## Quick Reference

A two-sample non-parametric test, equivalent to the **Wilcoxon rank-sum test**, introduced in 1947 by Mann and Whitney. It is assumed that the samples are random and come from populations (random variables *X* and *Y*) that have the same distribution after a translation of size *k*:

P(*X*<*x*)=P(*Y*<*x*+*k*), for all values of *x*.

The null hypothesis is that the random variables have the same distribution (i.e. *k*=0). See also test for equality of location.

With samples of sizes *m* and *n* (*m*≤*n*), the first stage is the replacement of the (*m*+*n*) observed values by their ranks in the combined sample. For the (smaller) sample of size *m*, denote the sum of the ranks by *R*. The distribution of *R* is approximately , so that the test statistic, *z*, is given by , where the ±½ is a continuity correction with sign chosen so as to reduce the absolute magnitude of the numerator.

For example, suppose that the marks obtained by a small random sample of statistics students were as follows:The question of interest is whether the data support the null hypothesis of a common mark distribution. The ranks areWorking with the (smaller) set of girls, *m* is 7 and *R* is 3+5 +…+16=63. Using *n*=11, the test statistic, *z*, is given by . Since |*z*|<1.96, we accept, at the 5% significance level, the hypothesis that the two sets of marks have come from the same distribution.

Boys

10, 22, 42, 59, 61, 63, 65, 83, 85, 90, 93

Girls

36, 53, 54, 56, 69, 84, 88.

Boys

1, 2, 4, 8, 9, 10, 11, 13, 15, 17, 18

Girls

3, 5, 6, 7, 12, 14, 16.

*Subjects:*
Probability and Statistics.

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