F-Statistic Calculator — F-Test for Equal Variances
Compute the F-statistic for comparing two sample variances and get the one- and two-tailed p-values. Enter each sample's standard deviation and size to test whether the variances differ significantly.
Two-tailed p-value: 0.0955 — no significant difference at α = 0.05
- 1
Variance 1 (s₁²)
12.5 × 12.5 = 156.25 - 2
Variance 2 (s₂²)
8.3 × 8.3 = 68.89 - 3
F statistic
156.25 ÷ 68.89 = 2.2681Larger variance goes in the numerator so F ≥ 1.
How does this calculator work?
The F-statistic is the ratio of two sample variances, F = s₁²/s₂² (larger in numerator). Degrees of freedom are n₁−1 and n₂−1. Under H₀ (equal variances), F follows an F-distribution; the two-tailed p-value tests whether the variances differ significantly. p < 0.05 suggests unequal variances.
Formula
How this is calculated
The F-test for equality of variances asks whether two independent samples could plausibly come from populations with the same variance. The test statistic is the ratio of the two sample variances: F = s₁²/s₂². By convention the larger variance goes in the numerator, so F is always ≥ 1. Under the null hypothesis that the population variances are equal, F follows an F-distribution with degrees of freedom df₁ = n₁ − 1 and df₂ = n₂ − 1.
The p-value is computed from the upper tail of the F-distribution using the regularised incomplete beta function. Because the larger variance is forced into the numerator, the one-tailed p-value is doubled for a two-tailed test (testing whether s₁² ≠ s₂² without specifying direction). A small p-value — typically below 0.05 — means the two variances are unlikely to be equal.
Important assumptions: both samples must be independent and drawn from approximately normal populations. The F-test is sensitive to non-normality — for non-normal data consider Levene's or Brown-Forsythe tests instead. The calculator returns a two-tailed p-value because most practical comparisons of variances are non-directional.
Frequently asked questions
Placing the larger variance on top forces F ≥ 1, so you only need the upper tail of the F-distribution to compute the p-value. Doubling that one-tailed probability gives the two-tailed result for the symmetric test H₀: σ₁² = σ₂².
A significant result (p < 0.05) suggests the two populations have different variances. This matters when choosing subsequent tests: equal-variance t-tests (pooled) assume homogeneity, while the Welch t-test does not. A significant F-test is a flag to use Welch rather than the pooled version.
No — the F-test for variance equality is quite sensitive to departures from normality, more so than the t-test for means. For skewed or heavy-tailed data, Levene's test (based on deviations from group medians) or Brown-Forsythe's test are more reliable alternatives.
Also known as
TG we-Calculate Editorial Team. (2026). F-Statistic Calculator — F-Test for Equal Variances [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/f-statistic-calculator
TG we-Calculate Editorial Team. "F-Statistic Calculator — F-Test for Equal Variances." TG we-Calculate. 2026. https://we-calculate.com/calculator/f-statistic-calculator.
TG we-Calculate Editorial Team, "F-Statistic Calculator — F-Test for Equal Variances," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/f-statistic-calculator
@misc{wecalculate_f_statistic_calculator, title = {F-Statistic Calculator — F-Test for Equal Variances}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/f-statistic-calculator}}, year = {2026}, note = {TG we-Calculate} }
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