F-Test Calculators

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The F-test is a statistical test that uses the F-distribution to compare variances or assess whether group means differ significantly. It is named after Ronald Fisher and is the foundation of analysis of variance (ANOVA). The F-statistic is the ratio of two variance estimates — between-group variance to within-group variance in ANOVA, or explained to unexplained variance in regression. A large F-value indicates that variance between groups is large relative to variance within groups, suggesting true group differences. The p-value from the F-distribution determines statistical significance.

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F-Statistic Formula

In one-way ANOVA: F = MS_between / MS_within

MS_between = SS_between / df_between (variance due to group differences). MS_within = SS_within / df_within (variance within groups, unexplained). df_between = k − 1 (k = number of groups). df_within = N − k (N = total observations).

If all group means are equal, F ≈ 1. Large F → reject the null hypothesis that all means are equal.

Interpreting F-Test Results

  • F = 1: between-group and within-group variance are equal — no evidence of group differences
  • F > 1: more variance between groups than within — potential group effect
  • Compare F to critical value from F-table at chosen α (0.05) and appropriate df numerator and denominator, OR use p-value directly
  • p < 0.05: statistically significant group differences

F-Test in Regression

In linear regression, the overall F-test evaluates whether the model explains significantly more variance than a model with no predictors: F = MS_regression / MS_residual. A significant F means at least one predictor is useful. R² = SS_regression / SS_total tells you the proportion explained.

Levene's F-Test for Equal Variances

Before ANOVA, test whether group variances are equal (homoscedasticity assumption) using Levene's test, which is itself an ANOVA on the absolute deviations from group medians. Significant Levene's test suggests Welch's ANOVA should be used instead of standard one-way ANOVA.

Glossary

F-Statistic
The ratio of two variance estimates: MS_between / MS_within in ANOVA; F ≈ 1 under the null hypothesis; large F indicates group differences; compared to F-distribution for p-value.
Mean Square (MS)
A variance estimate in ANOVA: MS = SS / df; MS_between measures variance due to group differences; MS_within measures variance within groups (unexplained noise).
Levene's Test
An F-test for homogeneity of variances; tests whether group variances are equal before ANOVA; significant result suggests using Welch's ANOVA instead of standard one-way ANOVA.

Frequently Asked Questions

The F-test compares two variance estimates using the F-statistic (ratio of variances). In one-way ANOVA, F = MS_between / MS_within. If group means are truly equal, both MS values estimate the same population variance and F ≈ 1. When group means differ, MS_between is inflated and F > 1. The p-value from the F-distribution tells you the probability of getting an F this large by chance if the null hypothesis (all means equal) is true.

A large F-statistic with a small p-value (typically p < 0.05) indicates statistically significant differences among group means. However, ANOVA only tells you that at least one group differs — it doesn't say which ones. Follow a significant ANOVA with post-hoc tests (Tukey HSD, Bonferroni, Dunnett) to identify which specific group pairs differ. F close to 1 with large p-value means you cannot reject the null hypothesis of equal means.

One-way ANOVA F-test assumes: (1) independent observations; (2) normally distributed residuals within each group; (3) homogeneity of variances (equal σ² across groups — tested with Levene's test). ANOVA is fairly robust to mild violations of normality, especially with large, equal sample sizes. For unequal variances, use Welch's ANOVA. For non-normal data with small samples, use the Kruskal-Wallis test (non-parametric alternative).

In ANOVA: F = MS_between / MS_within, testing whether group means differ. In linear regression: F = MS_regression / MS_residual, testing whether the overall model explains significantly more variance than a null model with no predictors. Both use the same F-distribution logic — ratio of explained to unexplained variance — but the context differs. In multiple regression, a significant overall F means at least one predictor is useful, but individual t-tests are needed to identify which specific predictors are significant.