Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N? Graphical Insights

Abstract

The Wilcoxon statistics are usually taught as nonparametric alternatives for the 1- and 2-sample Student-t statistics in situations where the data appear to arise from non-normal distributions, or where sample sizes are so small that we cannot check whether they do. In the past, critical values, based on exact tail areas, were presented in tables, often laid out in a way that saves space but makes them confusing to look up. Recently, a number of textbooks have bypassed the tables altogether, and suggested using normal approximations to these distributions, but these texts are inconsistent as to the sample size n at which the standard normal distribution becomes more accurate as an approximation. In the context of non-normal data, students can find the use of this approximation confusing. This is unfortunate given that the reasoning behind—and even the derivation of—the exact distributions can be so easy to teach but also help students understand the logic behind rank tests. This note describes a heuristic approach to the Wilcoxon statistics. Going back to first principles, we represent graphically their exact distributions. To our knowledge (and surprise) these pictorial representations have not been shown earlier. These plots illustrate very well the approximate normality of the statistics with increasing sample sizes, and importantly, their remarkably fast convergence.

• Graphics
• Mann-Whitney U statistic
• Nonparametrics
• Normal approximation
• Ranking methods
• Sampling distribution
• Wilcoxon rank-sum statistic
• Wilcoxon signed-rank statistic