Wilcoxon Signed-Ranks Test for Median of Single Population

Use the Wilcoxon signed-ranks test when data does not follow a normal distribution or when working with small samples. Unlike parametric tests, this nonparametric approach does not require normality assumptions, making it ideal for non-normal data. It accounts for both magnitude and direction of differences, providing a robust alternative for testing whether a population median differs from a specified value.

Begin by calculating the difference (d) between each observation and the hypothesized median. Rank the absolute values of these differences in ascending order, averaging ties. Assign each rank the original sign of its corresponding d-value to create signed ranks. This process establishes the foundation for computing the test statistic.

Sum the positive and negative signed ranks separately. The test statistic T is the smaller of these two sums (ignoring signs). For sample sizes smaller than 30, T is taken directly as this smallest sum. For larger samples, T is calculated using a formula based on the distribution of signed ranks.

Sample size (n) is the count of non-zero differences. When n is smaller than 30, the test statistic T is the smallest sum of signed ranks compared directly to critical values from standard tables. When n exceeds 30, T is calculated using a formula, and the critical Z-value is obtained from a Z-table for the given significance level.

The null hypothesis states that the sample median equals the hypothesized value. Reject the null hypothesis when the test statistic T is lower than the critical value at your chosen significance level. This conclusion indicates that the population median significantly differs from the specified value.

The sign test only considers the direction of differences from the median, ignoring magnitude. The Wilcoxon signed-ranks test accounts for both direction and magnitude by ranking absolute differences and assigning signs. This makes the Wilcoxon test more powerful and informative for detecting true differences in the population median.

The Wilcoxon signed-ranks test for a single population tests whether a median differs from a specified value. For comparing two related samples, use the wilcoxon signed ranks test for matched pairs instead. Both tests use signed ranks but apply to different research designs and hypotheses.