Mann-Whitney U Test Calculator — Rank-Based

The Mann-Whitney U Test. Two independent groups can differ without a clean bell curve. This rank test asks which group tends to produce the larger observations. Pool both samples, sort every observation, and assign ranks from smallest to largest. The original group labels stay attached throughout. Equal observations share a midrank. A two-way tie occupying ranks five and six gives both observations rank five point five. Add sample one's ranks to get R one. Subtract n one times n one plus one over two to obtain U one. U one also counts cross-group wins for sample one, with half a win for each tie. U two is every remaining comparison. Under identical distributions, U centers at n one times n two over two. Ties reduce its null variance through a specific correction. This page always uses the tie-corrected normal approximation and moves half a unit toward the null mean for continuity correction.

Independence is structural. No participant may appear in both groups, and observations within a group cannot be clustered or repeatedly measured. Ranks need at least ordered outcomes. Monotone transformations preserve their order, and one extreme value can move only one rank. The sharp null says both populations share one distribution. A rejection generally supports a distributional difference or a tendency toward larger values. Calling that difference a median or location shift needs another check: the two distributions should have reasonably similar shapes and spreads. Tail direction is defined relative to sample one. Right-tailed means sample one tends larger; left-tailed means it tends smaller. The rank-biserial effect is two U one over all cross-group pairs, minus one. Its sign flips when the sample labels swap. Paired outcomes, several independent groups, and clustered observations each need a structure-aware method instead of this two-group independent test.

Now compare two independent groups of nine skewed measurements each. Keep a two-sided alternative and alpha point zero five fixed. The pooled data contain two tied pairs. Their shared midranks are five point five and seven point five, producing tie term twelve. Sample one's rank sum is one hundred three. Subtract forty-five to get U one equals fifty-eight; U two equals twenty-three. Across eighty-one possible comparisons, the null mean is forty point five. The tie-corrected standard deviation is about eleven point three one three. Because U one is above its mean, subtract the half-unit correction. The resulting z score is about one point five zero three. The two-sided p-value is zero point one three two nine one nine. It exceeds alpha, so fail to reject identical distributions. The observed rank-biserial effect is zero point four three two. Sample one wins about seventy-one point six percent of cross-group comparisons.

When either group has fewer than eight observations, the calculator warns that the approximation is rough. It still does not switch to exact inference. If every pooled value is identical, rank variance becomes zero and the calculator stops. Equal ranks contain no ordering information. Swap the sample labels and U one becomes twenty-three while the effect becomes negative. The two-sided p-value itself stays unchanged. Specify group order, tail direction, and alpha before seeing results. Choosing a favorable tail afterward invalidates the advertised evidence. Heavy ties reduce rank resolution, unequal shapes complicate location language, and this page provides no shift estimate or confidence interval. Report both sample sizes, tail, tie handling, U one and U two, method, z, alpha, p-value, effect, warnings, and design limits. Rank two independent samples, inspect their cross-group wins, and run the matching calculator free at Distri Scope dot com.