Statistics
Perform one-way ANOVA to compare means across multiple groups
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F-Statistic
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P-value
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Significance (alpha=0.05)
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df (between)
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df (within)
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SS Between
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SS Within
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MS Between
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MS Within
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How it works
Analysis of variance tests whether three or more group means differ by more than sampling noise would explain. It compares the spread between the group means against the spread within each group, and reports that comparison as an F statistic with a p-value.
Formula
SS_between = Σ nⱼ(x̄ⱼ − x̄)² SS_within = Σ Σ (xᵢⱼ − x̄ⱼ)² F = (SS_between / (k − 1)) / (SS_within / (N − k))
Variables
knumber of groups being comparedNtotal observations across all groupsnⱼ, x̄ⱼsize and mean of group jx̄grand mean of every observationMSmean square: a sum of squares divided by its degrees of freedom
Worked example
Inputs: Three groups of four: 5, 7, 6, 6 | 8, 9, 7, 8 | 11, 9, 10, 10
- Group means are 6, 8, 10; grand mean is 8
- SS_between = 4(6 − 8)² + 4(8 − 8)² + 4(10 − 8)² = 16 + 0 + 16 = 32
- Each group has a within-group sum of squares of 2, so SS_within = 6
- Degrees of freedom: k − 1 = 2 between, N − k = 9 within
- MS_between = 32 / 2 = 16, MS_within = 6 / 9 = 0.6667
- F = 16 / 0.6667 = 24
Result: F(2, 9) = 24, p ≈ 0.00025, far past the 5% critical value of 4.26
Notes
- A significant F says at least one group differs, and nothing about which one. Follow it with Tukey HSD or a similar post-hoc test; running a separate t-test on every pair inflates the false-positive rate.
- Two-way ANOVA splits the between-groups variation into a main effect for each factor plus an interaction term. Read the interaction first, because when factors interact the main effects on their own can point in a misleading direction.
- The test assumes roughly equal variances across groups. A Levene test flags violations, and Welch ANOVA is the usual fix when group sizes are unequal too.
- F says nothing about effect size. Eta-squared, SS_between / SS_total, is 32 / 38 = 0.84 here, so group membership accounts for 84% of the total variation.
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