Calculate survey sample size for a proportion
Use this calculator when your survey question has a yes-or-no style answer and you want to know what share of a group agrees: the percentage who prefer a product, support a policy, or finished a task. It returns the number of completed responses needed for the precision you choose, or, if you already have responses, the margin of error they give you. It is built for estimating a proportion from a random sample, not for comparing two versions of a page or measuring the statistical power of an experiment.
Choose confidence, margin of error and expected proportion
Three choices drive the answer. The confidence level (90, 95 or 99 percent) describes how often intervals built this way would contain the true value across repeated samples; higher confidence needs more responses. The margin of error is the plus-or-minus width you can live with, in percentage points: ±5 pp means a result of 40 percent is reported as 35 to 45 percent. The expected proportion is your best guess of the answer you are measuring. The uncertainty is largest at 50 percent, so 50 is the conservative choice when you have no idea, and a guess nearer 0 or 100 needs fewer responses.
The formula is n₀ = z² × p × (1 − p) ÷ e², where z is the normal critical value for your confidence level (1.645, 1.960 or 2.576), p is the proportion and e is the margin of error as a fraction. Because the result is almost never a whole number, it is rounded up. At 95 percent confidence, ±5 pp and p = 50 percent, n₀ is 384.15, so you need 385 completed responses. At 90 percent it is 271 and at 99 percent it is 664; tightening to ±3 pp at 95 percent needs 1,068.
Adjust for a known finite population
If you are surveying a small, known group, such as 1,000 customers or 400 employees, you do not need as many responses, because each answer covers a larger share of everyone. The finite-population correction is n = n₀ ÷ (1 + (n₀ − 1) ÷ N). With N = 1,000 the 384.15 above becomes 277.73, rounded up to 278. The correction is applied to the unrounded n₀; rounding first can overshoot by a response. If the answer reaches the whole population, you have a census, which has no sampling error. Leave the population blank when it is unknown or very large; blank means large, never zero.
Estimate invitations from a response rate
Not everyone you invite will finish the survey. Divide the completed responses you need by your expected response rate and round up to get a planning figure for invitations. At a 60 percent rate you would invite 642 people to expect 385 completions. This is an expectation, not a guarantee; real response rates vary, and when the invitations needed exceed a finite population the calculator flags the plan as infeasible instead of quietly capping it.
Calculate margin of error from completed responses
Switch to Margin of error if the sample is already fixed. The margin is z × √(p × (1 − p) ÷ n), multiplied by √((N − n) ÷ (N − 1)) for a finite population. With 400 completed responses at 95 percent confidence and p = 50 percent the margin is ±4.90 pp. If fewer than 5 responses are expected in either answer group, the normal approximation is unreliable and the calculator says so rather than implying exact coverage.
Worked example: 385 responses and 642 invitations
You want to estimate the share of customers who would recommend a product within ±5 pp at 95 percent confidence, you have no prior estimate, and you expect 60 percent of invitees to respond. z = 1.9600, n₀ = 1.9600² × 0.5 × 0.5 ÷ 0.05² = 384.15, rounded up to 385 completed responses. Invitations: 385 ÷ 0.60 = 641.67, rounded up to 642. The table of ±3, ±5 and ±10 pp shows the trade-off: ±10 pp needs only 97 responses, while ±3 pp needs 1,068.
Assumptions: random sampling, normal approximation and independence
- Respondents are a random sample of the population. A narrow margin of error does not repair selection bias, self-selection or poor question wording.
- Responses are independent and the normal approximation holds. Clustered or repeated observations, such as consecutive trades in a trading journal, are usually dependent, and a large count of them does not prove a strategy is profitable. For an observed win rate, see the required win rate calculator and keep the independence caveat in mind.
- No design effect, survey weights, subgroup quotas or exact binomial recruitment guarantees. All values stay in your browser.
- Supported range: whole numbers up to 1,000,000,000. A larger requirement is reported as a limit, not truncated.
FAQ
How many survey responses do I need?
It depends on confidence, margin of error, the expected proportion and the population. For 95 percent confidence, ±5 pp and an unknown proportion, you need 385 completed responses in a large population.
What sample size do I need for 95% confidence and 5% margin of error?
385 completed responses when the population is large and you use 50 percent for the expected proportion. A known population of 1,000 reduces that to 278.
How do I calculate sample size for a known population?
Compute n₀ = z² p (1 − p) ÷ e², then n = n₀ ÷ (1 + (n₀ − 1) ÷ N), and round up. Enter the population size in the optional field and the calculator does both steps and shows them.
Why use 50% when the expected proportion is unknown?
Because p × (1 − p) is largest at 50 percent, so it gives the biggest required sample. If the true share turns out different, your margin of error will be as small as planned or smaller.
How many invitations should I send at a 60% response rate?
Divide the completed responses you need by 0.60 and round up: 385 completions means 642 invitations. Treat it as a planning figure, since actual response rates vary.
Is this an A/B test or statistical power calculator?
No. It sizes a survey that estimates one proportion. It does not compare two groups, calculate power or detect an effect size. To see where the critical values come from, use the normal distribution calculator; to work out the exam score you need in a course, try the final grade calculator.