# What is an F-test for the equality of variances?

Aug 17, 2016

An F-test uses critical values of the F-distribution to determine whether two variances are equal. The test uses a statistic which is the ratio of the two variances to achieve this.

#### Explanation:

The F-test can be used to test if the variances of two populations are equal. The statistic we define to test this is the ratio of the two variances:

$F = {s}_{1}^{2} / {s}_{2}^{2}$

Where ${s}_{1}$ and ${s}_{2}$ are the sample variances. The further this value deviates from 1, the more likely that the underlying variances are actually different. The F-distribution is used to quantify this likelihood for differing sample sizes and the confidence or significance we would like the answer to hold.

We define ${F}_{\alpha , {N}_{1} - 1 , {N}_{2} - 1}$ as the critical value of the F distribution with ${N}_{1} - 1$ and ${N}_{2} - 1$ degrees of freedom and a significance level of $\alpha$ where ${N}_{1}$ and ${N}_{2}$ are the number of samples from each of the two populations.

This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the variances are not equal.

The two tailed test is arranged as follows. Reject the null hypothesis if:

$F < {F}_{1 - \alpha / 2 , {N}_{1} - 1 , {N}_{2} - 1}$

or

$F > {F}_{\alpha / 2 , {N}_{1} - 1 , {N}_{2} - 1}$

The one-tailed versions only test in one direction, that is the variance from the first population is either greater than or less than (but not both) the second population variance.