Enumerating anything but small permutation distributions in unmanagable, so instead we take a random sample of the permutations
Ranks and the Wilcoxon Rank-Sum test
Given a sample X1, X2, ..., XN, of N observations, the rank of Xi among the N observations is denoted by R(Xi), and is given by:
R(Xi) = number of Xj's <= Xi
Let W be the sum of the ranks from one treatment group. The Wilcoxon rank-sum test is a two-sample permutation test based on W.
Revisiting our earlier example:
| New group | Old group | New group ranks | Old group ranks | New group rank sum |
| 7,15 | 2,4,8 | 3,5 | 1,2,4 | 8 |
| 8,15 | 2,4,7 | 4,5 | 1,2,3 | 9 |
| 2,15 | 4,7,8 | 1,5 | 2,3,4 | 6 |
| 4,15 | 2,7,8 | 2,5 | 1,3,4 | 7 |
| 7,8 | 2,4,15 | 3,4 | 1,2,5 | 7 |
| 2,7 | 4,8,15 | 1,3 | 2,4,5 | 4 |
| 4,7 | 2,8,15 | 2,3 | 1,4,5 | 5 |
| 2,8 | 4,7,15 | 1,4 | 2,3,5 | 5 |
| 4,8 | 2,7,15 | 2,4 | 1,3,5 | 6 |
| 2,4 | 7,8,15 | 1,2 | 3,4,5 | 3 |
Let's examine the permutation distribution of the new group rank sums:.
Steps in the Wilcoxon rank-sum test, Tables, adjustment for ties.