# Appendix 11.B Mann–Whitney (Rank Sum) Test for Two Independent Populations

The Mann–Whitney (M–W) test, like the runs test of Appendix 11.A, is designed to compare the identity of two population distributions by examining the characteristics of two independent random samples of sizes n_{1} and n_{2}, respectively, where n_{1} and n_{2} are taken to be “large.” Here, too, we need only assume that the population distributions are continuous and the observations are measured on an interval or ratio scale. However, unlike the runs test, the M–W procedure exploits the numerical ranks of the observations once they have been jointly arranged in an increasing sequence.

In this regard, suppose we arrange the n = n_{1} + n_{2} sample values in an increasing order of magnitude and assign them ranks 1, . . ., n while keeping track of the source sample from which each observation was selected for ranking, for example, an observation taken from sample 1 can be tagged with, say, letter a, and an observation selected from sample 2 gets tagged with letter b. (If ties in the rankings occur) simply assign each of the tied values the average of the ranks that would have been assigned to these observations in the absence of a tie.)

What can the rankings tell us about the population distributions? Let R_{1} and R_{2} denote the rank sums for the first and second samples, respectively. If the observations were selected from identical populations, then R_{1} and R_{2} should be approximately equal in value. However, if the data points ...

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