Variances and Confidence Intervals
Classical F test for comparing two population variances from independent normally distributed samples. Enter the two sample sizes and standard deviations to obtain the variance ratio, F statistic, degrees of freedom, p-value, and confidence interval.
The classical F test assumes independent random samples from normally distributed populations. The test is particularly sensitive to departures from normality.
The two-sample test for equality of variances is the classical F test. Under the null hypothesis that the two population variances are equal, the ratio of the unbiased sample variances follows an F distribution:
For a two-sided test, the p-value is twice the smaller of the lower- and upper-tail probabilities. For a one-sided alternative, the corresponding single tail is used.
The calculator also reports a 100(1 − α)% confidence interval for σ₁²/σ₂². If R = s₁²/s₂², the interval is:
A confidence interval containing 1 is consistent with equality of the population variances at the corresponding two-sided significance level. This test should be interpreted cautiously when the normality assumption is questionable because the F test is sensitive to non-normality.
A published example reports 12 observations in Group 1 with sample variance 40.24242 and 10 observations in Group 2 with sample variance 34.04444. Thus the corresponding standard deviations are approximately 6.3437 and 5.8348.
The implementation reproduces F = 1.18206 and p = 0.81475 for these inputs, matching the published example after rounding.
NCSS. PASS User's Guide III, Chapter 655: Inequality Tests for Two Variances. NCSS Statistical Software. The PASS documentation describes the F-distribution method for testing H₀: σ₁² = σ₂² against two-sided and one-sided alternatives.
A Review of Basic Concepts, Chapter 1, Figure 1.36: “Two Sample Test for Variances of SCORE within METHOD.” The worked example reports n₁ = 12, n₂ = 10, variances 40.24242 and 34.04444, F = 1.18, and Pr > F = 0.8148.