One-Way ANOVA Calculator (From Group Means & Variances)
Run a one-way ANOVA from each group’s mean, variance and size — get the F statistic, between/within degrees of freedom and a two-tailed p-value.
How it works
Grand mean x̄ = Σnᵢx̄ᵢ ÷ Σnᵢ · SSB = Σnᵢ(x̄ᵢ − x̄)² · SSW = Σ(nᵢ−1)s²ᵢ · F = (SSB/dfB) ÷ (SSW/dfW)
One-way ANOVA tests whether three or more group means are all equal. This simplified version works straight from each group’s summary statistics — its mean, sample variance and size — so you do not need the raw data. It splits total variation into a between-groups part (how far the group means sit from the grand mean) and a within-groups part (the pooled spread inside groups); a large ratio of the two is evidence the means differ.
Worked example
Three classes of 10 score means 82, 76 and 88 with variances 40, 55 and 35. The grand mean is 82, SSB = 720, SSW = 1,170, so MS between = 360 and MS within = 43.3. F = 360 ÷ 43.3 ≈ 8.3 on (2, 27) df, giving p ≈ 0.0016 — the class means differ significantly.
Frequently asked questions
What does ANOVA tell me that multiple t-tests do not?
Running many pairwise t-tests inflates the chance of a false positive. ANOVA gives a single test of "are any of the means different?" while holding the overall error rate at α. If it is significant, follow up with a post-hoc test to see which pairs differ.
What is η² (eta squared)?
Eta squared is the share of the total variation explained by group membership: SSB ÷ (SSB + SSW). It is a measure of effect size, from 0 (groups identical) toward 1 (groups explain everything). Roughly, 0.01 is small, 0.06 medium and 0.14 large.
Do the groups need equal sizes?
No. This calculator weights each group by its own size, so unequal group sizes are handled correctly as long as every group has at least two observations.
Related calculators
For education — confirm the test assumptions (independence, sample size, roughly normal data) before drawing conclusions from a p-value.