Mann-Whitney U Test Formula
Learn the Mann-Whitney U test — a non-parametric test that compares two independent groups without assuming a normal distribution.
Includes formula, worked example, and interpretation.
The Formula
The Mann-Whitney U test (also called the Wilcoxon rank-sum test) determines whether two independent samples come from the same distribution. Unlike the t-test, it does not assume that the data is normally distributed. It works by ranking all values across both groups combined, then comparing where each group's values tend to fall in that ranking.
The test statistic U represents the number of times a value from one group ranks ahead of a value from the other group. A small U (relative to its maximum possible value of n1 × n2) suggests the groups differ significantly.
Variables
| Symbol | Meaning |
|---|---|
| U1 | Mann-Whitney U statistic for group 1 |
| U2 | Mann-Whitney U statistic for group 2 (note: U1 + U2 = n1 × n2) |
| n1 | Sample size of group 1 |
| n2 | Sample size of group 2 |
| R1 | Sum of ranks assigned to group 1 observations |
| R2 | Sum of ranks assigned to group 2 observations |
| U (test statistic) | The smaller of U1 and U2 — compared to critical values |
Step-by-Step Procedure
- Combine all values from both groups into one list.
- Rank all values from smallest (rank 1) to largest. For tied values, assign the average of the ranks they would have received.
- Separately sum the ranks for each group to get R1 and R2.
- Calculate U1 and U2 using the formulas above.
- The test statistic U = min(U1, U2).
- Compare U to the critical value from a Mann-Whitney U table at your chosen significance level (α = 0.05 is typical).
- If U ≤ critical value, reject the null hypothesis — the groups differ significantly.
Example
Group A (treatment) scores: 85, 72, 90, 78. Group B (control) scores: 65, 70, 75, 68. Is there a significant difference? (n1 = 4, n2 = 4)
Step 1 — Combine and rank all 8 values: 65(1), 68(2), 70(3), 72(4), 75(5), 78(6), 85(7), 90(8)
Step 2 — Group A ranks: 85→7, 72→4, 90→8, 78→6. R1 = 7+4+8+6 = 25
Step 3 — Group B ranks: 65→1, 70→3, 75→5, 68→2. R2 = 1+3+5+2 = 11
Step 4 — U1 = 4×4 + 4(4+1)/2 − 25 = 16 + 10 − 25 = 1
U2 = 4×4 + 4(4+1)/2 − 11 = 16 + 10 − 11 = 15
Check: U1 + U2 = 1 + 15 = 16 = n1 × n2 = 4 × 4 ✓
Test statistic U = min(1, 15) = 1
At α = 0.05 with n1 = n2 = 4, the critical value is 1. Since U ≤ 1, we reject H0. Group A scores are significantly higher than Group B (p < 0.05).
When to Use the Mann-Whitney U Test
- Data is ordinal (ranked categories) rather than continuous
- The distribution is skewed or you cannot assume normality
- Small sample sizes where the Central Limit Theorem does not apply
- Comparing patient pain scores, Likert-scale survey responses, or ranked preferences
- Any time you would use a t-test but the normality assumption is violated
Large-Sample Approximation
When both samples have more than about 20 observations, the U statistic follows an approximately normal distribution:
The resulting z-score can then be compared to the standard normal distribution (z > 1.96 for α = 0.05, two-tailed).