Sample size for a one-proportion test

When to use it

Comparing an observed proportion with a fixed benchmark, such as a target coverage or a historical response rate.

Not for

Small samples or proportions near 0 or 1, where the normal approximation is poor. Use the exact binomial test.
Estimating the proportion rather than testing it. Use the prevalence under “Estimate precisely”.

Notation

$n$: Number of participants
$p_0$: Reference proportion under the null hypothesis
$p_1$: Expected true proportion
$\alpha$: 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

Normal approximation

n = \left\lceil \frac{\left[z_{1-\alpha/2}\sqrt{p_0(1-p_0)} + z_{1-\beta}\sqrt{p_1(1-p_1)}\right]^{2}}{(p_1 - p_0)^{2}} \right\rceil

The test statistic uses the null variance p₀(1 − p₀), while its spread under the alternative uses p₁(1 − p₁), which is why the two appear separately [1]. The calculator tests two-sided.

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

Expected proportion $p_1$

Values: Between 0 and 1

The proportion you expect to find, from pilot data or the literature.

Reference proportion $p_0$

Values: Between 0 and 1, different from p₁

The benchmark the study tests against.

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 a random sample, independent apart from any clustering specified.
n·p and n·(1 − p) are comfortably above 10 under both hypotheses, so the normal approximation holds.
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

Above a benchmark of one half

Expected 0.65 against a reference of 0.5: n = [1.96 × 0.5 + 0.8416 × 0.477]² / 0.15² = 84.8, rounded up to 85.

Sample size: 85

Protocol statement
To detect a proportion of 65% against a reference of 50% with 80% power at a two-sided significance level of 0.05, using a one-sample test of a proportion, the required sample size is 85 participants. Calculated with StatWisely (statwisely.com) using: Normal approximation: n ≈ [z_(1-α/tails)√(p₀q₀) + z_(1-β)√(p₁q₁)]² / (p₁-p₀)²; qᵢ=1-pᵢ. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for dropout

Expecting 10% to provide no outcome: 85 / 0.9 = 94.4, so 95 are recruited.

Needed for analysis: 85 · To recruit: 95

Protocol statement
To detect a proportion of 65% against a reference of 50% with 80% power at a two-sided significance level of 0.05, using a one-sample test of a proportion, the required sample size is 85 participants. Allowing for 10% dropout, 95 participants should be recruited. Calculated with StatWisely (statwisely.com) using: Normal approximation: n ≈ [z_(1-α/tails)√(p₀q₀) + z_(1-β)√(p₁q₁)]² / (p₁-p₀)²; qᵢ=1-pᵢ. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Sampling in clusters

Clusters of 20 with ICC 0.02: DE = 1.38, so 85 × 1.38 = 117.3, rounded up to 118, in 6 clusters.

Sample size: 118

Protocol statement
To detect a proportion of 65% against a reference of 50% with 80% power at a two-sided significance level of 0.05, using a one-sample test of a proportion, the test itself needs 85 participants. With clustering (ICC 0.02, an average of 20 per cluster; design effect 1.38), the required sample size is 118 participants, in 6 clusters. Calculated with StatWisely (statwisely.com) using: Normal approximation: n ≈ [z_(1-α/tails)√(p₀q₀) + z_(1-β)√(p₁q₁)]² / (p₁-p₀)²; qᵢ=1-pᵢ; clustered: n x DE, DE = 1 + (m-1)ρ. 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 one-proportion test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/proportions-one/

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

References

1.Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2003.
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

When to use it

Not for

Notation

Formulas

Normal approximation

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected proportionp1p_1p1​

Reference proportionp0p_0p0​

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Average cluster sizemmmOptional

Variation in cluster sizeCV\mathrm{CV}CVOptional

Expected loss of whole clustersℓ\ellℓOptional

Correction for few clustersFtF_tFt​Optional

Assumptions

Worked examples

Above a benchmark of one half

Allowing for dropout

Sampling in clusters

Validation

How to cite

References

Expected proportion $p_1$

Reference proportion $p_0$

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Average cluster size $m$ Optional

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

Expected loss of whole clusters $\ell$ Optional

Correction for few clusters $F_t$ Optional