The Mann Whitney U test, sometimes called the Mann Whitney Wilcoxon Test or the Wilcoxon Rank Sum Test, is used to test whether two samples are likely to derive from the same population (i.e., that the two populations have the same shape). Some investigators interpret this test as comparing the medians between the two populations. The Wilcoxon-Mann-Whitney test provides a better comparison, summarized in this snippet of output from the SAS/STAT procedure NPAR1WAY: Group N Mean Score ===== placebo 147 137.9 SHS67 148 158.1 Wilcoxon Two-Sample Test ===== Z -2.06 Two-Sided Pr > |Z| 0.0391 The higher mean rank for SHS67 (158 vs. 138) along with the 0.04 p-value supports the This is no longer an exponential distribution, but it has the same shape. That is a location-only shift. You can play this same sort of game with other PDFs by remembering that, for functions f(x) f ( x) in general, f(x − a) f ( x − a) is f(x) f ( x) shifted to the right by a a (so a left shift if a < 0 a < 0 ). How Wilcoxon-Mann-Whitney test works and why it's called "rank-sum" and "U". WMW-test in only 4 steps: 1. rank values of both samples from low to high no matter which group each value belongs to; sum the ranks for both samples separately, R 1 & R 2.This is where the rank-sum part of the name comes from. Which sample is R 1 is irrelevant; Calculate the test statistics: W-value for n In R, both the Wilcoxon and Mann-Whitney tests are carried out using the wilcox.test function. To implement the Wilcoxon test the paired argument should be set to TRUE, to implement the Mann-Whitney test paired is set to FALSE. Things to know about Wilcoxon Signed-Rank Test: For use with matched/paired data. So essentially it's as I said in comments; $\pi$ is in the ARE for the Wilcoxon-Mann-Whitney vs the two-sample t test, for the Wilcoxon signed rank test vs the one-sample t and the sign test vs the one-sample t test (in each case at the normal) quite literally because it appears in the normal density. Reference: J. L. Hodges and E. L. Lehmann .

what is wilcoxon mann whitney test