13.4 Test of Two Variances

Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.

In order to perform a F test of two variances, it is important that the following are true:

1. The populations from which the two samples are drawn are normally distributed.
2. The two populations are independent of each other.

Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p-values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here.

Suppose we sample randomly from two independent normal populations. Let ${\sigma }_1^2$ and ${\sigma }_2^2$ be the population variances and $s_1^2$ and $s_2^2$ be the sample variances. Let the sample sizes be n₁ and n₂. Since we are interested in comparing the two sample variances, we use the F ratio:

$F=\frac{\left[\frac{{(s_1)}^2}{{({\sigma }_1)}^2}\right]}{\left[\frac{{(s_2)}^2}{{({\sigma }_2)}^2}\right]}$

F has the distribution F ~ F(n₁ – 1, n₂ – 1)

where n₁ – 1 are the degrees of freedom for the numerator and n₂ – 1 are the degrees of freedom for the denominator.

If the null hypothesis is ${\sigma }_1^2={\sigma }_2^2$, then the F Ratio becomes $F=\frac{\left[\frac{{(s_1)}^2}{{({\sigma }_1)}^2}\right]}{\left[\frac{{(s_2)}^2}{{({\sigma }_2)}^2}\right]}=\frac{{(s_1)}^2}{{(s_2)}^2}$.

If the two populations have equal variances, then $s_1^2$ and $s_2^2$ are close in value and $F=\frac{{(s_1)}^2}{{(s_2)}^2}$ is close to one. But if the two population variances are very different, $s_1^2$ and $s_2^2$ tend to be very different, too. Choosing $s_1^2$ as the larger sample variance causes the ratio $\frac{{(s_1)}^2}{{(s_2)}^2}$ to be greater than one. If $s_1^2$ and $s_2^2$ are far apart, then $F=\frac{{(s_1)}^2}{{(s_2)}^2}$ is a large number.

Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or two-tailed.