Wilcoxon Rank-Sum Test

The Wilcoxon rank-sum test, also called the Mann-Whitney U test, is a nonparametric test that determines whether two independent populations have different medians. It compares the distributions of two independent samples without assuming normal distribution. This test is particularly useful when data are ordinal or when assumptions for parametric tests like the two-sample t-test are violated.

Data from both samples are combined into a single ranked list, with each value assigned a rank from smallest to largest. Tied values receive the average of the ranks for those positions. The ranks are then separated back into their respective groups, and the sum of ranks is calculated for each group independently to determine the test statistic.

When high or low ranks predominantly fall into one sample rather than being distributed evenly, this suggests the two samples may have different medians. This clustering pattern indicates a systematic difference in the underlying distributions of the two populations, which the test statistic evaluates for statistical significance.

For small sample sizes, typically when n is less than 20, critical values for U are obtained from statistical tables. For larger samples, a z-score approximation is applied, assuming a normal distribution. This approximation becomes more reliable and efficient as sample size increases, making it the preferred approach for larger datasets.

The test requires two independent samples that are randomly drawn from their populations. Data must be ordinal or capable of being converted to an ordinal scale so values can be ranked. There is no assumption that samples are normally distributed, making this test robust for non-normal data and a suitable alternative to parametric tests.

The Wilcoxon rank-sum test is two-tailed because the alternative hypothesis states that the distributions of the two populations are different, without specifying direction. The test statistic is compared against both positive and negative critical values, typically at the 5% significance level, to determine if a significant difference exists in either direction.

While the Wilcoxon rank-sum test is robust to outliers, it is prone to higher type-I error when data are biased, heteroscedastic (having different variance), or when sample distributions are extremely far from normal. Despite these limitations, the test remains reliable for most ordinal and non-normally distributed data in practical applications.