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Wilcoxon Signed-Rank Test — Paired & One-Sample

Non-parametric test for paired data or one-sample median. Uses ranks of differences. T = min(W+, W-).

Concept Fundamentals
Non-parametric
Test Type
Distribution-free
Rank-based comparison
Method
Mann-Whitney U equivalent
Equal medians
Null Hypothesis
Two independent samples
Ordinal / skewed data
Application
No normality assumed
Run Wilcoxon TestPaired or one-sample

Why This Statistical Analysis Matters

Why: When data are not normal or ordinal, Wilcoxon provides a robust alternative to the paired t-test.

How: Enter before/after pairs or data + hypothesized median. Get T, p-value, effect size.

  • ●T = min(W+, W-)
  • ●Normal approx for n≥20
  • ●Exact for small n
Sources:Wilcoxon (1945) — Original PaperNIST Engineering Statistics Handbook
W
STATISTICSInference & Tests

Wilcoxon Signed-Rank Test — Paired & One-Sample

Non-parametric test for paired data or one-sample median. Enter before/after or data + hypothesized median.

Mann-Whitney U (independent) →Paired t-Test →

Real-World Scenarios — Click to Load

Data Input

wilcoxon_signed_rank.sh
CALCULATED
$ wilcoxon_signed_rank --mode="paired" --tail="two-sided" --alpha=0.05
Decision
REJECT H₀
T statistic
0
p-value
0.002301
W⁺ / W⁻
0 / 55
μ_T
27.50
σ_T
8.6963
z
-3.1623
Effect size r
0.7071
Share:
Wilcoxon Signed-Rank Result
Paired Data
T = 0
p = 0.0023Significantr = 0.707
numbervibe.com/calculators/statistics/wilcoxon-rank-sum-test-calculator

Signed-Rank Distribution (W⁺ vs W⁻)

Difference Histogram (green = positive, red = negative)

Calculation Breakdown

DATA
Number of pairs (n)
10
Non-zero pairs (n')
10
ext{Exclude} ext{zero} ext{differences}
RANK SUMS
W⁺ (sum of positive ranks)
0
W⁻ (sum of negative ranks)
55
T = min(W⁺, W⁻)
0
ext{Test} ext{statistic}
NORMAL APPROXIMATION
μ_T = n'(n'+1)/4
27.5000
σ_T (with tie correction)
8.6963
z = (T − μ_T)/σ_T
-3.1623
ext{Standardized} ext{statistic}
p-value
0.002301
2 × min(Φ(z), 1−Φ(z))
EFFECT SIZE
Effect size r
0.7071
DECISION
REJECT H₀
α = 0.05

Quick Interpretation

The median difference differs significantly from zero (p < α). Negative differences dominate (values decreased). Effect size r = 0.707.

Step-by-Step Rank Table

PairBeforeAfterDiff|Diff|RankSigned Rank
174-335.5-5.5
285-335.5-5.5
363-335.5-5.5
496-335.5-5.5
585-335.5-5.5
674-335.5-5.5
763-335.5-5.5
885-335.5-5.5
974-335.5-5.5
1063-335.5-5.5

For educational and informational purposes only. Verify with a qualified professional.

📈 Statistical Insights

T

T = min(W+, W-)

— Test stat

W+

Sum of + ranks

— Ranks

n≥5

Min for approx

— Sample

Key Takeaways

  • Wilcoxon Signed-Rank test: Non-parametric alternative to paired t-test. Tests whether the median difference is zero.
  • Paired data: Compute dᵢ = xᵢ − yᵢ. Remove zeros. Rank |dᵢ|. W⁺ = sum of ranks where dᵢ > 0, W⁻ = sum where dᵢ < 0.
  • One-sample: dᵢ = xᵢ − m₀ (hypothesized median). Same ranking procedure.
  • Test statistic: T = min(W⁺, W⁻). Check: W⁺ + W⁻ = n'(n'+1)/2 where n' = non-zero pairs.
  • Large sample (n' ≥ 20): μ_T = n'(n'+1)/4, σ_T = √(n'(n'+1)(2n'+1)/24) with tie correction. z = (T − μ_T)/σ_T.
  • Effect size: r = z/√(2n) or matched-pairs rank-biserial correlation.
  • Interpretation: Significant T means the median difference differs from zero.

Did You Know?

📊Wilcoxon Signed-Rank is for paired/matched data. For independent samples, use Mann-Whitney U (Wilcoxon Rank-Sum).Source: Naming
🧪Common in pre/post designs: before vs after treatment, pre-test vs post-test, baseline vs follow-up.Source: Design
📐Uses both the sign and magnitude of differences (via ranks). More powerful than the sign test alone.Source: Power
💊Ideal for clinical trials with repeated measures when data are ordinal or non-normal.Source: Clinical
🎓Frank Wilcoxon (1945) introduced both the rank-sum (independent) and signed-rank (paired) tests.Source: History
⚠️Exclude zero differences from ranking. They provide no information about direction.Source: Zeros

Formulas

dᵢ = xᵢ − yᵢ (paired) or xᵢ − m₀ (one-sample)

Differences

W⁺ + W⁻ = n'(n'+1)/2

Rank sum check

μ_T = n'(n'+1)/4, σ_T = √(n'(n'+1)(2n'+1)/24)

Tie correction: subtract Σ(tᵢ³−tᵢ)/48

r = z/√(2n)

Effect size

Paired vs One-Sample Mode

Paired: Two columns (before, after). Each row is a matched pair. Tests whether the median difference is zero. One-sample: Single column of data. Tests whether the median equals a hypothesized value m₀. Use when you have one sample and want to compare its median to a known or theoretical value.

Frequently Asked Questions

When should I use Wilcoxon Signed-Rank instead of paired t-test?

When differences are non-normal, ordinal, or from small samples. The paired t-test assumes normality of the differences.

What happens to zero differences?

They are excluded from the ranking. Only non-zero pairs contribute to W⁺ and W⁻. n' is the number of non-zero pairs.

What does T = min(W⁺, W⁻) mean?

T is the test statistic. Small T suggests one direction dominates. Compare T to the critical value or use the normal approximation for p-value.

Can I use this for independent samples?

No. For two independent groups, use the Mann-Whitney U test (Wilcoxon Rank-Sum for independent samples).

What sample size for normal approximation?

n' ≥ 20 is typically safe. For smaller n', exact tables or permutation tests are more accurate.

Disclaimer: This calculator provides statistical guidance. For medical or clinical decisions, consult a qualified professional.

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