# Mann-Whitney U test

**U-test Mann-Whitney**

Comparison of central tendencies of two independent samples.

**Requirements:**

Dependent variable must be ordinal scaled (rank order scaled). Distribution free.

**Idea:**

The values of both samples are brought in one rank order. For H0 it is expected that 'x<y' occurrs as frequently as 'y<x'.

**Measure: **

U measures how often a y-value is smaller than an x-value. Definition:

if y<x, z=1,

if x<y, z=0.

U=Σ z.

**Calculation of U:**

T_{1} = sum of ranks for sample 1.

U^{’} = n_{1} * n_{2} - U = Σ(x < y)

For large sample sizes (n_{1} or n_{2} > 10) U is approximately normal distributed with:

and

Test:

If there are tied ranks, the standard deviation of U is corrected as follows:

Whereas

n = n_{1} + n_{2}

t_{i} = Number of subjects sharing rank i

k = Number of tied ranks

**Example of a Mann-Whitney U-test**

A physician is interested in the effect of an anaesthetic on reaction times. Two groups are compared, one with (A) and one without (B) taking the anaesthetic. Subjects had to react on a simple visual stimulus. Reaction times are not normally distributed in this experiment, so data is analysed with the Mann-Whitney U-Test for ordinal scaled measurements. The table below shows the rank-ordered data:

Mean RT | Rank |
Group |

131 |
1 |
B |

135 |
2 |
A |

138 |
3.5 |
B |

138 |
3.5 |
B |

139 |
5 |
A |

141 |
6 |
B |

142 |
8 |
B |

142 |
8 |
A |

142 |
8 |
B |

143 |
10 |
B |

144 |
11 |
A |

145 |
12 |
B |

156 |
13 |
B |

158 |
14 |
A |

165 |
15 |
A |

167 | 16 |
B |

171 |
17 |
A |

178 |
18 |
A |

191 |
19 |
B |

230 |
20 |
B |

244 |
21 |
A |

245 |
22 |
A |

256 |
23 |
A |

267 |
24 |
A |

268 |
25 |
A |

289 |
26 |
A |

Table showing Ranked Measures for each Group separately:

Group A |
Group B | |

2 |
1 | |

5 |
3.5 | |

8 |
3.5 | |

11 |
6 | |

14 |
8 | |

15 |
8 | |

17 |
10 | |

18 |
12 | |

21 |
13 | |

22 |
16 | |

23 |
19 | |

24 |
20 | |

25 |
||

26 |
||

Sum of Ranks |
231 |
120 |

Average Ranks |
16.5 |
10 |

The observed Z value is greater than the Z-value (5%, two-tailed). The anaesthetic group shows significantly slower reaction times than the non-anaesthetic group.

BrightStat output of Mann-Whitney U test example

This is a fictitious example

**Wiki link Mann-Whitney U test**

**References**

Mann, H.B. & Whitney, D.R. (1947). On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other.*Annals of Mathematical Statistics 18 * (1), 50 – 60. doi:10.1214/aoms/1177730491.

Bortz, J. (2005). *Statistik für Human- und Sozialwissenschaftler (6 ^{th} Edition).* Heidelberg: Springer Medizin Verlag.

Conover, W.J. (1999). *Practical nonparametric Statistics.(3 ^{rd} edition).* Wiley.