Survey planning
Survey sample size calculator
How results are checked
Named statistical review pending. Selected automated reference cases are published on the validation page.Completes needed
385for ±5%
95% confidence, p = 50%, large population
±3% needs 1,068
For a survey percentage, the required sample size is n₀ = z²·p(1 − p)/e²: at 95% confidence (z = 1.96), p = 50% and a ±5% margin of error that is 384.15, so you need 385 completes. Multiply by the design effect for weighted or clustered samples, and apply the finite population correction when the population is small. This calculator does all three, works backwards from n to the margin of error, and sizes two-group comparisons — free, no account, and nothing leaves your browser.
Calculate sample size online
Pick a tab: how many completes for a target margin of error, the margin of error you get from a sample you already have, or how many per group to detect a difference between two percentages.
50% gives the largest (safest) sample.
Applies the finite population correction.
1.0 for a simple random sample; weighting or clustering raise it.
Converts completes into invitations.
Required completes
385
A sample of 385 completes gives ±5.0% at 95% confidence (p = 50%).
n₀ = z² · p(1 − p) / e² = 1.960² × 0.500 × 0.500 / 0.050² = 384.15
n′ = n₀ × deff = 384.15 × 1 = 384.15
Rounded up: 385 completes
Margin of error at 95% confidence (p = 50%, simple random sample)
| Completes (n) | Margin of error |
|---|---|
| 100 | ±9.8% |
| 200 | ±6.9% |
| 400 | ±4.9% |
| 600 | ±4.0% |
| 1,000 | ±3.1% |
| 1,500 | ±2.5% |
| 2,000 | ±2.2% |
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Design effect, population size and response rate
The basic formula assumes a simple random sample. Weighting, clustering and quotas usually make estimates less precise; the design effect (deff) measures how much, and the effective sample size is n / deff. If you plan to weight by raking, try a few planning values (say 1.2, 1.5 and 2.0) to see how sensitive the requirement is; once you have weights, Kish's 1 + CV² of the weights gives a figure.
When the sample is a sizeable fraction of the population — a customer list of 2,000, a staff survey — the finite population correction lowers the requirement. Finally, divide completes by the expected response rate to see how many people to invite.
The method
Formula
n₀ = z² · p(1 − p) / e², n = n₀·deff / (1 + (n₀·deff − 1) / N)
where
- z
- critical value: 1.645, 1.960 or 2.576 for 90%, 95% or 99% confidence
- p
- expected proportion; 50% maximizes p(1 − p) and is the safe default
- e
- margin of error as a proportion (0.05 for ±5 points)
- deff
- design effect (1 for a simple random sample)
- N
- population size; omit it for a very large population
- MOE
- z · √(deff · p(1 − p) / n) · √((N − n)/(N − 1)) for a sample you already have
- Two groups
- n₁ = [z₁₋α/₂√(p̄q̄(1 + 1/k)) + z₁₋β√(p₁q₁ + p₂q₂/k)]² / (p₁ − p₂)², with n₂ = k·n₁ (Fleiss, Levin & Paik 2003)
Worked example
Margin of error target. For ±5% at 95% confidence with p = 50%: n₀ = 1.960² × 0.5 × 0.5 / 0.05² = 384.15, rounded up to 385 completes. A sample of 385 completes gives ±5.0% at 95% confidence (p = 50%).
Small population. If there are only 2,000 customers, n = 384.15 / (1 + 383.15 / 2,000) = 322.39, so 323 completes are enough.
Weighted design and invitations. With a design effect of 1.5 the requirement becomes 384.15 × 1.5 = 576.22 → 577 completes; at a 20% response rate that means about 2,885 invitations.
Margin of error from n. 1,000 completes give ±3.1% at 95% confidence (p = 50%).
Two groups. To detect 50% vs 60% with 80% power at α = 0.05 (two-sided), you need 388 per group (408 with the Fleiss continuity correction).
Using it well
When to use it
Use it when
- Use: Planning how many completes to buy or field for a survey estimate.
- Use: Checking the precision of a sample you already have, including weighted samples.
- Use: Sizing each cell of an A/B concept test, a wave-on-wave tracker comparison or a segment comparison.
Not the right tool when
- Avoid: Means of rating scales or continuous measures — those need the standard deviation, not p(1 − p).
- Avoid: Non-probability opt-in panels where the margin of error is a model-based approximation; report it as such.
- Avoid: Analysing results you already have — use the chi-square test or the survey crosstab tool to test the observed differences.
How to interpret it
Rule of thumb
Precision grows with the square root of n: halving the margin of error needs four times the sample. Subgroups have their own, larger margins of error — size the study for the smallest group you need to report, not just the total.
Questions
Frequently asked questions
How many survey responses do I need?
For a ±5% margin of error at 95% confidence on a percentage, 385 completes (assuming p = 50% and a large population). ±3% needs 1,068 and ±10% needs 97. Weighted or clustered designs need more, in proportion to the design effect.
Why use 50% as the expected proportion?
p(1 − p) is largest at p = 50%, so it gives the most conservative sample size. If you are confident the result will be near 10% or 90%, a smaller sample achieves the same margin of error for that question — but other questions in the survey may be closer to 50%.
When does population size matter?
Only when the sample is a noticeable share of the population — roughly more than 5%. For a national population the correction is negligible; for a list of 2,000 customers, ±5% at 95% needs 323 completes instead of 385.
What is a design effect?
The ratio of the actual variance of an estimate to the variance a simple random sample of the same size would give. Weighting, clustering and stratification change it; a deff of 1.5 means 600 completes are only as precise as 400 from a simple random sample (the effective sample size).
Should I use the continuity correction for two groups?
The uncorrected normal-approximation formula matches the uncorrected chi-square or z-test. If you will analyse with a continuity-corrected test (Yates), use the corrected figure, which is somewhat larger. Both are shown.
Sources and next steps
References & further reading
- [1]Cochran, W. G. (1977). Sampling Techniques (3rd ed.). New York: Wiley.
- [2]Kish, L. (1965). Survey Sampling. New York: Wiley.
- [3]Fleiss, J. L., Levin, B., & Paik, M. C. (2003). Statistical Methods for Rates and Proportions (3rd ed.). Hoboken, NJ: Wiley.
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