Wilcoxon Rank-Sum Test Calculator

The Wilcoxon rank-sum test is a non-parametric test for whether two independent samples were drawn from the same distribution. It is also called the Mann–Whitney U test, and it works by comparing the positions of the values in the combined list rather than their sizes.

Because it only uses ranks, it makes no assumption about the shape of the data. That makes it the right tool for ordinal scales like exam grades or pain scores, for heavily skewed measurements such as incomes, and for samples with outliers that would stretch a t-test beyond usefulness.

First every observation from both samples is ranked together, from 1 for the smallest to N for the largest. Where values tie, each one takes the average of the ranks it spans, so two equal values at positions 4 and 5 both receive rank 4.5.

The ranks of sample 1 are then summed to give W₁, and U₁ = W₁ − n₁(n₁ + 1)/2. Read as a count, U₁ is effectively the number of observations in sample 2 that fall below those in sample 1.

If both samples come from the same distribution, U is tightly determined: its mean is n₁n₂/2 and, without ties, its standard deviation is √(n₁n₂(N + 1)/12). A U far from its mean is evidence against that null.

The p-value comes from the exact null distribution of U where one can be tabulated — no ties and at most 20 observations per group — and from the normal approximation with tie and continuity corrections otherwise. The two methods agree closely where both are available.

The U statistic has no units. Larger values of U₁ mean sample 1 sits higher in the combined ranking, and the convention of quoting the smaller of U₁ and U₂ keeps critical values comparable across sample sizes.

The p-value is the probability of seeing a separation at least this extreme if the samples really did come from the same distribution. It is not the probability that the null hypothesis is true.

The rank-biserial correlation converts the statistic into a familiar effect size on the −1 to 1 scale, and the probability of superiority states it as a plain proportion: how often a random observation from sample 1 would be ranked above a random one from sample 2.

A non-significant result is not evidence of equality. With small samples the test has little power, so failing to reject the null is the expected outcome even when the medians differ noticeably.

The Method row matters when reading small p-values. The exact and approximate results are close for medium samples, but the exact one is never rounded away and is the safer figure to quote.

• Sample size. The standard deviation of U shrinks roughly as √(n₁n₂), so doubling both samples makes the test meaningfully sharper.
• Ties. Repeated values inflate the variance, which is why the tie correction exists, and they rule out the exact distribution entirely.
• The shape of the two distributions. The test detects any difference in distribution; it only reads as a difference in medians when the shapes match.
• Small sample sizes. The discreteness of U makes exact p-values lumpy, and a single extreme observation can flip the result.
• Outliers. Ranks limit the damage an outlier can do compared with a mean-based test, but a single wild value still pushes one sample’s whole rank block upward.

• Comparing two independent groups when normality is doubtful or the measurement is on an ordinal scale.
• Clinical or field data with small samples, where the normal approximation for a t-test would be unreliable.
• Skewed measures such as income, response times, waiting times or bacterial counts, where a mean is a poor summary anyway.
• Screening or quality data with natural ties, such as scores, defect counts or categories, provided the tie correction is applied.
• Teaching, as the clearest non-parametric test of two independent samples.

• The observations must be independent within and between groups. Clustered or repeated measurements violate this and make the test anti-conservative.
• Under the null the two distributions must have the same shape. If they differ in shape, a significant result does not identify where the difference lies.
• The measurement must be at least ordinal. A nominal scale with no meaningful ordering cannot be ranked.
• Exact p-values need no ties and small samples. Beyond that the normal approximation is used, and it is less reliable in the tails than the exact value.
• The test does not correct for multiple comparisons. Running it on twenty endpoints will produce a significant result by chance roughly once in five times.