F-Test Calculator

The F-test for equality of two variances is a statistical hypothesis test that determines whether two populations have the same variance. It computes the ratio of two sample variances (F = S₁²/S₂²) and compares it against a critical value from the F-distribution. A significant result suggests the variances are statistically different.

The F-statistic is the ratio of the two sample variances: F = S₁²/S₂². A value close to 1 suggests the variances are similar. Values much larger or smaller than 1 indicate a difference in variability between the two groups.

If the p-value is less than or equal to your chosen significance level (α), you reject the null hypothesis and conclude that the two population variances are significantly different. If the p-value exceeds α, there is insufficient evidence to reject the assumption of equal variances.

The F-test assumes that both samples are drawn independently from normally distributed populations. It is sensitive to departures from normality, so it is important to verify that your data does not strongly violate this assumption before relying on the result.

Use a two-tailed test when you simply want to detect any difference in variances (H₁: σ₁² ≠ σ₂²). Use a right-tailed test when you specifically hypothesize that Group 1 has a larger variance (H₁: σ₁² > σ₂²), and a left-tailed test when you expect Group 1 has a smaller variance (H₁: σ₁² < σ₂²).

The numerator degrees of freedom is df₁ = n₁ − 1 (from Group 1) and the denominator degrees of freedom is df₂ = n₂ − 1 (from Group 2). These values determine the shape of the F-distribution used to find the p-value and critical value.

Both the F-test for variances and ANOVA use the F-distribution, but they serve different purposes. The two-variance F-test specifically compares variability between exactly two groups, while ANOVA tests whether the means of three or more groups differ by comparing between-group and within-group variance.

If your data is not approximately normally distributed, the F-test for variances may produce unreliable results. In that case, consider using Levene's test or the Brown-Forsythe test, which are more robust alternatives for comparing variances when normality cannot be assumed.