Variance Estimates

$F$-Test of Equality of Variances

An $F$-test for the null hypothesis that two normal populations have the same variance is sometimes used; although, it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.

Notionally, any $F$-test can be regarded as a comparison of two variances, but the specific case being discussed here is that of two populations, where the test statistic used is the ratio of two sample variances. This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the $F$ distribution can be derived.

The Test

Let $X_1, \dots, X_n$ and $Y_1, \dots, Y_m$ be independent and identically distributed samples from two populations which each have a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. The test statistic is:

$\displaystyle F = \frac{S^{2}_{X}}{S^{2}_{Y}}$

It has an $F$-distribution with $n-1$ and $m-1$ degrees of freedom if the null hypothesis of equality of variances is true. The null hypothesis is rejected if $F$ is either too large or too small. The immediate assumption of the problem outlined above is that it is a situation in which there are more than two groups or populations, and the hypothesis is that all of the variances are equal.

Properties of the $F$ Test

This $F$-test is known to be extremely sensitive to non-normality. Therefore, they must be used with care, and they must be subject to associated diagnostic checking.

$F$-tests are used for other statistical tests of hypotheses, such as testing for differences in means in three or more groups, or in factorial layouts. These $F$-tests are generally not robust when there are violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the $F$-test is relatively robust. Although, if the normality assumption does not hold, it suffers from a loss in comparative statistical power as compared with non-parametric counterparts.