One group per line. An optional "name:" prefix is ignored; separate values with commas, spaces, tabs or semicolons.
SS_between = Σnᵢ(x̄ᵢ−x̄)², SS_within = ΣΣ(x−x̄ᵢ)²; F = MSB/MSW; p from the F-distribution upper tail via the regularized incomplete beta function.
About this tool
The One-Way ANOVA Calculator tests whether the means of two or more independent groups differ by more than chance. Paste one group per line — an optional label before a colon is ignored, so "Group A: 6, 8, 4" works just as well as bare numbers separated by commas, spaces, tabs or semicolons. Everything runs locally in your browser; nothing you enter is uploaded.
It splits the total variation into a between-groups and a within-groups part. The between-groups sum of squares is SS_between = Σ nᵢ(x̄ᵢ − x̄)² with df₁ = k − 1, and the within-groups sum of squares is SS_within = ΣΣ(x − x̄ᵢ)² with df₂ = N − k, where k is the number of groups and N the total sample size. Dividing each by its degrees of freedom gives the mean squares MSB and MSW, and the test statistic is F = MSB / MSW. The p-value is the upper-tail area of the F distribution, computed exactly from the regularized incomplete beta function Iₓ(df₂/2, df₁/2) rather than a lookup table.
The effect-size measure eta-squared (η² = SS_between / SS_total) reports the proportion of total variance explained by group membership. A result is flagged significant when p < 0.05, meaning you would reject the null hypothesis that all group means are equal. You need at least two groups and more observations than groups so the within-groups degrees of freedom is positive.
Frequently asked questions
How is the p-value calculated?
From the F distribution's upper tail: p = Iₓ(df₂/2, df₁/2) with x = df₂/(df₂ + df₁·F), where Iₓ is the regularized incomplete beta function evaluated with a Lentz continued fraction and a Lanczos log-gamma. This is exact, not a table interpolation.
What counts as significant?
By convention the tool flags significance at α = 0.05: if p < 0.05 you reject the null hypothesis that all group means are equal. A significant F tells you at least one group differs, but not which — use a post-hoc test like Tukey's HSD to find out.
What does eta-squared mean?
Eta-squared (η² = SS_between / SS_total) is the share of total variance explained by the grouping variable, from 0 to 1. Rough benchmarks are 0.01 small, 0.06 medium and 0.14 large. It is a descriptive effect size that complements the significance test.
How do I enter my groups?
One group per line. You may prefix a name and colon ("Control: 5, 7, 6") or just list the numbers; values can be separated by commas, spaces, tabs or semicolons. You need two or more groups and more total values than groups.