Wilcoxon Signed-Rank Test Calculator

The Wilcoxon signed-rank test is the non-parametric alternative to the paired t-test. Instead of comparing the mean of the paired differences, it ranks the sizesof those differences and asks whether the positive and negative ranks balance out — so skew, outliers, and ordinal data don't break it. It works on two paired columns (the same subjects measured twice) or on one sample against a hypothesised median μ₀.

1. Form the differences.For paired data, dᵢ = afterᵢ − beforeᵢ; for a one-sample test, dᵢ = valueᵢ − μ₀. Any dᵢ = 0 carries no direction, so the standard Wilcoxon rule discards it and reduces n. Pratt's (1959) option keeps zeros while ranking but still excludes them from W⁺ and W⁻.
2. Rank the absolute differences. Sort |dᵢ| from smallest to largest and assign ranks 1…n, giving tied magnitudes the average of the ranks they span (Wilcoxon, 1945). Two values tied for ranks 3 and 4 each become 3.5.
3. Sum the signed ranks. W⁺ = Σ ranks where dᵢ > 0 and W⁻ = Σ ranks where dᵢ < 0. As a check, W⁺ + W⁻ must equal n(n+1)/2. The reported statistic is W = min(W⁺, W⁻).
4. Exact p-value (small, tie-free samples). Under H₀ each rank is independently positive or negative with probability ½, so W⁺ ranges over the 2ⁿ equally likely sign assignments. This tool enumerates that null distribution and reads the tail directly: two-sided p = 2·min(P(W⁺ ≤ w), P(W⁺ ≥ w)), capped at 1.
5. Normal approximation (ties or larger samples). Otherwise it uses μ_W⁺ = n(n+1)/4 and σ²_W⁺ = n(n+1)(2n+1)/24 − Σ(tⱼ³−tⱼ)/48, where tⱼ is the size of each group of tied |dᵢ| (NIST e-Handbook). The z-score z = (W⁺ − μ_W⁺ ∓ ½)/σ_W⁺ — with an optional ±0.5 continuity correction toward the mean — is read against the standard-normal distribution.

The effect size is the matched-pairs rank-biserial correlation r = (W⁺ − W⁻)/(W⁺ + W⁻), bounded in [−1, 1] (Kerby, 2014): positive means values tended to rise. This module computes σ² straight from the ranks as ¼·Σrᵢ², which equals the NIST closed form exactly and is shown reconciled against it in the working. Every step runs client-side, so nothing you paste leaves your device.

• Wilcoxon, F. (1945) — Individual Comparisons by Ranking Methods (Biometrics Bulletin 1(6): 80–83)
• NIST/SEMATECH e-Handbook of Statistical Methods — normal approximation, tie correction, and continuity correction
• Kerby, D. S. (2014) — The simple difference formula: rank-biserial correlation (Comprehensive Psychology)

The formulas and all worked examples on this page were last cross-checked against these sources — and against SciPy's stats.wilcoxon reference cases — on 2026-07-10.