Test for Equal Variances

John C SmithMaster Black Belt

Using charts and information from minitab.com

Understand, Apply, and Interpret results from MiniTab’s Test for Equal Variances

Objectives

Confidence Intervals◦ The range of values that is likely to contain the

population parameter within some percent Confidence Intervals for Standard Deviations

◦ The range of values that is likely to contain the standard deviation within some percent

Definitions

Bonferroni Confidence Intervals for Standard Deviations◦ The upper boundary for a factor level is equal to (((n-1)

* var) / U)**0.5◦ where:◦ n = the sample size of the factor level◦ var = variance of the factor level◦ U = the inverse cumulative chi-square distribution

function for K with n - 1 degrees of freedom◦ K = (desired family error rate) / (2 * number of levels)◦ The lower boundary is calculated the same way, using L

instead of U, where L = inverse cumulative chi-square distribution function for 1 - K with n - 1 degrees of freedom.

◦ Calculate “by hand” – NERD ALERT!!

Definitions

F-Test◦ Fisher’s Test◦ Basic assumption is that data is normal.◦ Any statistical test in which the test statistic has

an F-distribution under the null hypothesis. Levene’s Test

◦ An inferential statistic used to assess the equality of variances in different samples.

◦ Test is robust to non-normal data.◦ Some common statistical procedures assume that

variances of the populations from which different samples are drawn are equal.

Definitions

According to Design and Analysis of Experiments, 6th edition, by Douglas C. Montgomery: "The modified Levene's test uses the absolute deviation of the observations in each treatment from the treatment median. It then evaluates whether or not the mean of these deviations are equal for all treatments. It turns out that if the mean deviations are equal, the variances of the observations in all treatments will be the same. The test statistic for Levene's test is simply the usual ANOVA F statistic for testing equality of means applied to the absolute deviations."

You can do this in Minitab by making a new column where each value is the absolute value of the response minus the median of that treatment. Then run One-Way ANOVA using the new column as the Response. The F statistic and p-value will be the test statistic and p-value for Levene's test. For an example, see the link, Calculating Levene's Test Using Oneway ANOVA, below.

Levene’s Test

Place response (data) in one column Place factor (descriptor) in another column

◦ Before or After◦ Treatment 1 or Treatment 2◦ Etc or etc

Select STAT > ANOVA > Test for Equal Variances

Application of Test

Interpret Results

Graphical

Graphical

Analy

tica

l

Interpret Results

If the p is Low, the Null must GO!

Not lower than .05, fail to reject the Null

Boxplot of data from Treatments 1 and 2. Follow standard boxplot graphic rules.

Interpret Results

This test compares the standard deviations, or spreads, of two or more sets of data.

Produces results for normal and non-normal data.

Produces graphical and analytical data for comparison.◦ Graphical: Bonferroni’s CI for StDev’s chart and

Boxplot of data◦ Analytical: p-value results

Summary