Mann–Whitney U test (Wilcoxon rank-sum)

A rank-based test comparing two independent groups; it tests whether values in one group tend to be larger than in the other.

U_1 = R_1 - \frac{n_1(n_1+1)}{2},\qquad U_2 = n_1 n_2 - U_1

U₁ counts the pairs (x, y) with x > y, ties counting ½.

$R_1$: Sum of the ranks of sample 1 in the pooled sample (average ranks for ties).
$n_1, n_2$: Group sizes.

z = \frac{U - n_1 n_2/2 - c}{\sigma_U},\qquad \sigma_U^2 = \frac{n_1 n_2}{12}\left[(N+1) - \frac{\sum (t^3 - t)}{N(N-1)}\right]

Normal approximation (used with ties or large samples); exact null distribution otherwise (no ties, both n ≤ 50).

$c$: Continuity correction ½ (default on).
$t$: Size of each group of tied values.
$N$: $n_1 + n_2$ .

When to use

Two independent groups with an ordinal outcome, or a quantitative outcome with small samples and strong skew or outliers (e.g. waiting times, delays).

Assumptions

Independent observations within and between groups.
Ordinal or continuous outcome. Exact p-values need no ties; with ties the engine uses the tie-corrected normal approximation.
To read it as a test of medians, the two distributions must have the same shape (one shifted relative to the other).

How to interpret

H₀: the two distributions are identical, so $P(X > Y) = \tfrac12$ . Report the rank-biserial correlation $r = 2U_1/(n_1n_2) - 1 = 2\hat P(X > Y) - 1$ and group medians.

Common mistakes

Describing it as "comparing means".
Claiming a difference in medians when the shapes differ (the test responds to any tendency for one group to be larger, including spread differences).
Using it for paired data (use the signed-rank test).

Example

x: 3, 5, 7, 9, 11; y: 1, 2, 4, 6. Pooled ranks of x: 3, 5, 7, 8, 9, so $R_1 = 32$ and $U_1 = 32 - 15 = 17$ (of 20 pairs, x is larger in 17). Exact two-sided $p = 0.111$ ; rank-biserial $r = 0.70$ . A large effect, but with 9 observations the evidence is weak.