Sample size calculator
How many respondents you need for a given accuracy.
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Plans the number of completed responses for estimating one proportion under simple random sampling. It reverses the normal confidence-interval approximation: confidence, a margin in percentage points and an anticipated proportion determine the sample size. With other settings fixed, p = 50% maximizes p·(1−p), making it conservative within this model. A known finite population N enables the correction for sampling without replacement. The calculation does not remove selection bias, nonresponse or measurement error, and is not a guarantee for every survey.
How it works
Formula and logic
n₀ = z²·p·(1−p)/e², where formula p and e are fractions, while fields use percentages and percentage points. For N > 0, n = N·n₀/(N−1+n₀); the final count is rounded upward. N is an integer from 0 to 9007199254740991: 0 means unknown or very large population, not its measured size. At N = 0 the sampling fraction is unavailable and shown as a dash. At p = 0 or 1 the formula formally gives zero: a degenerate model, not advice to obtain no responses. The normal approximation needs separate assessment near endpoint proportions.
Example
At 95% confidence, a 5-percentage-point margin and anticipated proportion 50%, the uncorrected n₀ ≈ 384.1459 rounds upward to 385 responses. For a known N = 500, the correction gives approximately 217.4872, hence 218 responses.
Fields and units
- Confidence level — list option
- Margin of error, % — unitless
- Expected proportion, % — unitless
- Population size (0 — infinite) — unitless
How to use
- — Margin of error is half the interval width: "±3 %" means 3.
- — If the expected proportion is unknown, 50% gives the largest size within this normal-approximation model for one proportion under simple random sampling. It does not guarantee adequacy for every study or remove selection bias.
- — Set population to zero when it is large or unknown — the correction then stays off.
- — The model assumes simple random sampling. Cluster, quota and other designs require a separate assessment of sampling, design effects and bias; there is no universal rule here that they need more people.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- Penn State STAT 506 §2.3: proportion sample size, finite correction and p = 0.5 OpenStax: normal approximation for one proportion, planning n and rounding upward
- Limitation
- Plans the number of completed responses for estimating one proportion under simple random sampling. The calculation does not remove selection bias, nonresponse or measurement error, and is not a guarantee for every survey.
FAQ
Why is the sample largest at 50 %?
p·(1−p) reaches its maximum 0.25 at p = 0.5. At p = 0.1 or 0.9 it is 0.09: without the finite correction, the unrounded sample size is 36% of the former size, not exactly one third. At 95% and a five-point margin, for example, 139 responses replace 385. This is conservatism within the model, not protection from survey bias.
Why does the sample barely depend on city size?
N is absent from the uncorrected formula. When a sample is appreciable relative to a known N and sampling is without replacement, the correction reduces the count: the N = 500 example gives 218 rather than 385. There is no universal exact threshold where city size suddenly starts to matter.
What does moving from 95 % to 99 % cost?
Without the finite correction and with p and e fixed, size scales with z²: moving from 95% to 99% multiplies the unrounded size by about 1.727. Halving the margin multiplies it by 4, a larger increase. With a 50% proportion and five-point margin, the rounded counts are 385 and 664.
Does this cover A/B tests?
No: this plans precision for one proportion. Comparing two variants requires an effect size relevant to the question, a significance level, statistical power and allocation between groups. Simply doubling this result is not an A/B power calculation.