# Kruskal-Wallis one-way analysis of variance

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In statistics, the Kruskal-Wallis one-way analysis of variance by ranks (named after William Kruskal and Allen Wallis) is a non-parametric method. Intuitively, it is identical to a one-way analysis of variance, with the data replaced by their ranks.

Since it is a non-parametric method, the Kruskal-Wallis test does not assume a normal population, unlike the analogous one-way analysis of variance.

## Method

1. Rank all data from all groups together.
2. The test statistic is given by: $K = (N-1)\frac{\sum_{i=1}^g n_i(\bar{r}_{i\cdot} - \bar{r})^2}{\sum_{i=1}^g\sum_{j=1}^{n_i}(r_{ij} - \bar{r})^2}$, where:
• $n_g$ is the number of observations in group $g$
• $r_{ij}$ is observation $j$ from group $i$
• $N$ is the total number of observations across all groups
• $\bar{r}_{i\cdot} = \frac{\sum_{j=1}^{n_i}{r_{ij}}}{n_i}$,
• $\bar{r}$ is the average of all the $r_{ij}$, equal to $(N+1)/2$.
Notice that the denominator of the expression for $K$ is exactly $(N-1)N(N+1)/12$.
3. Finally, the p-value is approximated by $\mathbf{P}(\chi^2_{N-g} \ge K)$. If some ni's are small the distribution of K can be quite different from this.