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.