Wilcoxon Signed-Rank Test#

The Wilcoxon signed-rank test is the nonparametric alternative to the paired t-test. It tests whether two related samples differ significantly without assuming normally distributed differences.

When to Use#

Use the Wilcoxon test when you want to:

Compare two paired measurements (e.g., before/after)
The data are at least ordinally scaled
The differences are not normally distributed or the sample is small
You need a nonparametric alternative to the paired t-test

Assumptions#

Paired (related) observations
At least ordinal scale of measurement
Symmetric distribution of differences around the median
Independence of pairs from one another

Formula#

For each pair, the difference $d_i = X_{i,1} - X_{i,2}$ is calculated. The differences are ranked by their absolute value. The test statistic is:

W = \sum_{i=1}^{n} \text{sgn}(d_i) \cdot R_i

where $R_i$ is the rank of the absolute difference $|d_i|$ and $\text{sgn}(d_i)$ is the sign of the difference.

Alternatively, $W$ can be computed as the smaller of the two rank sums:

W = \min(W^+, W^-)

where $W^+$ is the sum of positive ranks and $W^-$ is the sum of negative ranks.

Example#

Practical Example: Pain Therapy

A therapist wants to evaluate the effectiveness of a new pain therapy. 15 patients rate their pain intensity on a scale from 0–10 before and after treatment.

Measurement 1: Pain score before therapy
Measurement 2: Pain score after therapy

Since the pain scale is ordinal and the sample is small, the Wilcoxon test is used instead of the paired t-test to determine whether the therapy significantly reduces pain.

Effect Size#

The effect size $r$ is calculated from the standardized test statistic:

r = \frac{Z}{\sqrt{n}}

where $Z$ is the z-approximated value of the test statistic and $n$ is the number of pairs.

Effect Size	r
Small	0.1
Medium	0.3
Large	0.5