Sample Size Calculator
Determine the minimum number of respondents needed to estimate a population proportion at a chosen confidence level and margin of error. Optional finite-population correction tightens the requirement when the population is small.
Required sample size
385
- z-score
- 1.9600
- Margin of error
- ±5.00%
- Expected proportion (p)
- 50.0%
Survey at least 385 respondents for a ±5.0% margin at 95% confidence, assuming a population large enough that the finite-population correction is negligible.
How to use this calculator
Pick a confidence level — 95% is the default used in most surveys, market research and academic studies. Set the margin of error you can tolerate (±5% is typical for opinion polls; tighter studies use ±3% or ±1%). Set the expected proportion if you have a prior estimate of how the question will split; if you do not, leave it at 50% — that gives the largest, most conservative sample size and works for any actual response distribution. Optionally enter the population size: leave it at 0 (or any non-positive number) for infinite or unknown populations and the calculator returns the unadjusted Cochran sample size; enter a real population and it applies the finite-population correction, which can substantially reduce the requirement for small populations.
How the calculation works
The unadjusted sample size for estimating a proportion comes from inverting the normal-approximation confidence interval for a proportion: n₀ = z² · p · (1 − p) / E², where z is the two-sided standard-normal critical value for the chosen confidence level (1.96 for 95%, 2.576 for 99%), p is the expected proportion and E is the half-width of the confidence interval as a proportion (0.05 for ±5%). The variance term p · (1 − p) is maximised at p = 0.5, which is why 50% is the conservative default. When the population N is finite and small enough to matter, the calculator applies the finite-population correction n = n₀ / (1 + (n₀ − 1) / N) — for a 1,000-person population the requirement at 95% / ±5% drops from 385 to 278. Formulas follow Cochran (1977) Sampling Techniques §4.2 and the NIST/SEMATECH e-Handbook of Statistical Methods §7.2.4.2.
Worked example
You want a national opinion poll with ±5 percentage points of error at 95% confidence. With no prior estimate, set p = 50%. Then n₀ = 1.96² · 0.5 · 0.5 / 0.05² = 3.8416 · 0.25 / 0.0025 = 384.16, rounded up to 385. That matches the figure widely published by SurveyMonkey, Qualtrics and the major polling firms. If instead you are polling 1,000 members of a private club at the same confidence and margin, apply the finite-population correction: n = 384.16 / (1 + 383.16/1000) ≈ 277.74, rounded up to 278. A tighter ±1% margin at 95% needs n₀ = 9,604 — twenty-five times more interviews for one-fifth the margin, because the formula scales with 1 / E².
Frequently asked questions
What sample size do I need for ±5% margin of error at 95% confidence?
For a very large or unknown population, you need 385 respondents (the textbook Cochran figure, often quoted as 384). For a finite population the requirement is lower: 278 for a population of 1,000, 217 for 500, 80 for 100. The 385 figure assumes the most conservative expected proportion of 50%; if you genuinely know the response will be skewed (say 80/20), you can get away with fewer — about 246 at p = 80%.
Why is 50% the default expected proportion?
The variance term p · (1 − p) in the sample-size formula is maximised when p = 0.5, where it equals 0.25. Any other proportion makes the term smaller and shrinks the required sample. Using p = 0.5 therefore gives the largest, most conservative sample size — it works no matter how the responses actually split. If you have a credible prior estimate (from previous waves of the same survey, or a pilot study), plugging in the real p gives a tighter, cheaper sample, but for new surveys 50% is the safe choice.
When should I use the finite-population correction?
Apply it when your sample would otherwise be a meaningful fraction of the total population — roughly when n₀ exceeds 5% of N. For small populations (employees of a single company, members of a club, students in a school) the correction can dramatically reduce the required sample, because each respondent carries more information when the universe is bounded. For large populations (countries, large cities, customer bases above a few hundred thousand) the correction makes a negligible difference and can be ignored. The calculator applies it automatically when you enter a population size.
Does this calculator do power analysis for hypothesis tests?
No — this is the simpler problem of how many respondents you need to estimate a single proportion to within a given margin of error. Statistical power analysis for hypothesis tests (e.g. comparing two means, detecting a specified effect size at a chosen power) needs additional inputs — the effect size, the significance level α and the desired power 1 − β — and uses different formulas (often based on non-central t or non-central χ² distributions). For that, tools like G*Power, pwr (R) or statsmodels.power (Python) are appropriate.
What confidence level should I pick?
95% is the universal default for opinion polls, market research, social-science studies and most quality-control work. 90% is acceptable for exploratory or low-stakes work and gives a smaller sample. 99% is used in clinical trials, pharmaceutical work and high-stakes decisions where the cost of a wrong conclusion is large; it roughly doubles the sample relative to 95%. 99.9% is rare outside of regulated industries — it requires almost three times the 95% sample. Higher confidence is not "better" — it trades a wider interval and larger sample for stronger guarantees.
How do I sample for a mean rather than a proportion?
For estimating a mean μ to within a margin of error E at a given confidence level, the formula is n = (z · σ / E)², where σ is the population standard deviation. You need a prior estimate of σ from a pilot study or comparable data. The calculator on this page is specifically for proportions; for means, swap p · (1 − p) for σ² in the formula. The z-score and finite-population correction work the same way.