The paired test that ranks the differences
Wilcoxon's signed-rank test is the paired t-test's rank-based counterpart. Take the differences, rank them by size ignoring sign, then let the signs back in: sum the ranks of the gains (W+) and the losses (W−) separately. Under the null the two sums should be comparable; a lopsided split says one direction dominates. Zero differences are dropped: they carry no directional evidence.
The revision-week data (differences 6, 7, 5, 13, 14, 5, 6, 15, all gains): W+ = 36, W− = 0. That is the most one-sided result eight pairs can produce. The normal approximation (tie-corrected) gives z ≈ −2.46, p ≈ 0.014, significant, though less emphatic than the paired t-test's 0.0007 on the same data: ranking deliberately ignores how much bigger the big gains were, and that information was worth something.
When it beats the paired t-test
- Differences are skewed or outlier-prone: one wild gain should not carry the verdict.
- The measurement is ordinal-flavoured (rating scales) where "twice as much" is not meaningful.
- Samples are small and normality of differences is doubtful.
Checking normality of the raw before/after scores. The assumption lives on the differences, and for Wilcoxon, on their rough symmetry around their centre.
Frequently asked questions
What happens to pairs with zero difference?
They are excluded before ranking (the standard Wilcoxon convention), and the effective n shrinks. The page reports the pairs actually used.
Signed-rank or sign test?
Signed-rank, when difference sizes mean anything at all: it uses them and is more powerful. The sign test survives on direction alone.