Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it is a paired difference test). It can be used as an alternative to the paired Student's t-test, t-test for matched pairs, or the t-test for dependent samples when the population cannot be assumed to be normally distributed. A Wilcoxon signed-rank test is a nonparametric test that can be used to determine whether two dependent samples were selected from populations having the same distribution. A Wilcoxon rank sum test is a nonparametric test that can be used to determine whether two independent samples were selected from populations having the same distribution.

The test is named for Frank Wilcoxon (1892–1965) who, in a single paper, proposed both it and the rank-sum test for two independent samples (Wilcoxon, 1945). The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, ${\displaystyle W}$. In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T

Let ${\displaystyle N}$ be the sample size, i.e., the number of pairs. Thus, there are a total of 2N data points. For pairs ${\displaystyle i=1,...,N}$, let ${\displaystyle x_{1,i}}$ and ${\displaystyle x_{2,i}}$ denote the measurements.

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