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Mann-Whitney U Test
The Mann-Whitney U Test is a popular test for comparing two independent samples. It is a non-parametric test, as the analysis is undertaken on the rank order of the scores and so does not require the assumptions of a parametric test. It was originally proposed by Frank Wilcoxon in 1945 for equal sample sizes, but in 1947 H. B. Mann and D. R. Whitney extended it to unequal sample sizes (and also provided probability values for the distribution of U, the test statistic).
When the null hypothesis is true, and the ranks of the two samples are drawn from the same population distribution, one would expect the mean rank for the scores in one sample to be the same as the mean rank for the scores in the other sample. However, if there is an effect of the independent variable on the scores, then one would expect it to influence their rank order, and hence, one would expect the mean ranks to be different for the two samples.
This entry discusses the logic and calculation of the Mann-Whitney U test and the probability of U.
The Logic of the Test
The logic of the test can be seen by an example. A group of children sign up for a tennis camp during summer vacation. At the beginning of the course, a tennis expert examines the children on their tennis ability and records each child's performance. The children are then randomly allocated to one of two tennis coaches, either Coach Alba or Coach Bolt. At the end of the course, after 4 weeks of intensive coaching, the tennis expert again examines the children on the test of their tennis ability and records their performance. The amount of improvement in the child's tennis performance is calculated by subtracting their score at the beginning of the course from the one at the end. An interesting question arises: Is it better to be coached by Coach Alba or by Coach Bolt? Given that the children play on the same tennis courts and follow the same course of study, the only difference between the two groups is the coaching. So does one group improve more than the other?
As we are unsure whether the tennis expert's test scores satisfy parametric assumptions, a Mann- Whitney test is undertaken on the improvement scores to test the hypothesis. In this example, Alba coaches six students and Bolt coaches five. In Alba's group, Juan receives an improvement score of 23, Todd gets 15, Maria 42, Charlene 20, Brad 32, and Shannon 28. In Bolt's group, Grace receives an improvement score of 24, Carl gets 38, Kelly 48, Ron 45, and Danny 35. How do we decide whether one of the coaches achieves the better results? First, the results can be seen more clearly if they are put in a table in rank order, that is, listing the scores in order from least improved (at the bottom) to most improved (at the top):
| Rank | Student Name | Improvement Score | Coach |
| 11 | Kelly | 48 | Bolt |
| 10 | Ron | 45 | Bolt |
| 9 | Maria | 42 | Alba |
| 8 | Carl | 38 | Bolt |
| 7 | Danny | 35 | Bolt |
| 6 | Brad | 32 | Alba |
| 5 | Shannon | 28 | Alba |
| 4 | Grace | 24 | Bolt |
| 3 | Juan | 23 | Alba |
| 2 | Charlene | 20 | Alba |
| 1 | Todd | 15 | Alba |
In the final column of the table, it can be seen that five of Alba's students are in the bottom six places.
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