2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013

This is one of the following four articles on z-Tests in Excel

Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013

One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013

2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013

Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

 

Two-Independent

Sample, Unpooled z-Test

in 4 Steps in Excel

 

This hypothesis test evaluates two independent samples to determine whether the difference between the two sample means (x_bar₁ and x_bar₂) is equal to (two-tailed test) or else greater than or less than (one-tailed test) than a constant.

This is an unpooled test. An unpooled test can always be used in place of a pooled test. An unpooled test must be used when population variances are not similar. An unpooled test calculates the Standard Error using separate standard deviations instead of combining them into a single, pooled standard deviation as a pooled test does.

The t-Test is nearly always used to compare two independent samples. For this reason, only the unpooled, two-independent-sample z-Test will be covered. The pooled version of this z-Test will not be covered.

In the real world, the only the sample variances are known but the population variances are usually not known and therefore t-tests are nearly always used to perform a two-independent-sample hypothesis test of mean. For this reason, only the unpooled, two-independent-sample z-Test will be explained. This z-Test can always be used in place of the pooled z-Test that could be used if population variances were known to be similar enough.

x_bar₁ - x_bar₂ = Observed difference between the sample means

(Click On Image To See a Larger Version)

Note that this is the same formula for SE for the two-independent-sample, unpooled t-test except that the variance for the z-Test is the population variance as follows:

var₁ = σ₁²

var₂ = σ₂²

and not the sample variance used for the t-test as follows:

var₁ = s₁²

var₂ = s₂²

 
(Click On Image To See a Larger Version)

Null Hypothesis H₀: x_bar₁ - x_bar₂ = Constant

The Null Hypothesis is rejected if any of the following equivalent conditions are shown to exist:

1) The observed x_bar₁ - x_bar₂ is beyond the Critical Value.

2) The z Score (the Test Statistic) is farther from zero than the Critical t Value

3) The p value is smaller than α for a one-tailed test or α/2 for a two-tailed test.

 

Example of 2-Sample, 2-Tailed,

Unpooled z-Test in Excel

This problem is very similar to the problem solved in the t-test section for a two-independent-sample, two-tailed t-test. Similar problems were used in each of these sections to show the similarities and also contrast the differences between the two-independent-sample z-Test and t-test as easily as possible.

Two shifts on a production are being compared to determine if there is a difference in the average daily number of units produced by each shift. The two shifts operate eight hours per day under nearly identical conditions that remain fairly constant from day to day. A sample of the total number of units produced by each shift on a random selection of days is taken. Determine with a 95 percent Level of Confidence if there is a difference between the average daily number of units produced by the two shifts.

Note that when performing two-sample z-tests in Excel, always designate Sample 1 (Variable 1) to be the sample with the larger mean.

The results of the two-independent-sample z-Test will be more intuitive if the sample group with the larger mean is designated as the first sample and the sample group with the smaller mean is designated as the second sample.

Details about both data samples are shown as follows:

 

Summary of Problem Information

 

Sample Group 1 – Shift A (Variable 1)

x_bar₁ = sample₁ mean = 46.55

µ₁ (Greek letter “mu”) = population mean from which Sample 1 was drawn = Not Known

σ₁ (Greek letter “sigma”) = population standard deviation from which Sample 1 was drawn = 25.5

Var₁ = population₁ variance = σ₁² = 650.25

n₁ = sample₁ size = 40

 

Sample Group 2 – Shift B (Variable 2)

x_bar₂ = sample_s mean = 42.24

µ₂ (Greek letter “mu”) = population mean from which Sample 2 was drawn = Not Known

σ₂ (Greek letter “sigma”) = population standard deviation from which Sample 2 was taken = 11.2

Var₂ = population₂ variance = σ₂² = 125.44

n₂ = sample₁ size = 36

x_bar₁ - x_bar₂ = 46.55 – 42.24 = 4.31

Level of Certainty = 0.95

Alpha = 1 - Level of Certainty = 1 – 0.95 = 0.05

As mentioned, always designate Sample 1 (Variable 1) to be the sample with the larger mean when performing two-sample z-Tests in Excel.

The results of the unpooled z-Test will be more intuitive if the sample group with the larger mean is designated as the first sample and the sample group with the smaller mean is designated as the second sample.

Another reason for designating the sample group with the larger mean as the first sample is to obtain the correct result from the Excel data analysis tool for two-independent-sample, unpooled z-Tests called the z-Test:Two-Sample for Means. The test statistic (z in the Excel output, which stands for z Score) and the Critical z value (z Critical in the Excel output) will have the same sign (as they always should) only if the sample group with the larger mean is designated the first sample.

As with all Hypothesis Tests of Mean, we must satisfactorily answer these two questions and then proceed to the four-step method of solving the hypothesis test that follows.

 

The Initial Two Questions That Must be Answered Satisfactorily

What Type of Test Should Be Done?

Have All of the Required Assumptions For This Test Been Met?

 

The Four-Step Method For Solving All Hypothesis Tests of Mean

Step 1) Create the Null Hypothesis and the Alternative Hypothesis

Step 2 – Map the Normal or t-Distribution Curve Based on the Null Hypothesis

Step 3 – Map the Regions of Acceptance and Rejection

Step 4 – Perform the Critical Value Test, the p Value Test, or the Critical t Value Test

 

The initial two questions that need to be answered before performing the Four-Step Hypothesis Test of Mean are as follows:

 

Question 1) What Type of Test Should Be Done?

a) Hypothesis Test of Mean or Proportion?

This is a test of mean because each individual observation (each sampled shift’s output) within each of the two sample groups can have a wide range of values. Data points for tests of proportion are binary: they can take only one of two possible values.

 

b) One-Sample or Two-Sample Test?

This is a two-sample hypothesis test because two independent samples are being compared with each other. The two sample groups are the daily units produced by Shift A and the daily units produced by Shift B.

 

c) Independent (Unpaired) Test or Dependent (Paired) Test?

It is an unpaired test because data observations in each sample group are completely unrelated to data observations in the other sample group. The designation of “paired” or “unpaired” applies only for two-sample hypothesis tests.

 

d) One-Tailed or Two-Tailed Test?

The problem asks to determine whether there is a difference in the average number of daily units produced by Shift A and by Shift B. This is a non-directional inequality making this hypothesis test a two-tailed test. If the problem asked to determine whether Shift A really does have a higher average than Shift B, the inequality would be directional and the resulting hypothesis test would be a one-tailed test. A two-tailed test is more stringent than a one-tailed test.

 

e) t-Test or z-Test?

A z-Test is a statistical test in which the distribution of the Test Statistic under the Null Hypothesis can be approximated by the normal distribution.

The Test Statistic is distributed by the normal distribution if both samples are large and both population standard deviations are known. Both samples considered to be large samples because both sample sizes (n₁ = 40 and n₂ = 36) exceeds 30. Both population standard deviations (σ₁ = 25.5 and σ₂ = 11.2) are known.

Because both sample sizes (n₁ = 40 and n₂ = 36) exceeds 30, both sample means are therefore normal-distributed as per the Central Limit Theorem. The difference between two normally-distributed sample means is also normal-distributed. The Test Statistic is derived from the difference between the two means and is therefore normal-distributed. A z-Test can be performed if the Test Statistic is normal-distributed.

It should be noted that a two-independent-sample, unpooled t-Test can always be used in place of a two-independent-sample, unpooled. All z-Tests can be replaced be their equivalent t-Tests. As a result, some major commercial statistical software packages including the well-known SPSS provide only t-Tests and no direct z-Tests.

 

f) Pooled or Unpooled t-Test?

A pooled z-Test can be performed if the variances of both populations are similar, i.e., one population’s standard deviation is no more than twice as large as the other population’s standard deviation. An unpooled z-Test must be performed otherwise.

An unpooled z-Test can always be performed in the place of a pooled z-Test. Excel only provides a tool and formula for an unpooled z-test but not a pooled z-Test. For this reason the only type of two-independent-sample z-Test covered in this section will be the unpooled one.

t-Tests can always be performed in place of z-Tests. Excel does have separate tools and formulas for pooled and unpooled, two-independent-sample t-Tests.

This hypothesis test is a z-Test that is two-independent-sample, unpooled two-tailed hypothesis test of mean as long as all required assumptions have been met.

 

Question 2) Test Requirements Met?

a) Normal Distribution of Both Sample Means

The normal distribution can be used to map the distribution of the difference of the sample means (and therefore the Test Statistic, which is derived from this difference) only if the following conditions exist:

1) Both Population Standard Deviations, σ1 and σ2, Are Known

Those values are σ₁ = 25.5 and σ₂ = 11.2. Population standard deviation, σ, is one of the two required parameters needed to fully describe a unique normal distribution curve and must therefore be known in order to perform a z-Test (which uses the normal distribution).

and

2) Both samples sizes are large (n > 30).

Because both sample sizes (n₁ = 40 and n₂ = 36) exceeds 30, both sample means are therefore normal-distributed as per the Central Limit Theorem. The difference between two normally-distributed sample means is also normal-distributed. The Test Statistic is derived from the difference between the two means and is therefore normal-distributed.

The distributions of both samples and populations do not have to be verified because both sample means are known to be normal-distributed as a result of the large size.

The difference between the sample means and therefore the Test Statistic are normal-distributed because both samples are large and both population standard deviations are known.

 

We now proceed to complete the four-step method for solving all Hypothesis Tests of Mean. These four steps are as follows:

Step 1) Create the Null Hypothesis and the Alternative Hypothesis

Step 2 – Map the Normal or t-Distribution Curve Based on the Null Hypothesis

Step 3 – Map the Regions of Acceptance and Rejection

Step 4 – Determine Whether to Accept or Reject the Null Hypothesis By Performing the Critical Value Test, the p Value Test, or the Critical z Value Test

 

Proceeding through the four steps is done is follows:

 

Step 1 – Create the Null and Alternative Hypotheses

The Null Hypothesis is always an equality and states that the items being compared are the same. In this case, the Null Hypothesis would state that the average optimism scores for both sample groups are the same. We will use the variable x_bar₁-x_bar₂ to represent the difference between the means of the two groups. If the mean scores for both groups are the same, then the difference between the two means, x_bar₁-x_bar₂, would equal zero. The Null Hypothesis is as follows:

H₀: x_bar₁-x_bar₂ = Constant = 0

The Alternative Hypothesis is always in inequality and states that the two items being compared are different. This hypothesis test is trying to determine whether the first mean (x_bar₁) is different than the second mean (x_bar₂). The Alternative Hypothesis is as follows:

H₁: x_bar₁-x_bar₂ ≠ Constant

H₁: x_bar₁-x_bar₂ ≠ 0

The Alternative Hypothesis is non-directional (“not equal” instead of “greater than” or “less than”) and the hypothesis test is therefore a two-tailed test. It should be noted that a two-tailed test is more rigorous (requires a greater differences between the two entities being compared before the test shows that there is a difference) than a one-tailed test.

Parameters necessary to map the distributed variable, x_bar₁-x_bar₂, to the normal distribution are the following:

 

Step 2 – Map the Distributed Variable on a Normal Distribution Curve

H₀: x_bar₁-x_bar₂ = Constant = 0

n₁ = 40

n₂ = 36

Var₁ = σ₁² = (25.5)² = 650.25

Var₂ = σ₂² = (11.2)² = 125.44

 

Unpooled Population Standard Error

SE = SQRT[ (Var₁/n₁) + (Var₂/n₂) ]

SE = SQRT[ (650.25/40) + (125.44/36) ]

SE = 4.443

(Click On Image To See a Larger Version)

This non-standardized normal distribution curve has its mean set to equal the Constant taken from the Null Hypothesis, which is:

H₀: x_bar₁-x_bar₂ = Constant = 0

This non-standardized normal distribution curve is constructed from the following parameters:

Mean = 0

Standard Error = 4.443

Distributed Variable = x_bar₁-x_bar₂

 

Step 3 – Map the Regions of Acceptance and Rejection

The goal of a hypothesis test is to determine whether to reject or fail to reject the Null Hypothesis at a given level of certainty. If the two things being compared are far enough apart from each other, the Null Hypothesis (which states that the two things are not different) can be rejected. In this case we are trying to show graphically how different x_bar₁ is from x_bar₂ by showing how different x_bar₁-x_bar₂ (4.31) is from zero.

The non-standardized t-Distribution curve can be divided up into two types of regions: the Region of Acceptance and the Region of Rejection. A boundary between a Region of Acceptance and a Region of Rejection is called a Critical Value.

If the difference between the sample means, x_bar₁-x_bar₂ (4.31), falls into a Region of Rejection, the Null Hypothesis is rejected. If the difference between the sample means, x_bar₁-x_bar₂ (4.31), falls into a Region of Acceptance, the Null Hypothesis is not rejected.

The total size of the Region of Rejection is equal to Alpha. In this case Alpha, α, is equal to 0.05. This means that the Region of Rejection will take up 5 percent of the total area under this t-Distribution curve.

This 5 percent Alpha (Region of Rejection) is entirely contained in the outer right tail. The operator in the Alternative Hypothesis whether the hypothesis test is two-tailed or one-tailed and, if one tailed, which outer tail. The Alternative Hypothesis is the follows:

H₁: x_bar₁-x_bar₂ ≠ 0

A “not equal” operator indicates that this will be a two-tailed test. This means that the Region of Rejection is split between both outer tails.

The boundaries between Regions of Acceptance and Regions of Rejection are called Critical Values. The locations of these Critical Values need to be calculated.

Calculate the Critical Values

Two-Tailed Critical Values

Critical Values = Mean ± (Number of Standard Errors from Mean to Region of Rejection) * SE

Critical Values = Mean ± NORM.S.INV(1-α/2) * SE

Critical Values = 0 ± NORM.S.INV(1 - 0.05/2) * 4.443

Critical Values = 0 ± NORM.S.INV(0.975) * 4.443

Critical Values = 0 ± 8.708

Critical Values = -8.708 and 8.708

The Region of Rejection is therefore everything that is to the right of 8.708 and everything to the left of -8.708.

The following Excel-generated distribution curve with the blue Region of Acceptance and the yellow Regions of Rejection is shown is as follows:

 (Click On Image To See a Larger Version)

 

Step 4 – Determine Whether to Reject Null Hypothesis

The object of a hypothesis test is to determine whether to accept or reject the Null Hypothesis. There are three equivalent-Tests that determine whether to accept or reject the Null Hypothesis. Only one of these tests needs to be performed because all three provide equivalent information. The three tests are as follows:

 

1) Compare x_bar₁-x_bar₂ With Critical Value

Reject the Null Hypothesis if the sample mean, x_bar₁-x_bar₂ = 4.31, falls into the Region of Rejection. Fail to reject the Null Hypothesis if the sample mean, x_bar₁-x_bar₂ = 4.31, falls into the Region of Acceptance.

Equivalently, reject the Null Hypothesis if the sample mean, x_bar₁-x_bar₂, is further from the curve’s mean of 0 than the Critical Value. Fail to reject the Null Hypothesis if the sample mean, x_bar₁-x_bar₂, is closer than the curve’s mean of 0 than the Critical Value.

The Critical Values have been calculated to be +8.708 on the left and -8.708 on the right. x_bar₁-x_bar₂ (4.31) is closer to the curve mean (0) than the right Critical Value (+8.708). The Null Hypothesis would therefore not be rejected.

 

2) Compare the z Score with the Critical z Value

The z Score is the number of Standard Errors that x_bar₁-x_bar₂ (4.31) is from the curve’s mean of 0.

The Critical z Value is the number of Standard Errors that the Critical Value is from the curve’s mean.

Reject the Null Hypothesis if the z Score is farther from the standardized mean of zero than the Critical z Value. Fail to reject the Null Hypothesis if the z Score is closer to the standardized mean of zero than the Critical z Value.

Equivalently, reject the Null Hypothesis if the z Score is farther from the standardized mean of zero than the Critical z Value. Fail to reject the Null Hypothesis if the z Score is closer to the standardized mean of zero than the Critical z Value.

(Click On Image To See a Larger Version)

The Constant is the Constant from the Null Hypothesis (H₀: x_bar₁-x_bar₂ = Constant = 0)

Z Score (Test Statistic) = (4.31 – 0)/4.443

Z Score (Test Statistic) = 0.97

This means that the sample mean, x_bar₁-x_bar₂ (4.31), is 0.97 standard errors from the curve mean (0).

Two-tailed Critical z Values = ±NORM.S.INV(1-α/2)

Two-tailed Critical z Values = ±NORM.S.INV(1-0.05/2)

Two-tailed = ±NORM.S.INV(0.975) = ±1.9599

This means that the boundaries between the Region of Acceptance and the Region of Rejection are 1.9599 standard errors from the curve mean on each side since this is a two-tailed test.

The Null Hypothesis is not rejected because the z Score (+0.97) is closer to the standardized mean of zero than the Critical z Value on the right side (+1.9599).

 

3) Compare the p Value With Alpha

The p Value is the percent of the curve that is beyond x_bar₁-x_bar₂ (4.31). If the p Value is smaller than Alpha/2, the Null Hypothesis is rejected. If the p Value is larger than Alpha/2, the Null Hypothesis is not rejected.

p Value =MIN(NORM.S.DIST(z Score,TRUE),1-NORM.S.DIST(z Score,TRUE))

p Value =MIN(NORM.S.DIST(0.97,TRUE),1-NORM.S.DIST(0.97,TRUE))

p Value = 0.1660

The p Value (0.1660) is larger than Alpha/2 (0.025) Region of Rejection in the right tail and we therefore do not reject the Null Hypothesis.

The following Excel-generated graph shows that the red p Value (the curve area beyond x_bar₁-x_bar₂) is larger than the yellow Alpha, which is the 5 percent Region of Rejection split between both outer tails.

 (Click On Image To See a Larger Version)

 

Excel Data Analysis Tool Shortcut

This two-independent-sample, unpooled z-Test can be solved much quicker using the following Excel data analysis tool:

z-Test: Two Sample For Means. This tool uses the formulas for an unpooled, two-sample z-Test as are shown above. This tool can be accesses by clicking Data Analysis under the Data tab. The entire Data Analysis Toolpak is an add-in that ships with Excel but must first be activated by the user before it is available. This tool calculates the z Score and p Value using the same equations as shown.

Note that this tool requires that all data in each sample group be placed in a single column. In the following image, only the first 19 data points of each sample are showing.

 (Click On Image To See a Larger Version)

The completed dialogue box for this tool which produced the preceding output is as follows:

 (Click On Image To See a Larger Version)

 

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

• Histograms in Excel
  • Creating a Histogram With the Histogram Data Analysis Tool in Excel
  • Creating an Automatically Updating Histogram in 7 Steps in Excel With Formulas and a Bar Chart
• Bar Chart in Excel
  • Creating a Bar Chart in 7 Steps in Excel 2010 and Excel 2013
• Combinations & Permutations in Excel
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• Normal Distribution in Excel
  • Overview of the Normal Distribution
  • Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
  • Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
  • Solving Normal Distribution Problems in Excel 2010 and Excel 2013
  • Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
  • An Important Difference Between the t and Normal Distribution Graphs
  • The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
  • Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way
• t-Distribution in Excel
  • Overview of the t- Distribution
• Binomial Distribution in Excel
  • Overview of the Binomial Distribution in Excel 2010 and Excel 2013
  • Solving Problems With the Binomial Distribution in Excel 2010 and Excel 2013
  • Normal Approximation of the Binomial Distribution in Excel 2010 and Excel 2013
  • Distributions Related to the Binomial Distribution
• z-Tests in Excel
  • Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
  • One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
  • 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
  • Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013
• t-Tests in Excel
  • Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
  • 1-Sample t-Tests in Excel
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    • Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
    • 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
    • 1-Sample t-Test Power With G*Power Utility
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    • Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
    • Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
    • Two-Independent-Sample Pooled t-Test - All Excel Calculations
    • 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
    • 2-Sample Pooled t-Test Power With G*Power Utility
    • Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
    • 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups
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    • Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility
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    • Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
    • Excel Normality Testing of Paired t-Test Data
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    • Paired t-Test – Effect Size in Excel 2010, and Excel 2013
    • Paired t-Test – Test Power With G-Power Utility
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• Chi-Square Goodness-Of-Fit Tests in Excel
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• Confidence Intervals in Excel
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  • t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
  • Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
  • Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
  • Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013
• Simple Linear Regression in Excel
  • Overview of Simple Linear Regression in Excel 2010 and Excel 2013
  • Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
  • Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
  • Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
  • Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
  • All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
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• Multiple Linear Regression in Excel
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  • Multiple Linear Regression’s Required Residual Assumptions
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  • Evaluating the Excel Output of Multiple Regression
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  • Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
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  • Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
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  • Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
  • Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
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  • ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
  • ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
  • Power of Single-Factor ANOVA Test Using Free Utility G*Power
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• Two-Factor ANOVA Without Replication in Excel
  • Two-Factor ANOVA Without Replication in Excel 2010 and Excel 2013
• Randomized Block Design ANOVA in Excel
  • Randomized Block Design ANOVA in Excel 2010 and Excel 2013
• Repeated-Measures ANOVA in Excel
  • Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
  • Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
  • Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
  • Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
• ANCOVA in Excel
  • Single-Factor ANCOVA in 8 Steps in Excel 2010 and Excel 2013
• Normality Testing in Excel
  • Creating a Box Plot in 8 Steps in Excel
  • Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
  • Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
  • Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel
• Nonparametric Testing in Excel
  • Mann-Whitney U Test in 12 Steps in Excel
  • Wilcoxon Signed-Rank Test in 8 Steps in Excel
  • Sign Test in Excel
  • Friedman Test in 3 Steps in Excel
  • Scheirer-Ray-Hope Test in Excel
  • Welch's ANOVA Test in 8 Steps Test in Excel
  • Brown-Forsythe F Test in 4 Steps Test in Excel
  • Levene's Test and Brown-Forsythe Variance Tests in Excel
  • Chi-Square Independence Test in 7 Steps in Excel
  • Chi-Square Goodness-of-Fit Tests in Excel
  • Chi-Square Population Variance Test in Excel
• Post Hoc Testing in Excel
  • Tukey's HSD Post Hoc Test in Excel
  • Tukey-Kramer Post Hoc Test in Excel
  • Games-Howell Post Hoc Test in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
  • Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
  • Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
  • Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
  • Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
  • Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013
• Solving Problems With Other Distributions in Excel
  • Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
  • Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
  • Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
  • Solving Beta Distribution Problems in Excel 2010 and Excel 2013
  • Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
  • Solving Poisson Distribution Problems in Excel 2010 and Excel 2013
• Optimization With Excel Solver
  • Maximizing Lead Generation With Excel Solver
  • Minimizing Cutting Stock Waste With Excel Solver
  • Optimal Investment Selection With Excel Solver
  • Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
  • Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
  • Optimizing a Bond Portfolio With Excel Solver
  • Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers
• Chi-Square Population Variance Test in Excel
  • Overview of the Chi-Square Population Variance Test in Excel 2010 and Excel 2013
• Analyzing Data With Pivot Tables and Pivot Charts
  • Simplifying Excel Pivot Table and Pivot Chart Setup
• SEO Functions in Excel
  • Top 10 Excel SEO Functions - You'll Like These
• Time Series Analysis in Excel
  • Forecasting With Exponential Smoothing in Excel
  • Forecasting With the Weighted Moving Average in Excel
  • Forecasting With the Simple Moving Average in Excel
• VLOOKUP
  • Simplifying Excel Lookup Functions: VLOOKUP, HLOOKUP, INDEX, MATCH, CHOOSE, and OFFSET
  • VLOOKUP - Just Like Looking Up a Number in a Telephone Book
  • VLOOKUP To Look Up a Discount in a Distant Database
• Simplifying Useful Excel Functions
  • Simplifying Excel Functions: SUMIF, SUMIFS, COUNTIF, COUNTIFS, AVERAGEIF, and AVERAGEIFS

 

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