Calculate Sample Size for a Population Proportion

State the estimation problem

Before calculating a sample size, specify what the sample must estimate. This guide considers one population proportion, such as the fraction of inspected items meeting a defined condition. The calculation plans an approximate two-sided confidence interval with a chosen absolute margin of error. It is different from planning a test to detect a difference between groups. A calculator cannot determine the target population, sampling frame or outcome definition from a desired confidence percentage alone.

Choose three planning inputs

Select a confidence level, an expected proportion p and an absolute margin e. For 95% confidence, the local form uses the rounded normal critical value z=1.96. An expected proportion of 50% means p=0.5, and a margin of five percentage points means e=0.05. The public form fixes the expected proportion at 50%; enter 95 for confidence and 5 for margin. It has no editable proportion field. A five-point margin around 50% describes approximately 45% to 55%, not a five-percent relative change.

Calculate and round the first target

For a very large population, the normal-approximation formula is n₀=z²p(1−p)/e². Substituting 1.96, 0.5 and 0.05 gives n₀=384.16. Round upward to 385 complete observations, because rounding down would fall below the calculated target. Leave the optional population field blank to use this large-population calculation. The target is an approximation based on the chosen model and inputs, not a guarantee that every realized interval will have exactly the desired width.

Compare a supported prior proportion

Suppose credible prior information suggests the proportion is closer to 20%. With the same 95% confidence and five-point margin, n₀=1.96²×0.2×0.8/0.05²=245.8624, rounded up to 246. This is a separate formula calculation: the public form fixes the expected proportion at 50% and cannot directly enter 20%. This smaller target relies on the planning value being relevant. If the true proportion is closer to 50%, the variability is larger. Record the source of the expected proportion rather than choosing a convenient value solely to reduce the sample requirement.

Apply a finite-population correction when justified

For a known finite population of N=1000 sampled without replacement, the form adjusts n₀ by n=n₀/[1+(n₀−1)/N]. With the first example's unrounded 384.16, the corrected value is about 277.7408 and the integer target is 278. Keep the unrounded n₀ through this step. The correction assumes the stated finite population and sampling design; it should not be applied merely because a preliminary contact list contains 1000 names while the target population is much larger or not well defined.

Distinguish complete observations from invitations

If 385 usable responses are needed and the anticipated usable-response rate is 80%, a simple invitation planning calculation is ceil(385/0.8)=482. This allowance is separate from the form's sample estimate. It can help plan workload but does not correct nonresponse bias if responders differ systematically from nonresponders. Use a response-rate assumption supported by the collection method and population, and define what counts as a complete usable observation before applying the allowance.

Identify design features outside the formula

Cluster sampling, stratification, unequal weights and subgroup reporting can change the required design and standard error. A design-effect adjustment may be appropriate in some settings, but it must reflect the intended estimator and sampling plan. Detecting an effect with a specified power requires additional inputs and another calculation. The local form cannot make those decisions. Document the simple-random-sample estimate as a starting component, then use a design-appropriate method for the actual study.

Retain enough information to reproduce the plan

Report the target proportion, confidence level, absolute precision, prior p, population assumption and rounding rule together with the resulting count. Record any separate design and response adjustments. After data collection, calculate an interval appropriate to the observed counts rather than treating the planning margin as the actual result. With very rare outcomes or small expected counts, a normal approximation can be inadequate; interval-based or exact planning should be considered instead of assuming that this generic formula covers every case.

Related guides

Sample Size When the Population Size Is Unknown

Plan a proportion sample without a finite population count, distinguish unknown N from unknown prevalence, and test precision and prior assumptions.

How Large Should a Sample Be for Useful Precision?

Compare precision and confidence targets, distinguish estimation from power, and plan subgroup, design and response needs instead of using a fixed rule.

Confidence Interval Formula for a Population Proportion

Compute a sample proportion, compare Wald and Wilson intervals, and handle zero successes without claiming zero population uncertainty.