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 Siegel (1956) in his influential text book on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, . In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T.
Exclude pairs with . Let be the reduced sample size.
Order the remaining pairs from smallest absolute difference to largest absolute difference, .
Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let denote the rank.
Calculate the test statistic
, the absolute value of the sum of the signed ranks.
As increases, the sampling distribution of converges to a normal distribution. Thus,
For , a z-score can be calculated as .
If then reject
For , is compared to a critical value from a reference table.
If then reject
Alternatively, a p-value can be calculated from enumeration of all possible combinations of given .
The T statistic used by Siegel is the smaller of two sums of ranks of given sign; in the example given below, therefore, T would equal 3+4+5+6=18. Low values of T are required for significance. As will be obvious from the example below, T is easier to calculate by hand than W.