We have already discussed how to compare the means from two groups, when both groups are distributed according to a Gaussian distribution with the same variance. However, the nonparametric test requires no distributional assumption and works well almost every time. Of course, if both distributions are Gaussian with the same variance, then the regular t-test is better—this is derived from the fact that the t-test is uniformly the most powerful one.

The Mann-Whitney-Wilcoxon test is a nonparametric test that tests the null hypothesis that any element chosen at random from group A is equally likely to be greater or smaller than a respective random item from group B. A different way of posing this test is to think of it as a test of whether the distributions of group A and B are the same. The only strong assumption that this test requires is that the observations from both groups are independent.