Sample Size Calculator
Sample size grows with the confidence level and shrinks with the margin of error: a survey at 95 % confidence and ±5 % needs n = 385 respondents.
For a proportion, 0.05 means ±5 percentage points.
Use 0.5 if unknown: it gives the largest, safest sample.
385
- Before rounding up
- 384.146
- Critical value z
- 1.96
Step-by-step solution
- n₀ = z² p(1 − p) ÷ E² = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.146
- Round up: n = 385
Sample Size Formula
n = z² p (1 − p) ÷ E²
z critical value, p expected proportion, E margin of error; z²σ²/E² for a mean
Worked Example
A survey at 95 % confidence with a ±5 % margin and p = 0.5:
- n₀ = z² p(1 − p) ÷ E² = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.146
- Round up: n = 385
Critical value z: 1.96
Margin of error, confidence and the proportion to assume
The sample size grows with the square of the confidence z-value and falls with the square of the margin of error, so halving the margin of error needs four times the sample. When you have no estimate of the proportion, p = 0.5 gives the largest, safest sample. For a small population the finite population correction reduces the number needed. The result assumes a simple random sample; clustered or stratified designs and expected non-response need a larger sample.
Frequently asked questions
Why use p = 0.5 if I do not know the proportion?
p(1 − p) is largest at p = 0.5, so it gives the most conservative (largest) sample size.
What is the finite population correction?
For small populations, n = n₀ ÷ [1 + (n₀ − 1)/N]. Surveying a population of 10 000 at 95 % and ±5 % needs 370 instead of 385.
What is Cochran’s formula?
n₀ = z²p(1 − p)/E², the formula used here for proportions, proposed by W. G. Cochran.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Sample Size Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/sample-size-calculator/
IEEE
Scientific Converters, "Sample Size Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/sample-size-calculator/