Kruskal-Wallis test

The Kruskal-Wallis test is a non-parametric method that compares medians across three or more independent groups by ranking all observations together, used when ANOVA's normal-distribution assumption is not met.

ByHoang TruongUpdated July 5, 2026

See it move

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Three delivery teams' nine wait times are ranked together from 1 to 9, giving rank sums of 15, 24 and 6. The test statistic H = (12 ÷ 90) × (15²/3 + 24²/3 + 6²/3) − 30 works out to 7.2, which exceeds the chi-square critical value of 5.99 for 2 degrees of freedom, so the teams' distributions are not all equal.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicCommon Significance TestsAdvanced

The formula

LaTeX

H = \dfrac{12}{N(N+1)} \sum_{i=1}^{k}\dfrac{R_i^2}{n_i} - 3(N+1)

Variables

$N$: Total number of observations across all groups
$k$: Number of groups
$R_{i}$: Sum of ranks in group i
$n_{i}$: Number of observations in group i

Converts rank sums into a test statistic that follows a chi-square distribution with k − 1 degrees of freedom under the null hypothesis of equal distributions.