Sample Size Calculator

How This Calculator Works

Sample size determination finds how many respondents or observations a survey needs to estimate a population proportion within a chosen margin of error, at a chosen confidence level. Choose a confidence level, a desired margin of error, and (if you have one) an estimated proportion, and this calculator finds the minimum sample size before you collect a single response.

This is the question that comes before the Confidence Interval Calculator’s own question — that calculator tells you how wide an interval your data already produced; this one tells you how much data to collect in the first place so the resulting interval is no wider than you’re willing to accept.

If you have a known population size (the total number of people or items you could possibly survey), entering it applies the finite population correction, which can meaningfully shrink the required sample when the population itself is small. Leave it blank to assume a large or unknown population, which is the safer, more common case.

The Formula

$$n_0 = \frac{\vA{z}^2 \times \vB{p}(1-\vB{p})}{\vC{e}^2}$$

Where $\vA{z}$ is the critical value for the chosen confidence level, $\vB{p}$ is the estimated proportion (as a decimal), and $\vC{e}$ is the desired margin of error (as a decimal). The result is rounded up, since a sample can’t include a fraction of a respondent.

When a population size $\vD{N}$ is known, the finite population correction is applied:

$$n = \frac{n_0}{1 + \frac{n_0 - 1}{\vD{N}}}$$

$\vB{p} = 0.5$ (50%) is the most conservative choice when the true proportion is unknown — $p(1-p)$ is at its largest at exactly 50%, which maximizes (and therefore never underestimates) the required sample size.

Worked Example

At a 95% confidence level (z = 1.96), a desired margin of error of ±5%, and an estimated proportion of 50% (the conservative default):

1. Square z and multiply by p(1-p): $\vA{1.96}^2 \times \vB{0.5} \times (1 - \vB{0.5}) \approx 0.9604$. 2. Divide by e squared: $0.9604 \div \vC{0.05}^2 = 384.16$. 3. Round up, since a sample can’t include a fraction of a respondent: $384.16 \rightarrow 385$.

You’d need 385 respondents from a large or unknown population. If instead the true population is only 1,000 people, the finite population correction shrinks that to:

1. Subtract 1 from n0 and divide by the population: $(384.16 - 1) \div \vD{1{,}000} \approx 0.38316$.
2. Add 1: $1 + 0.38316 = 1.38316$.
3. Divide n0 by that result: $384.16 \div 1.38316 \approx 277.74 \rightarrow 278$.

Just 278 respondents are needed once the population itself is known to be that small.