One-way ANOVA

Tests whether k ≥ 2 group means are all equal by comparing variation between group means with variation within groups.

SS_B = \sum_{i=1}^{k} n_i(\bar y_i - \bar y)^2,\qquad SS_W = \sum_{i=1}^{k}\sum_{j=1}^{n_i}(y_{ij} - \bar y_i)^2,\qquad SS_T = SS_B + SS_W

$y_{ij}$: Observation j in group i.
$\bar y_i, n_i$: Mean and size of group i.
$\bar y$: Grand mean of all N observations.
$k$: Number of groups.

F = \frac{MS_B}{MS_W} = \frac{SS_B/(k-1)}{SS_W/(N-k)},\qquad df = (k-1,\ N-k)

p-value = upper tail of the F distribution.

$MS_B, MS_W$: Mean squares between and within groups.
$N$: Total number of observations.

When to use

A quantitative outcome compared across three or more independent groups (with two groups, ANOVA's F equals the pooled $t^2$ ). It replaces many pairwise t-tests, which would inflate the false-positive rate.

Assumptions

Independent observations and groups.
Roughly normal residuals within groups (robust with moderate, similar group sizes).
Equal population variances. If SDs differ markedly (largest/smallest above about 2) or sizes are unequal, use Welch's ANOVA; check with Levene/Brown–Forsythe and residual plots.

How to interpret

A small p-value is evidence that at least one mean differs — not that all differ, and not which. Follow with Tukey HSD or planned comparisons.
Effect sizes: $\eta^2 = SS_B/SS_T$ (share of variation explained in the sample) and $\omega^2$ (less biased).

Common mistakes

Concluding all means differ after a significant F.
Running ANOVA on repeated measurements of the same subjects as if groups were independent.
Treating a non-significant F as proof that the means are equal.

Example

Groups: (4, 5, 6, 5), (7, 8, 6, 7), (5, 6, 7, 6). Means 5, 7, 6; grand mean 6. $SS_B = 4(1 + 1 + 0) = 8$ , $SS_W = 2 + 2 + 2 = 6$ . $F = (8/2)/(6/9) = 6.0$ on (2, 9) df, $p = 0.022$ ; $\eta^2 = 0.571$ , $\omega^2 = 0.455$ . Tukey HSD then locates the difference (groups 1 and 2).