The Mann - Whitney - Wilcoxon Test
The Mann – Whitney – Wilcoxon test may be used to test the hypothesis that two independent samples A and B arise from the same population. The test is very nearly as powerful as the two sample t – test, and may be used where the t – test may not because of lack of normaility. The procedure is:
Pool the two samples and sort the combined data into ascending order, simultaneously keeping track of which data belongs to which group.
Allocate a rank to each data value with the smallest being given rank one. If two ranks are equal, allocate the average of the two ranks to each.
Add up the ranks with
and 
Notice that if A has n-A data points and B has n-B data points then
The Mann – Whitney – Wilcoxon test statistic is
Very small or large values for
imply rejection of the null hypothesis that the two samples are from the same population.- is compared with values of the null distribution of taken from a table.
Example: Suppose men and women run a 800m race.
The men finish in times 1.50.32,1.54.45,1.57.54,2.00.23,2.01.23 and 2.45.43.
The women finish in times 1.50.10.2.07.45,2.09.54,2.10.23,2.21.23 and 2.25.43.
Sorting these in order, with the labels M for men and W for women we have WMMMMMWWWWWM.
The sum of the ranks for the men is
so
At the 5% level the rejection region is
The null hypothesis is rejected.
