Quick answer: For 95% confidence and a ±5% margin with p = 50%, n = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.15, so you need 385 responses; from a population of 10,000 the correction brings it to 370.
How it's calculated
Cochran's formula for a proportion: n₀ = z²·p(1 − p) ÷ e², where z is the critical value for the confidence level (1.96 for 95%, 2.5758 for 99%; some calculators round it to 2.58, which gives a slightly larger sample), p the proportion you expect and e the margin of error, both as fractions. The result is always rounded up.
p(1 − p) is largest at p = 50%, so 50% gives the safest (largest) sample when you have no earlier estimate.
If the population is small, the finite population correction n = n₀ ÷ (1 + (n₀ − 1) ÷ N) lowers the number. For large populations it barely changes anything, which is why a national poll and a city poll need similar sample sizes. After collecting data, compute the actual confidence interval.
Worked examples
- 95%, ±5%, p = 50%: 384.15 → 385
- Same, population 10,000: 384.15 ÷ (1 + 383.15 ÷ 10,000) = 369.97 → 370
- 99%, ±3%, p = 50%: 2.5758² × 0.25 ÷ 0.03² = 1,843.03 → 1,844
Frequently asked questions
Why do people quote 385 or 384 for 95% and 5%?
Using z = 1.96 exactly gives 384.16, and with the precise z = 1.959964 it is 384.15. Either way you round up to 385 complete responses.
Does a bigger population need a bigger sample?
Hardly. Beyond a few tens of thousands the required sample is almost the same, because precision depends on the sample size, not the population size.
Is this for means as well?
This calculator is for percentages (proportions). For a mean, the formula is n = (z·σ ÷ e)², which needs an estimate of the standard deviation σ.
How many responses do I need for a 3% margin of error?
About 1,068 at 95% confidence, for a large population. Halving the margin of error roughly quadruples the sample you need.
What if most people don't reply?
Invite more. If you need 385 answers and expect one in five people to respond, send it to about 1,925.
How many people do I need from a company of 500?
About 218 for 95% confidence and ±5%. Enter the population size and the number comes down from the usual 385.