How to use
Choose 90%, 95% or 99% confidence, enter the desired margin of error in percentage points, and enter the proportion you expect to observe. If there is no prior estimate, 0.5 is the conservative default because it gives the largest sample under this formula. Add a population size only when sampling without replacement from a known finite population.
The pale 95%, 5 and 0.5 example can be calculated immediately and gives 385. Focusing the first numeric field clears all example numbers together; your later entries are preserved. Calculate validates the current form. Editing any value clears the confirmed result and disables copying until you calculate again. Clear removes numeric values but keeps the confidence selection.
Calculation and reading the result
For an effectively unlimited population, n₀ = z²p(1−p)/e², where e is the margin expressed as a decimal. The page uses established normal critical values: 1.6448536 for 90%, 1.9599640 for 95%, and 2.5758293 for 99%. These constants and the displayed intermediate values are rounded for reading, while the calculation retains their precision.
For a finite population N, the correction is n = n₀ / [1 + (n₀−1)/N]. Required sample is this value rounded upward. Unadjusted sample shows the result before that correction; adjusted sample shows the corrected value before ceiling. A result of 384.15 therefore requires 385 completed responses, not 384.
Worked examples
With 95% confidence, a 5-point margin and p = 0.5, n₀ is about 384.15 and the required sample is 385. For a population of 1,000, the finite-population correction lowers the adjusted value to about 277.74, so 278 completed responses are required.
With 90% confidence, a 5-point margin and p = 0.5, the result is 271. Changing confidence to 99% raises it to 664. A prior estimate of p = 0.2 lowers variance relative to 0.5; p = 0.8 gives the same sample because p(1−p) is symmetric.
Assumptions and limits
This is a precision estimate for one population proportion under simple probability sampling. It does not repair a biased frame, nonresponse, convenience sampling, duplicate responses or poor question design. The calculated count is the number of usable completed observations; invite more people when response loss is expected.
The normal approximation may be unsuitable when the expected number of successes or failures is very small. Design effects from clustering, weighting, stratification and repeated measurements are not added here. The finite correction matters only when the sample is a meaningful share of the population and sampling is without replacement.
What this result does not mean
Confidence is a long-run property of the interval procedure, not a probability that one completed interval contains the true value. The stated margin applies near the assumed p and excludes measurement error and systematic bias.
Do not use this page as a clinical-trial, A/B-test or experimental power calculator. Those tasks need an effect size, test direction, allocation, significance level and power. This result also does not guarantee that a sample represents every subgroup.
Common questions
Why use 0.5? It maximizes p(1−p), so it is a cautious choice when no credible prior proportion exists. Why round up? A fraction of a response cannot supply the planned precision. Can I enter 5% as 0.05? No: this field uses percentage points, so enter 5.
Should population be the number invited? No. Use the complete finite population from which a probability sample is drawn. Can I copy the result? Yes; Copy result includes the required and intermediate values. It does not save or transmit the inputs.