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Wilcoxon signed-rank test (paired)

The Wilcoxon signed-rank test compares paired measurements by ranking their absolute differences — the non-parametric counterpart to the paired t-test.

What is Wilcoxon signed-rank test (paired)?

Drop pairs with zero difference, rank the remaining |dᵢ| from smallest to largest, then sum the ranks of the positive differences (V). Under H₀ (median difference = 0) V is symmetric around n(n+1)/4; large deviations either way give small p.

Like Mann-Whitney, signed-rank is a test of location shift only when the differences have a symmetric distribution. With asymmetric differences it tests something subtler. The matched-pairs rank-biserial is the corresponding non-parametric effect size.

Prefer over the paired t-test when the differences are heavily skewed or n is small (n ≤ 15) and you can't safely invoke the CLT. For symmetric, larger samples the paired t-test is slightly more powerful and gives a more interpretable effect (mean difference).

When should I use Wilcoxon signed-rank test (paired)?

Paired ordinal data.
Paired continuous data with skewed differences.
Small n paired studies where the t-test's normality assumption is shaky.

What data does it need?

Two paired numeric columns.

What does it report?

V statistic + p + Hodges-Lehmann location shift + matched-pairs rank-biserial r.

What does it assume?

Independent pairs.
Differences are at least ordinal.
Symmetric distribution of differences (for 'median shift' interpretation).

How do I interpret the result?

The Hodges-Lehmann estimator (median of all pairwise differences) is the natural non-parametric effect-size summary — robust, reported on the original scale.

Matched-pairs rank-biserial: 0.1 small, 0.3 medium, 0.5 large.