Mann-Whitney U Test Walkthrough Cheat Sheet

Key Facts

• The Mann-Whitney $U$ test compares two independent groups using ranks rather than raw data values.
• The null hypothesis is usually $H_0$: the two population distributions are the same, while the alternative $H_a$ states that one distribution tends to have larger values or that the distributions differ.
• After combining both samples and ranking all observations, the rank sum for group $1$ is $R_1$ and the rank sum for group $2$ is $R_2$.
• The test statistics are $U_1 = R_1 - \frac{n_1(n_1+1)}{2}$ and $U_2 = R_2 - \frac{n_2(n_2+1)}{2}$.
• The two $U$ statistics satisfy $U_1 + U_2 = n_1n_2$, which is a useful arithmetic check.
• For many small-sample procedures, the reported statistic is $U = \min(U_1,U_2)$.
• For large samples with no ties, use $\mu_U = \frac{n_1n_2}{2}$ and $\sigma_U = \sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}$ to compute $z = \frac{U - \mu_U}{\sigma_U}$.
• A common rank-biserial effect size is $r_{rb} = 1 - \frac{2U}{n_1n_2}$ when $U$ is the smaller $U$ statistic.

Vocabulary

Mann-Whitney U test

A nonparametric test that compares two independent groups by analyzing the ranks of all observations.

Rank sum

The total of the assigned ranks for one group after all observations from both groups are combined and ordered.

Independent samples

Samples are independent when observations in one group are not paired with or directly related to observations in the other group.

Tie

A tie occurs when two or more observations have the same value and must receive the average of their rank positions.

Normal approximation

A large-sample method that converts the $U$ statistic to a $z$ score so a normal distribution can estimate the $p$-value.

Rank-biserial correlation

An effect size that describes the strength and direction of the difference between two groups based on rank dominance.