Sample Size for a Proportion Calculator

Enter the anticipated proportion as a decimal, the largest acceptable margin of error, and a two-sided confidence level. The result uses the standard normal approximation and always rounds upward, because a fractional participant or observation cannot satisfy the target precision. It is useful for planning surveys, polls, audits, experiments, and quality checks before collection begins.

Choose inputs that match the study question

Start by expressing every input as a decimal. A five-percentage-point margin is entered as 0.05, and a 95 percent confidence level is entered as 0.95. The estimated proportion is your best advance estimate of the share expected to have the characteristic of interest. It may come from a pilot study, a previous survey, operational records, or a closely comparable population. If no credible estimate exists, using 0.5 is the conservative conventional choice because the product p multiplied by one minus p is largest at 0.5. That choice therefore produces the largest sample under this formula for a fixed margin and confidence. Define the outcome before choosing the estimate: the proportion of customers who renew is not interchangeable with the proportion who recommend a product. Inputs should describe one binary outcome and one intended population. The calculator expects independent observations and a margin measured on the proportion scale, not a relative percentage of the expected result.

Understand the calculation and upward rounding

The calculator first converts the two-sided confidence level into its standard normal critical value, commonly called the z score. It then evaluates z squared times the estimated proportion times one minus that proportion, divided by the squared margin of error. This is the familiar planning formula for a proportion under the normal approximation. The computed quantity is rounded up to the next whole observation. Upward rounding matters: rounding to the nearest integer could leave the planned study just below its stated precision. The returned z score is shown for transparency, while the sample size is calculated from the unrounded internal value. The result is a minimum under the assumptions of the formula, not a promise that every realized sample will have exactly that error. Actual uncertainty depends on the observed proportion and the sampling process. The calculation also does not add allowances for nonresponse, invalid records, attrition, clustering, weighting, or repeated measurements; those design effects belong in the study plan after the statistical minimum is found.

Interpret the result and account for study design

Treat the returned sample size as the number of usable, independent responses or observations required for analysis. If only 80 percent of invited participants are expected to respond, divide the required usable count by 0.8 and round up to estimate how many invitations to send. Clustered samples, such as students within schools or patients within clinics, usually need an additional design-effect adjustment because observations inside a cluster are correlated. A known finite population can sometimes justify a finite population correction, but this calculator intentionally reports the standard large-population result so its assumptions remain clear and portable. Extremely rare or extremely common outcomes may also require methods tailored to interval coverage rather than this normal planning approximation. Before collecting data, document the chosen margin, confidence, anticipated proportion, expected response rate, and any design effect. That record makes the result reproducible and prevents a later change in assumptions from appearing to be a change in arithmetic. For automated planning, the API charge is $0.002 per request and returns the same deterministic calculation.