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

1 Answer
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.


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)#


#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.

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