Sample size for a sign test

When to use it

Paired outcomes recorded only as better or worse, or changes too irregular for any other test.

Not for

Changes measured on a scale: the signed-rank test uses their sizes and needs fewer pairs.

Notation

$n$: Number of pairs with a non-zero change
$p$: P(D > 0): chance a change is positive
$\alpha$: 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

Exact binomial power

\text{power}(n) = \Pr_{p}(X \le c - 1) + \Pr_{p}(X \ge n - c + 1), \qquad c = F^{-1}_{\mathrm{Bin}(n,\,1/2)}(\alpha/\tau)

The number of positive changes is binomial; under no effect p = ½. Because counts are whole numbers, power rises and falls with n, and the size reported is the one from which it stays above the target [1].

Dropout

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

Applied last, to the participants actually recruited.

Parameters

P(D > 0) $p$

Values: Between 0 and 1, not 0.5

The share of pairs expected to improve, among those that change.

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.

Standard deviation $\sigma$ 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

Pairs are independent; pairs with no change are excluded, so allow for them in recruitment.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Sixty-five per cent improve

p = 0.65, α = 0.05 two-sided, 80% power: power first reaches 80% at 90 pairs, and stays above it from 97.

Sample size: 97

Protocol statement
To detect a probability of 0.65 that a change is positive with 80% power at a two-sided significance level of 0.05, using an exact sign test, the required sample size is 97 pairs. Calculated with StatWisely (statwisely.com) using: n: exact binomial power of a sign test at p=P(D>0). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for dropout

Expecting 10% to have no usable pair: 108 are enrolled.

Needed for analysis: 97 · To recruit: 108

Protocol statement
To detect a probability of 0.65 that a change is positive with 80% power at a two-sided significance level of 0.05, using an exact sign test, the required sample size is 97 pairs. Allowing for 10% dropout, 108 pairs should be recruited. Calculated with StatWisely (statwisely.com) using: n: exact binomial power of a sign test at p=P(D>0). These are planning estimates; confirm the final design with a qualified statistician.

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 sign test. Method document, engine version 1.0.0, reviewed 2026-10-02. https://statwisely.com/methods/nonparametric-sign/

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

References

1.Chernick MR, Liu CY. The saw-toothed behavior of power versus sample size and software solutions: single binomial proportion using exact methods. The American Statistician. 2002;56(2):149–155.