Sample size for an exact binomial test

When to use it

A single-group study with few participants, or a proportion near 0 or 1, such as an early-phase trial’s response rate.

Not for

Large studies with a proportion away from 0 and 1, where the one-proportion test gives nearly the same answer more simply.

Notation

$n$: Number of participants
$X$: Number with the outcome, binomial(n, p)
$p_0,\ p_1$: Proportion under the null hypothesis and in truth
$R_n$: Rejection region: the counts that reject the null
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

Exact power

\Pr_{p_0}(X \in R_n) \le \alpha, \qquad \text{power}(n) = \Pr_{p_1}(X \in R_n) = \sum_{x \in R_n} \binom{n}{x} p_1^{x}(1-p_1)^{n-x}

The rejection region is enumerated from the binomial distribution, with α split equally between the two tails, so the power reported is exact rather than approximated.

Why the answer is not the first n that reaches the power

n^{*} = \min\{\,n : \text{power}(m) \ge 1-\beta \ \text{for every}\ m \text{ from } n \text{ to } n + 39\,\}

Because counts are whole numbers, power rises and falls as n grows: one more participant can shift the critical count and lose power. The calculator reports the size from which power stays above the target, and also the first size that reaches it [1]. The actual significance level is below α for the same reason.

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [2, 3].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [4].

All adjustments together

n_{\text{adj}} = \left\lceil n \mathrm{DE}\;F_t \right\rceil

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

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

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Reference proportion $p_0$

Values: Between 0 and 1

The value the study tests against, such as an unacceptable response rate.

Expected proportion $p_1$

Values: Between 0 and 1, different from p₀

The proportion worth detecting.

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.

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.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [5]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [3].

Expected loss of whole clusters $\ell$ Optional

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

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

Participants are independent apart from any clustering specified, with the same probability of the outcome.
The test is two-sided.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [2].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Seventy against fifty per cent

p₀ = 0.5, p₁ = 0.7, α = 0.05, 80% power: power first reaches 80% at n = 49 but dips below it again for some larger n, and stays above it from n = 54, where the actual significance level is 0.040.

Sample size: 54

Protocol statement
To detect a proportion of 70% against a reference of 50% with 80% power at a two-sided significance level of 0.05, using an exact binomial test, the required sample size is 54 participants. Calculated with StatWisely (statwisely.com) using: n: smallest n whose exact binomial rejection region reaches 1-β. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A rare outcome

p₀ = 0.05, p₁ = 0.15: first reached at n = 67, stable from n = 75.

Sample size: 75

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 an exact binomial test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/proportions-binomial/

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.
2.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
3.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
4.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
5.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013