Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used either to test the location of a population based on a sample of data, or to compare the locations of two populations using two matched samples. The one-sample version serves a purpose similar to that of the one-sample Student's t-test. For two matched samples, it is a paired difference test like the paired Student's t-test (also known as the "t-test for matched pairs" or "t-test for dependent samples"). The Wilcoxon test can be a good alternative to the t-test when population means are not of interest; for example, when one wishes to test whether a population's median is nonzero, or whether there is a better than 50% chance that a sample from one population is greater than a sample from another population. 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. The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics. Siegel used the symbol T for the test statistic, and consequently, the test is sometimes referred to as the Wilcoxon T-test. There are two variants of the signed-rank test. From a theoretical point of view, the one-sample test is more fundamental because the paired sample test is performed by converting the data to the situation of the one-sample test. However, most practical applications of the signed-rank test arise from paired data. For a paired sample test, the data consists of samples . Each sample is a pair of measurements. In the simplest case, the measurements are on an interval scale. Then they may be converted to real numbers, and the paired sample test is converted to a one-sample test by replacing each pair of numbers by its difference . In general, it must be possible to rank the differences between the pairs. This requires that the data be on an ordered metric scale, a type of scale that carries more information than an ordinal scale but may have less than an interval scale.