Wilcoxon signed-rank test
The Wilcoxon test compares two measurements of the same people (for example, before and after a program) when the differences aren’t normal or the scale is ordinal. It is the non-parametric alternative to the paired t-test. Paste the before scores in one box and the after scores in the other, in the same order.
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How it’s calculated
Each person’s difference is calculated: after − before (as in SPSS). Zero differences are dropped.
Differences are ranked by absolute value; ties get the average rank.
The ranks of positive differences (W+) and negative ones (W−) are added up. The T statistic is the smaller of the two.
Z = (T − n(n + 1) / 4) / σ, with σ corrected for ties. The asymptotic p-value comes from the normal distribution, two-tailed. With no ties or zeros and 50 pairs or fewer, the exact p-value is also calculated.
The effect size is r = |Z| / √n, with n the number of pairs with a non-zero difference.
How to read the result
- p < 0.05: there is a significant change between the two measurements. Look at how many went up (positive ranks) and down (negative ranks) to see the direction.
- With few pairs and no ties, report the exact p-value.
- Effect size r: 0.1 small, 0.3 medium and 0.5 large.
Frequently asked questions
- Does it match SPSS?
- Yes. Positive and negative ranks, Z and "Asymp. Sig. (2-tailed)" match Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples.
- Wilcoxon or paired t-test?
- If the before-after differences are normal, use the paired t-test. If not, or the scale is ordinal (Likert), use Wilcoxon.
- What about people who didn’t change?
- Zero differences say nothing about the direction of change, so they are left out. The calculator tells you how many there were.
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