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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.


  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.

See also


  • 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 de Kruskal-Wallis


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