Mann-Whitney Test

Analysis by Ranks

In 1946, an American statistician, Frank Wilcoxon, suggested that the use of ranks would be helpful in situations in which the assumptions underlying Student’s t test and other conventional procedures were not met. In a design in which the observations in two conditions are paired (such as a repeated-measures design), Wilcoxon argued that researchers could compute the difference scores between the two conditions and then rank the magnitude of the difference scores across the different cases; in this procedure, the direction of the difference was ignored in the ranking, but the total of the ranks for the positive differences was compared with the total of the ranks for the negative differences. The smaller of the two totals constituted a test statistic, which is usually denoted by the symbol T. Wilcoxon reported the probability of obtaining particular values of this statistic under the null hypothesis. This procedure became known as the Wilcoxon matched pairs signed-rank test.

In a design in which the observations in two conditions were unpaired (i.e., an independent-groups design), Wilcoxon proposed that researchers should rank the observations themselves, so that 1 refers to the smallest observation and N refers to the largest observation across both of the groups. The total of the ranks in each of the two groups is then calculated, and the smaller of the two totals is used as a test statistic, which is usually denoted by the symbol W. Wilcoxon once again reported the probability of obtaining particular values of this statistic under the null hypothesis of no difference between the groups. This is sometimes known as the Wilcoxon rank-sum test. Nevertheless, Wilcoxon’s account was [Page 1006]limited in two ways: First, he assumed that the two groups were of equal size and, second, he only provided a few points of the distribution of his statistic W under the null hypothesis.

Two other American statisticians, Henry B. Mann and D. Ransom Whitney, presented a more general account of this situation. Suppose that the numbers of cases in the two groups are n1 and n2 and that the totals of the ranks in the two groups are R1 and R2, respectively. A statistic U1 is defined as [n1 n2 + n1 (n1 + 1)/2−R1] and a statistic U2 is defined as [n1 n2 + n2 (n2 + 1)/2−R2]. Each of these statistics measures the separation of the two distributions of scores (i.e., the extent to which the cases in Group 1 tend to score less than the cases in Group 2 and vice versa). It is easy to show that U1 + U2 = n1n2, and hence, strictly speaking, one of the statistics is redundant. Mann and Whitney defined the statistic U as the smaller of U1 and U2. They calculated the exact probabilities of obtaining particular values of U or less for values of n1 and n2 between 3 and 8.

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