Statistics
ANOVA F-Test Guide
One-way ANOVA tests whether the means of three or more groups differ significantly by comparing variance between groups to variance within groups. This guide walks through the concept and provides an F-critical lookup table for common degrees of freedom.
How One-Way ANOVA Works
1
State hypotheses: H₀: all group means are equal (μ₁ = μ₂ = μ₃...). H₁: at least one group mean differs.
2
Calculate SSB (Sum of Squares Between): how much the group means vary from the grand mean.
3
Calculate SSW (Sum of Squares Within): how much individual values vary within each group.
4
Compute MSB and MSW: MSB = SSB / (k−1), MSW = SSW / (N−k), where k = number of groups, N = total observations.
5
F = MSB / MSW. A large F means between-group variance is large relative to within-group variance — evidence against H₀.
6
Compare F to F-critical at df₁ = k−1 (numerator), df₂ = N−k (denominator). If F > F-critical, reject H₀.
F = MSB / MSW = [SSB/(k−1)] / [SSW/(N−k)]
ANOVA assumes: independence of observations, approximately normal distribution within each group, and homogeneity of variance (Levene's test can verify this).
F-Critical Value Lookup
α = 0.05: —
α = 0.01: —
Quick Reference Table (α = 0.05)
| df₁ → | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| df₂ = 10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 |
| df₂ = 20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 |
| df₂ = 30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 |
| df₂ = 60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 |
| df₂ = 120 | 3.92 | 3.07 | 2.68 | 2.45 | 2.29 |