Mann-Whitney U test
The Mann-Whitney U test is the non-parametric alternative to the t-test for two independent groups. Use it when the scores aren’t normal, the groups are small, or with ordinal scales such as Likert. Instead of comparing means, it compares ranks: whether one group tends to have higher values than the other.
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How it’s calculated
All values are sorted together and ranked (1 for the smallest). Ties get the average rank.
U₁ = R₁ − n₁(n₁ + 1) / 2, where R₁ is the rank sum of group A. U₂ = n₁n₂ − U₁, and U = the smaller of the two is reported, as in SPSS.
Z = (U − n₁n₂ / 2) / σ, with σ corrected for ties. The asymptotic p-value comes from the normal distribution, two-tailed and without continuity correction, as in SPSS.
When there are no ties and the groups are small, the exact p-value is also calculated by counting every possible arrangement.
The effect size is r = |Z| / √N, with N the total number of values.
How to read the result
- p < 0.05: the two groups differ. Look at the mean ranks (or medians) to see which one tends to have higher values.
- With small groups (under 20 each) and no ties, report the exact p-value.
- Effect size r: 0.1 small, 0.3 medium and 0.5 large.
Frequently asked questions
- Does it match SPSS?
- Yes. U, Wilcoxon W, Z and "Asymp. Sig. (2-tailed)" match Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples.
- Why report the median and not the mean?
- Because the test doesn’t compare means. The median and mean rank describe each group better when the data aren’t normal.
- What if it’s the same people before and after?
- Then the samples are paired and you need the Wilcoxon signed-rank test, not Mann-Whitney U.
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