How sample size is calculated
The base sample size comes from Cochran's formula for estimating a proportion within a given margin of error at a chosen confidence level. If you have a known, finite population, a correction shrinks that number toward the population size.
- z — critical value for the confidence level (90% → 1.645, 95% → 1.96, 99% → 2.576)
- p — expected proportion of responses (use 50% if unknown — it maximizes required sample size)
- e — margin of error, as a decimal (5% → 0.05)
- N — population size; the correction is skipped when N is 0 (treated as unlimited)
More detail
Why 50% is the safest default for the response rate
p(1−p) is largest when p = 50%, so entering 50% when you don't know the true proportion gives the most conservative (largest) sample size — you won't undershoot. If you have prior data suggesting the real rate is, say, 20%, entering that shrinks the required sample because there's less variance to estimate.
When the finite population correction actually matters
For a population of 1,000 at 95% confidence and 5% margin of error, the uncorrected sample size is 385, but the corrected size drops to 278 — a population under roughly 20,000 is where the correction meaningfully reduces the number needed. Above that, the correction barely moves the result, since n0 stays small relative to N.
Margin of error drives the sample size the most. With the population left at 0 (unlimited), cutting the margin of error in half roughly quadruples the required sample size, since e is squared in the denominator. On that basis, tightening it from 5% to 2% takes 385 responses to 2,401 — roughly 6x. Enter a real population and the multiplier shrinks: at a population of 1,000 the same change goes from 278 to 707, only about 2.5x.
Frequently asked questions
What sample size do I need for a 95% confidence level with a 5% margin of error?
With p = 50% (the conservative default) and an unlimited population, n0 = 1.96² × 0.5 × 0.5 / 0.05² = 384.16, rounded up to 385. Enter these exact numbers above to confirm.
My population is only 1,000 people — do I still need 385 responses?
No — with a population of 1,000, the finite population correction applies: n = 384.16 / (1 + 383.16/1000) ≈ 277.7, rounded up to 278. Enter population size above to see the correction applied automatically.
Why does the calculator default the response rate to 50%?
p(1−p) is maximized at p = 50%, so it produces the largest (most conservative) required sample size when you have no prior estimate of the true proportion. If you have a reasonable estimate — say a past survey showed 20% — enter that instead to get a smaller, more targeted sample size.
How much does raising the confidence level from 95% to 99% change the required sample size?
With the population left at 0 (unlimited), the same 5% margin of error and p = 50%, 95% confidence needs 385 while 99% confidence (z = 2.576) needs 664 — about 1.72x, because a higher confidence level requires a larger z, which enters the formula squared. With a population of 1,000 the same step goes from 278 to 400, only about 1.44x.
Does this account for people who don't respond to my survey?
No — this is the number of completed, usable responses you need. If you expect a 40% response rate, you'd need to contact roughly 2.5x this many people to end up with enough completed responses.