How to use the Sample Size Calculator
- Set the margin of error you can live with. ±5% is the common survey standard, ±3% is newspaper-poll grade.
- Set the expected proportion; leave it at 50% when you have no prior estimate, since that is the most demanding case.
- Enter the population size if the group is small and countable, or leave it at 0 for a large population.
- Choose the confidence level.
- Use the returned figure as a target for completed responses, then divide by your expected response rate to size the invitation list.
How the calculation works
The formula inverts the margin-of-error expression for a proportion. Because the standard error carries √n in the denominator, sample size enters quadratically: the required n scales with the square of the confidence multiplier and with the inverse square of the margin. Tightening from ±5% to ±2.5% does not cost twice as many responses; it costs four times as many.
Setting the expected proportion to 50% is the conservative default because p(1 − p) is maximised there. If you have credible prior information that the true proportion is near 10% or 90%, using it cuts the required sample by nearly two thirds. Getting that prior wrong in the optimistic direction leaves you with a wider margin than you planned for, so only use it when the evidence is real.
The finite population correction matters when the sample is a meaningful fraction of the whole group. Surveying 400 people from a company of 500 is close to a census and the correction cuts the requirement sharply; surveying 400 from a city of a million changes nothing. This is why national polls in small and large countries need essentially the same number of respondents, a result that reliably surprises people.
n₀ = z²·p(1 − p) / e²; finite population: n = n₀ / (1 + (n₀ − 1)/N)Source: Cochran, Sampling Techniques, 3rd edition (1977), chapter 4; Krejcie & Morgan, 'Determining Sample Size for Research Activities', Educational and Psychological Measurement 30 (1970).
Worked example
A ±5% margin at 95% confidence, no prior estimate, population effectively unlimited.
- z = 1.96, p = 0.5, e = 0.05.
- n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 3.8416 × 0.25 ÷ 0.0025.
- n₀ = 0.9604 ÷ 0.0025 = 384.16, rounded up to 385.
- For a population of 5,000: 384.16 ÷ (1 + 383.16/5000) = 357.
385 completed responses for a large population, or 357 within a population of 5,000. The origin of the familiar 'about 400 people' rule of thumb.
Frequently asked questions
Why is 385 such a common answer?+
It is the exact result for ±5% at 95% confidence with p = 0.5 and an unlimited population, so it appears in almost every survey methodology note.
Does a bigger population need a bigger sample?+
Barely. Above about 20,000 the required sample is essentially flat, which is why national polls of 1,000 work everywhere.
Is this the number of invitations to send?+
No, it is the number of completed responses. Divide by your expected response rate, at 20%, needing 385 means inviting about 1,925 people.
What if I am measuring a mean rather than a proportion?+
Use n = (z·s/e)² with s as an estimated standard deviation and e as the margin in the same units.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.