---
title: "Wilcoxon signed-rank test"
method_id: SM.DIF.NPA.2DEP.WSR
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/wilcoxon_signed_rank.md
source: Clarus method library
---

# Wilcoxon signed-rank test

> A classical statistical method in the Clarus library (Difference family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** Wilcoxon matched-pairs signed-rank test; signed-rank test; non-parametric paired test; Wilcoxon T test.

## Hypotheses

- **Null (H0):** The distribution of the paired differences is symmetric about zero (median of the differences = 0; no systematic shift between {c1} and {c2}).
- **Alternative (H1):** The distribution of the paired differences is NOT centred at zero (median of the differences != 0; a systematic shift exists).

## When to use it

Use this method when your goal is to:
- compare two related measurements
- compare pre post within subjects
- compare matched pairs

## Data it expects

- **dependent:** 1 ordinal or continuous (rank-able)
- **independent:** 1 within-subject factor/2 levels
- **pairing:** paired

## Assumptions Clarus checks

- **symmetry of differences** (severity: warning; on violation: warn_only)
- **paired ordinal differences** (severity: critical; on violation: warn_only)
- **independence of pairs** (severity: design; on violation: warn_only)

## Effect size reported

- **rank biserial** (Kerby (2014); King, Rosopa & Minium — matched-pairs rank-biserial r_rb = (W_plus - W_minus) / sum_of_ranks (equivalently 1 - 2T/[n(n+1)/2]).)

## Honest limitations

- This test evaluates the WITHIN-unit change in rank/location; it says nothing about between-group differences.
- Pairs with a zero difference are dropped, which lowers the effective sample size — report how many pairs actually contributed.
- If the paired differences are markedly asymmetric, the result reflects a shift in the whole distribution (stochastic dominance), not specifically a median change.
- With very few non-zero pairs the exact test cannot reach significance even for a large effect, so a non-significant p-value is inconclusive rather than evidence of no change.
- A pre/post change is not causal unless the intervention between measurements was randomly assigned in a controlled design — maturation, history, practice, and regression-to-the-mean can all produce a shift.
- Results generalize only to the sampled population and to the specific interval between the two measurements.

## Primary sources

- Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83. https://doi.org/10.2307/3001968
- Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11.IT.3.1. https://doi.org/10.2466/11.IT.3.1
- Hollander, M., Wolfe, D. A., & Chicken, E. (2014). Nonparametric Statistical Methods (3rd ed.). Wiley.
- Field, A. (2013). Discovering Statistics Using IBM SPSS (4th ed.). Sage. Ch.6.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum. (r effect-size bands.)

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Wilcoxon signed-rank test", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/wilcoxon_signed_rank.md

**APA**

Clarus. (2026). Wilcoxon signed-rank test (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/wilcoxon_signed_rank.md

**BibTeX**

```bibtex
@misc{clarus-wilcoxon-signed-rank,
  author       = {Clarus},
  title        = {Wilcoxon signed-rank test},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/wilcoxon_signed_rank.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.DIF.NPA.2DEP.WSR · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library