Comparing two variances head-on
Sometimes the spread is the story: is the new machine less variable than the old, is one classroom's marks more scattered than another's? The F-test divides one sample variance by the other (larger over smaller by convention, so F ≥ 1) and reads the ratio on the F distribution with (n−1, n−1) degrees of freedom.
Bakery daily sales (s² = 22.57) against a class's quiz marks (s² = 7.93): F = 2.847 on (7, 7) df, two-tailed p = 0.1909. A variance nearly three times larger, and still not significant, because with eight observations each, variance estimates themselves wobble enormously. Spread is expensive to measure: that is the lesson this test teaches again and again.
The fragility warning, printed on the label
Of the common tests, this one trusts normality the most and forgives it the least. Heavy-tailed data can trigger "significant" variance differences that are really distribution shape. Modern practice reaches for Levene's test (which works on absolute deviations and shrugs at non-normality) when the conclusion matters. Use the F-test for coursework that asks for it and quick checks on well-behaved data.
Using this test as a gatekeeper for pooled vs Welch t-tests. The ritual "F-test first, then choose" performs worse than simply always using Welch, whose default status on this site's two-sample page reflects exactly that evidence.
Frequently asked questions
Why is the larger variance always on top?
Convention only: it lets one right-tail calculation serve the two-tailed test. The p-value accounts for both directions.
My variances differ 4x but p is 0.2. Really?
Really, if the samples are small. A variance from n = 8 carries a standard error of about half itself; ratios of 3–4 arise from equal populations more often than intuition suggests.