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What is the Wilcoxon Signed-Rank Test?

The Wilcoxon signed-rank test is a non-parametric statistical test used to compare two related samples, matched samples, or repeated measurements on a single sample. Developed by Frank Wilcoxon in 1945, it serves as the non-parametric alternative to the paired samples t-test. Instead of comparing means (which requires normally distributed data), the Wilcoxon test works with the ranks of the differences between paired observations, making it appropriate when your data violate normality assumptions or when you are working with ordinal data.

When to Use the Wilcoxon Signed-Rank Test

Use this test when you have paired or repeated-measures data and cannot assume normality. Common scenarios include pre-test/post-test designs where scores are not normally distributed, Likert scale data from surveys (ordinal data), small sample sizes where normality is difficult to verify, and any before/after study where you want a more robust analysis that is less sensitive to outliers.

Wilcoxon Signed-Rank Test vs. Paired T-Test

The key difference between these two tests lies in their assumptions. The paired t-test assumes that the differences between pairs are normally distributed, while the Wilcoxon signed-rank test only assumes that the distribution of differences is symmetric. This makes the Wilcoxon test more versatile, though when normality holds, the paired t-test is slightly more powerful (i.e., better at detecting real differences). As a rule of thumb: if your data are clearly normal, use the paired t-test; if there is any doubt about normality or your data are ordinal, use the Wilcoxon test.

Choosing Between Parametric and Non-Parametric Tests

Assumptions of the Wilcoxon Signed-Rank Test

While the Wilcoxon test has fewer assumptions than the paired t-test, it still requires certain conditions to be met:

Understanding the Rank-Biserial Correlation

The rank-biserial correlation (r) is the recommended effect size measure for the Wilcoxon signed-rank test. It ranges from −1 to +1, where values near ±1 indicate that nearly all pairs changed in the same direction, and values near 0 indicate no consistent direction of change. It is calculated as (W+ − W−) / (W+ + W−).

How to Report Wilcoxon Results in APA Format

According to APA 7th edition guidelines, Wilcoxon signed-rank test results should include the test statistic (W or T), z-approximation, p-value, effect size, and relevant descriptive statistics (medians). Here are templates you can use:

Note: Report W to one decimal place and z to two decimal places. Report p-values to three decimal places, except use p < .001 when the value is below .001. Always include an effect size measure (rank-biserial r).

Common Mistakes to Avoid

• Using the Wilcoxon test for independent samples: The Wilcoxon signed-rank test is only for paired or repeated-measures data. For two independent groups, use the Mann-Whitney U test instead.
• Confusing W+ and W− with the test statistic: Some software reports W as the sum of positive ranks, others as the smaller of W+ and W−. StatMate reports W = min(W+, W−), which is the standard for hypothesis testing.
• Ignoring tied or zero differences: Pairs with zero difference (pre = post) are excluded from the analysis. If many pairs have zero differences, the test's power is reduced and you should report how many pairs were excluded.
• Assuming the test compares medians directly: The Wilcoxon signed-rank test technically tests whether the distribution of differences is symmetric around zero, not whether the medians are equal. However, when the symmetry assumption holds, rejecting the null hypothesis does imply a median shift.
• Using it with very small samples without exact tables: The normal approximation (z-score) used here is accurate forn ≥ 10. For smaller samples, exact p-values from Wilcoxon tables or permutation methods are more appropriate.