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Friedman test

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The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns.


  1. Given data \{x_{ij}\}_{m\times n}, that is, a tableau with m rows (the blocks), n columns (the treatments) and a single observation at the intersection of each block and treatment, calculate the ranks within each block. Replace the data with a new tableau \{r_{ij}\}_{m \times n} where the entry r_{ij} is the rank of x_{ij} within block i.
  2. Find the values:
    • SS_t = n\sum_{j=1}^n (\bar{r}_{j} - \bar{r})^2,
    • SS_e = \frac{1}{m(n-1)} \sum_{i=1}^m \sum_{j=1}^n (r_{ij} - \bar{r})^2
    • \bar{r}_{j} = \frac{1}{m} \sum_{i=1}^m {r_{ij}}
    • \bar{r} = \frac{1}{mn}\sum_{i=1}^m \sum_{j=1}^n r_{ij}
  3. The test statistic is given by Q = \frac{SS_t}{SS_e}.
  4. Finally, the p-value is given by \mathbf{P}(\chi^2_{n-1} \ge Q).

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