Wilcoxon Rank-Sum Test Calculator

The Wilcoxon Rank-Sum test (also called the Mann-Whitney U test) is a non-parametric statistical test used to compare two independent groups. It tests whether one group tends to have higher values than the other, without assuming that the data follow a normal distribution. It ranks all observations from both groups together and compares the sum of ranks between groups.

Use the Wilcoxon Rank-Sum test when your data is ordinal rather than interval/ratio, when your samples are small and you cannot verify normality, or when your data contains significant outliers. It is the go-to non-parametric alternative when the assumptions of the independent samples t-test (normality, equal variance) are not met.

The U statistic counts the number of times an observation from Group 1 precedes an observation from Group 2 when all values are sorted. Two U values are computed (U₁ and U₂), and the smaller one is typically used. For large samples, U is converted to a Z score using the normal approximation to obtain a p-value.

The p-value represents the probability of observing a test statistic as extreme as the one computed, assuming the null hypothesis is true (the two groups have equal distributions). If the p-value is less than your chosen significance level α (e.g., 0.05), you reject the null hypothesis and conclude there is a statistically significant difference between the groups.

For sufficiently large samples (both n₁ and n₂ greater than about 10), the U statistic follows an approximately normal distribution. This calculator uses the normal approximation with an optional continuity correction to compute the Z score and p-value. For very small samples, exact tables or permutation methods would be needed.

The continuity correction adds or subtracts 0.5 from the U statistic before computing the Z score, improving the accuracy of the normal approximation for discrete data. It is generally recommended when using the normal approximation, especially for smaller samples. Most statistical software (like R's wilcox.test) applies it by default.

The Wilcoxon Rank-Sum test (this calculator) is for two independent samples — the subjects in Group A and Group B are different people or units. The Wilcoxon Signed-Rank test is for paired or related samples — the same subject is measured twice (e.g., before and after treatment). Using the wrong test will give misleading results, so always check your study design first.

When two or more observations share the same value, they receive averaged ranks (mid-ranks). This calculator handles ties automatically and applies a tie correction to the variance when computing the Z score, ensuring the p-value remains accurate even when tied values are present in your dataset.