What the F-test compares
Fisher’s F-test checks whether two independent samples come from populations with the same variance. It is used to compare the precision of two instruments or processes, and sometimes to choose between Student’s and Welch’s t-tests.
The formula
F = s₁² ÷ s₂²
with n₁ − 1 degrees of freedom in the numerator and n₂ − 1 in the denominator. If the variances are equal, F should be close to 1. The p-value comes from Fisher’s F distribution:
- right-tailed (variance 1 greater): p = P(F ≥ f);
- left-tailed (variance 1 smaller): p = P(F ≤ f);
- two-tailed: p = 2 × the smaller of the two tails.
Example
Sample 1: SD 12 with 25 values. Sample 2: SD 8 with 20 values.
- F = 144 ÷ 64 = 2.25, with 24 and 19 degrees of freedom;
- two-tailed p ≈ 0.075: not significant at α = 0.05;
- the two-tailed critical values are about 0.43 and 2.45; F falls between them.
Critical values
For a two-tailed test at level α, reject equal variances when F is below the lower critical value F(α/2) or above the upper one F(1 − α/2). Many printed tables only give the upper value, after putting the larger variance on top.
A test with caveats
The F-test is very sensitive to non-normality: with skewed or heavy-tailed data, it often signals a difference that is really a departure from the normal shape. For that reason:
- Levene’s test or the Brown–Forsythe test are preferred to check equal variances before an ANOVA;
- choosing between Student’s and Welch’s t-test on the basis of a preliminary F-test is discouraged; use Welch’s test by default.
Other uses of F
The same distribution appears in ANOVA, where F compares the variance between groups with the variance within groups, and in regression, where it tests whether a model explains more than chance. Use the F-distribution calculator to get p-values and critical values for any F statistic.
Tips
- Enter standard deviations, not variances; the calculator squares them.
- Sample sizes matter: with small samples, even a ratio of 3 or 4 can be compatible with equal variances.