Sample size for comparing two proportions

When to use it

A randomised trial or comparative study with two independent groups and a yes/no outcome.

Not for

The same people measured twice, or matched pairs. Use McNemar’s test.
Showing that a new treatment is not worse. Use non-inferiority for proportions.

Notation

$n$: Participants per group
$p_1,\ p_2$: Expected proportions in groups 1 and 2
$\bar{p}$: Average of the two proportions, (p₁ + p₂) / 2
$\delta$: Difference between the proportions, |p₁ − p₂|
$\alpha$: 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
$d_t,\ d_c$: Share of the treatment arm not taking treatment, and of the control arm receiving it
$q$: Expected dropout, overall or for each arm (q₁, q₂)

Formulas

Fleiss’s normal approximation

n = \left\lceil \frac{\left[z_{1-\alpha/2}\sqrt{2\bar{p}(1-\bar{p})} + z_{1-\beta}\sqrt{p_1(1-p_1) + p_2(1-p_2)}\right]^{2}}{\delta^{2}} \right\rceil

The null variance uses the pooled proportion p̄, as the test does; the alternative uses each group’s own [1]. The calculator uses equal groups and a two-sided test.

With the continuity correction

n_{cc} = \left\lceil \frac{n}{4}\left(1 + \sqrt{1 + \frac{4}{n\,\delta}}\right)^{2} \right\rceil

Optional, and on by default. It matches the analysis when that will be a continuity-corrected chi-square or Fisher’s exact test, and adds participants [2, 3].

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 [4, 5].

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 [6].

Non-adherence and crossover

n' = \frac{n}{(1 - d_t - d_c)^{2}}

An intention-to-treat analysis keeps everyone in the arm they were allocated to, so participants who do not receive their allocated treatment dilute the difference. Lachin’s correction restores the power [7].

All adjustments together

n_{\text{adj}} = \left\lceil n \mathrm{DE}\;F_t\,\frac{1}{(1 - d_t - d_c)^{2}} \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},i} = \left\lceil \frac{n_{\text{adj}}}{1 - q_i} \right\rceil, \qquad i = 1, 2

Applied last, to the participants actually recruited. The rate can differ between the arms; when it does not, q₁ = q₂ = q.

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 in group 1 $p_1$

Values: Between 0 and 1

Usually the control group, from routine data or previous studies.

Expected proportion in group 2 $p_2$

Values: Between 0 and 1, different from p₁

The proportion that would make the difference worth detecting; not an optimistic hope.

Continuity correction $cc$ Optional

Values: On or off

Turn it off only if the analysis will be an uncorrected chi-square or z test.

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.

Dropout in each arm $q_1,\ q_2$ Optional

Values: 0% up to, but not including, 100%, for each arm

When dropout is expected to differ between arms, for example higher with an unpleasant treatment. Each arm is then inflated by its own rate.

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 [8]; 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 [5].

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.

Non-adherence and crossover $d_t,\ d_c$ Optional

Values: 0% up to, but not including, 100%; the two together below 100%

The share of the treatment arm expected not to take it, and of the control arm expected to receive it, from comparable trials.

Assumptions

The groups are independent and equal in size, and participants are independent apart from any clustering specified.
Each group expects comfortably more than 10 participants with and without the outcome, 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 [4].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.
Non-adherence is corrected for an intention-to-treat analysis, with treatment received having its full effect and no effect otherwise.

Worked examples

Thirty against forty-five per cent

p₁ = 0.3, p₂ = 0.45, so p̄ = 0.375 and δ = 0.15; without the continuity correction: n = [1.96 × 0.685 + 0.8416 × 0.676]² / 0.15² = 162.4, rounded up to 163 per group.

Group 1: 163 · Group 2: 163 · Total: 326

Protocol statement
To detect a difference between proportions of 30% and 45% with 80% power at a two-sided significance level of 0.05, using a comparison of two proportions, the required sample size is 163 per group (326 in total). Calculated with StatWisely (statwisely.com) using: Fleiss normal approximation; optional Casagrande-Pike-Smith continuity correction. These are planning estimates; confirm the final design with a qualified statistician.

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With the continuity correction

The same comparison corrected for continuity: 176 per group.

Group 1: 176 · Group 2: 176 · Total: 352

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Allowing for non-adherence

With 10% of the treatment arm not taking it and 5% of controls receiving it: 163 / 0.85² = 225.6, so 226 per group.

Group 1: 226 · Group 2: 226 · Total: 452

Protocol statement
To detect a difference between proportions of 30% and 45% with 80% power at a two-sided significance level of 0.05, using a comparison of two proportions, the test itself needs 163 per group. With non-adherence of 10% in the treatment arm and crossover of 5% in the control arm, the required sample size is 226 per group (452 in total). Calculated with StatWisely (statwisely.com) using: Fleiss normal approximation; optional Casagrande-Pike-Smith continuity correction; non-adherence: n / (1 - d_t - d_c)^2. These are planning estimates; confirm the final design with a qualified statistician.

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A cluster-randomised trial

Clusters of 20 with ICC 0.02: DE = 1.38, so 163 × 1.38 = 224.9, rounded up to 225 per group, in 12 clusters per group.

Group 1: 225 · Group 2: 225 · Total: 450

Protocol statement
To detect a difference between proportions of 30% and 45% with 80% power at a two-sided significance level of 0.05, using a comparison of two proportions, the test itself needs 163 per group. With clustering (ICC 0.02, an average of 20 per cluster; design effect 1.38), the required sample size is 225 per group (450 in total), in 12 clusters per group. Calculated with StatWisely (statwisely.com) using: Fleiss normal approximation; optional Casagrande-Pike-Smith continuity correction; 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 comparing two proportions. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/proportions-two/

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.Casagrande JT, Pike MC, Smith PG. An improved approximate formula for calculating sample sizes for comparing two binomial distributions. Biometrics. 1978;34(3):483–486.
3.Fleiss JL, Tytun A, Ury HK. A simple approximation for calculating sample sizes for comparing independent proportions. Biometrics. 1980;36(2):343–346.
4.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
5.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
6.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
7.Lachin JM. Biostatistical Methods: The Assessment of Relative Risks. New York: John Wiley & Sons; 2000.
8.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

Fleiss’s normal approximation

With the continuity correction

Clustering

Few clusters

Non-adherence and crossover

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected proportion in group 1p1p_1p1​

Expected proportion in group 2p2p_2p2​

Continuity correctionccccccOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Dropout in each armq1, q2q_1,\ q_2q1​, q2​Optional

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

Non-adherence and crossoverdt, dcd_t,\ d_cdt​, dc​Optional

Assumptions

Worked examples

Thirty against forty-five per cent

With the continuity correction

Allowing for non-adherence

A cluster-randomised trial

Validation

How to cite

References

Expected proportion in group 1 $p_1$

Expected proportion in group 2 $p_2$

Continuity correction $cc$ Optional

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional

Dropout in each arm $q_1,\ q_2$ 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

Non-adherence and crossover $d_t,\ d_c$ Optional