When should I use the Wilcoxon signed-rank test?

Use it for paired or matched data (before/after, left/right) when the differences are not normally distributed or the sample is small. It tests whether the median difference is zero.

What is the test statistic for Wilcoxon signed-rank?

You rank the absolute differences, then sum the ranks with positive signs (W+) and negative signs (W−). The test statistic is the smaller of W+ and W−.

Does this calculator give exact p-values?

Yes, for up to 15 non-zero pairs it enumerates all sign combinations to give the exact two-sided p-value. For larger samples it uses the normal approximation with continuity correction.

What are the assumptions?

Data must be paired, differences should be at least ordinal and symmetrically distributed around the median, and pairs with zero difference are removed before ranking.

Wilcoxon Signed-Rank Test Calculator (Paired, Nonparametric)

Test whether the median difference between two paired samples is zero. Paste your “before / after” (or “A / B matched”) data below. The tool will remove zero differences, rank the absolute differences (with ties), compute W₊ and W₋, and give an exact two-sided p-value for n ≤ 15 non-zero pairs or a normal approximation for larger n.

How the Wilcoxon signed-rank test works

This is a non-parametric alternative to the paired t-test. It does not assume that the differences are normally distributed, only that they are symmetrically distributed.

Compute differences: d_i = B_i − A_i.
Remove pairs where d_i = 0.
Rank the absolute differences |d_i| from smallest to largest (average ranks for ties).
Assign each rank the sign of its original difference.
Sum positive ranks → W₊, sum negative ranks (absolute) → W₋.
Test statistic: T = min(W₊, W₋).

Exact vs. normal approximation

For very small samples the exact distribution is preferred. This tool enumerates all 2ⁿ sign patterns if n ≤ 15 and returns the exact two-sided p. For n > 15 it uses the usual normal approximation:

Mean: \( \mu_T = \frac{n(n+1)}{4} \)

Variance: \( \sigma_T^2 = \frac{n(n+1)(2n+1)}{24} \times \text{tie correction} \)

z: \( z = \frac{T - \mu_T - 0.5}{\sigma_T} \) (two-sided p = 2(1 − Φ(|z|)))

FAQs

What if I get p very close to 0?

It means the observed rank pattern is very unlikely under the null that median difference = 0.

Can I use it for Likert scales?

Yes, Wilcoxon is often used for ordinal paired data like Likert 1–5 before/after.

Full original guide (expanded)

If you report results academically, include: test used (Wilcoxon signed-rank), sample size after removing zeros, W value (often the smaller), and p-value (exact or approximate). Example: “A Wilcoxon signed-rank test showed that the training increased scores, T = 5, n = 10, p = 0.031 (two-tailed).”