Calculate the F-statistic with p-value, critical value, ANOVA summary table, effect sizes (eta-squared, omega-squared), and rejection region visualization.
The F-statistic tests whether group means are significantly different by comparing the variance between groups to the variance within groups. A large F indicates that the between-group differences are unlikely to be due to chance alone. It is the primary test statistic for ANOVA (Analysis of Variance) and for testing the overall significance of regression models.
This calculator computes the F-statistic from mean squares, derives p-values and critical values for the F-distribution, and generates a complete ANOVA summary table. It supports three test types: one-way ANOVA (comparing k independent groups), two independent samples (special case with k=2), and regression F-tests. Effect sizes (η² and ω²) quantify how much of the total variation is explained by group membership.
The rejection region visualization shows where your F-statistic falls relative to the critical value, and the reference table displays critical values at six common significance levels. Whether you're analyzing experimental data or checking textbook solutions, this calculator provides the complete inferential picture.
Computing the F-statistic by hand requires looking up critical values in printed tables that only cover a few degrees of freedom. This calculator handles arbitrary df₁ and df₂, provides exact p-values (not just "p < 0.05"), and generates the complete ANOVA summary table that research papers require.
The effect size calculations (η² and ω²) go beyond statistical significance to answer "how much does group membership matter?" — a question that p-values alone cannot answer. The multi-level critical value table lets you quickly see whether your result is significant at various alpha levels without recalculating.
F = MSB / MSW One-Way ANOVA: df₁ = k − 1, df₂ = N − k Two Samples: df₁ = 1, df₂ = N − 2 Regression: df₁ = p, df₂ = N − p − 1 η² = SSB / SST (eta-squared) ω² = (SSB − df₁ × MSW) / (SST + MSW) (omega-squared) where SSB = MSB × df₁, SSW = MSW × df₂, SST = SSB + SSW
Result: F(2, 27) = 3.5313, p = 0.0435, F-critical = 3.3541, η² = 0.2074
With 3 groups and N=30, the between-group variance is 3.53 times larger than the within-group variance. Since F = 3.53 exceeds the critical value of 3.35, we reject H₀ at α = 0.05. The p-value of 0.0435 confirms significance. Eta-squared of 0.207 indicates a large effect — about 20.7% of total variance is explained by group membership.
The F-distribution is a ratio of two independent chi-squared random variables, each divided by its degrees of freedom. It is right-skewed, with the shape depending on both df₁ and df₂. As both degrees of freedom increase, the distribution approaches normality. Unlike the t and z distributions, the F-distribution is not symmetric — critical values only exist in the right tail for the standard ANOVA F-test.
One-way ANOVA tests one factor. Two-way ANOVA tests two factors and their interaction, partitioning variance into main effects and interaction. Repeated measures ANOVA handles correlated observations. MANOVA extends to multiple dependent variables. ANCOVA adds continuous covariates. All use F-statistics, but with different degree-of-freedom formulas. This calculator handles one-way ANOVA, two-sample F-tests, and regression — the most common applications.
Statistical significance (p < α) tells you an effect exists; effect size tells you how big it is. Cohen's guidelines classify η² as small (0.01), medium (0.06), or large (0.14), but these are rough benchmarks. In applied research, even a "small" effect can be practically important if the outcome matters (medical treatments, safety interventions). Always interpret effect size in context.
The null hypothesis H₀ is that all group means are equal (μ₁ = μ₂ = ... = μₖ). The alternative hypothesis H₁ is that at least one mean differs. A significant F tells you that groups differ, but not which specific groups differ — you need post-hoc tests for that.
F is a ratio of two variances (MSB/MSW), and variances are always non-negative. Under H₀, F ≈ 1 because both estimates approximate the same population variance. Large F values (much greater than 1) suggest real group differences.
Eta-squared (η² = SSB/SST) is the proportion of variance explained but it's positively biased — it overestimates the population effect, especially with small samples. Omega-squared (ω²) corrects for this bias and is the preferred effect size measure for ANOVA.
The t-test compares exactly two groups; the F-test compares two or more. For exactly two groups, F = t² and the p-values are identical. Use ANOVA (F-test) when comparing three or more groups simultaneously to avoid inflating the Type I error rate that multiple t-tests would cause.
Independence of observations, normally distributed populations within each group, and homogeneity of variances (equal population variances across groups). Violations of normality matter less with large samples. For unequal variances, use Welch's ANOVA instead.
A non-significant F (p > α) means you fail to reject H₀ — there's not enough evidence to conclude that group means differ. This does not prove they're equal. Check statistical power: small samples may miss real effects (Type II error).