When to Use the Wilcoxon Test and What It Actually Is

May 2026·5 min read

When to use the Wilcoxon signed-rank test is a one-sentence answer: when you have a before/after design (paired data) but the difference scores are not normally distributed. It is the non-parametric replacement for the paired samples t-test and is common in intervention studies. This guide explains how it works, how to run it in SPSS, and how to report it in APA format.

What Is the Wilcoxon Test? The Paired t-Test Alternative

The Wilcoxon signed-rank test compares two related measurements from the same participants — typically a pre-test and a post-test — when the normality assumption of the paired t-test is not met.

Instead of comparing the means of difference scores (as the paired t-test does), Wilcoxon ranks the absolute values of those differences and tests whether positive ranks (improvements) systematically outweigh negative ranks (decrements).

Common thesis scenarios: pre/post intervention studies, or repeated measurements on the same participants under two different conditions.

Wilcoxon Signed-Rank vs. Wilcoxon Rank-Sum: The Difference

Students frequently confuse two tests that share the name ‘Wilcoxon’:

Wilcoxon signed-rank test: paired data, same participants measured twice. Non-parametric equivalent of the paired t-test.

Wilcoxon rank-sum test: two independent groups. This is mathematically identical to the Mann-Whitney U test — the same test, different name.

For thesis purposes: if your design is pre/post or matched pairs → Wilcoxon signed-rank. If your design is two separate groups → Mann-Whitney U.

[WARNING] Do not confuse Wilcoxon signed-rank (paired) with Wilcoxon rank-sum (independent groups). SPSS lists them separately: ‘Related Samples’ for signed-rank, ‘2 Independent Samples’ for rank-sum.

How to Run the Wilcoxon Signed-Rank Test in SPSS

SPSS path: Analyze → Nonparametric Tests → Legacy Dialogs → 2 Related Samples → move pre-test variable to Variable 1, post-test variable to Variable 2 → ensure Wilcoxon is ticked → OK.

• In the output, read:
• Negative Ranks, Positive Ranks, and Ties table: shows how many participants improved, declined, or stayed the same.
• Z statistic: the standardised test statistic.
• Asymptotic Significance (2-tailed): your p-value.

Get medians separately: Analyze → Descriptive Statistics → Frequencies → Statistics → tick Median.

Interpreting the Output and Calculating Effect Size r

A significant Wilcoxon test (p < .05) means the rank distribution of differences is unlikely to be due to chance — one measurement is systematically higher or lower than the other.

Effect size r = |Z| ÷ √N, where N = total number of observations (= 2 × number of participants for paired data). Benchmarks: r = .10 small, .30 medium, .50 large.

Example: Z = −3.12, 45 participants → N = 90 → r = 3.12 ÷ √90 = 3.12 ÷ 9.49 = .33 (medium effect).

APA Reporting Format for the Wilcoxon Signed-Rank Test

• Significant:
• "Stress scores decreased significantly from pre-test (Mdn = 27.0) to post-test (Mdn = 22.0), Z = −3.12, p = .002, r = .33."
• Non-significant:
• "No significant change in stress scores was found from pre-test (Mdn = 25.0) to post-test (Mdn = 24.0), Z = −0.87, p = .384, r = .09."

[INFO] Report medians (Mdn), not means, for Wilcoxon results. Include the Z statistic, p-value, and effect size r.