Overview

# Wilcoxon signed rank test

## Quick Reference

Non-parametric tests that extend the sign tests. The single-sample version (observations *x*_{1}, *x*_{2},…), suitable for a symmetric distribution, tests the null hypothesis that the population median has a specified value (*m*_{0}). The matched-pair (or paired-sample) version (observation pairs (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}),…) is concerned with the differences (*x*_{1} − *y*_{1}), (*x*_{2} − *y*_{2}),…. With the assumption that these differences are independent observations from a symmetric distribution, the null hypothesis is that this distribution has median zero.

To determine the value of the test statistic, *z*, the first step is to calculate the differences *d*_{1}, *d*_{2}, …, where *d** _{j}*=

*x*

*−*

_{j}*m*

_{0}(single sample) or

*d*

*=*

_{j}*x*

*−*

_{j}*y*

*(matched pairs). After zero differences have been discarded, the remaining*

_{j}*n*are arranged in ascending order of |

*d*

*|. The magnitudes are replaced by the corresponding ranks, with tied ranks where necessary. The signs of*

_{j}*d*

_{1},

*d*

_{2},…, are now attributed to the ranks, resulting in signed ranks. Let

*P*be the sum of the positive signed ranks and let

*T*be the smaller of

*P*and ½

*n*(

*n*+1)-

*P*. The test statistic, given by is an observation from the upper half of an approximate standard normal distribution. The ½ is a continuity correction.

As an example, suppose that a symmetric distribution is believed to have median 100. A random sample of eight observations is reported as consisting of the values 92.3, 57.6, 88.8, 110.5, 100.0, 181.0, 96.0, 105.7. The supposed median is subtracted from each observation to give −7.7, −42.4, −11.2, 10.5, 0, 81.0, −4.0, 5.7. The value 0 is discarded and the remainder are arranged in order of ascending absolute magnitude: −4.0, 5.7, −7.7, 10.5, −11.2, −42.4, 81.0. Retaining the signs while replacing the values by ranks gives −1, 2, −3, 4, −5, −6, 7, so that *p* = 13, ½*n*(*n*+1) − *P*=15 (since *n*=7) and *T*=13. The test statistic is

Comparing with the tables of upper-tail percentage points of the standard normal distribution (Appendix VI), we see that the null hypothesis that the median is 100 should not be rejected.