Kruskal-Wallis Test Reference Cheat Sheet

Key Facts

• Use the Kruskal-Wallis test to compare $k \ge 3$ independent groups when the response is ordinal or a continuous variable that does not meet ANOVA assumptions.
• The null hypothesis is $H_0$: all groups have the same distribution, often interpreted as equal population medians when group distributions have similar shapes.
• The alternative hypothesis is $H_a$: at least one group tends to have larger or smaller values than at least one other group.
• Rank all $N$ observations from smallest to largest across all groups, assigning average ranks to tied values.
• For group $i$, let $n_i$ be the group size and $R_i$ be the sum of ranks in that group.
• The Kruskal-Wallis statistic is $H = \frac{12}{N(N+1)}\sum_{i=1}^{k}\frac{R_i^2}{n_i} - 3(N+1)$.
• When ties occur, use the corrected statistic $H_c = \frac{H}{1 - \frac{\sum(t_j^3 - t_j)}{N^3 - N}}$, where $t_j$ is the size of tie group $j$.
• For large enough samples, compare $H$ or $H_c$ to $\chi^2_{k-1}$ and reject $H_0$ when the $p$-value is less than $\alpha$.

Vocabulary

Kruskal-Wallis test

A nonparametric hypothesis test used to compare three or more independent groups using ranks instead of raw data values.

Rank sum

The rank sum $R_i$ is the total of all ranks assigned to observations in group $i$.

Degrees of freedom

For the Kruskal-Wallis test, the degrees of freedom are $df = k - 1$, where $k$ is the number of groups.

Tie correction

A tie correction adjusts the $H$ statistic when equal data values create tied ranks.

Post hoc comparison

A post hoc comparison tests which specific group pairs differ after the overall Kruskal-Wallis test is significant.

Nonparametric test

A nonparametric test makes fewer assumptions about the population distribution than a parametric test such as ANOVA.