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What Is the Wilcoxon Rank-Sum Test?
The Wilcoxon rank-sum test is a non-parametric test for comparing two independent groups. It tests whether the distributions of two populations differ in location. The test is mathematically equivalent to the Mann-Whitney U test and is one of the most powerful non-parametric alternatives to the two-sample t-test.
The test procedure involves combining both samples, ranking all observations from smallest to largest, and then summing the ranks for each group. If one group consistently has higher ranks, the test statistic will be extreme, providing evidence that the groups differ. The test is robust to outliers and does not require normality.
Formula
Step-by-Step Procedure
- Combine both samples and rank all observations from 1 to N
- Sum the ranks for each group separately
- Compare the rank sum to the expected value under H0
- For large samples, use the normal approximation (Z-test)
Frequently Asked Questions
How are tied ranks handled?
When two or more observations have the same value, they receive the average of the ranks they would have been assigned. A correction factor can be applied to the variance formula for accuracy, though for small numbers of ties the effect is minimal.
What is the minimum sample size?
The Wilcoxon rank-sum test can be performed with very small samples (as few as 3-4 per group), but the power to detect differences increases with sample size. For very small samples, exact critical values from tables should be used instead of the normal approximation.