Tuesday, May 27, 2014

The Empirical Rule and Chebyshev’s Theorem in Excel

This is one of the following eight articles on the 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

 

Calculating How Much

of the Data Is a Certain

Distance From the Mean

There are two rules that can be used to calculate the proportion of data values that will be within a specified number of standard deviations from the mean. They are as follows:

For Normally-Distributed Data, Use the Empirical Rule

The Empirical Rule, a.k.a., the 68-95-99.7 Rule, states that when data is normal-distributed, the following is true:

68 percent of all data points have values that are within one standard deviation of the mean.

95 percent of all data points have values that are within two standard deviations of the mean.

97.5 percent of all data points have values that are within three standard deviations of the mean.

 

Empirical Rule Problem in Excel

Calculate the percent of values in a large normally-distributed data set of unknown distribution that will fall between 12 and 22 if the data’s set’s mean is 16 and its standard deviation is 2.

0.9759 = NORM.DIST(22,16,2,TRUE) - NORM.DIST(12,16,2,TRUE)

97.59 percent of all data values of a large, normally-distributed data set will fall between 12 and 22 if the data set’s mean is 16 and its standard deviation is 2.

 

For Data of Unknown Distribution,

Use Chevbyshev’s Theorem

Chebyshev’s Theorem states that (1 – 1/z2) of the data values will fall within z standard deviations of the mean as long as z is any value greater than 1.

According to Chebyshev’s Theorem, the following are true:

75 percent of data values will be within two standard deviations of the mean.

89 percent of all data points have values that are within three standard deviations of the mean.

94 percent of all data points have values that are within four standard deviations of the mean.

 

Chebyshev’s Theorem Problem in Excel

Use Chebyshev’s Theorem to calculate the percent of values in a large data set of unknown distribution that will fall between 12 and 22 if the data’s set’s mean is 16 and its standard deviation is 2.

12 is |12 – 16|/2 = 2 standard deviations to the left of the mean.

There will be at least 0.75 / 2 or 37.5 percent of all data values falling between the mean and 8.

22 is |22 – 16|/2 = 3 standard deviations to the left of the mean.

There will be at least 0.89 / 2 or 44.5 percent of all data values falling between the mean and 22.

82 percent = 37.5 percent + 44.5 percent

Chebyshev’s Theorem states that 82 percent of all data values will fall between 12 and 22 if of a large data set of unknown distribution if the data set’s mean is 16 and its standard deviation is 2.

 

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

 

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