Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is a non-parametric statistical test used to compare two related (paired) samples to determine whether their population mean ranks differ.

It’s the non-parametric equivalent of the paired t-test, used when the differences between pairs are not normally distributed.

In simple terms:

“The Wilcoxon test checks whether the median difference between paired observations is zero — without assuming the data follow a normal distribution.”

When to Use It

• Two sets of paired data - e.g., before/after, left/right, matched subjects
• Data are ordinal or continuous - but not necessarily normal
• Goal - Test if the median of the differences is zero

When not to Use It

• Independent groups - Use Mann–Whitney U test instead
• Normal differences - You can use a paired t-test instead

Example Scenario

“Did a training course significantly improve students’ test scores?”

Each student is tested before and after the course.
If the differences in scores aren’t normally distributed, use the Wilcoxon Signed-Rank Test instead of a paired t-test.

Hypotheses

• H₀ (Null Hypothesis) - The median difference between pairs = 0 (no change)
• H₁ (Alternative Hypothesis) - The median difference ≠ 0 (a change exists)

How It Works (Step-by-Step)

• Calculate the difference (d) for each pair:

• Ignore pairs where d_i ​= 0 (no change).
• Take the absolute value of each difference.
• Rank the absolute differences (1 = smallest).
• Assign the original signs (+/–) back to each rank.
• Compute:
• W₊ = sum of positive ranks
• W₋​ = sum of negative ranks
• The test statistic (W) is the smaller of W₊​ and W₋.
• Compare W to the critical value (from Wilcoxon table)
  or compute the p-value.

If p < 0.05, reject H₀ → the difference is statistically significant.

Example in Python

Let’s test if a meditation program reduced stress levels (lower scores = less stress):

import numpy as npfrom scipy.stats import wilcoxon​# Stress levels before and after a programbefore = np.array([72, 65, 78, 70, 69, 75, 80, 74, 77, 73])after  = np.array([68, 60, 74, 66, 65, 72, 78, 70, 73, 69])​# Perform Wilcoxon Signed-Rank Teststat, p = wilcoxon(before, after, alternative='two-sided')​print(f"Wilcoxon Statistic: {stat:.3f}")print(f"P-value: {p:.4f}")

Interpretation:

• If p < 0.05, reject H₀ → the program significantly reduced stress levels.
• If p ≥ 0.05, fail to reject H₀ → no significant change.

Alternative Options:

wilcoxon(before, after, alternative='greater')

• 'greater' → test if after > before
• 'less' → test if after < before
• 'two-sided' → test for any change (default)

If most ranks are negative (after < before),
the sum of negative ranks (W₋) will be much larger — indicating a significant decrease.

Advantages

• Doesn’t require normality
• Handles outliers well
• Works with small samples
• Simple and intuitive (rank-based)

Limitations

• Only works for paired data
• Less powerful than the paired t-test when data are normal
• Assumes data are symmetrically distributed about the median difference

Python code

import pingouin as pgimport numpy as npimport pandas as pd​# Example: same subjects measured before and after treatmentnp.random.seed(42)df = pd.DataFrame({    'Col3': np.random.normal(loc=50, scale=5, size=20),    'Col4': np.random.normal(loc=52, scale=5, size=20) })​# Run Wilcoxon signed-rank test (paired samples)results = pg.wilcoxon(x=df['Col3'], y=df['Col4'], alternative='two-sided')print(results)

Output:

          W-val alternative     p-val       RBC  CLESWilcoxon   73.0   two-sided  0.245487 -0.304762  0.42