What Is the Wilcoxon Signed-Rank Test?

Developed by Frank Wilcoxon in 1945, the signed-rank test compares a sample of paired measurements against a hypothesized median (usually zero). It is the standard non-parametric counterpart to the paired t-test: rather than assuming normality, it ranks the absolute differences and asks whether positive and negative ranks occur in roughly equal proportions.

How the Calculation Works

Zero differences are discarded. The remaining absolute differences are ranked from smallest to largest, with ties receiving the average rank. The ranks of positive differences are summed to give W⁺, and the ranks of negative differences give W⁻. Under the null hypothesis of no shift, W⁺ follows a known distribution centered at n(n+1)/4.

This calculator computes the exact two-tailed p-value by enumerating every possible sign assignment of the observed ranks — the same logic as the textbook exact test — so results are correct even for small samples where the normal approximation would be unreliable.

Interpreting the Result

If the p-value falls below your significance level, the median difference is significantly different from zero. The smaller of W⁺ and W⁻ is the reported statistic, matching the convention used in most tables. Report the sample size and the direction of the shift (which tail is larger) alongside the p-value.