Kruskal–Wallis test

A rank-based alternative to one-way ANOVA that tests whether k independent groups come from the same distribution.

H = \frac{\dfrac{12}{N(N+1)}\displaystyle\sum_{i=1}^{k} n_i\left(\bar R_i - \frac{N+1}{2}\right)^2}{1 - \dfrac{\sum (t^3 - t)}{N^3 - N}},\qquad df = k - 1

p-value = upper tail of χ² with k − 1 df (large-sample approximation). The denominator is the tie correction.

$\bar R_i$: Mean rank of group i in the pooled sample (average ranks for ties).
$n_i, N, k$: Group size, total size, number of groups.
$t$: Size of each group of tied values.

\varepsilon^2 = \frac{H}{N - 1}

Effect size (epsilon squared), between 0 and 1.

$H$: The Kruskal–Wallis statistic.

When to use

Three or more independent groups with an ordinal outcome or skewed quantitative data with small groups (e.g. delays by airline).

Assumptions

Independent observations; ordinal or continuous outcome. Interpreting it as comparing medians requires similarly shaped distributions. The χ² approximation is rough when groups have fewer than about 5 observations.

How to interpret

A small p-value is evidence that at least one group tends to have larger values than another. Follow up with pairwise Mann–Whitney tests (or Dunn's test) with a multiple-comparison adjustment.

Common mistakes

Calling it "ANOVA on medians" without checking shapes.
Forgetting the follow-up adjustment for pairwise comparisons.

Example

Groups (2.1, 3.4, 2.8, 3.0), (4.2, 3.9, 4.8, 3.6), (3.1, 2.5, 3.3, 2.7). Mean ranks 4.5, 10.5, 4.5 (overall mean rank 6.5). $H = \frac{12}{12 \cdot 13}\cdot 4(4 + 16 + 4) = 7.38$ on 2 df, $p = 0.025$ ; $\varepsilon^2 = 0.67$ . The second group tends to be higher (approximation rough with 4 per group).