Wilcoxon Rank-Sum Test Calculator

Also known as Mann-Whitney U test, the Wilcoxon rank-sum test compares two independent samples. Use this calculator to test for significant differences between distributions.

Last updated: March 2026

Test Calculator

Sample 1 (comma or space separated)

Sample 2 (comma or space separated)

What is the Wilcoxon Rank-Sum Test?

The Wilcoxon rank-sum test, also called the Mann-Whitney U test, is a non-parametric statistical test used to determine whether two independent samples come from the same distribution. Unlike the t-test, it does not assume normality and works with ranked data rather than raw values.

This test is particularly useful when you have small sample sizes, non-normal data, or ordinal (ranked) data. It answers the question: "Are these two groups significantly different?" by comparing the ranks of observations rather than their actual values.

The test produces a U-statistic and converts it to a z-score for p-value calculation. A p-value less than 0.05 typically indicates a statistically significant difference between the groups.

How to Use the Wilcoxon Rank-Sum Test

Step-by-Step Process

Step 1: Enter values for Sample 1 (comma or space separated)

Step 2: Enter values for Sample 2 (comma or space separated)

Step 3: Click "Run Test" to calculate U-statistic and p-value

Step 4: Interpret results: p < 0.05 suggests samples are from different distributions

Important Notes

Both samples must have at least 2 observations
Tied values are handled via average ranking (e.g., ranks 3,4,5 become 4 each)
Variance calculation does not include tie correction; p-values may be less accurate when many ties are present
The calculator uses normal approximation for p-values
This is a two-tailed test

Worked Example

Scenario: Two groups completed a treatment program. Group A scores: [4, 7, 9, 12, 15]. Group B scores: [3, 5, 8, 10, 11].

Calculation Summary

Combined and ranked: [3(1), 4(2), 5(3), 7(4), 8(5), 9(6), 10(7), 11(8), 12(9), 15(10)]

R₁ = 2 + 4 + 6 + 9 + 10 = 31

U₁ = 31 - (5×6/2) = 16

U₂ = 5×5 - 16 = 9

U = min(16, 9) = 9

μ = (5×5)/2 = 12.5, σ = √(275/12) ≈ 4.787

z = (9 - 12.5) / 4.787 ≈ -0.73, p ≈ 0.46 (not significant)

Frequently Asked Questions

When should I use Wilcoxon rank-sum test instead of a t-test?

Use the rank-sum test when your data is non-normally distributed, has small sample sizes, contains outliers, or consists of ordinal (ranked) data. It's more robust and doesn't assume normality.

What does a p-value of 0.05 mean?

A p-value of 0.05 means there's a 5% probability of observing your data (or more extreme) if the null hypothesis (no difference) were true. This is the conventional significance threshold.

Can I use this test for paired data?

No. The Wilcoxon rank-sum test is for independent samples. For paired data, use the Wilcoxon signed-rank test instead.

How are tied values handled?

Tied values are assigned the average of the ranks they would occupy. For example, if values are tied for ranks 3, 4, 5, each gets rank 4 (the average).

What sample sizes does this calculator support?

Theoretically unlimited, but the normal approximation is most accurate for samples with at least 5-10 observations each. For very small samples, consider exact tests.

Is continuity correction applied?

No. The calculator uses the standard large-sample normal approximation without continuity correction. For small samples (< 5 per group), consider exact tests for more accurate p-values.

Wilcoxon Rank Sum Test Calculator