The Wilcoxon Signed Rank test assumes that the population of interest is both continuous and symmetric (not necessarily normal). Since the mean and median are the same (for symmetrical distribution), the hypothesis tests on the median are the same as the hypothesis test on the mean.
The Wilcoxon test is performed by ranking the non-zero deviations in order of increasing magnitude (that is, the smallest non-zero deviation has a rank of 1 and the largest deviation has a rank of $n$). The ranks of the deviations with positive and negative values are summed.
These sums are used to determine whether or not the deviations are significantly different from zero. Wilcoxon Signed Rank Test is an alternative to the Paired Sample t-test.
One-Tailed Test
$H_0: \mu = \mu_0\quad $ vs $\quad H_1: \mu < \mu_0$
Test Statistics: $T^-$: an absolute value of the sum of the negative ranks
Two-tailed Test
$H_0: \mu = \mu_0 \quad$ vs $\quad H_1:\mu \ne \mu_0$
Test Statistics: $min(T^+, T^-)$
Because the underlying population is assumed to be continuous, ties are theoretically impossible, however, in practice ties can exist, especially if the data has only a couple of significant digits.
Two or more deviations having the same magnitude are all given the same average rank. The deviations of zero are theoretically impossible but practically possible. Any deviations of exactly zero are simply thrown out and the value of $n$ is reduced accordingly.
Single Sample Wilcoxon Signed Rank Test
The Wilcoxon Signed Rank Test is important for researchers as it fills a critical gap in statistical analysis.
- Non-normal data: Most of the statistical tests, like the dependent samples t-test, assume that the data follows a normal distribution (bell curve). The Wilcoxon Signed Rank Test supersede the assumption of normality, making it ideal for analyzing data that is skewed, ranked, or ordinal (like survey responses on a Likert scale Questions).
- Robust against outliers: Outliers (very large or small observations in the data) can significantly skew the results of some statistical tests. The Wilcoxon Signed Rank Test focuses on the ranks of the differences, making it less sensitive to extreme values (outliers) in the data compared to tests that rely on raw numbers.
- Focuses on changes within subjects: The Wilcoxon Signed Rank Test is designed for paired data (dependent samples), to look at the same subjects before and after situation (like a treatment) or under two different conditions.