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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

Rank all data from all groups together.
The test statistic is given by: , 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 $(N-1)N(N+1)/12$ .
Finally, the p-value is approximated by $\mathbf{P}(\chi^2_{N-g} \ge K)$ . If some n_i's are small the distribution of K can be quite different from this.

References

William H. Kruskal and W. Allen Wallis. Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association 47 (260): 583–621, December 1952.es:Prueba de Kruskal-Wallis

nl:Kruskal-Wallis

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Kruskal-Wallis one-way analysis of variance

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