One-Way ANOVA Calculator

Formula

How this is calculated

Enter your data with one group per line, separating the numeric observations within a line by spaces or commas. The calculator reads k groups, computes each group mean x̄ⱼ over its nⱼ values, and the grand mean x̄ across all N observations. It then partitions the total variability into a between-groups sum of squares SSB = Σ nⱼ(x̄ⱼ − x̄)² and a within-groups (error) sum of squares SSW = ΣΣ(x − x̄ⱼ)².

The degrees of freedom are df_between = k − 1 and df_within = N − k. Dividing each sum of squares by its degrees of freedom gives the mean squares MSB = SSB/(k−1) and MSW = SSW/(N−k). The test statistic is F = MSB / MSW; under the null hypothesis that all group means are equal it follows an F distribution with (df_between, df_within) degrees of freedom. The reported p-value is the upper-tail probability P(F ≥ observed), evaluated with the regularized incomplete beta function.

One-way ANOVA assumes independent observations, approximately normal residuals, and roughly equal variances across groups. You need at least two groups and at least one more total observation than there are groups (so df_within > 0). A large F with a small p-value (commonly p < 0.05) indicates at least one group mean differs; ANOVA itself does not say which pair differs, so follow a significant result with a post-hoc test.