Before running a survey, it helps to know how many respondents are actually needed. This works out the required sample size for a given margin of error and confidence level.
How it works
Required sample size for a proportion
n = z² p(1−p) / e²
z is the critical value for your confidence level, p is the expected proportion (use 50% if unknown), e is the margin of error. If a population size is supplied, a finite-population correction pulls the requirement down for smaller populations.
Why 50% is the “safe” default
p(1−p) is largest — meaning it demands the largest sample — exactly when p is 50%. Using 50% when the true proportion is genuinely unknown gives the sample size guaranteed to be large enough regardless of what the real proportion turns out to be. If a reasonable estimate already exists — from a pilot survey, say — using it instead gives a smaller, still statistically valid required sample.
Precision is expensive, and the reason is a square
The margin of error appears squared in the denominator, so halving it — say, from a 5% margin to 2.5% — roughly quadruples the required sample, not doubles it. This is the same squared-scaling relationship the area converter’s own documentation explains for a linear dimension: demanding twice the precision costs roughly four times the respondents, which is why very tight margins get expensive to survey for very quickly.
How to use this calculator
- Choose a confidence level and margin of error.
- Enter an expected proportion, or leave it at 50% if you have no estimate.
- If surveying a small, known population, enter its size for a more precise (usually smaller) requirement.
Frequently asked questions
Why does population size matter for the required sample?
For a very large or effectively infinite population, sampling more of it barely improves precision beyond a certain point. For a small population, sampling a meaningful fraction of the whole group does improve precision faster, which is what the finite-population correction accounts for.
What if I have no idea what proportion to expect?
Leave it at 50% — it is the value that produces the largest, and therefore safest, required sample size, guaranteed to be sufficient whatever the true proportion turns out to be.
Why does a tighter margin of error need so many more respondents?
Because margin of error is squared in the formula. Going from a 10% margin to a 5% margin is a factor of two in precision, but a factor of four in the required sample — not a proportional trade.
Is this the right formula for a survey about an average, not a proportion?
No — this specific formula is for estimating a proportion (a yes/no style question). Estimating a numeric average (like a mean income) uses a related but different formula involving the expected standard deviation of the values themselves, not p(1−p).
Does a bigger sample always mean better data?
A bigger sample reduces margin of error, but only if the sample is genuinely representative — a large but biased sample (self-selected respondents, for instance) can still give a misleading result regardless of size. Sample size and sample quality are separate questions.
