Wilcoxon Signed-Rank Test

Nonparametric alternative to paired t-test for comparing paired samples.

Test Overview

What it tests:

A non-parametric alternative to the paired t-test that compares median differences between paired observations without requiring normal distribution assumptions.

How it works:

Ranks the absolute differences between paired values, then assigns signs based on the direction of change, testing whether positive and negative ranks are equally distributed.

Advantages:

• • No normality assumption required
• • Robust to outliers
• • Works with small sample sizes
• • More powerful than sign test
• • Uses magnitude of differences

When to Use:

• • Paired/matched data
• • Non-normal differences
• • Ordinal or continuous data
• • Before/after comparisons
• • Small sample sizes

Test Statistic & Procedure

Step-by-Step Process:

1. Calculate Differences

For each pair: di=Xi-Yi (exclude zero differences)

2. Rank Absolute Differences

Rank |di| from smallest to largest (handle ties by averaging ranks)

3. Assign Signs

Apply original sign of di to each rank

4. Calculate Test Statistic

W = sum of positive ranks (or negative ranks, whichever is smaller)

Wilcoxon signed-rank test statistic

= rank of |di|

sgn(di)

= sign of the i-th difference

= number of non-zero differences

Normal approximation for large samples

= sum of signed ranks (or one-sided rank sum)

Used when n is large (commonly n ≥ 25); a continuity correction of 0.5 is typically applied.

Effect size

= standardized test statistic

= total number of observations (2 x pairs)

Reading the Wilcoxon Signed-Rank Plot

Interpretation Guidelines

Decision Rule:

No Significant Difference (p ≥ 0.05):

• • No evidence of systematic change
• • Positive and negative ranks balanced
• • Median difference likely zero
• • Consider effect size magnitude

Significant Difference (p < 0.05):

• • Evidence of systematic change
• • Median difference significantly ≠ 0
• • Direction indicated by rank sums
• • Calculate confidence interval

Effect Size Interpretation (r):

Small Effect

r ≈ 0.10

Minimal practical difference

Medium Effect

r ≈ 0.30

Moderate practical difference

Large Effect

r ≈ 0.50

Substantial practical difference

Sample Size Considerations:

• • Small samples (n < 25): Use exact critical values
• • Large samples (n ≥ 25): Normal approximation appropriate
• • Ties: Reduce effective sample size and power
• • Power: Generally 95% as powerful as paired t-test

sixsigmacenter.com Implementation

Example Data Layout for Wilcoxon Signed-Rank Test

Paired data showing before/after measurements with calculated differences, ranks, and signed ranks for the Wilcoxon test. Sample Size Recommendation: Minimum 10-15 paired observations for adequate power in Wilcoxon signed-rank test - nonparametric alternative when normality assumptions are violated.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.