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Wilcoxon Signed-Rank Test Calculator
The Wilcoxon signed-rank test is the non-parametric counterpart of the paired t-test. It uses the differences between paired measurements, ranks their absolute sizes, and compares the sum of ranks with positive signs to the sum with negative signs. It is appropriate when the differences are not normally distributed, when the sample is small, or when the data are ordinal.
Enter the two paired samples in matching order. Pairs with a zero difference are dropped, as in the standard (Wilcoxon) treatment. The calculator lists each difference, its absolute value, rank and sign, then reports W₊, W₋, the test statistic W = min(W₊, W₋), and the p-value: exact when there are 50 or fewer non-zero differences and no ties, otherwise from the normal approximation with tie correction. A continuity correction can be switched on to match R’s default.
A significant result means the differences are not symmetric around zero, so one condition tends to give larger values than the other. For two independent groups, use the Mann-Whitney U test.
Separate values with commas, spaces or new lines.
Same order as sample 1. Pairs with zero difference are dropped.
Result and step-by-step solution
- State the hypothesesH₀: the median difference is zero. H₁: the paired differences are not centred on zero (the two conditions differ).
- Differences and ranks of |d| (zero differences dropped: 1)
d = sample 1 − sample 2 |d| Rank Sign 15 15 7 + −7 7 3 − 5 5 1.5 + 20 20 9 + −9 9 4 − 17 17 8 + −12 12 6 − 5 5 1.5 + −10 10 5 − - Signed-rank sumsW₊ = sum of ranks with positive d = 27; W₋ = 18 (check: W₊ + W₋ = n(n + 1)/2 = 45). Test statistic W = min(W₊, W₋) = 18.
- p-valueNormal approximation: μW = n(n + 1)/4 = 22.5; σW = √[n(n + 1)(2n + 1)/24 − Σ(t³ − t)/48] = 8.4336. z = (W₊ − μW) ÷ σW = 0.5336; p = 0.5936.
- Decisionp = 0.5936 ≥ α = 0.05
| Median of the differences | 5 |
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Frequently asked questions
- What is the null hypothesis of the Wilcoxon signed-rank test?
- That the paired differences are symmetrically distributed around zero, which for practical purposes means the median difference is zero.
- What happens to pairs with no difference?
- They are removed before ranking and n is reduced accordingly. This is the Wilcoxon zero method used by SciPy and R.
- Which statistic do I report, W₊ or W?
- Tables use W = min(W₊, W₋). Software often reports W₊ (called V in R). The p-value is the same for a two-tailed test.
- Does the normal approximation match R?
- R applies a continuity correction by default and SciPy does not. Choose the continuity correction option to reproduce R’s p-value.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.