Enter your target margin of error, confidence level, and optionally population size to compute the minimum required sample size for a statistically valid survey or study.
How to use this calculator
Select your desired confidence level (95% is standard).
Enter your acceptable margin of error in percent.
Use 50% for estimated proportion unless you have a prior estimate.
Enter population size if finite (leave 0 for large/unknown populations).
Read the minimum sample size required.
Formula & method
Sample size formula
n₀ = (z* × p × (1−p)) / E² · With finite population: n = n₀N / (n₀ + N − 1)
z* = confidence level critical value, p = estimated proportion, E = margin of error (decimal), N = population size.
Example
95% CI (z*=1.96), MOE=5% (E=0.05), p=50%: n₀ = (1.96² × 0.25) / 0.0025 = 384.16 → n = 385.
Key insights
The classic result: for a 95% CI and ±5% MOE, you need n ≥ 385 (infinite population).
Sample size grows with the square of z* and shrinks with the square of MOE — halving MOE quadruples required n.
Using p = 0.5 maximises required sample size, providing a conservative estimate.
Finite population correction reduces sample size for small populations: n_adj = n₀N / (n₀ + N − 1).
How to interpret your result
Margin of error vs sample size trade-off
A smaller MOE requires a much larger sample. Going from ±5% to ±2.5% quadruples the required sample.
Conservative 50% estimate
When you have no prior estimate of the proportion, use p = 0.5. This maximises variance and gives the largest required sample.
Frequently asked questions
How many respondents do I need for a 95% CI and ±5% margin of error?
For an unknown/large population: 385 respondents. This is the standard minimum for most surveys.
Does population size always matter?
Only for small populations. For populations above ~10,000, the finite correction has little effect and the infinite formula is sufficient.
Why use 50% as the estimated proportion?
p = 0.5 maximises p(1−p) = 0.25, giving the largest required sample size and the most conservative (safe) estimate.
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