Calculation details
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What does this calculator estimate?
A sample size calculator tells you how many people or observations you need to survey for a target precision. Enter the confidence level, margin of error, expected proportion, and population size to get the required n.
- n₀ = z²p(1−p) ÷ e²
- p = 0.5 is the most conservative (largest n)
- Finite populations need a small correction
Why sample size matters
A sample that's too small can't support precise conclusions; one that's too large wastes money and time. The formula balances the confidence level, the margin of error you can tolerate, and the expected variability of the thing you're measuring.
Limitations to watch for
The formula assumes a simple random sample. Real surveys use design effects (clustering, stratification) that raise the required n. The expected proportion p is a guess — using 0.5 is safest but yields the largest n. Non-response means you should plan to contact more people than the answer.
How to use it in practice
Decide the margin of error you can live with before collecting data. Use p = 0.5 when you don't know the true proportion. If your population is small (under ~10,000), include it for the finite-population adjustment.
['Choose the confidence level and margin of error.', 'Enter your expected proportion (0.5 if unsure).', 'Optionally enter the population size for the finite correction.']
How this calculator works
Formula
Required sample size n₀ = z² × p(1 − p) ÷ e², where z is the critical value for the confidence level (1.645/1.96/2.576), p is the expected proportion, and e is the margin of error. With a finite population, n = n₀ ÷ (1 + (n₀ − 1) ÷ N).
Worked example
For 95% confidence (z = 1.96), a ±5% margin (e = 0.05), and p = 0.5: n₀ = 1.96² × 0.25 ÷ 0.0025 ≈ 384.16 → 385. For a population of 10,000, the finite correction gives about 370.
Assumptions to verify
- Simple random sampling with independent observations.
- The proportion p is close to the expected value.
- Non-response and design effects are not included.
Frequently asked questions
How many people do I need to survey?
It depends on the confidence level, margin of error, and expected proportion. For 95% confidence and ±5% margin, you need roughly 385 respondents (fewer with a small population).
What is the formula?
n₀ = z² × p(1 − p) ÷ e², where z is the z-value (1.96 for 95%), p the expected proportion, and e the margin of error. A finite population applies n = n₀ ÷ (1 + (n₀ − 1) ÷ N).
Why is p = 0.5 conservative?
p(1 − p) peaks at 0.25 when p = 0.5, giving the largest sample size. If you don't know the true proportion, using 0.5 guarantees enough precision.
What is a good margin of error?
±5% is common for market research; ±3% is tighter and more expensive; ±10% is acceptable for quick directional reads. Smaller margins need bigger samples.
Does population size matter?
Only for small populations. Above ~10,000–20,000 the finite-population correction barely changes the answer; below that it reduces the required n.
Does this include non-response?
No — the answer is the number of completed responses. If you expect a 30% response rate, contact roughly 3.3× as many people.
Is random sampling assumed?
Yes. Non-random samples (convenience, self-selected) aren't fixed by a bigger n — the formula only protects against sampling error.
Cite this tool
BoringToolsKit. “Sample Size Calculator.” boringtoolskit.com/sample-size-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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