Wilcoxon Signed-Rank Test Calculator — Paired

The Wilcoxon Signed-Rank Test. Paired measurements can shift without normally distributed differences. This test asks whether positive and negative differences carry balanced rank mass around zero. Keep every pair intact and subtract sample two from sample one. The subtraction order defines every sign, tail, and effect that follows. An exactly zero difference contributes no direction. This page drops every zero before ranking and records how many informative pairs remain. Take absolute values of the nonzero differences, rank them from smallest to largest, then restore each difference's original sign. Equal absolute differences share a midrank. Their positive or negative signs stay attached while the tie term reduces the null variance. Add positive ranks into W plus and negative ranks into W minus. Together they equal n times n plus one over two. This page always uses a tie-corrected normal approximation. It standardizes W plus without a continuity correction and never switches automatically to exact inference.

Pairing is the design. Row i in sample one and row i in sample two must describe the same person or deliberately matched unit. Different pairs must remain independent. Households, clinics, or one person contributing several pairs require a model that represents that dependence. Difference magnitudes must be meaningful because larger absolute differences receive larger ranks. If only directions are trustworthy, use the sign test instead. The sharp null says the paired-difference distribution is symmetric about zero. It is not a direct test of the mean paired difference. Calling a rejection a median or location shift needs reasonably symmetric differences. Strong skew can move signed ranks for another reason. Tail direction follows sample one minus sample two. Right-tailed means positive differences dominate; left-tailed means negative differences dominate. The matched rank-biserial compares positive and negative rank mass. Standardized r divides z by the square root of nonzero n; keep them separate.

Now compare nine matched measurements under two conditions. Keep sample one minus sample two, a two-sided alternative, and alpha point zero five fixed. The nine differences contain seven positives, two negatives, and no zeros. All nine pairs therefore enter the rank calculation. The absolute ranks in pair order are eight, three, nine, four, seven, six, five, two, and one. None are tied. Positive ranks sum to forty. Negative ranks sum to five, and together they recover all forty-five units of rank mass. Under symmetry, W plus centers at twenty-two point five. With no ties, its standard deviation is about eight point four four one. Subtract the null mean and divide by that spread, with no half-unit adjustment. The resulting z score is about two point zero seven three. The two-sided p-value is zero point zero three eight one five two, below alpha. Reject symmetry around zero, while reporting rank-biserial zero point seven seven eight and standardized r zero point six nine one.

With fewer than ten nonzero pairs, the calculator warns that the approximation is rough. The nine-pair example still does not become exact. In a diagnostic set, one zero exits first. Equal absolute differences then share ranks, and the tie correction reduces the variance. Unequal sample lengths or fewer than two original pairs fail validation. If every difference is zero, no rank signal remains and the calculator stops. Swap the sample order and W plus becomes five, z and both effects turn negative, but the two-sided p-value stays unchanged. Lock pair order, subtraction direction, tail, and alpha before results. Choosing a favorable method or tail afterward invalidates the evidence. Report original and nonzero pairs, zeros, ties, W plus, W minus, method, z, p, both effects, symmetry limits, and design boundaries. No shift estimate or confidence interval is provided. Rank meaningful paired differences, inspect their signed rank mass, and run the Wilcoxon signed-rank calculator free at Distri Scope dot com.