Sample size for a Wilcoxon signed-rank test

When to use it

Before-and-after or matched measurements whose changes are skewed.

Not for

Changes that are roughly normal: the paired t-test needs fewer pairs.

Notation

$n$: Number of pairs
$p$: P(Dᵢ + Dⱼ > 0) for two independent within-pair changes
$\Phi$: Standard normal distribution function
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\tau$: Number of tails: 2 for a two-sided test, 1 for one-sided
$q$: Expected dropout

Formulas

Noether’s formula

n = \left\lceil \frac{(z_{1-\alpha/\tau} + z_{1-\beta})^{2}}{3\,(p - \tfrac{1}{2})^{2}} \right\rceil

The signed-rank statistic estimates the probability that the average of two changes is positive [1].

p from a difference and SD

p = \Phi\!\left(\frac{\Delta\sqrt{2}}{\sigma_d}\right)

For normal changes with SD σ_d.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Parameters

P(Dᵢ + Dⱼ > 0) $p$

Values: Between 0 and 1, not 0.5

The chance that, for two pairs, the average of their changes is positive. 0.5 is no effect.

Difference, instead of p $\Delta$ Optional

Values: Not zero

With the SD, p is derived assuming normal data, so the parametric test’s cost can be shown alongside.

SD of the changes $\sigma_d$ Optional

Values: Above 0

Given with the difference.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

One- or two-sided $\tau$

Values: 2 (two-sided) or 1 (one-sided)

Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Assumptions

Participants are independent; the outcome is continuous, so ties are rare. Many ties make these approximations optimistic.
Noether’s formulas are large-sample approximations and slightly conservative: they ask for a few percent more than a simulated trial needs [1].
The changes are symmetric about their centre.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

A probability of 0.65

p = 0.65: n = 7.849 / (3 × 0.15²) = 116.3, so 117 pairs.

Sample size: 117

Protocol statement
To detect a probability of 0.65 that the average of two within-pair changes is positive with 80% power at a two-sided significance level of 0.05, using a Wilcoxon signed-rank test, the required sample size is 117 pairs. Calculated with StatWisely (statwisely.com) using: n = [z_(1-α/2)+z_(1-β)]²/[3(p-½)²], p=P(Dᵢ+Dⱼ>0). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

From a standardised change of 0.5

Mean change 0.5 with SD 1: p = Φ(0.5 × √2) = 0.760, so 39 pairs, against 34 for a paired t-test.

Sample size: 39

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.

How to cite

StatWisely. Sample size for a Wilcoxon signed-rank test. Method document, engine version 1.0.0, reviewed 2026-10-02. https://statwisely.com/methods/nonparametric-signed-rank/

Cite the original references for the formulas your design uses as well.

References

1.Noether GE. Sample size determination for some common nonparametric tests. Journal of the American Statistical Association. 1987;82(398):645–647.