the scatterplot.</p> <p><font color="#0000cc"><font face="Arial, serif"><b>9) The Least-Squares Lines For Covariate/Dependent Variable Data Have Similar Slopes For Each Sample Group</b></font></font> Once again this can be quickly observed on a scatterplot of Covariate/Dependent data points. A least-squares line for data points from one sample group (one of the training methods) can be compared with the least-square line for data points of the other sample group. Least-squares lines for data from different sample groups should have similar slopes. This ensures that the categorical variable (the training method used in this case) did not have an effect on the relationship between the Covariate and Dependent data.</p> <p>The overall purpose of this chapter is to demonstrate the differences between performing single-factor ANOVA and performing ANCOVA. For this reason we will forego validation of the above required assumptions. This section of this manual covering Single-Factor ANOVA provides detailed discussion on how to perform normality and variance testing of ANOVA sample groups. These same methods can be directly applied to ANCOVA sample groups for the dependent variable.</p> <p>A scatterplot of Covariate/Dependent data pairs can be quickly created in Excel to validate assumptions 8 and 9.</p> <p><a href="http://lh6.ggpht.com/-iyrdfb5aZzA/VItTp5p70mI/AAAAAAAA3vE/lrW98k8m9XE/s1600-h/ANCOVA_15_Scatterplot_1_RGB_72%25255B2%25255D.jpg"><img title="ANCOVA in Excel - Scatterplot for Method 1" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Scatterplot for Method 1" src="http://lh4.ggpht.com/-YBbzUmB5bS0/VItTqKcgbVI/AAAAAAAA3vM/Xk9tXLhFHnY/ANCOVA_15_Scatterplot_1_RGB_72_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="219" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh6.ggpht.com/-1lHeb4ikB74/VItTq2VH4UI/AAAAAAAA3vU/7JXtKPiPeHo/s1600-h/ANCOVA_15_Scatterplot_2_RGB_72%25255B2%25255D.jpg"><img title="ANCOVA in Excel - Scatterplot for Method 2" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Scatterplot for Method 2" src="http://lh3.ggpht.com/-gVf61_KaMwU/VItTrYIqQkI/AAAAAAAA3vc/mdMuZHIcKXc/ANCOVA_15_Scatterplot_2_RGB_72_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="197" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The slopes of both least squares lines are similar enough to verify that the categorical variable (the type of training method) does not affect the relationship between the Covariate Variable (X – Prior Years of Experience) and the Dependent Variable (Y – Monthly Sales Increase)</p> <p>When the required assumptions of ANCOVA have been validated, ANCOVA can be performed in Excel as follows:</p> <h3><u>ANCOVA in Excel - The 8 Steps </u></h3> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Run Single-Factor ANOVA on the Dependent Variable Data</font></b></font></font></font></p> <p>Run Excel Single-Factor ANOVA on the Y (dependent variable) data in order to obtain SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(Y)</b></font></font></sub></font>, SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(Y)</b></font></font></sub></font>, and SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>between(Y)</b></font></font></sub></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Run Single-Factor ANOVA on the Covariate Variable Data</font></b></font></font></font></p> <p>Run Excel Single-Factor ANOVA on the X (covariate) data in order to obtain SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(X)</b></font></font></sub></font> and SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(X)</b></font></font></sub></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Calculate r</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">total</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><sup><font size="4"><span style="font-weight: normal">2</span></font></sup></font></font></font></p> <p>Calculate r<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font><sup><b>2</b></sup> for Y<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font>. r<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font><sup><b>2</b></sup> is derived from r<sub>total</sub>, which is the overall correlation between X and Y</p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">total(Y)</font></sub></b></font></font></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 5) Calculate r</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">within</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><sup><font size="4"><span style="font-weight: normal">2</span></font></sup></font></font></font></p> <p>Obtain r<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font><sup><b>2</b></sup> for Y<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font> by first calculating SC<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 6)</font></b></font></font></font> <font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">within(Y)</font></sub></b></font></font></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 7)</font></b></font></font></font> <font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">between(Y)</font></sub></b></font></font></font></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 8) Calculate the p Value</font></b></font></font></font></p> <p>Arrange all of the above data on the ANOVA Excel output table and calculate the new p Value</p> <p>Here is a detailed description of the performance of each step</p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 1) Run Single-Factor ANOVA on the Dependent Variable Data</font></b></font></font></font></p> <p>Run Excel Single-Factor ANOVA on the Y (Dependent Variable) data in order to obtain SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(Y)</b></font></font></sub></font>, SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(Y)</b></font></font></sub></font>, and SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>between(Y)</b></font></font></sub></font></p> <p>The first step is to arrange the Dependent Variable data with each sample group in a different but contiguous (touching) column as shown. After the data is correctly arranged on the Excel worksheet, use the Excel data analysis tool <b>ANOVA: Single-Factor</b> to perform single-factor ANOVA on the two sample groups of Dependent data as follows:</p> <p><a href="http://lh4.ggpht.com/-NlVDo8UEF30/VItTr8OdIqI/AAAAAAAA3vk/7Xc6Adu3rFY/s1600-h/ANCOVA_2_Arranging_Data_RGB_72g%252520-%252520Copy%25255B3%25255D.jpg"><img title="ANCOVA in Excel - Preparing Dependent Variable Data For ANOVA" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Preparing Dependent Variable Data For ANOVA" src="http://lh6.ggpht.com/-pWoxyPmjUWk/VItTsVL5JgI/AAAAAAAA3vs/PTI6_foSgjw/ANCOVA_2_Arranging_Data_RGB_72g%252520-%252520Copy_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="212" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><a href="http://lh4.ggpht.com/-XvZ0Kh3VrYU/VItTs3HpRXI/AAAAAAAA3v0/O62yYFWFXlc/s1600-h/ANCOVA_3_1-Way_ANOVA_1_RGB_72.jpg"><img title="ANCOVA in Excel - ANOVA Output For Dependent Variable" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - ANOVA Output For Dependent Variable" src="http://lh5.ggpht.com/-KXAz-D791yE/VItTtSLQozI/AAAAAAAA3v8/HQK9OgtQC1g/ANCOVA_3_1-Way_ANOVA_1_RGB_72_thumb.jpg?imgmax=800" width="400" height="294" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>between(Y)</b></font></font></sub></font> = 6.05</p> <p>SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(Y)</b></font></font></sub></font> = 662.5</p> <p>SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(Y)</b></font></font></sub></font> = 668.55</p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 2) Run Single-Factor ANOVA on the Covariate Variable Data</font></b></font></font></font></p> <p>Run Excel Single-Factor ANOVA on the X (covariate) data in order to obtain SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(X)</b></font></font></sub></font> and SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(X)</b></font></font></sub></font></p> <p>The first step is to arrange the Covariate Variable data with each sample group in a different but contiguous (touching) column as shown. After the data is correctly arranged on the Excel worksheet, use the Excel data analysis tool <b>ANOVA: Single-Factor</b> to perform single-factor ANOVA on the two sample groups of Covariate data as follows:</p> <p><a href="http://lh4.ggpht.com/-KLBX01nZQaU/VItTt0U9JNI/AAAAAAAA3wE/tD01NQJd8RM/s1600-h/ANCOVA_4_Arranging_Data_2_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Preparing Covariate Variable Data For ANOVA" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Preparing Covariate Variable Data For ANOVA" src="http://lh3.ggpht.com/-dxk0mfMc4S0/VItTuXOW7gI/AAAAAAAA3wM/PqnW18CQk5o/ANCOVA_4_Arranging_Data_2_RGB_72_thumb.jpg?imgmax=800" width="400" height="217" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh3.ggpht.com/-zpcgS9s5WTk/VItTvHn3N1I/AAAAAAAA3wU/Fek0_4rwjbA/s1600-h/ANCOVA_5_1-Way_ANOVA_2_RGB_72.jpg"><img title="ANCOVA in Excel - ANOVA Output For Covariate Variable" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - ANOVA Output For Covariate Variable" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRZaluqqdB7Qfm8immewRf5xjZZ1DSYVyySztaAStOwU9E3BtqNfHEAhiCB10vU_7dcXnGAOJtUIJycOxDFfqYwl3e3Lucdrw5NtNxOmcuD7brfHGZMGMXD3W5dcOKVcgCoIqpGMiWmoY-/?imgmax=800" width="400" height="369" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within(X)</b></font></font></sub></font> = 788.9</p> <p>SS<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total(X)</b></font></font></sub></font> = 908.5</p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 3) Calculate R</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">total</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><sup><font size="4"><span style="font-weight: normal">2</span></font></sup></font></font></font></p> <p>Calculate R<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font><sup><b>2</b></sup> for Y<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font>. R<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font><sup><b>2</b></sup> is derived from r<sub>total</sub>, which is the overall correlation between X and Y</p> <p><a href="http://lh6.ggpht.com/-O0GTtmUNuUY/VItTwEz2pZI/AAAAAAAA3wk/yxF8Dsd2RQk/s1600-h/ANCOVA_6_Calculating_r2_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calculating R Square For Y total" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calculating R Square For Y total" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVuBlNkDDCbgYFIIsblCVN-v7MaXD2cLOaZv4BhIaT8v-ZeN4p2_ZopBRJMstgJVAMWgtfc0-TOMeLHOUmn6u5CKge-vybmrMh0seAis6LD7npuIzyrokvCJIZyW3J6VA13QfJFp1vbzKE/?imgmax=800" width="275" height="480" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><font color="#000080"><font face="Arial, serif"><font size="3"><b><font color="#00000a">R</font></b></font></font></font><font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>total</b></font></font></sub></font><sup><b>2</b></sup> = 0.645</p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 4) Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">total(Y)</font></sub></b></font></font></font></p> <p> </p> <p><a href="http://lh6.ggpht.com/-Zc50OgtHTo0/VItTxPGOqSI/AAAAAAAA3w0/FniQjAo4KII/s1600-h/ANCOVA_7_Calculate_SS_1_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calculating Adjusted SS y total" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calculating Adjusted SS y total" src="http://lh6.ggpht.com/-WyUovI-xfw0/VItTxmiAl_I/AAAAAAAA3w8/DXI2iVpWl-Q/ANCOVA_7_Calculate_SS_1_RGB_72_thumb%25255B1%25255D.jpg?imgmax=800" width="400" height="219" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 5) Calculate r</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">within</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><sup><font size="4"><span style="font-weight: normal">2</span></font></sup></font></font></font></p> <p>Obtain r<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font><sup><b>2</b></sup> for Y<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font> by first calculating SC<font color="#000080"><sub><font face="Arial, serif"><font size="3"><b>within</b></font></font></sub></font></p> <p> </p> <p><a href="http://lh6.ggpht.com/-rBxqshAj6Q8/VI3NYNEoWiI/AAAAAAAA3zE/bo0tREfRec4/s1600-h/ANCOVA_8_Calculating_r2_2_RGB_72%25255B6%25255D.jpg"><img title="ANCOVA_8_Calculating_r2_2_RGB_72" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA_8_Calculating_r2_2_RGB_72" src="http://lh5.ggpht.com/-2e9BaReONW0/VI3NYguJvlI/AAAAAAAA3zM/T-XKrUX3vnI/ANCOVA_8_Calculating_r2_2_RGB_72_thumb%25255B3%25255D.jpg?imgmax=800" width="400" height="329" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><a href="http://lh4.ggpht.com/-qRYP3h8F9jg/VI3QmOr2KzI/AAAAAAAA3zo/PUXGWxejeZE/s1600-h/ANCOVA_9_Calcuating_r2_3_RGB_72%25255B10%25255D.jpg"><img title="ANCOVA_9_Calcuating_r2_3_RGB_72" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA_9_Calcuating_r2_3_RGB_72" src="http://lh3.ggpht.com/-lSZcCxXOiiI/VI3QmnljAkI/AAAAAAAA3zs/c55vdAICgmI/ANCOVA_9_Calcuating_r2_3_RGB_72_thumb%25255B5%25255D.jpg?imgmax=800" width="400" height="300" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p><a href="http://lh5.ggpht.com/-vJWeZLFQ_tU/VItT0ANtfwI/AAAAAAAA3xk/WSevV6MegsM/s1600-h/ANCOVA_10_Calucating_r2_4_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calculating R Square within" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calculating R Square within" src="http://lh3.ggpht.com/-BmKAwtALicI/VItT0j-5SYI/AAAAAAAA3xs/SfisdLvB3WQ/ANCOVA_10_Calucating_r2_4_RGB_72_thumb.jpg?imgmax=800" width="285" height="480" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 6)</font></b></font></font></font> <font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">within(Y)</font></sub></b></font></font></font></p> <p> </p> <p><a href="http://lh3.ggpht.com/-J2BonHXuMsw/VItT1dfOrgI/AAAAAAAA3x0/nX4bm45oSfM/s1600-h/ANCOVA_11_Calculting_SS2_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calcuating Adjusted SS within(Y)" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calcuating Adjusted SS within(Y)" src="http://lh5.ggpht.com/-mEzs0oDJ4Hg/VItT1pqjc2I/AAAAAAAA3x8/fu4oZW4lwSM/ANCOVA_11_Calculting_SS2_RGB_72_thumb.jpg?imgmax=800" width="400" height="201" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 7)</font></b></font></font></font> <font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Calculate </font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">(adjusted)</font></sub></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">SS</font></b></font></font></font><font color="#000080"><font face="Arial, serif"><font size="5"><b><sub><font size="4">between(Y)</font></sub></b></font></font></font></p> <p> </p> <p><a href="http://lh5.ggpht.com/-Fe8DjN3ZYXc/VItT2Q0sqvI/AAAAAAAA3yE/RLJxB2kHc7E/s1600-h/ANCOVA_12_Calculating_SS3_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calcuating Adjusted SS between(Y)" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calcuating Adjusted SS between(Y)" src="http://lh3.ggpht.com/-QKfcusdEEzU/VItT2n6Nw6I/AAAAAAAA3yM/aeB14TUcEO0/ANCOVA_12_Calculating_SS3_RGB_72_thumb.jpg?imgmax=800" width="400" height="270" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <p><font color="#000080"><font face="Arial, serif"><font size="5"><b><font size="4">Step 8) Calculate the p Value</font></b></font></font></font> </p> <p>Arrange all of the above data on the ANOVA Excel output table and calculate the new p Value.</p> <p>Calculate MS<sub><b>between</b></sub> and MS<sub><b>within</b></sub> from <sub><b>(adjusted)</b></sub>SS<sub><b>between</b></sub><sub>(Y) </sub>, <sub>(</sub><sub><b>adjusted)</b></sub>SS<sub><b>within(Y)</b></sub><sub> </sub>, and the degrees of freedom of each.</p> <p>Calculate the F Value from MS<sub><b>between</b></sub> and MS<sub><b>within</b></sub>.</p> <p>Calculate the p Value from the F Value and both degrees of freedom.</p> <p><a href="http://lh4.ggpht.com/-BPcC__kuYP4/VItT3BHUiOI/AAAAAAAA3yU/5RAWNwxCa2E/s1600-h/ANCOVA_13_ANCOVA_Table_RGB_72%25255B1%25255D.jpg"><img title="ANCOVA in Excel - Calculating Final p Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calculating Final p Value" src="http://lh3.ggpht.com/-rHuf98Z86QQ/VItT3tktX6I/AAAAAAAA3yc/Dh8wemnCRYw/ANCOVA_13_ANCOVA_Table_RGB_72_thumb%25255B1%25255D.jpg?imgmax=800" width="400" height="200" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> <a href="http://lh4.ggpht.com/-z67uqO-8fHw/VIt1aWqTXeI/AAAAAAAA3ys/qAB04xUnLJA/s1600-h/ANCOVA_14%25252BANCOVA_Output_RGB_72%25255B6%25255D.jpg"><img title="ANCOVA in Excel - Calculating Final p Value" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA in Excel - Calculating Final p Value" src="http://lh5.ggpht.com/-XmDWOz2zt8k/VIt1bPXxNGI/AAAAAAAA3y0/kvAexMlx_rE/ANCOVA_14%25252BANCOVA_Output_RGB_72_thumb%25255B4%25255D.jpg?imgmax=800" width="400" height="307" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The original ANOVA test did not detect a difference in monthly sales increases from either training method. This ANOVA was performed in Step 1 and produced a p Value = 0.68994.</p> <p>ANCOVA did detect a significant difference in monthly sales increases between the two training methods after removing the effects of the Covariate variable (years of prior experience). The very low p Value = 0.00092 indicates that Training method A significantly outperformed Training Method B. The high r<sub><b>within</b></sub><sup><b>2</b></sup><b> </b>indicates that the years of experience of each salesperson had a very substantial effect on the salesperson’s ability to translate either training program into increased sales following the training.</p> <p> <br /></p> <h2>Derivation of the 8 Steps of ANCOVA</h2> <p>The most intuitive way to understand the derivation of the steps needed to perform ANCOVA is to work backwards from the very last step – the calculation of the p Value for the ANCOVA. This reverse derivation is performed as follows:</p> <p> <br /></p> <h3><font face="Arial, serif">Calculating </font><sub><font face="Arial, serif">adjusted</font></sub><font face="Arial, serif"> p Value</font></h3> <p>The p Value for single-factor ANOVA is calculated in Excel as follows:</p> <p>p Value = F.DIST.RT(F value, df<sub>between</sub>, df<sub>within</sub>)</p> <p>The p Value for ANCOVA is the p Value of single-factor ANOVA adjusted by removing the effects of the Confounding Variable. This adjusted p Value is defined in Excel as follows:</p> <p><sub>adjusted </sub>p Value = F.DIST.RT(<sub>adjusted</sub> F value, <sub>adjusted</sub> df<sub>between</sub>, <sub>adjusted</sub> df<sub>within</sub>)</p> <p><sub>adjusted</sub> df<sub>between</sub> = df<sub>between</sub></p> <p>There is no adjustment necessary for df<sub>between</sub> because the number of groups has not changed from ANOVA to ANCOVA.</p> <p><sub>adjusted</sub> df<sub>within</sub> = df<sub>within</sub> – 1</p> <p>In ANCOVA the within-group degrees of freedom is reduced from ANOVA by 1 to account for the fact that the covariance portion of within-groups has been removed from the analysis.</p> <p> <br /></p> <h3><font face="Arial, serif">Calculating </font><sub><font face="Arial, serif">adjusted</font></sub><font face="Arial, serif"> F Value</font></h3> <p>F Value = MS<sub>between</sub> / MS<sub>within</sub></p> <p><sub>adjusted </sub>F Value = <sub>adjusted </sub>MS<sub>between</sub> / <sub>adjusted </sub>MS<sub>within</sub></p> <p> <br /></p> <h3><font face="Arial, serif">Calculating </font><sub><font face="Arial, serif">adjusted</font></sub><font face="Arial, serif"> MS</font></h3> <p>MS<sub>between</sub> = SS<sub>between</sub> / df<sub>between</sub></p> <p><sub>adjusted </sub>MS<sub>between</sub> = <sub>adjusted </sub>SS<sub>between</sub> / <sub>adjusted </sub>df<sub>between</sub></p> <p> <br /></p> <p>MS<sub>within</sub> = SS<sub>within</sub> / df<sub>within</sub></p> <p><sub>adjusted </sub>MS<sub>within</sub> = <sub>adjusted </sub>SS<sub>within</sub> / <sub>adjusted </sub>df<sub>within</sub></p> <p> <br /></p> <p>We now need to calculate <sub>adjusted </sub>SS<sub>between</sub> and <sub>adjusted </sub>SS<sub>within</sub> to complete the derivation.</p> <p> <br /></p> <h3><font face="Arial, serif">Calculating </font><sub><font face="Arial, serif">adjusted</font></sub><font face="Arial, serif"> SS</font><sub><font face="Arial, serif">within</font></sub></h3> <p>SS<sub>total(Y)</sub> is found within the output of Excel single-factor ANOVA performed on the sample groups of the Dependent Variable Y, which is the Monthly Sales Increase.</p> <p>r<sub>XY</sub> = Correlation between the X-Y data pairs. X is the Covariate variable which is the Prior Years of Experience. Y is the Dependent variable, which is the Monthly Sales Increase.</p> <p>r<sub>XY</sub> = CORREL( X data array, Y data array)</p> <p>R Square = (r<sub>XY</sub>)<sup>2</sup> and represents the portion of total variance (SS<sub>total</sub>) that is explained by the Dependent Variable Y. The following relationship is therefore correct:</p> <p>SS<sub>total(Y)</sub> = Portion of SS<sub>total</sub> attributed to X = (r<sub>XY</sub>)<sup>2</sup> * SS<sub>total(Y)</sub></p> <p><sub>adjusted</sub> SS<sub>total(Y)</sub> = SS<sub>total(Y)</sub> - Portion of SS<sub>total</sub> attributed to X</p> <p><sub>adjusted</sub> SS<sub>total(Y)</sub> = SS<sub>total(Y)</sub> - (r<sub>XY</sub>)<sup>2</sup> * SS<sub>total(Y)</sub></p> <p> <br /></p> <p>SS<sub>total(Y)</sub> = SS<sub>between(Y)</sub> + SS<sub>within(Y)</sub></p> <p><sub>adjusted</sub> SS<sub>total(Y)</sub> = <sub>adjusted</sub> SS<sub>between(Y)</sub> + <sub>adjusted</sub> SS<sub>within(Y)</sub></p> <p><sub>adjusted</sub> SS<sub>within(Y)</sub> = SS<sub>within(Y)</sub> – Portion of SS<sub>within(Y) </sub>attributed to X</p> <p> <br /></p> <p>SS<sub>within(Y)</sub> is found within the output of Excel single-factor ANOVA performed on the sample groups of the Dependent Variable Y, which is the Monthly Sales Increase.</p> <p> <br /></p> <p>Portion of SS<sub>within(Y) </sub>attributed to X = r<sub>within</sub><sup>2</sup> * SS<sub>within(Y)</sub></p> <p>r<sub>within</sub> = SC<sub>within</sub> / SQRT( SS<sub>within(X)</sub> * SS<sub>within(Y)</sub> ) </p> <p> <br /></p> <p>This equation is derived as follows:</p> <p>Correlation between X and Y = r<sub>xy</sub> </p> <p>r<sub>xy</sub> = cor(x,y) </p> <p>= cov(x,y) / σ<sub>X</sub> σ<sub>Y</sub> </p> <p>= σ<sub>XY</sub> / σ<sub>X</sub> σ<sub>Y</sub> </p> <p>= σ<sub>XY</sub> / SQRT( σ<sub>XX </sub>* σ<sub>YY</sub> )</p> <p>= SC<sub>total</sub> / SQRT( SS<sub>total(X)</sub> * SS<sub>total(Y)</sub> )</p> <p> <br /></p> <p>Because </p> <p>σ<sub>X</sub> = standard deviation of X</p> <p>σ<sub>Y</sub> = standard deviation of Y</p> <p>σ<sub>XY</sub> = cov(x,y)</p> <p>σ<sub>XX </sub>= Variance of X = SS<sub>total(X)</sub> and is found in the output of Excel single-factor ANOVA performed on the sample groups of the Covariate Variable X, which is the Years of Prior Experience.</p> <p>σ<sub>YY </sub>= Variance of Y = SS<sub>total(Y)</sub> and is found in the output of Excel single-factor ANOVA performed on the sample groups of the Dependent Variable Y, which is the Monthly Sales Increase.</p> <p>Since we now have r<sub>XY</sub>, we can now <sub>adjusted</sub> SS<sub>total(Y)</sub> calculate as follows:</p> <p><sub>adjusted</sub> SS<sub>total(Y)</sub> = SS<sub>total(Y)</sub> - (r<sub>XY</sub>)<sup>2</sup> * SS<sub>total(Y)</sub></p> <p> <br /></p> <p>Because of the above equation </p> <p>r<sub>xy</sub> = SC<sub>total</sub> / SQRT( SS<sub>total(X)</sub> * SS<sub>total(Y)</sub> )</p> <p>The following equation is valid:</p> <p>r<sub>within</sub> = SC<sub>within</sub> / SQRT( SS<sub>within(X)</sub> * SS<sub>within(Y)</sub> ) </p> <p>In order to calculate r<sub>within</sub> we still need to calculate SC<sub>within</sub> </p> <p> <br /></p> <p>SC<sub>within</sub> = SC<sub>within(A)</sub> + SC<sub>within(B)</sub></p> <p>A designates data associated with Training method 1</p> <p> <br /></p> <p>B designates data associated with Training method 2</p> <p>SC<sub>within</sub> = Covariance(X,Y) which is defined by the following formula:</p> <p> <br /></p> <p>cov(X,Y) = Σ (X<sub>i</sub>Y<sub>i</sub>) - µ<sub>X</sub>µ<sub>Y</sub> </p> <p>cov(X,Y) = Σ (X<sub>i</sub>Y<sub>i</sub>) – [ (ΣX<sub>i</sub>)(ΣY<sub>i</sub>) / N ]</p> <p> <br /></p> <p>SC<sub>within</sub> = Σ (X<sub>i</sub>Y<sub>i</sub>) – [ (ΣX<sub>i</sub>)(ΣY<sub>i</sub>) / N ]</p> <p>SC<sub>within(A)</sub> and SC<sub>within(B)</sub> are therefore defined as follows:</p> <p>SC<sub>within(A)</sub> = Σ (X<sub>Ai</sub>Y<sub>Ai</sub>) – [ (ΣX<sub>Ai</sub>)(ΣY<sub>Ai</sub>) / N<sub>A</sub> ]</p> <p>SC<sub>within(B)</sub> = Σ (X<sub>Bi</sub>Y<sub>Bi</sub>) – [ (ΣX<sub>Bi</sub>)(ΣY<sub>Bi</sub>) / N<sub>B</sub> ]</p> <p> <br /></p> <p>We can now calculate the following:</p> <p>SC<sub>within</sub> = SC<sub>within(A)</sub> + SC<sub>within(B)</sub></p> <p> <br /></p> <p>Which means we can now calculate the following:</p> <p>r<sub>within</sub> = SC<sub>within</sub> / SQRT( SS<sub>within(X)</sub> * SS<sub>within(Y)</sub> ) </p> <p> <br /></p> <p>Which allows us to calculate the following:</p> <p>Portion of SS<sub>within(Y) </sub>attributed to X = r<sub>within</sub><sup>2</sup> * SS<sub>within(Y)</sub></p> <p>We can now calculate <sub>adjusted</sub> SS<sub>within(Y)</sub> as follows:</p> <p><sub>adjusted</sub> SS<sub>within(Y)</sub> = SS<sub>within(Y)</sub> – Portion of SS<sub>within(Y) </sub>attributed to X</p> <p> <br /></p> <h3><font face="Arial, serif">Calculating </font><sub><font face="Arial, serif">adjusted</font></sub><font face="Arial, serif"> SS</font><sub><font face="Arial, serif">between</font></sub></h3> <p><sub>adjusted</sub> SS<sub>total(Y)</sub> = <sub>adjusted</sub> SS<sub>between(Y)</sub> + <sub>adjusted</sub> SS<sub>within(Y)</sub></p> <p><sub>adjusted</sub> SS<sub>between(Y) </sub>= <sub>adjusted</sub> SS<sub>total(Y)</sub> - <sub>adjusted</sub> SS<sub>within(Y)</sub></p> <p> <br /></p> <p>Now that we have <sub>adjusted</sub> SS<sub>within(Y)</sub>, <sub>adjusted</sub> SS<sub>between(Y)</sub>, <sub>adjusted </sub>df<sub>within</sub>, and <sub>adjusted </sub>df<sub>between</sub></p> <p>We can calculate <sub>adjusted </sub>MS<sub>between</sub> and <sub>adjusted </sub>MS<sub>within</sub> as follows:</p> <p><sub>adjusted </sub>MS<sub>between</sub> = <sub>adjusted </sub>SS<sub>between</sub> / <sub>adjusted </sub>df<sub>between</sub></p> <p><sub>adjusted </sub>MS<sub>within</sub> = <sub>adjusted </sub>SS<sub>within</sub> / <sub>adjusted </sub>df<sub>within</sub></p> <p> <br /></p> <p>Now that we have <sub>adjusted </sub>MS<sub>between</sub> and <sub>adjusted </sub>MS<sub>within</sub></p> <p>We can calculate <sub>adjusted </sub>F Value as follows:</p> <p><sub>adjusted </sub>F Value = <sub>adjusted </sub>MS<sub>between</sub> / <sub>adjusted </sub>MS<sub>within</sub></p> <p> <br /></p> <p>Now that we have <sub>adjusted </sub>F Value, <sub>adjusted </sub>df<sub>between</sub>, and <sub>adjusted </sub>df<sub>within</sub></p> <p>We can calculate <sub>adjusted </sub>p Value in Excel as follows:</p> <p><sub>adjusted </sub>p Value = F.DIST.RT(<sub>adjusted</sub> F value, <sub>adjusted</sub> df<sub>between</sub>, <sub>adjusted</sub> df<sub>within</sub>)</p> <h2>Calculating Effect Size in ANCOVA</h2> <p>Effect size is a way of describing how effectively the method of data grouping allows those groups to be differentiated. A simple example of a grouping method that would create easily differentiated groups versus one that does not is the following.</p> <p>Imagine a large random sample of height measurements of adults of the same age from a single country. If those heights were grouped according to gender, the groups would be easy to differentiate because the mean male height would be significantly different than the mean female height. If those heights were instead grouped according to the region where each person lived, the groups would be much harder to differentiate because there would not be significant difference between the means and variances of heights from different regions.</p> <p>Because the various measures of effect size indicate how effectively the grouping method makes the groups easy to differentiate from each other, the magnitude of effect size tells how large of a sample must be taken to achieve statistical significance. A small effect can become significant if a larger enough sample is taken. A large effect might not achieve statistical significance if the sample size is too small.</p> <p>The two most common measures of effect size of one-way ANCOVA are the following:</p> <h4><font color="#0000cc">Eta Square (η<sup>2</sup>)</font></h4> <p>(Greek letter “eta” rhymes with “beta”) Eta square quantifies the percentage of variance in the dependent variable (the variable that is measured and placed into groups) that is explained by the independent variable (the method of grouping). If eta squared = 0.35, then 35 percent of the variance associated with the dependent variable is attributed to the independent variable (the method of grouping).</p> <p>Eta square provides an overestimate (a positively-biased estimate) of the explained variance of the population from which the sample was drawn because eta squared estimates only the effect size on the sample. The effect size on the sample will be larger than the effect size on the population. This bias grows smaller is the sample size grows larger.</p> <p>Eta square is affected by the number and size of the other effects.</p> <p>η<sup><b>2</b></sup> = SS<sub><b>Between_Groups</b></sub> / SS<sub><b>Total</b></sub><sub> </sub>These two terms are part of the ANOVA calculations found in the Single-factor ANOVA output.</p> <p>Jacob Cohen in his landmark 1998 book <i>Statistical Analysis for the Behavior Sciences</i> proposed that effect sizes could be generalized as follows:</p> <p>η<sup><b>2</b></sup> = 0.01 for a small effect. A small effect is one that not easily observable.</p> <p>η<sup><b>2</b></sup> = 0.05 for a medium effect. A medium effect is more easily detected than a small effect but less easily detected than a large effect.</p> <p>η<sup><b>2</b></sup> = 0.14 for a large effect. A large effect is one that is readily detected with the current measuring equipment.</p> <p>Eta squared is sometimes called the nonlinear correlation coefficient because it provides a measure of strength of the curvilinear relationship between the dependent and independent variables. If the relationship is linear, eta squared will have the same value as r squared.</p> <p>Eta squared is calculated in Excel with the formula</p> <p>η<sup><b>2</b></sup> = SS<sub><b>Between_Groups</b></sub> / SS<sub><b>Total</b></sub><sub> </sub></p> <h4><font color="#0000cc">Omega Squared (ώ<sup>2</sup>)</font></h4> <p>Omega squared is an estimate of the population’s variance that is explained by the treatment (the method of grouping).</p> <p>Omega squared is less biased (but still slightly biased) than eta square and is always smaller the eta squared because eta squared overestimates the explained variance of the population from which the sample was drawn. Eta squared estimates only the effect size on the sample. The effect size on the sample will be larger than the same effect size on the population.</p> <p>Magnitudes of omega squared are generally classified as follows: Up to 0.06 is considered a small effect, from 0.06 to 0.14 is considered a medium effect, and above 0.14 is considered a large effect. Small, medium, and large are relative terms. A large effect is easily discernible but a small effect is not.</p> <p>Omega Square is implemented in Excel as follows:</p> <p><a href="http://lh3.ggpht.com/-ULp_8BEmTrg/VJHhKfEUxLI/AAAAAAAA310/y6_7mUYepf8/s1600-h/ANCOVA_17_Omega_Square_RGB_72%25255B8%25255D.jpg"><img title="ANCOVA_17_Omega_Square_RGB_72" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA_17_Omega_Square_RGB_72" src="http://lh6.ggpht.com/-43BNDap0t3E/VJHhK7NBNGI/AAAAAAAA318/P8Tfs7WnT94/ANCOVA_17_Omega_Square_RGB_72_thumb%25255B4%25255D.jpg?imgmax=800" width="400" height="58" /></a> </p> <p class="western"><a name="_GoBack"></a>After the adjusted ANCOVA table is created for the dependent variable (Measured Post-Training Sales Increase), both of the preceding Effect Size formulas can be quickly calculated with figures taken directly from the adjusted ANCOVA table as follows:</p> <p class="western"><a href="http://lh3.ggpht.com/-7YsHKUL8ez8/VJHhLblMR7I/AAAAAAAA32E/fen0sOqDRhI/s1600-h/ANCOVA_16_Effect_Size_Calculations_RGB_72%25255B4%25255D.jpg"><img title="ANCOVA_16_Effect_Size_Calculations_RGB_72" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="ANCOVA_16_Effect_Size_Calculations_RGB_72" src="http://lh5.ggpht.com/-1K3bLSuiUlo/VJHhL6SIAtI/AAAAAAAA32M/MKgAtWgQySs/ANCOVA_16_Effect_Size_Calculations_RGB_72_thumb%25255B2%25255D.jpg?imgmax=800" width="400" height="255" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>An eta-square value of 0.448 would be classified as a very large size effect.</p> <p>An omega-square value of 0.442 would also be classified as a very large size effect.</p> <p>A large effect is one that is readily observable (after the effects of the Covariate variable have been removed).</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com27tag:blogger.com,1999:blog-3568555666281177719.post-62759283173009013892014-06-03T12:43:00.001-07:002015-03-24T19:50:03.149-07:00Creating an Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013<h1>Creating an Interactive <br /> <br />Statistical Distribution <br /> <br />Graph in Excel</a> 2010 and <br /> <br />Excel 2013</h1> <p>This is one of the following eleven articles on creating user-interactive graphs of statistical distributions in Excel</p> <p><a href="http://blog.excelmasterseries.com/2014/06/interactive-statistical-distribution.html" target="_blank">Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-normal.html" target="_blank">Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-chi-square.html" target="_blank">Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-t-distribution-in.html" target="_blank">Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/t-distributions-pdf-in-excel-2010-and.html" target="_blank">Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/t-distributions-cdf-in-excel-2010-and.html" target="_blank">Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/interactive-graph-of-binomial.html" target="_blank">Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-exponential.html" target="_blank">Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-beta-distribution.html" target="_blank">Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-gamma-distribution.html" target="_blank">Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/interactive-graph-of-poisson.html" target="_blank">Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013</a></p> <p>Interactive area charts are very useful for demonstrating how changes made to a statistical distribution’s parameters affect the distribution’s shape, scale, and location. </p> <p>This section shows how to create a user-interactive normal distribution graph in Excel that instantaneously adjusts its shape, scale, location, and size of outer tails when a user changes any of the yellow cells that contain the graph’s parameters, label and legend text, and percentage of curve area in each outer tail. The following user-controlled settings produce the Excel area chart that is shown here:</p> <p><a href="http://lh3.ggpht.com/-rQDIcdL6gAk/U44kJfBv2bI/AAAAAAAAuNs/TDjh_zat3nw/s1600-h/clip_image001%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-lAO0rTp34_c/U44kKPPv6mI/AAAAAAAAuN0/O6ZwwDl5ngk/clip_image001_thumb%25255B1%25255D.jpg?imgmax=800" width="412" height="326" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Changes to each of the nine user-controlled settings instantly update the chart as shown here:</p> <p><a href="http://lh4.ggpht.com/-jCi9Eg9odJk/U44kK-em7cI/AAAAAAAAuN8/EMe3B7-RF3E/s1600-h/clip_image002%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-kLwl9lXb2Qc/U44kMZ1b1TI/AAAAAAAAuOE/Zf8gHE_aI30/clip_image002_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="325" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This section will provide instructions to create this interactive Excel area chart.</p> <p>The X and Y values used to create this chart are based upon z Score(X), the z Score of each X value. The z Score is the number of standard deviations that a specific point (X value) is from the mean. </p> <p>The z Score provides the basis of every element on this chart and therefore should be constructed first. 99.7 percent of all values in a normally distributed population will be within 3 standard deviations from the mean. A sufficient range of z Score values would include z Scores from 3 standard deviations below the mean (z Score = -3) to 3 standard deviations above the mean (z Score = +3).</p> <p>The z Score data will be presented as a list of z Scores starting at -3 and increasing incrementally to a value of +3. The increments need to be small enough so that Excel graph based upon these z Scores will be smooth. The list of z Scores will increase by increments of 0.10 from -3 to +3. This means that there be 60 z Score values and therefore 60 points graphed in this Excel chart.</p> <p>As mentioned the z Score will start at -3 and increase in 0.1 increments until the z Score has reached +3. There are many ways in Excel to incrementally increase the values in cells on the way down a column. ROW() generates the number of the current row and is used in the following fashion to incrementally increase z Score value:</p> <p><a href="http://lh6.ggpht.com/-mBjm3XBzsYM/U44kNcrquaI/AAAAAAAAuOM/Ca28WmV3VtY/s1600-h/clip_image003%25255B2%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-Ws0Vyok_-xE/U44kOUyKSbI/AAAAAAAAuOU/smKLtiNI9qw/clip_image003_thumb%25255B2%25255D.jpg?imgmax=800" width="401" height="445" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Excel Split View tool was used to simultaneously view the beginning and the end of a data column. The beginning and end values of data columns are usually the most important parts when creating interactive Excel graphs. The Split View tool is found under the View tab and is labeled Split. The horizontal and vertical partitions of the Split can be dragged-and-dropped to any location on the worksheet by the user. </p> <p>The next set of graph parameters that should be created are the X-axis values. The values of the X-axis are calculated from the z Score, the population mean (µ) and the population standard deviation (σ). The calculation of the X values on the horizontal axis is given as follows:</p> <p>X = µ + (z Score(X) * σ)</p> <p>z Score data has already been created. The population mean, µ, and population standard deviation, σ, will be tied to user-inputs on the Excel worksheet.</p> <p>Changing the Excel graph’s user inputs µ and σ changes the values of the X-axis. The graph’s shape remains the same but the values of the X axis are shifted and scaled based upon changes made to the user inputs µ and σ. The two graphs at the beginning of this section show the changes in the X-axis location and scale based upon changes to user inputs for µ and σ. The following image shows how X-axis values on the Excel worksheet are calculated based upon z Scores and the user inputs µ and σ:</p> <p><a href="http://lh5.ggpht.com/-X6bHMVjtvGA/U44kO_ZPreI/AAAAAAAAuOc/91rBi3egDvE/s1600-h/clip_image004%25255B4%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-5sUUU2yx1ww/U44kPlKx4ZI/AAAAAAAAuOk/2aAgDsj6RVU/clip_image004_thumb%25255B4%25255D.jpg?imgmax=800" width="389" height="572" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>After creating the column of z Scores and the column of X-axis values tied to the z Scores, the next step is to create a column of PDF data. The z Score data is the fundamental basis of the entire graph and each PDF value in the PDF column is based upon the z Score that is directly across from it in the z Score column.</p> <p>These PDF values are calculated in Excel as follows:</p> <p>PDF(X) </p> <p>= f(X,µ,σ)</p> <p>=NORM.DIST(X, µ,σ,FALSE) </p> <p>= NORM.S.DIST(z Score(X),FALSE)</p> <p>= NORM.S.DIST((X - µ)/σ,FALSE)</p> <p>This is implemented in the Excel worksheet as follows:</p> <p><a href="http://lh5.ggpht.com/-vO3UsWGVPJ0/U44kQ63Hq6I/AAAAAAAAuOs/WSUS0OM2tPI/s1600-h/clip_image005%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-CVIEOFMJldE/U44kR6DxLMI/AAAAAAAAuO0/sXgsNc-wgXM/clip_image005_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="411" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The next data set that should be created are the values of the normal distribution’s CDF (Cumulative Distribution Function) at each z Score increment. This is calculated in Excel as follows:</p> <p>CDF(X) </p> <p>= F(X,µ,σ)</p> <p>=NORM.DIST(X, µ,σ,TRUE) </p> <p>= NORM.S.DIST(z Score(X),TRUE)</p> <p>= NORM.S.DIST((X - µ)/σ,TRUE)</p> <p><a href="http://lh3.ggpht.com/-4QUnZMje5WQ/U44kT1IdMOI/AAAAAAAAuO8/P4lbIwShcaY/s1600-h/clip_image006%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/--I955Ud6blU/U44kUjJPTNI/AAAAAAAAuPE/aciuWD-PS_g/clip_image006_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="369" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The data sets created so far include z Score values, X-axis values, and the PDF and CDF values at each z Score. The final data set to be created are the Y-axis values.</p> <p>This Excel area chart is a graph of the normal distribution’s bell-shaped PDF curve. The Y value at each point on this curve will therefore have the same value as the normal distribution’s PDF at that point. Setting the Y-values to equal the PDF values for each z Score increment is implemented as follows:</p> <p><a href="http://lh5.ggpht.com/-CMbr8uzP6jo/U44kVZtdUUI/AAAAAAAAuPM/XqTHQStlf6E/s1600-h/clip_image007%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-xxKH3-hzD54/U44kWLvpfeI/AAAAAAAAuPU/cuZ11rjF3OE/clip_image007_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="423" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Y-values are given the label Series 1 because this represents the first data series created for the area chart.</p> <p>Series 1 can now be graphed because both the X and Y values are now available. Creating this graph in Excel is done as follows:</p> <p>The first step is to select the entire column of data including the label for Series 1 as shown. </p> <p>With that selected data highlighted, insert an area chart into the worksheet. That chart is inserted by going under the Insert tab and then selecting Area Charts / 2-D Area Chart as follows:</p> <p><a href="http://lh5.ggpht.com/-gnUCftDzvqs/U44kWwV7jjI/AAAAAAAAuPc/hw-5WZN5S8A/s1600-h/clip_image008%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-LbGKnPHVjjg/U44kX29qqoI/AAAAAAAAuPk/dT3Rf9SVLtQ/clip_image008_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="493" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The following graph of the normal distribution’s PDF is automatically created as follows:</p> <p><a href="http://lh4.ggpht.com/-G2DGDn4cLww/U44kY736AbI/AAAAAAAAuPs/GDirhluFOm4/s1600-h/clip_image009%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-moBW9xzjB9A/U44kZk9vFII/AAAAAAAAuP0/zBsCEoZj3qY/clip_image009_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="278" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The default color of the first data series graphed is blue. This can be changed if desired by right-clicking on blue bell-shaped curve and selecting Format Graph Area. </p> <p>The area chart has been created based upon the column of Y values. No specific values have been assigned to the X-Axis. The default X values are created from the numbering of the points starting at 1 on the left.</p> <p>The next step is to assign specific X-values to the data points. The X-value data set column has been created.</p> <p>To assign specific X-values to the graph’s data points, click anywhere on the graph and choose Select Data from the pop-up short-cut menu as follows:</p> <p><a href="http://lh5.ggpht.com/-EY8lb9F4cgc/U44kaG3y7uI/AAAAAAAAuP8/Sdr7JxpLvQc/s1600-h/clip_image010%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-rfuRQD06PKE/U44ka3GYBtI/AAAAAAAAuQE/gUGLKjBCq18/clip_image010_thumb%25255B1%25255D.jpg?imgmax=800" width="413" height="358" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This brings up the Select Data Source dialogue box. Select the Edit button on the right side under Horizontal (Category) Axis Labels.</p> <p><a href="http://lh5.ggpht.com/-OSW3bhjPpqQ/U44kbqNQBeI/AAAAAAAAuQM/9Fsjv5Lp6rc/s1600-h/clip_image011%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-VEk-yHKQkVU/U44kcLENGUI/AAAAAAAAuQU/p0rd1yPGQQA/clip_image011_thumb%25255B1%25255D.jpg?imgmax=800" width="417" height="232" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will bring up the Axis Label dialogue box. Highlight the data in the X-Axis data column on the worksheet and hit OK as follows:</p> <p><a href="http://lh3.ggpht.com/-rDbg2MK8j8w/U44kd_TFRGI/AAAAAAAAuQc/VUYLYc5PcXQ/s1600-h/clip_image012%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-Im-gHQ9eZfU/U44keq0BLrI/AAAAAAAAuQk/KX3JBnEIRFU/clip_image012_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="432" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Excel graph will now attach the selected X values to the data points as follows:</p> <p><a href="http://lh3.ggpht.com/-JbknYz0b1EU/U44kfHdcTuI/AAAAAAAAuQs/c11DhYjOt3E/s1600-h/clip_image013%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-7N6EeR1bbRE/U44kf_zB5XI/AAAAAAAAuQ0/2XvzxrfWdWM/clip_image013_thumb%25255B1%25255D.jpg?imgmax=800" width="419" height="230" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>These X-values change instantly when the user inputs for µ and σ are changed. This will be demonstrated shortly.</p> <p>The values in the X-axis should have additional formatting applied to them. The X axis can be made more legible if the X-axis values have their font weight changed to Bold, are set to show at least one decimal place, and become spaced out in larger increments than the current crowded setting of displaying than every third X value. These changes can be made to the X–axis values as follows:</p> <p>Right-click anywhere on the X-axis to bring up both the short-cut font menu and the short-cut pop-up menu. Select B on the short-cut font menu and the X values will have their font weight changed to bold. </p> <p>Right-click again on the numbers on the X-axis to bring up both menus again. Select Format Axis as follows:</p> <p><a href="http://lh4.ggpht.com/-tdLQqc5gTIE/U44kg-02KaI/AAAAAAAAuQ8/o6neddLAyjA/s1600-h/clip_image014%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-KZy9brp7mDw/U44khw2Fw7I/AAAAAAAAuRE/4GOvKRBz0fk/clip_image014_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="369" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will bring up the Format Axis dialogue box. Select the first option on the left which is Axis Options. The following settings shown here will display every tenth X-axis value instead of every third value:</p> <p><a href="http://lh6.ggpht.com/-teugV5o9kPc/U44kidsqy_I/AAAAAAAAuRM/bLFskzxe1Ks/s1600-h/clip_image015%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-Nusnp9B91DI/U44kjKu0cuI/AAAAAAAAuRU/0Ey2_DUjDpc/clip_image015_thumb%25255B1%25255D.jpg?imgmax=800" width="417" height="452" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The Number category brings up the Number dialogue page. The following setting will configure the X values to display one decimal place:</p> <p><a href="http://lh4.ggpht.com/-dZJ8LFLqbaU/U44kkEV19kI/AAAAAAAAuRc/4Bdv_ocer08/s1600-h/clip_image016%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/-l0bZ1tIsVMI/U44klISiSZI/AAAAAAAAuRg/qaxnduzJIkE/clip_image016_thumb%25255B1%25255D.jpg?imgmax=800" width="413" height="459" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>These settings configure values on the X axis to be displayed as follows:</p> <p><a href="http://lh4.ggpht.com/-0yRQJ7cgrYQ/U44kmDqdYoI/AAAAAAAAuRs/kDdaPnXPEw8/s1600-h/clip_image017%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-FoPOd_dXAMk/U44knAjZfAI/AAAAAAAAuR0/YrFBKfGJHyA/clip_image017_thumb%25255B1%25255D.jpg?imgmax=800" width="414" height="225" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>As mentioned, this graph is user-interactive. Changing user-inputs µ and σ produce the following changes in the Excel graph:</p> <p><a href="http://lh3.ggpht.com/-iuWsEGwrz1Y/U44knn5i_bI/AAAAAAAAuR8/8lIoJdt6H3M/s1600-h/clip_image018%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-5yLJ4QFuEmA/U44koU1MrAI/AAAAAAAAuSE/oWvyOXloeTU/clip_image018_thumb%25255B1%25255D.jpg?imgmax=800" width="424" height="226" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The graph’s shape did not change. The location of the mean was shifted to 50 and the graphed was scaled so that its standard deviation is now equal to 20. These changes are all implemented in the X axis.</p> <p>Excel graphs can be configured to link the text of the chart title and axis labels to external cells that are user inputs. The first step to implementing this is to create the external cells that will be the user inputs. Place these inputs in the following locations:</p> <p><a href="http://lh6.ggpht.com/-D7oijPc34pk/U44koyNM8JI/AAAAAAAAuSM/ydX_NS7iMDs/s1600-h/clip_image019%25255B3%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-Y4VyBJCx-Mg/U44kpWEa9tI/AAAAAAAAuSU/_77MLxj1AsQ/clip_image019_thumb%25255B3%25255D.jpg?imgmax=800" width="415" height="99" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The current generic Excel area chart does not have axis labels. The labels for the X-axis and the Y-axis have to created. </p> <p>A generic X-axis label is created by first clicking anywhere on the chart. This will bring up the Chart Tools menu. From this menu select the Layout tab / Axis Titles / Primary Horizontal Axis Title / Title Below Axis. </p> <p><a href="http://lh6.ggpht.com/-B8QywkUK0kc/U44kqAm3LII/AAAAAAAAuSc/-_oWNWhIRqA/s1600-h/clip_image020%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-SyQaPJ79a7c/U44kq02hQ4I/AAAAAAAAuSk/I8D2pw7Q7N0/clip_image020_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="406" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will create a generic X-axis title on the bottom of the graph shown in the next image of the graph.</p> <p>A generic Y-axis label is also created by first clicking anywhere on the chart. This once again bring up the Chart Tools menu. From that menu select the Layout tab / Axis Titles / Primary Vertical Axis Title / Horizontal Title. </p> <p><a href="http://lh5.ggpht.com/-IyGV38K96ms/U44krgtksKI/AAAAAAAAuSs/JBSr_MMxugY/s1600-h/clip_image021%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/--Y14OnRqANg/U44kszYTG2I/AAAAAAAAuS0/zf2TOgjuwXI/clip_image021_thumb%25255B1%25255D.jpg?imgmax=800" width="419" height="407" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will create a generic Y-axis title on the left side of the graph that has horizontal text that is shown in the next image of the graph.</p> <p>The Excel chart is now showing generic X-axis and Y-axis labels, a chart title, and a legend for data Series 1.</p> <p><a href="http://lh3.ggpht.com/-zwFHm5KY9cQ/U44ktZoHUqI/AAAAAAAAuS8/cZ7mxxmA3bw/s1600-h/clip_image022%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-NA-30ERONcU/U44kubLfMyI/AAAAAAAAuTE/Vsr2EdFmevs/clip_image022_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="413" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Linking the text in the chart title to text in an external user-input cell is fairly simple. Just click anywhere on the chart title and then type an equal sign in the formula bar as follows:</p> <p><a href="http://lh4.ggpht.com/-6yogEEhYu68/U44ku4XumCI/AAAAAAAAuTM/MpT67aUa8qY/s1600-h/clip_image023%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-hOJv2aAO7uA/U44kv7W-rxI/AAAAAAAAuTU/wAhDye3lCFQ/clip_image023_thumb%25255B1%25255D.jpg?imgmax=800" width="423" height="362" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>After typing the equal sign in the formula bar, click on the external cell that contains text. The address of this cell will appear in the formula bar including the worksheet’s name as follows:</p> <p><a href="http://lh5.ggpht.com/-o09Dx_4xKjs/U44kwcu5VOI/AAAAAAAAuTc/anhjhPEkEAI/s1600-h/clip_image024%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-eNYa-D-ln9I/U44kxNJ8v6I/AAAAAAAAuTk/XDsiF91Tduo/clip_image024_thumb%25255B1%25255D.jpg?imgmax=800" width="419" height="362" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The result is that the chart title now displays the text contained in the external cell as follows:</p> <p><a href="http://lh4.ggpht.com/-aGjXwmvVOZM/U44kxiMLv2I/AAAAAAAAuTs/wSyGqbtudp8/s1600-h/clip_image025%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-YIv1wrKsJzw/U44kyUNu8HI/AAAAAAAAuT0/CBZ7jFyTgfI/clip_image025_thumb%25255B1%25255D.jpg?imgmax=800" width="417" height="323" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>If the text in the external cell is changed, the updated text will be immediately displayed in the Excel chart.</p> <p>That exact same procedure can be used to link the X and Y-axis labels to external cells that are controlled by the user. Performing this procedure for both labels produces the following result:</p> <p><a href="http://lh6.ggpht.com/-00PSeiZjj1A/U44k062Qv_I/AAAAAAAAuT8/Xp8R4LVWk9Y/s1600-h/clip_image026%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-X_iHwdOIYMs/U44k1or_iJI/AAAAAAAAuUE/NylYF8lDvx0/clip_image026_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="314" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The next step is to link the legend of data Series 1 to an external cell. To do this click anywhere on the chart to bring up the short-cut menu. From this menu, choose Select Data as follows:</p> <p><a href="http://lh5.ggpht.com/-voN05Bdpvks/U44k2vu8ZHI/AAAAAAAAuUM/Nv4-141AOqw/s1600-h/clip_image027%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-cGyuouTSlfM/U44k3XWz27I/AAAAAAAAuUQ/B_1cKmWho9U/clip_image027_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="297" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will bring up the Select Data Source dialogue box. Select Series 1 on the left side and then click the Edit button also on the left side as follows:</p> <p><a href="http://lh5.ggpht.com/-IXdCrNOzhNs/U44k4GRysQI/AAAAAAAAuUc/WPIJNLFpNog/s1600-h/clip_image028%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-XzgrkuTn1X0/U44k4j1ZHrI/AAAAAAAAuUk/ZwGhDs94g1c/clip_image028_thumb%25255B1%25255D.jpg?imgmax=800" width="424" height="239" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will bring up the Edit Series dialogue box. Under the Series name input shown below in this dialogue box, select the external cell containing the text for the legend as follows:</p> <p><a href="http://lh5.ggpht.com/-2xboQ6eyKZ8/U44k5est6uI/AAAAAAAAuUs/S89eF7KRxrw/s1600-h/clip_image029%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/-Oc3zUNFNkPM/U44k527vdMI/AAAAAAAAuU0/I7Dt8zNQGQE/clip_image029_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="222" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The preceding steps have produced an Excel area chart that has the chart title, the labels for the X and Y axes, and the legend for Series 1 linked to external cells that can each be changed by the user as follows:</p> <p><a href="http://lh5.ggpht.com/-1CsgwY9KzhM/U44k6qX09QI/AAAAAAAAuU8/kPsxeMy4C78/s1600-h/clip_image030%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/--znl1ckotbM/U44k7N8hZ2I/AAAAAAAAuVE/Hd_Mv5G5f-w/clip_image030_thumb%25255B1%25255D.jpg?imgmax=800" width="410" height="283" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Any changes in these user inputs are instantly reflected in the Excel graph as follows:</p> <p><a href="http://lh3.ggpht.com/-aMCIDr_zoBs/U44k78_Rq1I/AAAAAAAAuVM/Enq539bRW6I/s1600-h/clip_image031%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-GJpi3Rv0bww/U44k8Vgb0VI/AAAAAAAAuVU/rnvANkOx6Ig/clip_image031_thumb%25255B1%25255D.jpg?imgmax=800" width="422" height="294" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The next step is to create an additional data series that will display the outer tails of the graph. This data series will have its data values controlled by the user inputs. These inputs will control the curve area displayed in each of the outer tails of the graph.</p> <p>The first step in creating this data series (Series 2) is to add new user inputs as follows:</p> <p><a href="http://lh3.ggpht.com/-2zPb24-f0ik/U44k87oB53I/AAAAAAAAuVc/rwyof1dEYoQ/s1600-h/clip_image032%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh3.ggpht.com/-HVM4kkrtztw/U44k9loC3zI/AAAAAAAAuVk/CKGfujHXa2E/clip_image032_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="158" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The data values for Series 2 can now be created. </p> <p>The Y values of Series 2 will also be set to equal the PDF values at each point on the graph as follows:</p> <p><a href="http://lh4.ggpht.com/-f1lxH2GdRjk/U44k-buITAI/AAAAAAAAuVs/WFCYxoXSVyc/s1600-h/clip_image033%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/-TcbaIH25Mc8/U44k_BuiDiI/AAAAAAAAuV0/idJIPzCuYPY/clip_image033_thumb%25255B1%25255D.jpg?imgmax=800" width="418" height="327" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Now that the column of Y values for Series 2 has been created, Series 2 can be added to the area chart and linked to the column of Y values. To do that click anywhere on the chart to bring up the pop-up short-cut menu. Choose Select Data on the menu as follows:</p> <p><a href="http://lh4.ggpht.com/-iu9eLCPftAs/U44k_1IGaDI/AAAAAAAAuV8/wbzCy1rpaNk/s1600-h/clip_image034%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/-MWRrU3eibaU/U44lAsfEnQI/AAAAAAAAuWE/VxCWinaLFnQ/clip_image034_thumb%25255B1%25255D.jpg?imgmax=800" width="417" height="365" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>This will bring up the following Select Data Source dialogue box. Select Add on the left side under Legend Entries (Series).</p> <p><a href="http://lh4.ggpht.com/-aVIJDT4iJz4/U44lBClZGkI/AAAAAAAAuWM/UQ2GSC7xcsU/s1600-h/clip_image035%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-w1inlOHoPsc/U44lFcnq1dI/AAAAAAAAuWU/U7gn-DMYlwE/clip_image035_thumb%25255B1%25255D.jpg?imgmax=800" width="424" height="238" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>For the Series name data input box, click on the external cell that contains the text that will be the legend for Series 2.</p> <p><a href="http://lh6.ggpht.com/-OGnjNZtEUGI/U44lGPIrv3I/AAAAAAAAuWc/ekS4VmTZxCA/s1600-h/clip_image036%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-BPZRG7g8YY8/U44lG7PidQI/AAAAAAAAuWk/oGkjDQRlgqk/clip_image036_thumb%25255B1%25255D.jpg?imgmax=800" width="422" height="308" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>From the Series Values input box, select the column of Series 2 Y values on the worksheet as follows:</p> <p><a href="http://lh3.ggpht.com/-Sq_wMGuMMeU/U44lHgNGvRI/AAAAAAAAuWs/7K8pV6F2z0k/s1600-h/clip_image037%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/--cjxiceXNEI/U44lIBynHkI/AAAAAAAAuW0/4I7l69RV7MU/clip_image037_thumb%25255B1%25255D.jpg?imgmax=800" width="418" height="325" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Any data series that is added to an Excel area chart will sit on top of all previously-added ones. Series 2 (default color is red) sits on top of Series 1 (default color is blue) and completely covers Series 1 on the chart as follows:</p> <p><a href="http://lh4.ggpht.com/-1TdslyfDSm4/U44lJN-Z_xI/AAAAAAAAuW8/x-FgiKjTXKg/s1600-h/clip_image038%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-KfjjKImr0QQ/U44lJwsmlCI/AAAAAAAAuXE/LRragSGwDJ0/clip_image038_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="330" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Series 2 sits on top of and covers Series 1 on the chart. The only way to see any of the blue Series 1 is to zero out Y values of Series 2.</p> <p>For example, if the following outer values of Series 2 were set to zero:</p> <p><a href="http://lh6.ggpht.com/-xs98D4PpiDU/U44lKdveSsI/AAAAAAAAuXM/FsIwMpqtbQg/s1600-h/clip_image039%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-FEmKzWp1on8/U44lLFNlsXI/AAAAAAAAuXU/NZ04hhirRPk/clip_image039_thumb%25255B1%25255D.jpg?imgmax=800" width="409" height="580" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The blue Y values of Series 1 underneath are uncovered and show through as the outer tails on each side. The resulting chart from the above data is shown as follow:</p> <p><a href="http://lh6.ggpht.com/--B9ZsTFR6sA/U44lLjghd1I/AAAAAAAAuXc/vg9CNoFmONo/s1600-h/clip_image040%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh4.ggpht.com/-pzpkr_ERzxg/U44lMsPjm9I/AAAAAAAAuXk/mZ-ySJXUQz4/clip_image040_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="198" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>It is possible to create a user input that automatically determines how many outer Y values on each side of Series 2 will be set to zero. The number of Y values set to zero on each side of Series 2 determines how much curve area each blue outer tail in the graph will contain.</p> <p>Each outer tail is addressed separately and has a separate user input assigned to it. Each of the two user inputs specify the percentage of the total curve area that will be displayed as a blue outer tail on its respective side.</p> <p>The upper user input set to the value of 0.05 controls the outer left tail. The user-controlled value of 0.05 specifies that the blue outer left tail will contain 5 percent of the total area under the curve. </p> <p>Each of the Y values in Series 2 is calculated in Excel by an If-Then-Else statement. The If-Then-Else statement sets the Y value to zero if the user input (currently set at 0.05 for both tails) is equal or greater than its respective CDF value (in the left tail) or 1 – CDF (in the right tail).</p> <p>When the above condition is not met, the If-Then-Else statement sets the Y value to its respective PDF value. The Y value will either be set to zero (if the condition is met) or set to its PDF value (if the condition is not met).</p> <p>The If-Then-Else formula is changed when the z Score reaches zero. This formula change ensures that the user input for each of the two outer tails will always be compared to the area in that outer tail. </p> <p>The CDF(X) states the percentage of the total bell-shaped normal curve is to the left of X. If point X is in the left tail, the curve area outside of point X in the left tail is specified by CDF(X). If point X is in the right tail, the curve area outside of point X in the right tail is specified by 1 – CDF(X). </p> <p>The following image shows the formula change at the point that the z Score equals zero. The Excel formulas for the Y values of Series 2 change when the z Score reaches zero. Y values in the left tail have the tan background. Y values in the right tail have the light blue background.</p> <p><a href="http://lh3.ggpht.com/-oNVfmdMzoos/U44lNdA7zHI/AAAAAAAAuXo/M5IfyQff7Z0/s1600-h/clip_image041%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-UbFan8eBzm4/U44lOCaff7I/AAAAAAAAuXw/JhE-kyseo6g/clip_image041_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="317" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>As a result, the settings shown below automatically configure the graph to display blue outer left and right tails that each contain 5 percent of the total area under the normal curve. </p> <p>This was implemented by setting the Y values of Series 2 to zero in the outer 5 percent of each tail. This enables the blue Y values of Series 1 underneath to be displayed in those outer tails.</p> <p><a href="http://lh3.ggpht.com/-tPF0Duz-eWE/U44lO31VpPI/AAAAAAAAuX4/oJJ_ZXESFL8/s1600-h/clip_image0021%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh6.ggpht.com/-qFGRxNgX3P8/U44lPm5t5MI/AAAAAAAAuYE/ehIVxLsumh8/clip_image0021_thumb%25255B1%25255D.jpg?imgmax=800" width="421" height="326" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Changes to any of the yellow user inputs are automatically reflected in the Excel graph. </p> <p>The following graph shows how the changes made to all nine user inputs are instantly and automatically reflected in the Excel area graph as follows:</p> <p><a href="http://lh3.ggpht.com/-7ZPrxLeeSSo/U44lQjrGYzI/AAAAAAAAuYM/4DdaPA78gwo/s1600-h/clip_image0011%25255B1%25255D.jpg"><img title="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="statistics, graph, excel excel 2010 graph, excel 2010, excel 2013" src="http://lh5.ggpht.com/-CrRH5rWcC-g/U44lRRGFQ6I/AAAAAAAAuYU/_6WjY128niM/clip_image0011_thumb%25255B1%25255D.jpg?imgmax=800" width="426" height="337" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com8tag:blogger.com,1999:blog-3568555666281177719.post-59964866807556756032014-06-03T12:22:00.001-07:002015-03-24T19:55:01.741-07:00Overview of the Normal Distribution<h1>Overview of the Normal <br /> <br />Distribution</h1> <p>This is one of the following eight articles on the normal distribution in Excel</p> <p><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution_3.html" target="_blank">Overview of the Normal Distribution</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-pdf-in-excel-2010.html" target="_blank">Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-cdf-in-excel-2010.html" target="_blank">Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/solving-normal-distribution-problems-in.html" target="_blank">Solving Normal Distribution Problems in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/standard-normal-distribution-in-excel.html" target="_blank">Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/an-important-difference-between-t-and.html" target="_blank">An Important Difference Between the t and Normal Distribution Graphs</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-empirical-rule-and-chebyshevs.html" target="_blank">The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-demonstrate-central-limit.html" target="_blank">Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way</a></p> <p>The normal distribution is a very useful and widely-occurring distribution. The normal distribution curve has the well-known bell shape and is symmetrical about a mean. The normal distribution is actually a family of distributions with each unique normal distribution being fully described by the following two parameters: the population mean, µ (Greek letter “mu”), and the population standard deviation, σ (Greek letter “sigma”). </p> <p>The following Excel-generated image is an example of the PDF (Probability Density Function) of a normal distribution curve whose population mean, µ, equals 10 and population standard deviation, σ, equals 5.</p> <p><a href="http://lh3.ggpht.com/-2w_Sl4wJB2I/U44gRXGHPfI/AAAAAAAAuYc/3lajo1DK_1A/s1600-h/clip_image001%25255B1%25255D.jpg"><img title="normal distribution, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics" src="http://lh5.ggpht.com/-887Xwhr3Lrc/U44gSRydILI/AAAAAAAAuYk/QZph94gGohI/clip_image001_thumb%25255B1%25255D.jpg?imgmax=800" width="422" height="264" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>Following is image of the PDF of a different normal distribution curve with a population mean, µ, equals 0 and population standard deviation, σ, equals 1. This is a special normal distribution known as the Standard Normal Distribution as shown in the following Excel-generated image:</p> <p><a href="http://lh5.ggpht.com/-s1H7ymzZkbQ/U44gTXvdLFI/AAAAAAAAuYs/_tQX8Qs4_ok/s1600-h/clip_image002%25255B1%25255D.jpg"><img title="normal distribution, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics" src="http://lh3.ggpht.com/-67H8tHXY5dw/U44gUcYzZdI/AAAAAAAAuYw/BsyVS5bBKaI/clip_image002_thumb%25255B1%25255D.jpg?imgmax=800" width="427" height="269" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>You may notice that the shapes of both normal distribution PDF graphs are the same but the x axis has been shifted and scaled. The entire family of normal distributions is symmetrical about a mean and contains approximately <b><u>68 percent</u></b> of the entire area under the curve within one standard deviation of the mean. Approximately <b><u>95 percent</u></b> of the total area under the curve lies within two standard deviations of the mean, and approximately <b><u>99.7 percent</u></b> of the total curve area within three standard deviations of the mean. </p> <p>If PDF curves of different normal distributions are created with the mean and standard deviation in the same locations on the horizontal axis, the curves will be in the same place and have exactly the same shape. The two previous images demonstrate this.</p> <p>If the PDF curves of different normal distributions are placed on the same horizontal and vertical axes, each curve will be shifted left or right so it is symmetrical about its population mean, µ, and its width will be scaled (widened or narrowed) depending on the size of its population standard deviation, σ. </p> <p>The PDF graph of a normal distribution with a smaller population standard deviation would appear thinner and taller than the PDF graph of a normal distribution with a larger population standard deviation, <b><u>if both curves were graphed on the same set of horizontal and vertical axes</u></b>.</p> <p> </p> <h2>Uses of the Normal Distribution</h2> <p>Many real-valued variables are modeled using the normal distribution. An example of normally-distributed natural phenomenon would be the velocities of the molecules in an ideal gas. Test scores for large numbers of people follow the normal distribution. Random variation in output of a machine barring special causes is often normal-distributed.</p> <p>In other cases it is often a variable’s logarithm that is normal-distributed. Biological sciences provide many such examples such as the length of appendages and blood pressure. A variable whose logarithm is normal-distributed is said to have a log-normal distribution.</p> <p>An outcome that is the result of a number of small variables that are additive, independent, and similar in magnitude is often normal-distributed. Measurement error of a physical experiment might be one example of such an outcome.</p> <p>In some fields the changes in the logarithm of a variable are assumed to be normal-distributed. In the financial fields, changes in the logarithms of exchange rates, price indices, and the stock market indices are modeled by the normal distribution.</p> <p> </p> <h2>Useful Normal Approximations of <br /> <br />Other Distributions</h2> <p>Several distributions can be approximated by the normal distribution under certain conditions. When the normal approximation is valid, tools such as normal-distribution-based hypothesis testing and hypothesis testing can be conveniently applied directly to samples from that-Distribution.</p> <p> </p> <h3>Normal Approximation of the Binomial Distribution</h3> <p>The normal approximation of the binomial distribution is of particular importance. Each unique binomial distribution, B(n,p), is described with the following two parameters: n (sample size) and p (probability of each binary sample having a positive outcome of the two possible outcomes). A binomial distribution is approximately normal-distributed with a mean np and variance np(1-p) if n is large an p is not too close to zero. A unique normal distribution, N(µ,σ), is completely described by the following two parameters: µ (population mean) and σ (population standard deviation). When the normal approximation of the binomial distribution is appropriate, normal-distribution-based hypothesis testing and confidence intervals can be performed with binomially-distributed data by applying the following substitution: N(µ,σ) = N<b>(</b>np, SQRT<b>(</b>np(1-p)<b>)</b><b>)</b>.</p> <p>The normal approximation of the binomial distribution enables the convenient analysis of population proportions with normal-distribution-based hypothesis testing and confidence intervals. A well-known example of this type of analysis would be the estimation of the percentage of a population along with a percent margin of error that will vote for or against a particular political candidate. This is known as a confidence interval of a population proportion. </p> <p>The t-Distribution (sometimes called the Student’s t-Distribution) is closely related to the normal distribution in function and appearance. The t-Distribution is used to analyze normally-distributed data when the sample size is small (n< 30) or the population standard deviation is not known. The t-Distribution is always centered about its mean of zero and closely resembles the standard normal curve. The standard normal curve is the unique normal distribution curve whose mean equals 0 and standard deviation equals 1. The t-Distribution has a lower peak and slightly thicker outer tails than the standard normal distribution, but converges to the exact shape of the standard normal distribution as sample size increases. </p> <p>In the real world, normally-distributed data are much more frequently analyzed with t-Distribution-based tools than normal-distribution-based tools. The reason is that the t-Distribution more correctly describes the distribution normally-distributed data in the common occurrences of small sample size and unknown population standard deviation. t-Distribution-based tools are also equally appropriate for analysis of large samples of normally-distributed data because the t-Distribution converges to nearly an exact match of the standard normal distribution when sample size exceeds 30.</p> <p><b></b></p> <h3>Normal Approximation of the Poisson Distribution</h3> <p>Of lesser importance is the normal distribution’s approximation of the Poisson and chi-Square distribution under certain conditions. Each unique Poisson distribution curve is completely described by a single parameter λ. When λ is large, data distributed according to the Poisson distribution can be analyzed with normal-distribution-based tools by making the following substitution: </p> <p>N(µ,σ) = N(λ, SQRT(λ)).</p> <p>Each unique Chi-Square distribution curve is completely described by the single parameter that is its degrees of freedom, k. When k is large, data distributed according to the Chi-Square distribution can be analyzed with normal-distribution-based tools by making the following substitution: </p> <p>N(µ,σ) = N(k, SQRT(2k)). </p> <p>One of the most useful modeling applications of the normal distribution is due to the fact the means of large random samples from a population are approximately normal-distributed no matter how the underlying population is distributed. This property is described by the Central Limit Theorem. Normal-distribution based analysis tools such as hypothesis testing and confidence intervals can therefore be applied to the means of samples taken from populations that are not normal-distributed. </p> <p> </p> <h2>Statistical Tests That Require <br /> <br />Normality of Data or Residuals</h2> <p>Statistical tests that as classified a <i>parametric </i>tests have a requirement that the sample data or the residuals are normal-distributed. For example, linear regression requires that the residuals be normal-distributed. t-tests performed on small samples (n < 30) require that the samples are taken from a normally-distributed population. ANOVA requires normality of the samples being compared.</p> <p>When normality requirements cannot be met, <i>nonparametric</i> tests can often be substituted for parametric tests. Nonparametric tests do not requirements that data or residuals follow a specific distribution. Nonparametric tests are usually not as powerful as parametric tests.</p> <p> </p> <h2>History of the Normal Distribution</h2> <p>The first indirect mention of the normal distribution is credited to French mathematician and friend of Isaac Newton, Abraham De Moivre. De Moivre developed such repute as a mathematician that Newton often referred questions to him, saying “Go to Mr. de Moivre. He knows these things better than I do.”</p> <p>In 1738 De Moivre published the second edition of his <i>The Doctrine of Chances</i> that contained a study of binomial coefficients which is considered to the first reference, albeit, indirect, to the normal distribution. That particular book became highly valued among gamblers of the time. The book noted that the distribution of the number of times that a binary event produces a positive outcome, such as a coin toss resulting in heads, becomes a smooth, bell-shaped curve when the number of trials is high enough. This bell-shaped curve approaches the normal curve. DeMoivre was alluding to what we know today as the normal distribution’s approximation of the binomial distribution. DeMoivre did not speak of the normal distribution in terms of a probability density function and therefore does not receive full credit for discovering the normal distribution. </p> <p>De Moivre, who spent his adult life in London, remained somewhat poor because he was unable to secure a professorship at any local university, partially due to his French origins. He earned a substantial part of his living from tutoring mathematics and being a consultant to gamblers. One day when De Moivre became older, he noticed that he required more sleep every night. He determined that he was sleeping an extra 15 minutes every night. He correctly calculated the date his own death to be November 27, 1754, the date that the total required sleep time would reach 24 hours. </p> <p>The first true mention of the normal distribution was made by German mathematician Carl Friedrich Gauss in 1809 as a way to rationalize the nonlinear method of least squares. The normal distribution curve is often referred to as the Gaussian curve as a result.</p> <p>Gauss was one of the greatest mathematicians who ever lived and is sometimes referred to as “the Prince of Mathematicians” and “the greatest mathematician since antiquity.” Gauss was a child prodigy and made some of his groundbreaking mathematical discoveries as a teenager. Gauss was an astonishingly prolific scientist in many fields. Here is a link to a partial list of over 100 scientific topics named after him:</p> <p><a href="http://en.wikipedia.org/wiki/List_of_things_named_after_Carl_Friedrich_Gauss">http://en.wikipedia.org/wiki/List_of_things_named_after_Carl_Friedrich_Gauss</a></p> <p>Significant contributions to the normal distribution were made by another of the greatest mathematicians of all-time, Frenchman Pierre Simon LePlace, who was sometimes referred to as the French Newton. LePlace is credited with providing the first proof of one of statistics’ most important tenets related to the normal distribution, the Central Limit Theorem. LePlace has an interesting life and was appointing by Napoleon to be the French Minister of the Interior shortly after Napoleon seized power in a coup. The appointment lasted about six weeks until nepotism took over and the post was given to Napoleon’s brother.</p> <p>The distribution’s moniker, the “normal distribution,” was made popular by Englishman Karl Pearson, another giant in the field of mathematics. Karl Pearson is credited with establishing the discipline of mathematical statistics and founded the world’s first university statistics department in 1911 at the University College of London. Many of topics covered in all basic statistics courses such a p Values and correlation are the direct result of Karl Pearson’s work. </p> <p>Some of Pearson’s publishings, particularly his book <i>The Grammar of Science</i>, provided a number of themes that Einstein would weave into several of his most well-known theories of relativity. Pearson postulated that a person traveling faster than the speed of light would experience time being reversed. Pearson also discussed the concept that the operation of physical laws depended on the relative position of the observer. These are central themes in Einstein’s relativity theories.</p> <p> </p> <h2>Properties of the Normal <br /> <br />Distribution</h2> <p>This graph of the standard normal distribution’s PDF is once again presented to assist in the understanding of each listed property of the normal distribution.</p> <p><a href="http://lh6.ggpht.com/-yMfh2envyhY/U44gVPDv1KI/AAAAAAAAuY8/YVASBw1PLv0/s1600-h/clip_image0021%25255B1%25255D.jpg"><img title="normal distribution, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics" src="http://lh4.ggpht.com/-MOhdzpS3mwY/U44gVnA3iiI/AAAAAAAAuZE/3DDe2HtLhXA/clip_image0021_thumb%25255B1%25255D.jpg?imgmax=800" width="416" height="262" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p>The normal distribution’s probability density function, f(x), has the following properties:</p> <p>1) It is symmetric about its population mean µ. Half of the values of a normally-distributed population will be less than (to the left of) the population mean and the other half of the population’s value will be greater than (to the right of) the population mean.</p> <p>2) Its mode and median are equal to the population mean µ.</p> <p>3) It is unimodal. This means that it has only one peak, i.e., only one point that is a local maximum.</p> <p>4) The total area under the normal distribution’s PDF is equal to 1.</p> <p>5) Each unique normal distribution curve is entirely defined by its two parameters population mean µ and population standard deviation σ.</p> <p>6) The density of the normal distribution’s PDF is the highest at its mean and always decreases as distance from the mean increases.</p> <p>7) 50 percent of values of a normally-distributed population are less than the population mean and 50 percent of the values are greater than the mean.</p> <p>8) Approximately 68 percent of the total area <b>under</b> the PDF curve resides within one σ from the mean, approximately 95 percent of the total resides within two σs, and approximately 99.7 percent of the total area resides within three σs from the mean. This is sometimes known as the <b><u>Empirical Rule or the 68-95-99.7 Rule</u></b>.</p> <p>9) f(x) is infinitely differentiable.</p> <p>10) The first derivative of f(x) is positive for all x < µ and negative for all x > µ.</p> <p>11) The second derivative of f(x) has two inflection points which are located one population standard deviation above and below the population mean. These inflection points are located at x = µ ± σ. An inflection point occurs at the point that the 2<sup>nd</sup> derivative equals zero and changes sign as x continues.</p> <p>12) It is log-concave. A function f(x) is log-concave if its natural log, ln[f(x)], is concave. A log-concave function does not have multiple separate maxima and its tails are not “too thick.” Other well-known distributions that are log-concave include the following:</p> <p>- exponential distribution</p> <p>- uniform distribution over any convex set</p> <p>- logistic distribution</p> <p>- gamma distribution if its shape parameter is >=1</p> <p>- Chi-Square distribution if the number of degrees of freedom >=2</p> <p>- beta distribution if both shape parameters are >=1</p> <p>- Weibull distribution if the shape parameter is >=1</p> <p>The following well-known distributions are non-log-concave for all parameters:</p> <p>- t-Distribution</p> <p>- log-normal distribution</p> <p>- F-distribution</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com3tag:blogger.com,1999:blog-3568555666281177719.post-14936071206445793202014-06-03T12:01:00.001-07:002015-03-24T20:01:05.617-07:00Normal Distribution’s PDF in Excel 2010 and Excel 2013<h1><a name="_Toc379362745">Normal Distribution’s PDF <br /> <br />(Probability Density <br /> <br />Function)</a> in Excel</h1> <p>This is one of the following eight articles on the normal distribution in Excel</p> <p><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution_3.html" target="_blank">Overview of the Normal Distribution</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-pdf-in-excel-2010.html" target="_blank">Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-cdf-in-excel-2010.html" target="_blank">Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/solving-normal-distribution-problems-in.html" target="_blank">Solving Normal Distribution Problems in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/standard-normal-distribution-in-excel.html" target="_blank">Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/an-important-difference-between-t-and.html" target="_blank">An Important Difference Between the t and Normal Distribution Graphs</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-empirical-rule-and-chebyshevs.html" target="_blank">The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-demonstrate-central-limit.html" target="_blank">Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way</a></p> <p>The normal distribution is a family of distributions with each unique normal distribution being fully described by its two parameters µ (“mu,” population mean) and σ (“sigma,” population standard deviation). The population mean, µ, is a <b><u>location parameter</u></b> and the population standard deviation, σ, is a <b><u>scale parameter</u></b>. When two different normal distribution curves are plotted on the same set of horizontal and vertical axes, the means determine how shifted one curve is to the left of right of the other curve. The standard deviations detail how much wider or more narrow the first normal curve is than to the second.</p> <p>The normal distribution is often denoted as <i>N(µ,σ<sup>2</sup>)</i>. σ<sup>2</sup> equals the population variance. When a random variable X is normal-distributed with a population mean µ and population variance <i>σ<sup>2</sup></i>, it is written in the following form:</p> <p><i>X ~ N(µ, σ<sup>2</sup>)</i></p> <p>It is important to note that the two parameters of the normal distribution are population parameters, not measurements taken from a sample. Sample statistics would provide only estimates of population parameters. The t-Distribution is used to analyze normally-distributed data when only sample statistics and/or population parameters are not known. In the real world it is much more common to analyze normally-distributed data with t-Distribution based tests than normal-distribution-based tests because only data from small samples (n<30) are available.</p> <p>As with all distributions, the normal distribution has a PDF (Probability Density Function) and a CDF (Cumulative Distribution Function).</p> <p>The normal distribution’s PDF (Probability Density Function) equals the probability that sampled point from a normal-distributed population has a value <b><u>EXACTLY EQUAL TO</u></b> X given the population’s mean, µ, and standard deviation, σ.</p> <p>The normal distribution’s PDF is expressed as f(X,µ,σ).</p> <p>f(X,µ,σ) = the probability that a randomly-sampled point taken from normally-distributed population with a mean µ and standard deviation σ has the value of X. It is given by the following formula:</p> <p><a href="http://lh4.ggpht.com/-R6E6fQ2MPHw/U44bYNHReNI/AAAAAAAAuMY/55LKVZd43BE/s1600-h/clip_image001%25255B1%25255D.png"><img title="normal distribution pdf in excel 2010 and excel 2013 statistical distribution" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution pdf in excel 2010 and excel 2013 statistical distribution" src="http://lh3.ggpht.com/-BF5Ke-X6lDY/U44bY4phhxI/AAAAAAAAuMg/ixaMSNtKdr8/clip_image001_thumb%25255B1%25255D.png?imgmax=800" width="370" height="66" /></a> <br /><i>(Click On Image to See a Larger Version)</i></p> <p>exp refers to the value of the mathematical constant e which is the base of the natural logarithm. e is equal to 2.71828 and is the limit of (1 + 1/n)<sup>n</sup> and n approaches infinity. e<b><sup>a</sup></b> would be expressed in Excel as =exp(a).</p> <p>The mathematical constant π (“pi”) is equal to 3.14159 and is the ratio of a circle’s circumference to its diameter.</p> <p>In Excel 2010 and beyond, the normal distribution’s PDF can be calculated directly by the following Excel formula:</p> <p>f(X,µ,σ) = NORM.DIST(X,µ,σ,FALSE)</p> <p>The Excel formula parameter “FALSE” indicates that the formula is calculating the normal distribution’s PDF (Probability Density Function) and not its CDF (Cumulative Distribution Function)</p> <p>Prior to Excel 2010, the normal distribution’s PDF was calculated in Excel by this formula:</p> <p>f(X,µ,σ) = NORMDIST(X,µ,σ,FALSE)</p> <p>Statistical formulas that worked in Excel versions prior to 2010 will also work in Excel 2010 and 2013.</p> <p>The following Excel-generated graph shows the PDF of a normal distribution that has a population mean of 10 and population standard deviation equal to 5. </p> <p><a href="http://lh5.ggpht.com/-DsyIDspihho/U44baXFpYkI/AAAAAAAAuNM/TB3lBjst3Ow/s1600-h/clip_image002%25255B5%25255D.jpg"><img title="normal distribution, pdf, probability density function, excel, excel 2010, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, pdf, probability density function, excel, excel 2010, excel 2013, statistics" src="http://lh4.ggpht.com/-aEUn46MLu7c/U44ba_keZuI/AAAAAAAAuNU/8Ryl8pmiJsY/clip_image002_thumb%25255B5%25255D.jpg?imgmax=800" width="411" height="263" /></a> <br /><i>(Click On Image to See a Larger Version)</i></p> <p> </p> <h2>Normal Distribution PDF <a name="_Toc379362746">Example in <br /> <br />Excel</a></h2> <p>Determine the probability that a randomly-selected variable X taken from a normally-distributed population has the value of 5 if the population mean equals 10 and the population standard deviation equals 5. The preceding Excel-generated image shows a normal distribution PDF curve with the population mean equaling 10 and the population standard deviation equaling 5.</p> <p>X = 5</p> <p>µ = 10</p> <p>σ = 5</p> <p>f(X,µ,σ) = NORM.DIST(X,µ,σ,FALSE)</p> <p>f(X=5,µ=10,σ=5) = NORM.DIST(5,10,5,FALSE) = 0.04834</p> <p>There is a 4.834 percent chance that randomly-selected X = 5 if X is taken from a normally-distributed population with a population mean µ = 10 and population standard deviation σ = 5. The PDF diagram of this normal distribution curve also shows the probability of X at X = 5 to that value.</p> <p>Performing the same calculation in Excel using the full normal distribution PDF formula as shown as follows:</p> <p><a href="http://lh4.ggpht.com/-giQ0l_dwnFg/U44bbpLHdGI/AAAAAAAAuNc/zMHsu7iuwis/s1600-h/clip_image0011%25255B4%25255D.png"><img title="normal distribution, pdf, probability density function, excel, excel 2010, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, pdf, probability density function, excel, excel 2010, excel 2013, statistics" src="http://lh5.ggpht.com/-QlVTpkg70ds/U44bcUUVDTI/AAAAAAAAuNk/qDhIm2CAuM4/clip_image0011_thumb%25255B4%25255D.png?imgmax=800" width="370" height="66" /></a> <br /><i>(Click On Image to See a Larger Version)</i></p> <p>f(X=5,µ=10,σ=5) =<strong><font color="#ff0000">(</font></strong>1/<b><font color="#0000ff">(</font></b>SQRT<b><font color="#008000">(</font></b>2*3.14159*5^2<b><font color="#008000">)</font></b><b><font color="#0000ff">)</font></b><b><font color="#ff0000">)</font></b>*EXP<b><font color="#ff0000">(</font></b>(-1)*<b><font color="#0000ff">(</font></b>(5-10)^2<b><font color="#0000ff">)</font></b>/<b><font color="#0000ff">(</font></b>2*5^2<b><font color="#0000ff">)</font></b><b><font color="#ff0000">)</font></b></p> <p>= 0.04834</p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com2tag:blogger.com,1999:blog-3568555666281177719.post-85066210312302764282014-06-03T11:52:00.001-07:002015-03-24T20:12:07.721-07:00Normal Distribution’s CDF in Excel 2010 and Excel 2013<p>This is one of the following eight articles on the normal distribution in Excel</p> <p><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution_3.html" target="_blank">Overview of the Normal Distribution</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-pdf-in-excel-2010.html" target="_blank">Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-cdf-in-excel-2010.html" target="_blank">Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/solving-normal-distribution-problems-in.html" target="_blank">Solving Normal Distribution Problems in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/standard-normal-distribution-in-excel.html" target="_blank">Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/an-important-difference-between-t-and.html" target="_blank">An Important Difference Between the t and Normal Distribution Graphs</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-empirical-rule-and-chebyshevs.html" target="_blank">The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-demonstrate-central-limit.html" target="_blank">Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way</a></p> <h1>The Normal Distribution’s </h1> <h1>CDF (Cumulative </h1> <h1>Distribution Function)in </h1> <h1>Excel</h1> <p>The normal distribution’s CDF (Cumulative Distribution Function) equals the probability that sampled point from a normal-distributed population has a value <b><u>UP TO</u></b> X given the population’s mean, µ, and standard deviation, σ.</p> <p>The normal distribution’s CDF is expressed as F(X,µ,σ).</p> <p><a href="http://lh4.ggpht.com/-vBdKsbRnPUA/U44ZOUxI9pI/AAAAAAAAuZM/3IKkhMLUdNA/s1600-h/clip_image001%25255B1%25255D.png"><img title="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" src="http://lh5.ggpht.com/-5Kbnp2f49pw/U44ZPuBggqI/AAAAAAAAuZU/MlCg2D9mNJY/clip_image001_thumb%25255B1%25255D.png?imgmax=800" width="416" height="159" /></a> <br /><i>(Click on Image To See a Larger Version)</i></p> <p>Unlike the normal distribution’s PDF, the CDF has no convenient closed form of its equation, which is the integral just shown.</p> <p>In Excel 2010 and beyond, the normal distribution’s CDF must be calculated by the following Excel formula:</p> <p>F(X,µ,σ) = NORM.DIST(X,µ,σ,TRUE)</p> <p>The Excel formula parameter “TRUE” indicates that the formula is cumulative, i.e., it is calculating the normal distribution’s CDF (Cumulative Distribution Function) and not its PDF (Probability Density Function).</p> <p>Prior to Excel 2010, the normal distribution’s PDF was calculated in Excel by this formula:</p> <p>F(X,µ,σ) = NORMDIST(X,µ,σ,TRUE)</p> <p>Statistical formulas that worked in Excel versions prior to 2010 will also work in Excel 2010 and 2013.</p> <p>Note that the CDF has asymptotic values of 0 as X decreases and an asymptotic value of 1 as X increases as shown in the following Excel-generated image:</p> <p><a href="http://lh4.ggpht.com/-0I574Pz9h5g/U44ZQNIR1mI/AAAAAAAAuZc/rUFKs0ELmGA/s1600-h/clip_image002%25255B1%25255D.jpg"><img title="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" src="http://lh6.ggpht.com/-mZBn31CIlmM/U44ZRJXvH1I/AAAAAAAAuZk/negHIL-fdqI/clip_image002_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="278" /></a> <br /><i>(Click on Image To See a Larger Version)</i></p> <p>The normal distribution’s CDF has a value of exactly 0.5 when X equals the population mean. This indicates that 50 percent of the entire area under the normal distribution’s PDF is contained under the curve before X reaches a value of the population mean. The underlying meaning is that a randomly-sample point from a normally-distributed population has a 50 percent chance of having a value less than or equal the population mean. This can be seen in the Excel-generated graph of the normal distribution’s PDF as follows:</p> <p><a href="http://lh6.ggpht.com/-3MLIe6d7MKg/U44ZStIpYoI/AAAAAAAAuZs/Gm23amjDUPk/s1600-h/clip_image003%25255B1%25255D.jpg"><img title="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" src="http://lh6.ggpht.com/-CUQS17mfKBA/U44ZTBLDu0I/AAAAAAAAuZ0/z-yeXKp00pU/clip_image003_thumb%25255B1%25255D.jpg?imgmax=800" width="422" height="264" /></a> <br /><i>(Click on Image To See a Larger Version)</i></p> <p>This CDF always has a value of 0.15866 when X is one standard deviation below the mean. This indicates that 15.866 percent of the area under the normal distribution’s PDF curve occurs before X reaches the value of the point that is one standard deviation below the population’s mean. The underlying meaning is that a randomly-sample point from a normally-distributed population has a 15.866 percent chance of having a value less than or equal the value that is one standard deviation below the population mean.</p> <p>This CDF always has a value of 0.84135 when X is one standard deviation above the mean. This indicates that 84.135 percent of the area under the normal distribution’s PDF curve occurs before X reaches the value of the point that is one standard deviation below the population’s mean. The underlying meaning is that a randomly-sample point from a normally-distributed population has an 84.135 percent chance of having a value less than or equal the value that is one standard deviation below the population mean.</p> <p>A randomly-selected point from a normally-distributed population has a 68.269 percent chance of having a value X that is within one standard deviation above or below the mean (84.135 – 15.866 = 68.269). In other words, 68.269 percent of normally-distributed data lie within one standard deviation of the mean. Similar analysis shows that approximately 95 percent of all normally-distributed data lie within two standard deviations of the mean and 99.7 percent of the data are within three standard deviations of the mean. This rule is often referred to as the Empirical Rule or the 68-95-99.5 Rule.</p> <p> </p> <h2>Normal Distribution CDF <a name="_Toc379362748">Example in <br /> <br />Excel</a></h2> <p>Determine the probability that a randomly-selected variable X taken from a normally-distributed population has the value of <b><u>UP TO</u></b> 5 if the population mean equals 10 and the population standard deviation equals 5.</p> <p>X = 5</p> <p>µ = 10</p> <p>σ = 5</p> <p>F(X,µ,σ) = NORM.DIST(X,µ,σ,TRUE)</p> <p>F(X=5,µ=10,σ=5) = NORM.DIST(5,10,5,TRUE) = 0.15866</p> <p>There is a 15.866 percent chance that randomly-selected X equals <b><u>UP TO</u></b> 5 if X is taken from a normally-distributed population with a population mean µ = 10 and population standard deviation σ = 5. The CDF diagram of this normal distribution curve also shows the probability of X at X = 5 to that value in the following Excel-generated image.</p> <p><a href="http://lh5.ggpht.com/-_zhsg7-sRuk/U44ZT6MzK8I/AAAAAAAAuZ8/OsGDFX_D_Kg/s1600-h/clip_image004%25255B1%25255D.jpg"><img title="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" src="http://lh4.ggpht.com/-YUdqb7BZF2g/U44ZUsl7jYI/AAAAAAAAuaE/j0jFk8tY2to/clip_image004_thumb%25255B1%25255D.jpg?imgmax=800" width="418" height="276" /></a> <br /><i>(Click on Image To See a Larger Version)</i></p> <p>The following PDF diagram of this normal distribution curve shows 15.866 percent of the total area under the bell-shaped curve that is to the left of X = 5.</p> <p>.<b><a href="http://lh5.ggpht.com/-YfDiog8f1OY/U44ZWK3quMI/AAAAAAAAuaM/FBpmYKiJfYg/s1600-h/clip_image005%25255B1%25255D.jpg"><img title="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, cdf, cumulative distribution function, excel, excel 20101, excel 2013, statistics" src="http://lh3.ggpht.com/-5fLHUiYkE-g/U44ZWsoCcjI/AAAAAAAAuaU/nWLEp0s_NbY/clip_image005_thumb%25255B1%25255D.jpg?imgmax=800" width="424" height="265" /></a></b> <br /><i>(Click on Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com26tag:blogger.com,1999:blog-3568555666281177719.post-72710571489185874732014-06-03T11:43:00.001-07:002015-03-24T20:16:30.313-07:00Solving Normal Distribution Problems in Excel 2010 and Excel 2013<p>This is one of the following eight articles on the normal distribution in Excel</p> <p><a href="http://blog.excelmasterseries.com/2014/06/overview-of-normal-distribution_3.html" target="_blank">Overview of the Normal Distribution</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-pdf-in-excel-2010.html" target="_blank">Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/normal-distributions-cdf-in-excel-2010.html" target="_blank">Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/solving-normal-distribution-problems-in.html" target="_blank">Solving Normal Distribution Problems in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/standard-normal-distribution-in-excel.html" target="_blank">Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013</a></p> <p><a href="http://blog.excelmasterseries.com/2014/06/an-important-difference-between-t-and.html" target="_blank">An Important Difference Between the t and Normal Distribution Graphs</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-empirical-rule-and-chebyshevs.html" target="_blank">The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean</a></p> <p><a href="http://blog.excelmasterseries.com/2014/05/how-to-demonstrate-central-limit.html" target="_blank">Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way</a></p> <h1>Solving Normal <br /> <br />Distribution Problems in <br /> <br />Excel</h1> <p> </p> <h2>Problems Using NORM.<font color="#ff0000">DIST</font>() To <br /> <br />Calculate F(X,µ,σ)</h2> <p>F(X,µ,σ) = NORM.DIST(X,µ,σ,TRUE)</p> <p> </p> <h3>Problem 1</h3> <p>A certain brand of automobiles has normally-distributed fuel consumption. The brand’s mean fuel consumption is 27 miles-per-gallon and standard deviation is 5 miles-per-gallon. What percentage of automobiles of this brand can be expected to have fuel consumption between 25 and 30 miles-per-gallon?</p> <p>0.3812 = NORM.DIST(30,27,5,TRUE) - NORM.DIST(25,27,5,TRUE)</p> <p>0.3812 = 0.7257 – 0.3446</p> <p>38.12 percent of autos of the brand have fuel consumption between 25 and 30 miles-per-gallon as shown in the following Excel-generated graph:</p> <p><a href="http://lh6.ggpht.com/--GWcxCps_kU/U44XMkqtw9I/AAAAAAAAuac/fvbcQ0ZFJII/s1600-h/clip_image001%25255B1%25255D.jpg"><img title="normal distribution, statistics, excel, excel 20101, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics, excel, excel 20101, excel 2013" src="http://lh3.ggpht.com/-B1XLyIa6yMo/U44XNeu77cI/AAAAAAAAuak/8wOF3TDhpTI/clip_image001_thumb%25255B1%25255D.jpg?imgmax=800" width="419" height="264" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h3>Problem 2</h3> <p>A company packages potatoes into knitted net bags for shipment to supermarkets. The weight of the filled bags of potatoes is normal-distributed with a mean of 20 pounds and a standard deviation of 1 pound. What is the probability that a randomly selected bag of potatoes will either weigh more than 22 pounds or less than 18.5 pounds?</p> <p>Probability that potatoes weigh less than 18.5 pounds =</p> <p>NORM.DIST(18.5,20,1,TRUE)</p> <p>Probability that potatoes weigh less than 18.5 pounds = 0.0668 = 6.68 %</p> <p>Probability that potatoes weigh less than 22 pounds = NORM.DIST(22,20,1,TRUE)</p> <p>Probability that potatoes weigh more than 22 pounds = 1 - NORM.DIST(22,20,1,TRUE)</p> <p>Probability that potatoes weigh more than 22 pounds = 1 - 0.9773 = 0.0227 = 2.27 %</p> <p>Probability that the potatoes weigh less than 18.5 pounds OR more than 22 pounds equals</p> <p>Probability that the potatoes weigh less than 18.5 pounds = 0.0668</p> <p>PLUS</p> <p>Probability that the potatoes weigh more than 22 pounds = 0.0227</p> <p>Which equals </p> <p>NORM.DIST(18.5,20,1,TRUE) + [1 - NORM.DIST(22,20,1,TRUE) ]</p> <p>Which equals </p> <p>0.0668 + 0.0227 = 0.0896 = 8.96 %</p> <p>8.96 percent of bags of potatoes have weights outside of 18.5 pounds and 22 pounds as shown in the following Excel-generated graph:</p> <p><a href="http://lh5.ggpht.com/-Hv1GNF4QjW8/U44XORMVOII/AAAAAAAAuas/4bu5mEdLnUI/s1600-h/clip_image002%25255B1%25255D.jpg"><img title="normal distribution, statistics, excel, excel 20101, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics, excel, excel 20101, excel 2013" src="http://lh6.ggpht.com/-UC34nZUVX7c/U44XPMHv7GI/AAAAAAAAua0/lgM3VmhSkfI/clip_image002_thumb%25255B1%25255D.jpg?imgmax=800" width="418" height="261" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h2>Problems Using NORM.<font color="#ff0000">INV</font>() To <br /> <br />Calculate X Given F(X,µ,σ)</h2> <p>NORM.INV[ F(X,µ,σ), µ, σ ] = X</p> <p> </p> <h3>Problem 1</h3> <p>A tire company makes a tire with a normally-distributed tread life. The mean tread life is 39,000 miles and the standard deviation of tread life is 5,300 miles. What tread life would be exceeded by only 3 percent of all tires?</p> <p>48968 = NORM.INV(0.97,39000,5300)</p> <p>97 percent of all tires will wear out before they are driven 48,968 miles hours as shown in the following Excel-generated graph:</p> <p><a href="http://lh4.ggpht.com/-ENr9CYzLrjs/U44XSZKALnI/AAAAAAAAua8/SQs95O1r_WM/s1600-h/clip_image003%25255B1%25255D.jpg"><img title="normal distribution, statistics, excel, excel 20101, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics, excel, excel 20101, excel 2013" src="http://lh6.ggpht.com/-PHXlMRbxzB4/U44XSzYExUI/AAAAAAAAubE/_8FvPxxwTO0/clip_image003_thumb%25255B1%25255D.jpg?imgmax=800" width="420" height="264" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p> </p> <h3>Problem 2</h3> <p>A company’s package delivery time is normal-distributed and has a mean of 10 hours and a standard deviation of 3 hours. What delivery time will be beaten by only 2.5 percent of all deliveries?</p> <p>4.120 = NORM.INV(0.025,10,3)</p> <p>Only 2.5 percent of all deliveries are made quicker than 4.120 hours as shown in the following Excel-generated graph:</p> <p><a href="http://lh5.ggpht.com/-47jLExi1Ojo/U44XTyXvA6I/AAAAAAAAubM/nXAhAuM4cEo/s1600-h/clip_image004%25255B1%25255D.jpg"><img title="normal distribution, statistics, excel, excel 20101, excel 2013" style="border-left-width: 0px; border-right-width: 0px; border-bottom-width: 0px; display: inline; border-top-width: 0px" border="0" alt="normal distribution, statistics, excel, excel 20101, excel 2013" src="http://lh3.ggpht.com/-A6yRVaYbeQk/U44XUxEBc9I/AAAAAAAAubU/R2QcFFoHE7M/clip_image004_thumb%25255B1%25255D.jpg?imgmax=800" width="422" height="264" /></a> <br /><i>(Click On Image To See a Larger Version)</i></p> <p align="center"> </p> <p align="center"><span style="font-size: x-large; color: red"><strong>Excel Master Series Blog Directory</strong></span></p> <p align="center"> </p> <p align="center"><a href="http://blog.excelmasterseries.com/2015/03/blog-directory-of-statistics-topics-and.html" target="_blank"><span style="font-size: x-large; text-decoration: underline; font-family: arial, helvetica, sans-serif; color: navy; text-align: center"><strong>Click Here To See a List Of All <br /> <br />Statistical Topics And Articles In <br /> <br />This Blog</strong></span> </a></p> <p align="center"> </p> <p align="center"><strong>You Will Become an Excel Statistical Master!</strong></p> Anonymoushttp://www.blogger.com/profile/02423338645515400885noreply@blogger.com14