Wilcoxon Signed Rank Test
The Wilcoxon Signed Rank Test compares two related measurements (typically before-and-after on the same subjects) without assuming the differences are normally distributed. Example: comparing patient pain scores before and after treatment when score differences are highly skewed.
Step 1 — Choose the input method
| Method | When to use |
|---|---|
| Manual Data Input | Paste Before and After values side by side |
| Excel Upload | Upload your file and select the two paired columns |
Step 2 — Provide your data
In manual mode, fill two text areas (Before and After) with the same number of values.
In Excel mode, a Generate Random Data button is also available.
Step 3 — Calculate
Click the calculate button. On success, results appear with a "New Analysis" button to reset.
Step 4 — Read the results
| Field | What it shows |
|---|---|
| W Statistic | Sum of positive ranks |
| Z-score | Standardised Z with continuity correction |
| P-value (NEJM) | Significance |
| Effect Size (r) | Rank-biserial with interpretation label |
| Medians | Median of Before and After |
| Rank table | All paired differences with signed ranks |
Interpretation includes significance and direction statements.
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Statistical methods used
Test statistic
| Component | Formula |
|---|---|
| Pair differences | dᵢ = afterᵢ − beforeᵢ |
| Zero-difference handling | Pairs with dᵢ = 0 are excluded |
| Ranking | |
| W statistic | Sum of positive ranks (W⁺) |
Normal approximation with continuity correction
| Component | Formula |
|---|---|
| μ_W | n(n + 1) / 4 |
| σ_W | √(n(n + 1)(2n + 1) / 24) |
| Z-score | (W_corrected − μ_W) / σ_W (continuity correction applied) |
| p-value | `2 × (1 − Φ( |
Hypothesis statement
- H₀: The median of paired differences is 0
- H₁: The median of paired differences is non-zero (two-tailed)
- Significance level: α = 0.05
Effect size — Rank-biserial correlation (r)
r = |Z| / √n (n = number of non-zero differences)
| |r| value | Label |
|---|---|
| < 0.10 | Negligible effect |
| 0.10 – 0.30 | Small effect |
| 0.30 – 0.50 | Medium effect |
| ≥ 0.50 | Large effect |
Error handling
| Condition | Error message |
|---|---|
| All differences are 0 | "All differences are zero. Cannot perform Wilcoxon signed-rank test." |