Sample Size

Finding the Minimum Sample Size for a Survey

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.

Key Factors to Consider

• A smaller desired margin of error requires a disproportionately larger sample size, not a proportionally larger one. Because margin of error appears squared in the denominator of the formula, cutting the margin of error in half roughly QUADRUPLES the required sample size — this is exactly why very tight margins of error (like ±1%) demand dramatically more respondents than a more typical ±5%.
• A higher confidence level also increases the required sample, though less dramatically than tightening the margin of error. Moving from 90% to 95% to 99% confidence increases the critical z-value used in the formula, which raises the required sample size — the exact tradeoff between confidence level and sample size is a genuine planning decision, not a fixed rule.
• This formula estimates sample size for a single proportion (like “what percent of people prefer X”), not every kind of statistical question. Estimating a numeric average (like average spending) or detecting a difference between two groups both use related but distinct sample-size formulas — this calculator is scoped specifically to the proportion-estimation case, the most common survey scenario.
• The finite population correction only meaningfully shrinks the required sample when the population itself is relatively small. For a large population (in the tens of thousands or more), the correction has a negligible effect on the required sample size — this is why survey methodology commonly treats “large population” and “unknown population” as functionally the same case, both defaulting to the uncorrected formula.

Common Mistakes

• Entering the population size as the number of people you plan to contact, not the total population you’re drawing from. The finite population correction only applies to the TOTAL group you could possibly survey (e.g. all 1,000 employees at a company) — entering how many responses you hope to collect defeats the purpose of the calculation.
• Assuming a smaller margin of error only needs a slightly larger sample. Because margin of error is squared in the formula, cutting it in half roughly quadruples the required sample — planning for a tight margin without budgeting for the sample-size cost is a common surprise.
• Confusing this calculator with the Confidence Interval Calculator. This one answers “how many respondents do I need before collecting data”; the other answers “how precise was the data I already collected” — they run in opposite directions and aren’t interchangeable.

Useful to Know

• Already collected your survey responses and want to know how precise your results actually are? Confidence Interval Calculator calculates the resulting margin of error from real data.
• Need to calculate other summary statistics from your collected responses, like the mean or standard deviation? Statistics Calculator covers that directly.
• Curious how sample size relates to the odds of a particular outcome? Probability Calculator covers single-event and combined-event probability calculations more generally.