Statistics and research
Sample size calculator
Sample size to estimate a proportion or a mean within a chosen margin of error, with the finite population correction for small populations.
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Results
- Critical z
- 1.959964
- Without population correction
- 384.14588 → 385
- Formula
- n₀ = z² · p(1 − p) / E² = 1.95996² × 0.5 × 0.5 / 0.05²
- Required sample size (rounded up)
- 385
Round up, then divide by the expected response rate to get the number to invite (for example 385 / 0.6 ≈ 642).
Method and limits
How to use it
Pick whether you are estimating a proportion (most surveys) or a mean, then the confidence level and the margin of error you can accept. For a small population, enter its size to apply the finite population correction.
Formulas
- Proportion: n₀ = z² p(1 − p) / E², with E the margin as a fraction (5 points = 0.05).
- Mean: n₀ = (z σ / E)², with E in the units of the data.
- Finite population correction (Cochran 1977): n = n₀ / (1 + (n₀ − 1) / N). Always round up.
- z is the two-sided critical value: 1.645 for 90%, 1.960 for 95%, 2.576 for 99%.
Assumptions and limits
- Simple random sampling. For cluster or stratified designs multiply by the design effect.
- This is the number of completed responses: divide by the expected response rate to get the number to invite.
- p = 50% gives the largest n; use a smaller expected proportion only with good prior evidence.
- To detect a difference between groups (rather than estimate one value), use a power calculation instead.
Worked example
95% confidence, ±5 points, p = 50%: n₀ = 1.95996² × 0.5 × 0.5 / 0.05² = 384.15, so 385 responses. With a population of 1,000: 384.15 / (1 + 383.15 / 1000) = 277.7, so 278. For a mean with σ = 15 and E = 5: (1.95996 × 15 / 5)² = 34.6, so 35.
References
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