How this instrument works
The Wilcoxon rank-sum test checks whether two independent samples tend to come from the same distribution, using ranks instead of raw values — which makes it a standard nonparametric alternative to the independent-samples t-test when you can't confidently assume your data is normally distributed. It was introduced by Frank Wilcoxon in 1945, predating the closely related Mann-Whitney U test, which is why the same underlying method carries both names in common use today.
The calculation pools both samples into one combined list, ranks every value from smallest to largest (tied values share the average rank they'd span), and sums the ranks that landed in sample 1 — that sum is W. A small W means sample 1's values tended to cluster at the low end of the pooled ranking; a large W means they clustered at the high end; a W near the middle of its possible range suggests the two samples are well mixed together, with no strong tendency for either group to rank systematically higher or lower.
This calculator's engine is the exact same rank computation this site's separate u-test calculator uses — W and the Mann-Whitney U statistic are two different summaries of the identical pooled-rank data, related by W = U1 + n1(n1+1)/2, not two competing formulas. This page exists because 'Wilcoxon rank-sum test' and 'Mann-Whitney U test' are both long-established, independently searched names for the same procedure, and each is reported with its own conventional statistic — W here, U on the other page.
- Enter the first group's raw values into Sample 1 — separated by commas, spaces or new lines.
- Enter the second group's raw values into Sample 2 — separated by commas, spaces or new lines.
- Read Wilcoxon rank-sum statistic (W, sample 1) — the sum of sample 1's ranks once both samples are pooled and ranked together.
- Read Large-sample normal approximation z — useful for judging significance once both samples are reasonably large (commonly cited as roughly n1, n2 ≥ 8-10 each).
Worked example — checkout wait times, branch A vs. branch B
Suppose branch A's customers waited 1, 2, and 3 minutes, while branch B's customers waited 4, 5, and 6 minutes. Enter 1, 2, 3 into Sample 1 and 4, 5, 6 into Sample 2. Pooling all six wait times and ranking them from shortest to longest gives branch A's three values the ranks 1, 2, and 3 (since every one of them is shorter than every branch B wait time), and branch B's values the ranks 4, 5, and 6.
Wilcoxon rank-sum statistic (W, sample 1) reads 6.0000 exactly — the sum of branch A's ranks (1+2+3). Large-sample normal approximation z reads -1.963961. This is the same underlying pooled-ranking result as this site's u-test calculator's matching example, where U1 = W − n1(n1+1)/2 = 6 − 6 = 0 — two different statistics summarizing the identical rank computation, not two different answers.
Questions
What's the difference between this and the U-test calculator on this site?
None, mathematically — the Wilcoxon rank-sum test and the Mann-Whitney U test are the same procedure on the same pooled, ranked data (NIST's e-Handbook describes the rank-sum test as 'often called the Mann-Whitney U test'). This page reports W, the direct sum of sample 1's ranks; the separate u-test page reports U1 and U2, derived from W by U1 = W − n1(n1+1)/2. Both pages exist because each name is independently well-known and searched for, and each statistic has its own conventional critical-value tables in textbooks — reporting them on separate, clearly-labeled pages avoids forcing one name's audience to learn the other's terminology.
What does a low or high W value mean?
W is the sum of sample 1's ranks once both samples are pooled together — so a low W means sample 1's values tended to land toward the low end of the combined ranking (its values were generally smaller), and a high W means they tended toward the high end. Its minimum possible value is n1(n1+1)/2 (every sample 1 value ranks lowest) and its maximum is n1(2n2+n1+1)/2 (every sample 1 value ranks highest); values in between reflect varying degrees of overlap between the two groups.
Why does the test use ranks instead of the raw values?
Because ranks don't depend on the data actually being normally distributed, or even on the values being measured on a strict numeric scale — only on being sortable. That makes rank-based tests like this one robust to outliers, skewed data, and small samples where you can't reliably verify normality, situations where a standard t-test's assumptions would be shaky and its p-values less trustworthy.
How are tied values handled when computing W?
Tied values across the two pooled samples receive the average of the ranks they would have spanned had they not been tied — for instance, two identical values that would otherwise occupy ranks 2 and 3 both receive rank 2.5 instead. This averaging convention keeps the test from arbitrarily favoring one sample over another just because of how ties happened to be broken.
Is Wilcoxon rank-sum the same as the Wilcoxon signed-rank test?
No — despite the similar name, they're different tests for different situations. The Wilcoxon rank-sum test (this calculator) compares two independent, unrelated samples. The Wilcoxon signed-rank test is a separate procedure for paired or matched data, such as before-and-after measurements on the same subjects, and uses a different computation based on the signed differences within each pair rather than pooled ranks across two independent groups.