For a single prevalence estimate, Cochran's formula converts confidence, expected prevalence and absolute precision into a base sample. The correct target also depends on sampling design, population size, non-response and whether prevalence is truly the primary objective.
1. Use the single-proportion formula for the right question
For a large population, n₀ = z²p(1−p)/e², where z is the confidence-level critical value, p is expected prevalence and e is the desired absolute margin of error. At 95% confidence, 5 percentage-point precision and p = 0.50, the base is 384.16 and is rounded up to 385.
The formula estimates one proportion. Comparing groups, testing a hypothesis, estimating a mean, fitting a regression or accounting for repeated measures needs a different design-specific calculation.
2. Justify confidence, precision and prevalence
A narrower margin of error increases the required sample sharply because precision is squared in the denominator. Use an expected prevalence from relevant prior evidence when credible; 50% gives the largest variance and is conservative when no estimate is available.
Do not select inputs merely because they produce an affordable answer. If resources cannot support the desired precision, change the design or state the achievable precision transparently.
3. Apply design and population adjustments in a stated order
For clustered sampling, multiply the base by a defensible design effect. For a known finite population, the adjusted value can use n = n₀/[1 + (n₀−1)/N]. State whether the finite-population correction was applied before or after the design effect so another reviewer can reproduce the number.
Allow for non-response by dividing the analysis target by the expected response proportion. Adding ten percent and dividing by 0.90 are not identical; the latter targets a ten-percent non-response rate directly.
- Primary outcome and formula
- Confidence level and precision
- Expected prevalence source
- Sampling design and design effect
- Finite population and non-response
4. Work through the recruitment target
Suppose the base is 385, the design effect is 1.5, the eligible population is 5,000 and expected non-response is 10%. The design-adjusted value is 577.5; applying the stated finite-population correction gives about 518; dividing by 0.90 gives a recruitment target of about 576 after rounding up.
The final protocol should preserve the unrounded calculation, state the rounding convention and distinguish the number to recruit from the number required for analysis.
5. Review what the calculator cannot decide
A sample-size number does not validate the sampling frame, instrument, outcome definition or planned analysis. Complex surveys may need strata, unequal weights, cluster counts and intraclass correlation considered explicitly.
Ask for biostatistical review when the study has multiple primary outcomes, subgroup claims, small clusters, rare outcomes or a model-based objective.
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