# Introduction to Equal Variance Assumption

## Equal Variance Assumption Basics

5. Parametric tests vs non-parametric tests
There are non-parametric alternatives to both the two-sample t-test and ANOVA. However, it should be noted these also assume that the underlying distributions are symmetric, with the same shape. Stating that the distributions have the same shape is tantamount to stating that they have the same variance. Since both the parametric and the non-parametric tests assume basically the same thing, the parametric tests are preferred, since they are uniformly more powerful.

6. Differences in variance are also desirable
One of the major goals of every project should be to reduce the variation in the CTQ, or Y. Don’t forget that the two-sample t-test and ANOVA are both methods for detecting changes in the mean of Y. If changes in the variation are observed, these are important, perhaps more important than the changes in the mean one is testing for. These differences should be studied to determine if they are consistent.

What is the bottom line? If you use a statistical tool that assumes equal variance, you can and probably should test this assumption. Remember that if the sample sizes are equal, or nearly equal, this assumption can be relaxed a great deal. Also, as a rule of thumb, even when the sample sizes are not nearly equal, there is usually no problem provided the largest sample standard deviation is not more than twice the size of the smallest sample standard deviation. When there is a problem, remember that what may be a “problem” as far as testing for differences in the mean of Y can also be a “solution” for determining ways to reduce the variation in Y. Be sure to check whether the variances increase as the size of the data increases. This is not at all uncommon, and can be remedied easily with a log or Box-Cox power transformation of the data. If all else fails, remember that the p-value you see will be smaller than it actually should be. This is only cause for concern when the p-value is marginally significant. You might want to run the non-parametric alternative (if there is one) and see if the results agree. And, consider the practical significance as well as the statistical significance.

How it Fits With the Breakthrough Strategy (DMAIC)

Analyze PhaseIn the Analyze Phase of a Black Belt project, the Black Belt must isolate variables which exert leverage on the CTQ. These leverage variables are uncovered through the use of various statistical tools designed to detect differences in means, differences in variances, patterns in means, or patterns in variances, in the case where the CTQ is continuous.

Two of the prominent tools for detecting differences in means are the two-sample t-test and ANOVA. The assumptions for both of these are the same, since the two-sample t-test is just a special case of one-way ANOVA. Both of these tools also have non-parametric alternatives, at least in some cases. The t-test also has an alternative where equal variances are not assumed. There are non-parametric alternatives for one-way and two-way ANOVA. As mentioned earlier, one should use good sense and judgement when deciding which test is more appropriate.

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