Wilcoxon signed-rank test calculator

A clear answer. Here’s how.

Each difference is the first value minus the second (or the single value given), minus the hypothesised median μ, which is 0 for the usual test. The absolute differences are ranked from 1, tied values sharing the average of the ranks they span, and V is the sum of the ranks of the positive differences, as R’s wilcox.test reports it. If the differences are centred on μ, each rank is equally likely to carry a plus or a minus sign, so V has mean (sum of the ranks) ÷ 2 and variance (sum of the squared ranks) ÷ 4, which are n(n + 1) ÷ 4 and n(n + 1)(2n + 1) ÷ 24 without ties or zeros. Ties and zero differences are handled in one of two ways, which follow the two recent versions of R’s wilcox.test. R up to 4.5 drops zero differences before ranking (Wilcoxon’s method) and uses the exact p-value only when there were no zeros and there are no ties; otherwise it uses the normal approximation. R 4.6 uses the exact p-value even with ties and zeros, as the permutation distribution of the signed ranks, found by counting all 2^m sign patterns of the m non-zero differences with exact integer arithmetic; the zeros stay in the ranking, where they take the lowest ranks but add nothing to V (Pratt’s treatment). With no ties and no zeros the two settings are the same. The exact p-value is used for fewer than 50 pairs, as in R. Otherwise, and in both versions, the normal approximation drops the zero differences, ranks the rest and uses z = (V − mean − c) ÷ sd, where c is a continuity correction of 0.5 (the sign of V − mean for a two-sided test, +0.5 for “first larger”, −0.5 for “first smaller”) and sd includes the correction for ties; the two-sided p-value doubles the smaller normal tail. The development version of R, the future 4.7, also keeps zeros in the normal approximation and rounds differences to seven significant digits before ranking; this tool does not follow that. Differences are computed exactly from the decimals you type, so 0.4 − 0.3 and 0.3 − 0.2 tie; R compares binary floating-point numbers, in which they do not, so R’s answer can differ where rounding noise would break a tie. The pseudo-median is the Hodges–Lehmann estimate: the median of all Walsh averages (dᵢ + dⱼ) ÷ 2 with i ≤ j, over the differences used in the test, before subtracting μ. Its two-sided interval is shown for fewer than 50 pairs with no ties or zeros, and runs from the q-th smallest to the q-th largest Walsh average, where q is the smallest integer with P(V ≤ q) ≥ (1 − level) ÷ 2, as in R’s exact interval; it is conservative, so the level it actually reaches is shown. With ties, zeros or 50 or more pairs R finds an interval and estimate by a numerical search that this tool does not reproduce, so none is given. The rank-biserial correlation is (V⁺ − V⁻) ÷ (V⁺ + V⁻), from the ranks of the non-zero differences (Kerby’s simple difference formula): +1 when every pair goes up, −1 when every pair goes down. The test assumes independent pairs and, for its median interpretation, differences that are roughly symmetric about their median; it does not test means, compare independent groups (use the Mann–Whitney tool) or compute power. It is a calculator, not a substitute for a statistician when decisions depend on the answer.