Hypothesis Tests

Mann-Whitney U Test

Nonparametric alternative to the two-sample t-test for independent samples.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Nonparametrics > Mann-Whitney

Description

The Mann-Whitney U test (also Wilcoxon rank-sum test) is a nonparametric test that evaluates whether two independent samples come from the same distribution. It is based on the combined ranks of both samples and is used when the normality assumption of the t-test is not met. For large samples, a normal approximation is used for the p-value.

Formulas

U = min (U_{1}, U_{2}), U_{i} = R_{i} - \frac{n _{i} ( n _{i} + 1 )}{2}

U statistic: minimum of the two rank-sum-based values

z = \frac{U - μ _{U}}{σ _{U}}, μ_{U} = \frac{n _{1} n _{2}}{2}, σ_{U} = \frac{n _{1} n _{2} ( n _{1} + n _{2} + 1 )}{12}

z approximation for large samples

H₀

H_{0} : F_{1} (x) = F_{2} (x)

Null hypothesis: Both populations have the same distribution

Assumptions

Samples are independent
Observations are continuous (at least ordinal)
Both distributions have the same shape (shift model)

Limitations

Less powerful than the t-test for truly normal data
Many ties can affect the 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
Hollander, M. & Wolfe, D. A. (2013), Nonparametric Statistical Methods, 3rd Ed.