Sample Size for Estimating a Proportion

Finds how many responses are needed to measure a proportion to a given precision. The sample size is z² × p(1 − p) ÷ margin², which peaks at p = 50%, so 50% is the safe choice when there is no prior estimate.

Surveys that measure a proportion — a support rate, an audience share, a defect rate — need a sample size worked out the same way as for a mean: set the confidence interval no wider than the margin of error and solve.

The spread of a proportion is governed by $p(1 - p)$ , which takes the place of the standard deviation.

n = \dfrac{z^{2} \, p(1 - p)}{E^{2}}

$n$ is the required sample size
$z$ is the critical value for the confidence level, 1.96 at 95%
$p$ is the expected proportion
$E$ is the margin of error

$p(1 - p)$ peaks at 0.25 when $p = 0.5$ and shrinks towards 0 as the proportion approaches either extreme. A near-even split therefore demands the most data. Turned around: with no idea what the proportion will be, computing at 50% guarantees enough whatever the answer turns out to be.

Example

Take the defaults: an expected proportion of 30%, a margin of error of 3%, and 95% confidence.

$1.96^{2}$ is about 3.8416, $0.3 \times 0.7$ is 0.21, and $0.03^{2}$ is 0.0009. Then $3.8416 \times 0.21 \div 0.0009$ comes to 896.4, rounded up to 897 respondents.

Assuming nothing and using 50% instead gives $3.8416 \times 0.25 \div 0.0009$ , or 1067.1, rounded up to 1068. This is why opinion polls so often sample a little over a thousand people: 1068 responses deliver a 3% margin at 95% confidence no matter what the true proportion is.

Population size does not appear

There is no population size in the formula. A town of ten thousand voters and a nation of a hundred million both need the same 1068 responses for the same precision.

That feels wrong, but it is correct. Tasting a pot of soup does not require more soup because the pot is bigger; one spoonful suffices provided it is stirred. What matters in sampling is the number of observations and whether they were drawn without bias, not what fraction of the whole they represent.

Points to watch

Random selection is assumed. This formula only says how many to collect. If the collection is biased, 1068 responses and 100,000 responses are equally wrong. How the sample is chosen matters far more than how large it is.

Small populations are the one exception: if the sample would exceed about 5% of the population, a finite population correction can reduce it.

The count is of responses received, not questionnaires sent.