Perform Mann-Whitney U test (Wilcoxon rank-sum) with automatic ranking, tie correction, z-approximation, effect size, and detailed rank tables.
The Mann-Whitney U Test Calculator performs the non-parametric alternative to the independent-samples t-test. Enter raw data for two groups and the tool automatically ranks all observations, calculates U-statistics, applies tie corrections, and provides exact z-scores and p-values.
The Mann-Whitney U test (also called the Wilcoxon rank-sum test) compares two independent groups without assuming normal distributions. Instead of comparing means, it tests whether one group tends to produce larger values than the other. This makes it ideal for ordinal data, skewed distributions, small samples, or scenarios where parametric assumptions don't hold.
The calculator handles tied ranks automatically using average ranking and corrects the standard error accordingly. The effect size r = z/√N translates the result into a standardized metric comparable to Cohen's d. Group summary statistics include medians, mean ranks, and rank sums for complete reporting. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
The Mann-Whitney U test is the most widely used non-parametric test for comparing two independent groups. It's robust against non-normality, outliers, and unequal variances — making it safer than the t-test when assumptions are questionable. Many journals now recommend it as the default for small-sample comparisons.
This calculator automates the tedious ranking process, handles ties correctly, and provides the complete suite of outputs needed for publication: U-statistics, z-score, p-value, effect size, group summaries, and the full ranking table. Presets for common scenarios let you explore immediately.
U₁ = R₁ − n₁(n₁+1)/2, U₂ = n₁n₂ − U₁. z = (U − μᵤ) / σᵤ, where μᵤ = n₁n₂/2 and σᵤ = √[n₁n₂(n₁+n₂+1)/12 − tie correction].
Result: U₁ = 7.0, U₂ = 218.0, z = −4.39, p < 0.001, r = 0.80
Group 2 values are consistently higher, producing very low U₁. The z-score of −4.39 gives p < 0.001. Effect size r = 0.80 indicates a large effect. Group 2 has significantly higher values than Group 1.
Non-parametric tests like Mann-Whitney make fewer assumptions about the data. They don't require normality, work with ordinal data, and are resistant to outliers. The trade-off is slightly lower power when parametric assumptions ARE met. For moderate-to-large samples with non-normal data, the power loss is minimal.
The U-statistic has a beautiful interpretation: it counts the number of times a randomly chosen observation from Group 1 is less than a randomly chosen observation from Group 2. If U₁ = 0, every Group 1 value is less than every Group 2 value — perfect separation. If U₁ = n₁n₂/2, the groups are perfectly mixed.
The Kruskal-Wallis test extends Mann-Whitney to three or more groups (analogous to one-way ANOVA). The Wilcoxon signed-rank test handles paired data. The Brunner-Munzel test is a modern alternative that doesn't assume continuous distributions and handles ties more naturally.
Use Mann-Whitney when: (1) data is ordinal, not interval/ratio, (2) distributions are non-normal and sample sizes are small, (3) variances are very different between groups, or (4) data has outliers. For large normal samples, the t-test and Mann-Whitney give similar results.
Tied values receive the average of the ranks they would have occupied. For example, if values at positions 3, 4, 5 are tied, each gets rank 4. The standard error is corrected for ties using the tie correction factor, which reduces σᵤ.
U counts the number of times a value from one group precedes (is less than) a value from the other group. U₁ + U₂ always equals n₁ × n₂. A small U indicates strong separation between groups.
Report r = z/√N. Guidelines: r < 0.1 is negligible, 0.1-0.3 is small, 0.3-0.5 is medium, > 0.5 is large. You can also report the common language effect size (probability that a random observation from one group exceeds one from the other).
Yes. The Mann-Whitney U test and Wilcoxon rank-sum test are mathematically equivalent. They use different test statistics (U vs W) but produce identical p-values. The U form emphasizes pair comparisons; the W form emphasizes rank sums.
The z-approximation is reliable when both groups have at least 8-10 observations. For smaller samples (n < 8), exact tables or permutation tests are more accurate. This calculator uses the normal approximation with continuity correction.