## The Mann Whitney U Test

The Mann Whitney U test allows the comparison of two independent random samples (1 and 2).

The Mann Whitney U statistic is defined as:

- where samples of sizeandare pooled and Ri are the ranks of sample 1.

U can be resolved as the number of times observations in one sample precede observations in the other sample in the ranking.

In most circumstances a two sided test is required; here the alternative hypothesis is that 1 values tend to be distributed differently to 2 values. For a lower side test the alternative hypothesis is that 1 values tend to be smaller than 2 values. For an upper side test the alternative hypothesis is that 1 values tend to be larger than 2 values.

Assumptions of the Mann-Whitney test:

random samples from populations

independence within samples and mutual independence between samples

measurement scale is at least ordinal

A confidence interval for the difference between two measures of location is provided with the sample medians. The assumptions of this method are slightly different from the assumptions of the Mann-Whitney test:

random samples from populations

independence within samples and mutual independence between samples

two population distribution functions are identical apart from a possible difference in location parameters

Example

The following data represent fitness scores from two groups of boys of the same age, those from homes in the town and those from farm homes.

Farm Boys | Town Boys |

14.8 11.1 | 12.7 18.9 |

12.2 | 17.4 |

| 14.2 |

Pooling and sorting gives 11.1, 12.2, 12.7, 14.2, 14.8, 17.4, 18.9.

The Farm Boys have ranks1, 2 and 5.andso

Comparison with values in statistical tables causes us not to reject the null hypothesis of no difference at the 10% level.