One-Way ANOVA

Every value's distance from the grand mean splits into two parts: its group's distance from the grand mean and its own distance from the group mean. Summed and squared these are SS_between = Σ nᵢ(x̄ᵢ − x̄)² and SS_within = Σ(x − x̄ᵢ)², adding to SS_total. Divided by their degrees of freedom (k − 1 and N − k) they become mean squares, and F = MS_between ÷ MS_within. If the group means are all equal, F is around 1; means that differ push it up.

The p-value is the upper tail of the F distribution with (k − 1, N − k) degrees of freedom, computed through the regularised incomplete beta function; the critical F at your α is found by inverting the same function. η² = SS_between ÷ SS_total is the share of the variance explained by group membership; ω² corrects its optimism for small samples. The classic test assumes roughly normal groups with similar spread — the page reports the ratio of the largest to smallest SD and flags it above 2 — and independent observations.

A significant F says at least one mean differs, not which. As a simple post-hoc step the page runs a Welch t-test on every pair (no equal-variance assumption) and multiplies each p by the number of pairs (Bonferroni), a conservative correction. Tukey HSD, the usual alternative, needs the studentized range distribution and is not computed here. With two groups the test reduces to a t-test (F = t²). Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.