Last updated: 2026-06-23

Sample Size (Proportion)

Before a survey or poll is fielded, one question has to be answered: how many people need to respond? Collect too few and the result carries a margin of error so wide it cannot distinguish between meaningfully different answers; collect too many and time and budget are wasted on precision no one needs. When the quantity being estimated is a proportion — the share who approve, who would buy, who answer "yes" — the required sample size follows directly from three choices: how confident you want to be, how precise you want to be, and a rough guess at the proportion itself.

The formula

For estimating a population proportion, the sample size needed to hit a target margin of error is

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

Each term plays a distinct role:

• $z_{\alpha/2}$ is the critical value from the standard normal distribution for the chosen confidence level. It is 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%. Higher confidence means a larger critical value, and therefore a larger sample.
• $p$ is the expected (assumed) proportion you are estimating, written as a fraction between zero and one. The factor $p\,(1 - p)$ is the variance of a single yes/no response.
• $E$ is the margin of error — the half-width of the confidence interval, the $\pm$ precision you want around the estimate. A smaller margin sits in the denominator squared, so halving the margin of error roughly quadruples the sample.

Why 50% is the safe default

The term $p\,(1 - p)$ is what makes the proportion matter, and it behaves in a convenient way. It is zero when the proportion is 0 and when it is 1, and it reaches its maximum of 0.25 exactly when the proportion is 0.5. In other words, a proportion near one half is the hardest to estimate — responses are maximally variable — and a proportion near the extremes is easier.

If you already have a credible estimate of the proportion from a pilot study or past research, use it: a value away from 0.5 yields a smaller, cheaper sample. But when you have no reliable prior, assuming a proportion of 0.5 is the conservative choice. It maximizes $p\,(1 - p)$, produces the largest required sample, and therefore guarantees your margin of error will be met no matter what the true proportion turns out to be.

Worked example

Suppose you want to estimate a population proportion at 95% confidence with a margin of error of 5 percentage points, and you have no prior estimate, so you assume a proportion of 0.5. With a critical value of 1.96:

Rounding up gives 385 respondents — the familiar figure behind countless "±3%" and "±5%" polls. Demanding 99% confidence instead raises the critical value to 2.576 and pushes the requirement to 664. If a pilot suggested the true proportion were closer to 30%, then $p\,(1 - p)$ equals 0.21 instead of 0.25, and at 95% confidence the requirement falls to 323.

Finite population correction

The formula above assumes the population is effectively infinite, which is a good approximation when your sample is a small slice of a large group. When the sample would be a large fraction of a small, known population $N$ — a common rule of thumb is more than about 5% of it — apply the finite population correction:

$n_{\text{adj}} = \dfrac{n}{1 + \dfrac{n - 1}{N}}$

This always lowers the requirement, sometimes substantially. Surveying 385 people out of a town of 2,000, for instance, the correction trims the needed sample to roughly 323. For large populations the adjustment is negligible and can be ignored.

Rounding up

Sample size must be a whole number of respondents, and the formula almost never lands on an integer. Always round up rather than to the nearest value: rounding down would leave the achieved margin of error slightly wider than you specified, while rounding up guarantees the precision you asked for. The same logic governs the companion Sample Size Calculator for a Mean calculator, which sizes a study for estimating an average instead of a proportion, and the Margin of Error Calculator calculator, which runs the relationship in reverse to report the precision a given sample delivers.