13.8 : Wilcoxon Signed-Ranks Test for Median of Single Population

The Wilcoxon signed-rank test for the median of a single population is a nonparametric test used to evaluate whether the median of a population differs from a specified value. Unlike parametric tests, it does not require data to follow a normal distribution, making it suitable for non-normal or small samples. The test begins by calculating the difference (d) between each observation and the hypothesized median. The absolute values of these differences are ranked in ascending order, with ties averaged. Each rank is then assigned the original sign of the corresponding d-value, creating a set of signed ranks.

The next step is to separately sum the positive and negative signed ranks. The test statistic is based on the smaller of these two sums (absolute value), which reflects the degree of symmetry around the hypothesized median. The sample size (n) is the number of non-zero d-values (differences that are not exactly zero). Based on n and the distribution of signed ranks, the test statistic is evaluated against critical values for a given significance level to determine whether to reject the null hypothesis that the sample median equals the hypothesized value. The Wilcoxon signed-rank test is particularly useful for data that deviates from normality, as it accounts for both the magnitude and direction of differences, unlike the simpler sign test, which only considers direction

1. If n is smaller than 30, the test statistic T is taken as the smallest of the two sums obtained;
2. If n is higher than 30, the test statistic T is calculated using the formula

In both cases, the critical Z-value is obtained from its table for a particular significance level and sample size n. The null hypothesis is rejected if the test statistic, T, is lower than the critical value.