📊 Non-Parametric Test · Hypothesis Testing
Wilcoxon Signed-Rank Test Calculator
Free online tool for paired non-parametric hypothesis testing. Enter your before/after or matched pairs data to get W-statistic, z-score, p-value, effect size r, APA 7th edition results, and publication-ready charts — instantly.
Wilcoxon Signed-Rank Test
Non-Parametric
Paired Data
Free Online Calculator
Effect Size r
APA 7th Reporting
Generated by STATS UNLOCK — statsunlock.com
📥 Data Input — Before & After (Paired Values)
✓ 15 valid values
✓ 15 valid values
Supports .csv, .txt, .xlsx, .xls — headers detected automatically. Column 1 = Before, Column 2 = After.
| # | Before / Group 1 | After / Group 2 |
|---|
⚙️ Test Configuration
📊 Test Results — Summary
🔢 Ranked Differences Table
✅ Assumption Checks
📈 Visualizations
Paired Differences — Distribution
Before vs After — Connected Pairs
💬 Interpretation of Results
A. Formulas Used
Step 1 — Compute differences:
dᵢ = x₂ᵢ - x₁ᵢ (After − Before)
Step 2 — Exclude zero differences (standard method):
n* = count of non-zero dᵢ
Step 3 — Rank absolute differences:
Rank |dᵢ| from 1 to n*; average ranks for ties
Step 4 — Assign signed ranks:
R⁺ᵢ = rank if dᵢ > 0; R⁻ᵢ = rank if dᵢ < 0
Step 5 — Compute sums:
T⁺ = Σ R⁺ᵢ (sum of positive ranks)
T⁻ = Σ R⁻ᵢ (sum of negative ranks)
W = min(T⁺, T⁻) [test statistic for small n]
Step 6 — Z-approximation (n* > 25 or for continuous p-value):
μ_T = n*(n*+1)/4
σ_T = √[ n*(n*+1)(2n*+1)/24 − Σtⱼ(tⱼ²-1)/48 ]
(tie correction: Σtⱼ(tⱼ²-1)/48 where tⱼ = size of j-th tied group)
z = (T⁺ − μ_T − 0.5) / σ_T [continuity correction]
Step 7 — Effect size r:
r = |z| / √(2 × n*)
Where: r < 0.1 trivial, 0.1–0.3 small, 0.3–0.5 medium, > 0.5 large
Step 8 — Confidence interval for median difference:
Walsh averages: wᵢⱼ = (dᵢ + dⱼ)/2 for all i ≤ j
Sort Walsh averages; pick critical indices from Wilcoxon table or z-approx
B. Technical Notes
- No normality assumption: The Wilcoxon test makes no distributional assumption about the data itself, only that the differences are symmetrically distributed around the median.
- Z-approximation: This tool always uses the z-approximation with tie correction and continuity correction for a continuous p-value, even for small n.
- Pratt method: Retains zero differences in the ranking but excludes them from the final signed-rank sums. Slightly more conservative than standard exclusion.
- Power consideration: The Wilcoxon test has ~95% power efficiency relative to the paired t-test under normality; it is preferred when normality is violated.
- Recommended follow-up: If significant, report descriptive statistics for both groups (median, IQR), and consider the Sign Test as a robustness check.
This free Wilcoxon signed-rank test tool is designed for researchers, students, and data analysts who need to compare two related measurements without assuming normality. It is the non-parametric alternative to paired t test and is especially suited for small sample sizes or ordinal/skewed data.
- ✔ Data are paired (same subjects measured twice, or matched pairs)
- ✔ Dependent variable is at least ordinal (ranked, interval, or ratio)
- ✔ You cannot assume normality of differences (small n or skewed)
- ✔ Observations within pairs are independent of each other
- ✖ Do NOT use for two independent (unpaired) groups → use Mann-Whitney U
- ✖ Do NOT use when data are purely nominal → use McNemar's Test
- ✖ Do NOT use when normality holds and n ≥ 30 → consider Paired t-test for more power
Real-World Examples
🏥 Medical / Clinical
Testing whether a pain-relief treatment reduces VAS pain scores from baseline to 4 weeks post-treatment in the same patients.
Testing whether a pain-relief treatment reduces VAS pain scores from baseline to 4 weeks post-treatment in the same patients.
🧠 Psychology / Therapy
Measuring anxiety scores (GAD-7) before and after 8 sessions of Cognitive Behavioural Therapy (CBT) in a small clinical sample.
Measuring anxiety scores (GAD-7) before and after 8 sessions of Cognitive Behavioural Therapy (CBT) in a small clinical sample.
🎓 Education Research
Comparing student test scores under a standard curriculum versus a flipped-classroom method, using the same students in a crossover design.
Comparing student test scores under a standard curriculum versus a flipped-classroom method, using the same students in a crossover design.
🌿 Ecology / Biology
Comparing species richness counts in paired plots (treated vs control) when the count distribution is skewed or overdispersed.
Comparing species richness counts in paired plots (treated vs control) when the count distribution is skewed or overdispersed.
Decision Tree
Paired / Repeated-Measures Data?
├─ YES → Differences normally distributed?
│ ├─ YES (or n ≥ 30) → Paired t-test (more power)
│ └─ NO (or n < 30) → ★ WILCOXON SIGNED-RANK TEST ★
│
└─ NO (Independent groups)?
├─ Two groups → Mann-Whitney U Test
└─ Three+ groups → Kruskal-Wallis Test
Sample Size Guidance: Minimum n = 6 for this test to be meaningful. For 80% power at medium effect (r = 0.40, α = 0.05 two-tailed), you need approximately n = 25 pairs. For small effects (r = 0.20), n ≈ 100 pairs.
- Enter your data: Paste Before and After values as comma-separated numbers (e.g., 52, 48, 55, 61, 47, ...) into the two text areas. Both groups must have the same number of observations.
- Or load a sample dataset: Click the dropdown to select one of 5 built-in examples — pain scores, blood pressure, reaction times, exam results, or anxiety ratings.
- Or upload a file: Switch to the Upload tab. Drop a .csv, .txt, or .xlsx file. The tool auto-detects columns, shows a preview, and lets you select two numeric columns.
- Or use Manual Entry: Switch to the Manual Entry tab. Fill in values row by row. Click "Add Row" for more rows. Click "Use This Data" when done.
- Set significance level (α): Choose 0.01, 0.05, or 0.10. The confidence level for the CI is set automatically (1 − α).
- Choose test direction: Two-tailed tests whether median differences ≠ 0. Left-tailed tests if After < Before. Right-tailed tests if After > Before.
- Choose zero-handling and tie correction: Standard is to exclude zero differences and apply the tie correction to σ_T.
- Click "Run Wilcoxon Signed-Rank Test": The tool computes T⁺, T⁻, W, z-score, p-value, effect size r, and median difference with 95% CI instantly.
- Review results: Read the ranked differences table, assumption checks, significance banner, and two charts (distribution + paired lines).
- Export: Click Download Doc for a plain-text report or Download PDF to save a print-ready version with all results, tables, and interpretation.
📌 Worked Example: A physiotherapist measures knee pain (VAS 0–100) in 15 patients before and after a 4-week exercise programme. She enters the Before values (52,48,55,61,47,63,44,58,51,67,45,59,53,50,62) and After values (48,45,50,55,43,57,41,52,49,60,42,54,49,47,56), selects two-tailed α = 0.05, and clicks Run. The test returns W = 0, z = −3.41, p < .001, r = 0.62 (large effect), confirming a statistically significant reduction in pain.
❓ Frequently Asked Questions
The following references support the statistical methods used in this Wilcoxon signed-rank test calculator, covering effect size interpretation, non-parametric hypothesis testing, and best practices in APA 7th edition reporting.
- Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80–83. https://doi.org/10.2307/3001968
- Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
- American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
- Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
- Hollander, M., Wolfe, D. A., & Chicken, E. (2014). Nonparametric statistical methods (3rd ed.). Wiley. https://doi.org/10.1002/9781119196037
- Pratt, J. W. (1959). Remarks on zeros and ties in the Wilcoxon signed rank procedures. Journal of the American Statistical Association, 54(287), 655–667. https://doi.org/10.1080/01621459.1959.10501526
- 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
- Conover, W. J. (1999). Practical nonparametric statistics (3rd ed.). Wiley.
- Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.). McGraw-Hill.
- Corder, G. W., & Foreman, D. I. (2014). Nonparametric statistics for non-statisticians. Wiley. https://doi.org/10.1002/9781118165881
- R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
- NIST/SEMATECH. (2023). e-Handbook of statistical methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/
- Tomczak, M., & Tomczak, E. (2014). The need to report effect size estimates revisited. An overview of some recommended measures of effect size. Trends in Sport Sciences, 1(21), 19–25.
