F-Test for Equality of Two Variances Calculator

The F-test for equality of two variances is a classical statistical procedure used to determine whether two independent populations have the same variance. Named after Sir Ronald A. Fisher, the test is widely used as a diagnostic check before applying a two-sample t-test, which assumes that the two groups have equal population variances. If the F-test rejects that assumption, the Welch t-test — which does not require equal variances — should be used instead. The test statistic is the ratio of the two sample variances: F = s₁² / s₂². By convention, the larger sample variance is placed in the numerator so that F ≥ 1, which restricts all critical mass to the upper tail of the F-distribution and simplifies interpretation. The null hypothesis H₀ states that the population variances are equal (σ₁² = σ₂²), while the alternative hypothesis H₁ states that they differ (σ₁² ≠ σ₂²). The degrees of freedom are df₁ = n₁ − 1 (numerator) and df₂ = n₂ − 1 (denominator), where n₁ and n₂ are the respective sample sizes. To evaluate significance, the calculated F-value is compared to the F-distribution with (df₁, df₂) degrees of freedom. For a two-tailed test, the p-value equals 2 × P(F > F_obs). If the p-value is less than or equal to the chosen significance level α (typically 0.05 or 0.01), H₀ is rejected and the variances are declared significantly different. The critical F-value at the chosen α provides an equivalent decision boundary: if F_obs > F_crit, reject H₀. The F-test has broad practical applications. In manufacturing, it verifies whether two production lines produce parts with equal dimensional variability — a prerequisite for quality-control procedures that assume uniform processes. In clinical research, it checks whether two treatment groups have similar response variability, which affects both study design and interpretation. In finance, it compares the volatility of two assets or portfolios, informing risk assessment and diversification strategies. In agriculture, it evaluates whether two fertilisers produce crops with equal yield consistency. Despite its power, the F-test has an important limitation: it is highly sensitive to departures from normality. Both samples must come from normally distributed populations for the test to be valid. When normality is uncertain, analysts often prefer the more robust Levene's test or the Brown–Forsythe test, which replace raw deviations from the mean with absolute deviations or median deviations. This calculator uses the exact F-distribution CDF via the regularised incomplete beta function, producing p-values consistent with R (var.test), Python (scipy.stats.levene), and SPSS.