Wilcoxon Signed-Rank Test
Also known as: Wilcoxon matched-pairs signed-rank test, signed-rank test, Wilcoxon İşaretli Sıra Testi
The Wilcoxon signed-rank test is the nonparametric alternative to the paired t-test, comparing two related measurements on the same subjects to decide whether their typical difference is zero. It was introduced by Frank Wilcoxon in 1945 and works on continuous or ordinal data without assuming normality.
Key highlights
- Does not require the differences to be normally distributed, making it robust for small or non-normal paired samples.
- Works with ordinal as well as continuous outcomes.
- Uses the size of the changes (via ranks), so it is more powerful than the sign test when the symmetry assumption holds.
Intuition
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How it works
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When to use it
Use it to compare two related (paired or repeated) measurements on the same subjects when the outcome is continuous or at least ordinal and the differences are not normally distributed. The test assumes the pairs are dependent observations and that the distribution of the differences is symmetric. At least about ten pairs are recommended; with fewer pairs, or when the differences are clearly skewed, the sign test is more reliable.
Strengths & limitations
- Does not require the differences to be normally distributed, making it robust for small or non-normal paired samples.
- Works with ordinal as well as continuous outcomes.
- Uses the size of the changes (via ranks), so it is more powerful than the sign test when the symmetry assumption holds.
- Assumes the distribution of the paired differences is symmetric; asymmetry undermines validity.
- Limited to comparing exactly two related measurements.
- With very few pairs (fewer than about ten) the rank statistic lacks the power to discriminate.
Common pitfalls
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Applications
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Frequently asked
How is this different from the paired t-test?
Both compare two related measurements on the same subjects, but the paired t-test assumes the differences are normally distributed and works on their means, while the Wilcoxon signed-rank test ranks the differences and makes no normality assumption. Use the signed-rank test when the differences are non-normal or the outcome is ordinal.
When should I use the sign test instead?
The signed-rank test assumes the distribution of the paired differences is symmetric. If the differences are clearly skewed (for example |skewness| > 1.0), or if you have very few pairs, the sign test is more reliable because it only uses the direction of each change, not its rank.
How many pairs do I need?
At least about ten paired observations are recommended. With fewer pairs the rank statistic cannot separate signal from noise well, and the sign test is the safer alternative.
How is the effect size reported?
A common effect size is r = Z / √N, where Z is the standardized test statistic and N is the number of pairs. Reporting it together with a 95% confidence interval conveys the magnitude of the change, not just its statistical significance.
Sources
- 1.Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80–83.
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Cite this page
ScholarGate. (2026, June 1). Wilcoxon signed-rank test. ScholarGate. https://scholargate.app/statistics/wilcoxon-signed-rank