PUBLIC LIBRARY / CALCULATOR

Sample size for comparing two proportions

Estimates participants needed per group for a comparative trial or cohort to detect the expected difference between two proportions.

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Check the population, units and version. The result shows the formula or classification; it does not make an automatic treatment decision.

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Method · Limits of application

Independent cohorts, Charan/Biswas 2013 p. 123, m=1; normal quantiles to six decimal places; no continuity correction; WHO 1991 not directly checked

This approximation calculates sample size per group for two independent binary proportions, with equal allocation and a two-sided test. Specify a clinically relevant difference, expected proportions, significance level and power; do not treat the desired difference as a known result. It does not incorporate matching, clusters, repeated measurements or unequal allocation, which need their own methods. Allowing for losses increases recruitment but does not correct bias or an unsuitable design. The full WHO 1991 manual has not been checked. The equation corresponds to the independent-cohort form in Charan and Biswas (2013, p. 123), with m = 1; the trial form using entirely pooled variance on p. 124 is different. The normal quantiles were checked mathematically and rounded to six decimal places; this does not validate the choice of study design. The WHO 1991 citation remains without direct verification of the manual.

Conditions of use

Check the population, units, inclusion and exclusion criteria, and version in the original source. A result alone does not establish a diagnosis, discharge decision or prescription.

Documented parameters

Expected proportion in group 1 (e.g., control) · %
Expected proportion in group 2 (e.g., intervention) · %
Significance level (two-sided)
Statistical power
Expected losses (optional) · %

3/3 reference cases checked. Numerical tests are not clinical validation.

Sources and documentation

Documented formula

Transcribed from the source or a recorded revision; this is not a clinical recommendation.

n per group = [z_α/2 × √(2 p̄ q̄) + z_β × √(p₁q₁ + p₂q₂)]² / (p₁ − p₂)²; p̄ = (p₁ + p₂)/2; q = 1 − p.

The implementation uses coefficients with six decimal places: z_α/2 = 1.959964 (α 5%) or 2.575829 (α 1%); z_β = 0.841621 (power 80%) or 1.281552 (power 90%). Equal-sized groups, without continuity correction.

The equation corresponds to the independent-cohort form in Charan and Biswas (2013, p. 123), with m = 1; the trial form using entirely pooled variance on p. 124 is different. The normal quantiles were checked mathematically and rounded to six decimal places; this does not validate the choice of study design. The WHO 1991 citation remains without direct verification of the manual.

References

Identifier: tamanho-amostral-duas-proporcoes

Original clinical content: Brazilian Portuguese.

Method version

Independent cohorts, Charan/Biswas 2013 p. 123, m=1; normal quantiles to six decimal places; no continuity correction; WHO 1991 not directly checked

tamanho-amostral-duas-proporcoes@native-2026-10-05+fc378cb3d872

Source for the active edition

The bibliographic edition identifies the source. By itself, it proves neither full conformity of the formula nor clinical approval.

Synthetic cases checked (3)

Synthetic cases for checking the implementation, with no patient data.

Case 1

reference-001

Inputs

Expected proportion in group 1 (e.g., control): 20 %
Expected proportion in group 2 (e.g., intervention): 10 %
Significance level (two-sided): α = 5%
Statistical power: 80%

{
  "p1": 20,
  "p2": 10,
  "alfa": "5",
  "poder": "80"
}

Expected result

{
  "n": 198.97,
  "perGroup": 199,
  "total": 398
}

Actual result

{
  "n": 198.96338254023922,
  "perGroup": 199,
  "total": 398
}

Case 2

reference-002

Inputs

Expected proportion in group 1 (e.g., control): 20 %
Expected proportion in group 2 (e.g., intervention): 10 %
Significance level (two-sided): α = 5%
Statistical power: 90%

{
  "p1": 20,
  "p2": 10,
  "alfa": "5",
  "poder": "90"
}

Expected result

{
  "n": 265.87,
  "perGroup": 266
}

Actual result

{
  "n": 265.8560593204369,
  "perGroup": 266,
  "total": 532
}

Case 3

reference-003

Inputs

Expected proportion in group 1 (e.g., control): 30 %
Expected proportion in group 2 (e.g., intervention): 30 %
Significance level (two-sided): α = 5%
Statistical power: 80%

{
  "p1": 30,
  "p2": 30,
  "alfa": "5",
  "poder": "80"
}

Expected result

{
  "error": true
}

Code and reproducibility

This tool’s code on GitHub

Download code and tests

Sources, version and reproduction instructions

README

Local installation

Code license: Apache-2.0

Guidelines and conditions

This repository’s license applies to the code. Third-party instruments, questionnaires, data and models remain subject to their owners’ conditions.

The code tests are technical checks. Clinical approval and professional translation review are recorded separately.

Relationship with cancer research

Support for research — context required

Statistical planning to compare proportions. The study must justify the expected effect, power, losses and appropriateness of the design.

Applications: Clinical trials and methods · Epidemiology