Sample Size Calculator

For a large or unknown population, the calculator uses Cochran’s formula for estimating a proportion:

n₀ = z² × p × (1 − p) / e²

When you enter a population size N, it applies the finite population correction to the unrounded n₀:

n = n₀ / [1 + (n₀ − 1) / N]

The same correction can be written as n = N·n₀ / (N + n₀ − 1). Only the final n is rounded up to a whole response.

z

The two-sided standard normal value for the confidence level: 1.959964 for 95%, 1.644854 for 90% and 2.575829 for 99%. The calculator does not round it to 1.96.

p

The expected proportion as a decimal, such as 0.5 for 50%. It is the share of respondents expected to give the answer you are measuring.

e

The margin of error as a decimal of the whole scale: ±5 percentage points is 0.05. It is an absolute margin, not 5% of the estimate.

N

The number of people in the population you are sampling from, if known.

n

The minimum number of completed responses.

Each row is calculated with the same code as the calculator above. The standard case, 95% confidence, ±5% and p = 50%, gives n₀ = 1.959964² × 0.5 × 0.5 / 0.05² = 384.15, which rounds up to 385.

The chart in the calculator plots the same calculation across a range of margins, so you can see how quickly the requirement grows as the margin narrows.

For the staff survey, n = 384.15 / [1 + 383.15 / 1,000] = 277.73, so 278 completed responses. If you rounded n₀ to 385 first, you would get 279; rounding once at the end avoids that.

The result is the smallest number of completed responses that gives the stated margin of error at the stated confidence level, if the assumptions below hold. It is a floor for precision, not a guarantee.

• It does not make a sample representative. A large convenience sample can still be biased.
• It does not give statistical power to detect a difference or test a hypothesis.
• It applies to the whole sample. If you need ±5% for each of four departments, each department needs its own calculation.
• For a questionnaire with many questions, size it for the most important estimate. When in doubt, use p = 50%, which needs the largest sample.

To estimate invitations, divide the completed responses by the expected response rate and round up. For example, 385 completed responses at a 30% response rate needs about 385 ÷ 0.30 = 1,284 invitations.

Check that number against the population. In a staff of 1,000, a target of 278 completed responses at a 25% response rate would need 1,112 invitations: at that response-rate assumption, the target is not feasible within the population. A higher response rate is only a fair assumption if you have evidence for it, so reassess the precision target or the contact strategy.

Report response rates using the AAPOR Standard Definitions, and remember that inviting more people does not remove nonresponse bias when people who respond differ from those who do not.

• Simple random sampling. Every member of the population has an equal probability of selection. For a large or unknown population, the formula treats responses as independent draws. When you enter a known population size, the finite population correction assumes simple random sampling without replacement, in which selections are not independent; the correction accounts for that. Cluster, stratified or multistage designs need design-based methods; cluster designs usually need more responses.
• Normal approximation. The formula uses the normal approximation to the binomial distribution. It works poorly when p is close to 0% or 100% or when few responses are expected in one answer group; the calculator warns in those cases. Exact or Wilson-type intervals behave better there (Agresti and Coull, 1998).
• One proportion. Means, differences between groups, correlations, regression and diagnostic accuracy studies need different formulas.
• Small populations. The finite population correction matters once the sample is a sizeable share of the population. If the calculator asks for half or more of the population, consider a census.
• Input ranges. Margin of error 0.1 to 50 percentage points, expected proportion 0.1% to 99.9%, population a whole number up to 10 billion, response rate 1% to 100%.

Use the copyable methods statement in the calculator as a starting point. It records the confidence level, margin of error, expected proportion, population size, formula and rounding rule, plus the response-rate assumption if you entered one.

Journals and reporting guidelines ask authors to state how the sample size was determined and which assumptions were used. See our guides to writing the methods section and the SAMPL statistical reporting guidelines.

What is the minimum sample size for a survey?

With 95% confidence, a ±5 percentage-point margin of error, an expected proportion of 50% and a large or unknown population, the minimum is 385 completed responses. It is a minimum only under those assumptions: it does not guarantee that the respondents represent the population, and it is not a power calculation for testing a hypothesis.

Why use 50% when I do not know the expected proportion?

The term p(1 − p) is largest when p is 50%, so 50% gives the largest, most conservative sample size for a given margin of error. If earlier data suggest a proportion such as 20%, using it gives a smaller sample, but the margin will be wider than planned if the true proportion is closer to 50%.

How much does population size change the result?

Only noticeably when the sample is a sizeable share of the population. At 95% confidence and ±5%, the minimum is 278 completed responses for a population of 1,000, 377 for 20,000 and 385 when the population is large or unknown.

Is the sample size the number of people I should invite?

No. The result is the number of completed, usable responses. If you need 385 completed responses and expect a 25% response rate, you would invite about 385 ÷ 0.25 = 1,540 people. Check that against the population: 278 completed responses from a staff of 1,000 at a 25% response rate would need 1,112 invitations, so that target is not feasible unless the response rate is higher or the precision target is relaxed. Inviting more people also does not remove nonresponse bias if the people who answer differ from those who do not.

Can I use this calculator for a clinical trial or to compare two groups?

No. This calculator sets the precision of one estimated proportion. Comparing groups or testing a hypothesis needs a power calculation that specifies the effect size, significance level, statistical power and study design. Use software built for that design and consult a statistician.

Why do other sample size calculators give slightly different numbers?

Common reasons are using z = 1.96 instead of the precise 1.959964, rounding the infinite-population size before applying the finite population correction (385 corrected for a population of 1,000 gives 279 rather than 278), using a different correction such as n₀/(1 + n₀/N), or using a different formula altogether. Check which assumptions a tool states before comparing results.

Does this use the same method as SurveyMonkey’s sample size calculator?

SurveyMonkey’s published explanation of sample size uses the same proportion formula, with z = 1.96 for 95% confidence and p = 0.5 when the proportion is unknown, and rounds 384.16 up to 385 completed responses. This calculator also lets you change the expected proportion, adds an optional response rate and shows each step. Journal Metrics is not affiliated with SurveyMonkey.

What if my sample is not a simple random sample?

The formula assumes simple random sampling, in which every member of the population has an equal probability of selection; with a known population size, the finite population correction assumes sampling without replacement. Cluster or multistage designs usually need a larger sample, often by multiplying by a design effect, and convenience or volunteer samples cannot be fixed by sample size alone.

Are my inputs stored or sent anywhere?

No. The calculator runs in your browser and does not ask for any research data, file or account. Your inputs are not put into the page address and are not saved.

• PRISMA flow diagram generator: build and check a systematic review selection diagram.
• PRISMA-ScR flow diagram for scoping reviews.
• Journal search: compare journals before you submit.

• Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977. ISBN 978-0-471-16240-7. Source of the proportion formula and the finite population correction.
• Lwanga SK, Lemeshow S. Sample size determination in health studies: a practical manual. Geneva: World Health Organization; 1991. ISBN 92-4-154405-8.
• Israel GD. Determining Sample Size (PEOD6). University of Florida IFAS Extension. Explains Cochran’s formula, the correction and inflation for nonresponse.
• Agresti A, Coull BA. Approximate is better than “exact” for interval estimation of binomial proportions. Am Stat. 1998;52(2):119–126.
• American Association for Public Opinion Research. Standard Definitions (10th edition, 2023), with definitions for final disposition codes and survey outcome rates.
• SurveyMonkey. How survey sample size works. Cited only for the FAQ comparison; SurveyMonkey is a trademark of its owner.

Content updated October 15, 2026. Examples are illustrative and use no real survey data.