Wilcoxon Signed-Rank Test

Use it when the same people are measured twice on an ordinal or skewed outcome, such as a rating scale, and you want to know whether scores tend to go up or down. It works on the size and direction of each person's change, so it suits ratings, and changes with a few extreme values, where an average would mislead. Skewed scores are fine; what matters is the shape of the changes (see Symmetric Differences). Unlike the sign test, it has to rank the sizes of people's changes, not just tell whether each person went up or down; on an ordinal scale, that means treating the sizes of score differences as comparable. The scale's points needn't be exactly evenly spaced, but a bigger change must mean a bigger change (see Comparable Differences); if you can't defend that, the sign test uses only the direction of each change. It also works on a single set of scores, as a one-sample test of whether they center on a value chosen in advance (roughly, a hypothesized median): subtract that value from every score and test the differences. For example:

• Does a brisk walk take the edge off smokers' cravings?
• Do patients rate their pain lower on a 0-to-10 scale after a hypnosis session than before it?
• Do clients' 1-to-7 satisfaction ratings center above the scale's neutral midpoint of 4?

• Why. Walking and sitting happened on different days in random order, so the drop can reasonably be credited to the walk, but not to the exercise itself rather than the fresh air, the distraction or the time away from cigarettes.
• How many points. The test compares ranks, so its verdict isn't in points, but two summaries are. The median change, −1, is the middle of the 30 changes. The Hodges-Lehmann estimate, −0.5 points (95% CI −1.5 to just under 0, from all 30 changes as R computes it; dropping the 4 zeros first gives −1), is the one that matches the test: the median of the averages of every pair of changes, which R prints as the "(pseudo)median" with conf.int = TRUE. With coarse 0-to-10 ratings the two can differ by half a point.
• Whether it matters. Significance doesn't make a 1-point drop on a 0-to-10 rating enough to keep someone from lighting up.
• That there's no change, if p had been above .05. A non-significant result would have meant the data were inconclusive, not that walking does nothing for craving.

The matched-pairs rank-biserial correlation, r_{rb}, is the share of the rank evidence pointing one way minus the share pointing the other (Kerby, 2014). It runs from −1 (every change was a drop) to 1 (every change was a rise), with 0 meaning rises and drops balance. It has no units and doesn't grow with the number of people, so it can be compared across scales, but it still depends on the design, how varied the sample is, and how reliably things were measured.

You'll also see r = z/\sqrt{N}, which divides z by the number of people or, in some books, by the number of scores. It isn't the same quantity as r_{rb} (here −.42 with N = 30, or −.29 with N = 60), so say which one you report.

Of the 351 rank points, 263 (75%) belonged to smokers whose craving dropped after walking and 88 (25%) to smokers whose craving rose. 25% − 75% = −.50: the evidence leans clearly toward lower craving.

Rounded, −.50 sits right on the benchmarks' “large” line, but the unrounded −.4986 falls just short of it, and the 95% confidence interval, [−.76, −.11], runs from large to small. That interval is our own calculation: the standard-error formula R's effectsize package uses, on Fisher's z scale, but counting only the 26 smokers whose craving changed. effectsize itself (version 1.0.3) counts all 30 and gives [−.74, −.14]. Both are approximations; say which you used. Read the estimate as that three-to-one split rather than as a firm label. For size in the outcome's own terms, look at the changes: the median change was −1 point on the 0-to-10 scale, a 1-point decrease, and 18 of the 30 smokers craved less after walking. Whether a point less craving keeps anyone from smoking is a question about smoking, not statistics.

p answers one narrow question: if the changes were really balanced around zero, with a drop of any size as likely as a rise of the same size, how often would the ranks come out at least this lopsided, in either direction? Below the conventional α = .05, the result is called statistically significant.

If walking made no difference to craving, the 26 smokers whose craving changed would show rises and drops at least this unbalanced about 2.3% of the time, roughly 1 in 44. That's below .05, so it's significant.

Zero differences and software versions matter. This page's hand calculation uses Wilcoxon's convention: drop the zero differences, then rank the remaining absolute differences. Its normal approximation, with the standard error adjusted for ties, gives p = .023; that's what SPSS reports and what the R and Python code below asks for. Since version 4.6.0, R instead uses Pratt's convention by default, keeping the zero differences when assigning ranks, and in samples this size computes an exact conditional p that allows for ties and zeros: .026 here. The two differ in both the zero convention and how p is worked out, so state which convention and which method you used. With about 20 or more people who changed, the normal approximation is usually close; with fewer, use an exact p (R 4.6's default, or Exact… in SPSS's dialog). Say which you used.

p is not the probability that the null hypothesis is true, and it says nothing about how large or important the change is. That's what the median change and the effect size are for.

Power is the chance that a study detects an effect that's really there. A real effect doesn't guarantee a significant result: with too few people, it can easily come out non-significant. Power depends on how big the true effect is, how many pairs you have, α, whether the test is one- or two-sided, and, for a rank test, the shape of the changes. The same effect size gives different power for different shapes.

A practical way to plan is to work out how many pairs a paired t-test would need, then adjust. When the changes are roughly normal, the signed-rank test needs only slightly more pairs; with heavy tails or outliers it can need fewer. In large samples its efficiency relative to the t-test never falls below 0.864 (Hodges & Lehmann, 1956), so adding about 16% to the t-test's number (dividing it by 0.864) covers the worst case for continuous scores in large samples. With coarse ratings, ties and zero changes, that isn't guaranteed; simulate.

Suppose we expected craving to drop by half a standard deviation of the changes (d_z = 0.5), with roughly normal changes. A paired t-test with 34 smokers would come out significant about 81% of the time. In a simulation, the signed-rank test with 34 smokers did so about 79% of the time, and with 36 about 81%.

Researchers usually aim for at least 80% power. Plan it before collecting data, using the smallest effect that would matter. The calculator below plans in d_z, as for a paired t-test, inflated by the asymptotic efficiency factor 1 ÷ 0.864. That's a conservative planning approximation for continuous, symmetric shifts, not a guarantee for every distribution or sample size. For rating scales with many ties or zeros, simulate data that look like what you expect and count how often the test comes out significant. Try it: