F-Test Calculator

The F-test compares two sample variances to decide whether they could plausibly come from populations with the same variance. The test statistic is F = s1^2 / s2^2. Convention is to put the larger variance in the numerator, so F >= 1 always.

Under the null hypothesis that both populations have equal variance, F follows an F-distribution with degrees of freedom df1 = n1 - 1 and df2 = n2 - 1. A large F means the two variances are quite different. How large is too large depends on the sample sizes.

The p-value is the probability of observing an F this large or larger by chance if the variances really are equal. This calculator uses a two-tailed test, so the reported p-value accounts for the possibility that either group could have the larger variance. For a two-tailed test, the critical F at significance level alpha is the value where the upper-tail probability is alpha/2.

Assumptions: both samples must come from approximately normal populations. The F-test is sensitive to non-normality, more so than the t-test. If normality is in doubt, Levene’s test is a more robust alternative.

A significant result tells you the variances differ, not by how much. Practical significance depends on context. Two manufacturing lines where one produces three times the variance of the other may be a serious quality issue even if the F-test p-value barely crosses the threshold.

The F-distribution is right-skewed and defined only for positive values. Its exact shape depends on both degrees of freedom. With large samples, the distribution becomes narrower and the critical value needed to reject the null gets closer to 1.

This test is the building block for ANOVA, where the same F-ratio logic compares between-group variance to within-group variance across multiple groups.