ANOVA Calculator — One-Way ANOVA from Summary Statistics

公式

How this is calculated

For each line you enter the sample size nⱼ, group mean x̄ⱼ, and sample standard deviation sⱼ. The calculator first computes the weighted grand mean x̄ = Σ(nⱼ × x̄ⱼ) / N. Then it partitions the total variability: the between-group sum of squares SSB = Σ nⱼ(x̄ⱼ − x̄)² captures how far each group mean lies from the grand mean; the within-group (error) sum of squares SSW = Σ(nⱼ − 1)sⱼ² pools the within-group variance across all groups using the sample variances you supplied.

Dividing each sum of squares by its degrees of freedom gives the mean squares MSB = SSB/(k−1) and MSW = SSW/(N−k). The F statistic is MSB/MSW; under the null hypothesis that all group means are equal it follows an F distribution with (k−1, N−k) degrees of freedom, and the p-value is the upper-tail probability. The eta-squared effect size η² = SSB / (SSB + SSW) indicates what fraction of the total variability is explained by the group factor (small ≈ 0.01, medium ≈ 0.06, large ≈ 0.14 by Cohen's conventions).

This approach assumes that supplying n, mean, and SD is equivalent to having the raw data for within-group variance (i.e. the sample SD reflects actual within-group spread). It requires nⱼ ≥ 2 for each group so that within-group variance is estimable. One-way ANOVA additionally assumes independent observations, approximate normality of residuals, and homogeneity of variance across groups.