What determines how many responses you need
Four things set the sample size, and population size is the least important of them.
The precision you want. The margin of error appears squared in the denominator, so precision is expensive: halving the margin from 5 points to 2.5 quadruples the sample, from 385 to 1,537. This single relationship explains why almost every national poll lands between 1,000 and 1,500 respondents — that is where the marginal cost per point of precision starts climbing steeply.
The confidence level. Moving from 95% to 99% multiplies z from 1.959964 to 2.575829, and because z is squared, it multiplies the sample by (2.575829/1.959964)² = 1.727. Going the other way, 90% confidence needs (1.644854/1.959964)² = 0.704 of the 95% sample.
How variable the answer is. For a percentage this is captured by p(1−p), which peaks at 0.25 when p = 50%. If you already know the answer is near 10%, the variance term drops to 0.09 and you need 0.09/0.25 = 36% of the sample. If you have no idea, assume 50% and you can never be caught short.
The population size. Only through the finite population correction, and only when your sample is a meaningful fraction of it. For a country, it makes no difference at all. For a 400-person company, it makes an enormous one.
Cochran's formula and the population adjustment
Start from the margin of error you want to be able to quote: e = z√(p(1−p)/n). Everything except n is something you choose or estimate, so solve for n and you get Cochran's formula, n₀ = z²p(1−p)/e². That is the whole derivation. The subscript zero marks it as the answer for an effectively infinite population.
When the population is finite, the standard error carries the correction factor √((N−n)/(N−1)), and solving the same equation with that factor included gives n = n₀N/(N + n₀ − 1), which is more commonly written n₀/(1 + (n₀−1)/N). Notice what it does: the adjusted n can never reach N, and it collapses towards n₀ whenever N is much larger than n₀. At N = 10,000 with n₀ = 385, the adjustment saves you 15 responses. At N = 500 it saves you 167, bringing the requirement down to 218.
Response rate enters last, and it is arithmetic rather than statistics. If you need 385 completes and you historically get 30% of invitations returned, send ⌈385/0.30⌉ = 1,284 invitations. The response rate does not change the sample size; it changes the effort required to reach it. What it does change is your exposure to non-response bias, which is a separate and usually larger problem — see the margin of error calculator for what the margin does and does not cover.
One consequence people find surprising: because n appears alone in the formula and N only appears in the correction, a sample of 1,000 delivers essentially the same precision for a city of 200,000 as for a nation of 200 million. Precision is bought with absolute sample size, not with a percentage of the population.
Worked example: an 8,500-employee engagement survey
You are running an engagement survey across 8,500 employees. You want to report percentages to within 5 points at 95% confidence, you have no prior estimate so you assume 50%, and your last three surveys returned about 35% of invitations.
- Critical value. At 95%, z = 1.959964, so z² = 3.841459.
- Cochran n₀. 3.841459 × 0.5 × 0.5 / 0.05² = 0.960365 / 0.0025 = 384.146.
- Finite population adjustment. n = 384.146 × 8,500 / (8,500 + 384.146 − 1) = 3,265,241 / 8,883.146 = 367.58, so you need 368 completed responses.
- Invitations. 368 / 0.35 = 1,051.4, so invite 1,052 employees.
- Margin you actually get. √(0.25/368) = 0.0260643, the correction is √(8,132/8,499) = 0.978171, so E = 1.959964 × 0.0260643 × 0.978171 = 0.04997, or 5.00 points — just inside target, as rounding up guarantees.
Now sanity-check the subgroups you plan to report. If you intend to break results out by five departments and expect roughly even representation, each department cell holds about 368/5 = 74 responses, whose margin of error is 1.959964 × √(0.25/74) × 100 = 11.4 points. If departmental comparison is the point of the survey, you need to size for the smallest cell you intend to report, not for the total.
How to use the number you get
Treat it as a floor on completed responses, not on invitations. The sample size formula counts people who actually answered; everything about invitation volume is downstream of your response rate assumption, and that assumption should come from your own history rather than an industry figure.
Size for the smallest reporting cell. The headline number covers the whole-sample estimate. Any crosstab you intend to publish — by region, by tenure, by product line — has its own margin of error driven by its own count. Work backwards from the breakdowns you promised.
Do not over-interpret a small margin of error. Reaching 385 responses means your random sampling error is about 5 points. If your sampling frame excluded contractors, or the survey went out by email to a workforce half of which does not have a desk, the estimate can be far more than 5 points away from the truth and the formula will never tell you.
If your required sample approaches the population, stop and consider a census. Once the correction is pulling the requirement above about half the population, the incremental cost of surveying everyone is usually small, and a census removes sampling error entirely while also removing awkward questions about who was left out.
Completed responses needed at 95% confidence, p = 50%
| Target margin | N = 1,000 | N = 10,000 | N = 100,000 | Very large N |
|---|---|---|---|---|
| ±1% | 906 | 4,900 | 8,763 | 9,604 |
| ±2% | 707 | 1,937 | 2,345 | 2,401 |
| ±3% | 517 | 965 | 1,056 | 1,068 |
| ±5% | 278 | 370 | 383 | 385 |
| ±7% | 165 | 193 | 196 | 196 |
| ±10% | 88 | 96 | 96 | 97 |
Read across a row to see how little the population size matters once N is large, and down a column to see the squared cost of precision.
Assumptions and limits
- It assumes a simple random sample. Stratified designs usually need fewer responses for the same precision; cluster designs usually need more, sometimes several times more. The multiplier is called the design effect, and it is not in this formula.
- It assumes the normal approximation holds. With very small samples or an expected proportion near 0% or 100%, use exact binomial methods instead.
- It sizes for one estimate, not for a test. If your goal is to detect a difference between two groups rather than to estimate one number, size for power: use the A/B test sample size calculator or the statistical power calculator.
- It says nothing about bias. Coverage error, non-response bias and measurement error are unaffected by sample size, and in most modern surveys they dominate.
- It sizes for a percentage. If your key metric is an average — mean spend, mean satisfaction on a 1–10 scale — the variance term is σ² rather than p(1−p); use the sample size for a mean estimate calculator.
Related tools and where the formula comes from
The three calculators in this family invert one equation. This page fixes the margin of error and solves for n; the margin of error calculator fixes n and solves for the margin; the confidence interval for a proportion reports the result as limits around your observed percentage once the data are in.
If you are running an experiment rather than a survey, the sample size question changes shape. You are no longer asking “how precisely can I estimate one number” but “how small a difference can I reliably detect”, which brings in statistical power and the size of the effect you care about. Start with the A/B test sample size calculator.
The formula is set out in William Cochran's Sampling Techniques, the standard reference for survey sampling theory, where the finite population correction and the n₀ adjustment are derived from first principles. Taro Yamane's simplified version, n = N/(1 + Ne²), has the same shape but bakes in p = 0.5 and z = 2 rather than 1.96, and drops the −1 from the denominator. It therefore asks for slightly more than Cochran's formula even at 95% confidence: for N = 10,000 at ±5 points it gives 10,000/(1 + 10,000 × 0.0025) = 10,000/26 = 385, against the 370 this calculator returns. Use it as a quick mental estimate, not as a check on the arithmetic here.