When should a rank test be used instead of a t test?

When the data are ordinal, the sample is small, or the distribution is skewed or has outliers, since rank tests stay valid without normality and are more robust, at a modest cost in efficiency when normality does hold.

Is the Mann-Whitney test the same as the Wilcoxon rank-sum test?

Yes. They are algebraically equivalent two-sample procedures derived independently; both compare the locations of two distributions using the ranks of the pooled data.

Rank-Based Methods

Rank-based methods replace the data by their order, producing tests whose null behavior holds for any continuous distribution and that resist outliers.

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Definition

Rank-based methods are statistical procedures that use only the ranks of the observations rather than their numerical values, yielding distribution-free tests valid for any continuous data-generating distribution.

Scope

This topic covers the sign test, the Wilcoxon signed-rank test for paired data, the Wilcoxon rank-sum and equivalent Mann-Whitney test for two samples, the Kruskal-Wallis test for several samples, Spearman and Kendall rank correlations, the general theory of linear rank statistics and their asymptotic normality, and the asymptotic relative efficiency of rank tests compared with their normal-theory counterparts.

Core questions

Why is the null distribution of a rank statistic free of the underlying continuous distribution?
How do the Wilcoxon and Kruskal-Wallis tests compare locations without normality?
What is asymptotic relative efficiency, and how do rank tests compare with the t and F tests?
How are rank correlations used to measure monotone association?

Key theories

Distribution-free rank tests: Because ranks are invariant under monotone transformations, the null distribution of a rank statistic depends only on the sample sizes, giving exact distribution-free tests of location and association.
Asymptotic relative efficiency: Rank tests lose little efficiency under normality and can be far more efficient for heavy-tailed data; the Wilcoxon test, for instance, never falls below about 86 percent efficiency relative to the t test at the normal model.

Clinical relevance

Rank tests are the default for ordinal scales, small samples, and skewed or outlier-prone data in clinical, psychological, and ecological research, where their validity without a normality assumption makes them safer than the t and F tests.

History

Wilcoxon proposed the signed-rank and rank-sum tests in 1945, Mann and Whitney gave the equivalent two-sample test in 1947, and Kruskal and Wallis extended it to several groups in 1952, establishing the core of distribution-free testing.

Key figures

Frank Wilcoxon
Henry Mann
Donald Whitney
Maurice Kendall

Seminal works

hollander2013

Frequently asked questions

When should a rank test be used instead of a t test?: When the data are ordinal, the sample is small, or the distribution is skewed or has outliers, since rank tests stay valid without normality and are more robust, at a modest cost in efficiency when normality does hold.
Is the Mann-Whitney test the same as the Wilcoxon rank-sum test?: Yes. They are algebraically equivalent two-sample procedures derived independently; both compare the locations of two distributions using the ranks of the pooled data.