Kruskal-Wallis Test

The non-parametric analogue of one-way ANOVA — comparing three or more groups using ranks rather than raw values.

Hypothesis test — rejection region and test statistic

✓ Fail to reject H₀ — p-value ≈ 0.0719 > α=0.05

α = 0.05

z = 1.80

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Definition

The Kruskal-Wallis test is the nonparametric analog of one-way ANOVA. It tests whether $k \geq 2$ independent groups have the same distribution, without assuming normality.

Null hypothesis: all $k$ groups have the same distribution (or equivalently, the same population median).

Procedure:

Pool all observations and assign ranks (smallest = 1, handle ties by averaging)
Compute the mean rank for each group: $\bar{R}_j = R_j / n_j$
Test statistic:

$H = \frac{12}{N(N+1)} \sum_{j=1}^k \frac{R_j^2}{n_j} - 3(N+1)$

Under $H_0$ , $H \approx \chi^2_{k-1}$ for large samples

Key properties

Reduces to the Mann-Whitney U test exactly when $k=2$ groups
Uses only rank information, so it's robust to outliers and doesn't require normality
Tests equality of entire distributions, not just medians — differences in spread or shape can also trigger rejection
$H$ is always non-negative, with larger values indicating more uneven rank distribution across groups

Common mistakes

Interpreting a significant result as "the medians differ": the null hypothesis is equality of distributions — groups with equal medians but different variances or shapes can still trigger rejection
Skipping multiplicity correction for post-hoc comparisons: a significant omnibus result only says some group differs; pairwise follow-ups need correction (Dunn's test or Bonferroni-adjusted Mann-Whitney)

Comparing three teaching methods

Three groups of students (same size $n_j = 5$ , total $N = 15$ ). After pooling and ranking:

Method A: ranks $\{2, 5, 7, 9, 11\}$ , $R_A = 34$
Method B: ranks $\{1, 3, 6, 8, 10\}$ , $R_B = 28$
Method C: ranks $\{4, 12, 13, 14, 15\}$ , $R_C = 58$

$H = \frac{12}{15\cdot16}(34^2/5 + 28^2/5 + 58^2/5) - 3(16) = \frac{12}{240}(925.8) - 48 \approx 6.3$

Compare to $\chi^2_2 = 5.99$ ( $\alpha = 0.05$ ): $6.3 > 5.99$ , reject $H_0$ .

Try it

The Kruskal-Wallis test tells you some groups differ. What must you do to determine which groups differ?

Solution

Run post-hoc tests — pairwise Mann-Whitney U tests between groups. But with $k$ groups, there are $\binom{k}{2}$ pairs, leading to multiple testing concerns. You must apply a correction such as Bonferroni (adjust $\alpha$ by dividing by number of comparisons) or Dunn's test (a procedure specifically designed for post-hoc comparisons after Kruskal-Wallis, using the original ranks).

Dunn's test is preferred because it uses the pooled variance from the full Kruskal-Wallis ranking, making it more powerful than separate Mann-Whitney tests with Bonferroni correction.

Related concepts

Statistics· Statistical Tests

ANOVAAnalysis of variance — comparing means across three or more groups by partitioning total variability into between-group and within-group components.

Statistics· Statistical Tests

Mann-Whitney TestA non-parametric test comparing two groups by ranking all observations — valid without normality assumptions.

Statistics· Inference

Hypothesis TestingA formal framework for deciding whether data provides enough evidence to reject a default assumption.