F-Test Calculator

How to Use This Calculator

1. Enter the two sample variances and sample sizes for an equality-of-variances test.
2. The larger variance goes in Variance 1 (calculator will auto-detect).
3. Use F in ANOVA tab if you have MS between/within from an ANOVA table.
4. Use F in Regression tab to test overall model significance from SS regression and residual.
5. Professional tab computes 95% CI for the variance ratio and flags robustness concerns.

Formula

Example

Frequently Asked Questions

• What is the F-test and what does it test? ▼
  The F-test is a statistical hypothesis test that uses the F-distribution to compare two variances or assess overall model fit. In its simplest form — the variance equality F-test — it tests whether two population variances are equal by computing the ratio of the larger sample variance to the smaller: F = s₁² / s₂². Under the null hypothesis that both populations have equal variance, this ratio follows an F-distribution with (n₁−1) and (n₂−1) degrees of freedom. The F-distribution is always non-negative and right-skewed. Large F-values (far from 1.0) suggest the variances differ significantly. In ANOVA, the F-test compares mean square between groups to mean square within groups, testing whether any group means differ. In linear regression, the F-test assesses whether the model as a whole explains a statistically significant portion of the variance in the outcome. The F-statistic is named after R.A. Fisher, though George Snedecor formalized the F-distribution in 1934.
• How is the F-test different from a t-test? ▼
  The t-test and F-test address fundamentally different hypotheses. A t-test compares means: 'Are the average values in these groups the same?' An F-test for variance equality compares spread: 'Is the variability in these groups the same?' These are separate questions and require separate tests. For example, two student cohorts might have the same average exam score (t-test not significant) but very different variability — one cohort scores consistently around 75 while the other ranges from 40 to 100 (F-test significant). There is a mathematical relationship: a t-statistic squared equals an F-statistic with df₁=1, which is why an F-test with one degree of freedom in the numerator is equivalent to a two-tailed t-test. In ANOVA, the F-test is the natural generalization of the t-test to more than two groups. But when comparing exactly two group means with equal variances, the t-test and F-test give identical p-values. In regression, the F-test assesses overall model significance while t-tests assess individual predictor significance.
• When do I use F-test for variance equality vs ANOVA F-test? ▼
  These are two distinct applications of the F-distribution. The variance equality F-test is a preprocessing step or standalone test: use it before a two-sample t-test to decide whether to apply Student's (equal variance) or Welch's (unequal variance) version. You're testing a structural assumption about the data-generating process. The ANOVA F-test is the main inferential test: it determines whether group means differ. It uses MS_between / MS_within rather than a direct variance ratio. In ANOVA, you assume variances ARE equal (homoscedasticity) — that's an assumption you check with Levene's test, not with the F-test for variances. Conflating these two uses is a common error. The variance equality F-test produces a ratio of two sample variances. The ANOVA F-test produces a ratio of mean squares representing different sources of variance. Both follow F-distributions but with different degrees of freedom and different null hypotheses.
• Why is the F-test sensitive to non-normality? ▼
  The F-test for variance equality is notoriously sensitive to departures from normality — far more so than the t-test. This is because the ratio of two chi-squared random variables follows an F-distribution only when the underlying data is normally distributed. Even mild skewness or kurtosis in the data can cause the F-test to produce highly inflated Type I error rates (false rejections of H₀). A Monte Carlo simulation by Box in 1953 showed that F-tests for variance equality with non-normal data can have actual Type I error rates of 20–40% even when the nominal α is 0.05. For this reason, statisticians now generally recommend Levene's test (which tests whether absolute deviations from the group median differ) or the Brown-Forsythe test (using medians instead of means) as more robust alternatives. Bartlett's test, while more powerful under normality, is even more sensitive to non-normality than the classical F-test. In practice, the ANOVA F-test for means is much more robust to non-normality than the variance equality F-test.
• What's a typical F-statistic value that indicates significance? ▼
  There is no single universal critical F-value — it depends on the degrees of freedom in the numerator and denominator, and the chosen significance level α. For the variance equality F-test with n₁=15, n₂=12 (df₁=14, df₂=11) at α=0.05 (two-tailed, so α/2=0.025 per tail), the critical value is approximately 3.09. For ANOVA with k=3 groups and N=30 total observations (df₁=2, df₂=27) at α=0.05, the critical F is approximately 3.35. For a regression model with 3 predictors and 46 error degrees of freedom at α=0.05, critical F ≈ 2.80. As degrees of freedom increase, critical F values decrease toward 1.0. F=1.0 means between-group and within-group variances are equal — exactly what the null hypothesis predicts. Rules of thumb: F > 4 is often significant for small studies; F > 2 may be significant for large studies with many degrees of freedom. Always compute the exact p-value rather than relying on a fixed threshold.

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