Kruskal-Wallis Test#

The Kruskal-Wallis test (also: H-test) is the nonparametric alternative to one-way analysis of variance (ANOVA). It tests whether the central tendency of three or more independent groups differs significantly.

When to Use#

Use the Kruskal-Wallis test when you want to:

Compare three or more independent groups
The dependent variable is at least ordinally scaled
The assumptions of ANOVA (normality, homogeneity of variance) are not met
The samples are small

Assumptions#

Independence of observations
At least ordinal scale of measurement for the dependent variable
Similar distribution shape across all groups (for interpretation as median comparison)
Random sampling

Formula#

All observations are ranked together. The test statistic $H$ is calculated as:

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

where $N$ is the total number of observations, $k$ is the number of groups, $n_j$ is the number of observations in group $j$ , and $R_j$ is the rank sum of group $j$ .

A correction is applied for ties:

H_{\text{corr}} = \frac{H}{1 - \frac{\sum (t_i^3 - t_i)}{N^3 - N}}

Example#

Practical Example: Customer Satisfaction

A company wants to compare customer satisfaction (scale 1–5) across three different store locations:

Store A (n=25): City center
Store B (n=30): Shopping mall
Store C (n=22): Suburb

Since the satisfaction scale is ordinal and the normality assumption is not tenable, the Kruskal-Wallis test is used. If the result is significant, pairwise comparisons follow (e.g., Dunn's test).

Effect Size#

Eta-squared ( $\eta^2$ ) based on the H statistic:

\eta^2_H = \frac{H - k + 1}{N - k}

Effect Size	$\eta^2$
Small	0.01
Medium	0.06
Large	0.14