Sample size for a proportion depends on three things: the confidence level, the margin of error you will accept, and how variable the responses are. Variability peaks when the population splits evenly, so assuming fifty percent gives the safest, largest answer — which is what most survey planning uses.
The result surprises people because it barely depends on population size. Polling a nation of seventy million needs almost exactly the same sample as polling a city of two hundred thousand. Only when your sample becomes a sizeable fraction of the whole population does the finite population correction pull the requirement down, which is why surveying a small company or club needs far fewer responses than intuition suggests.
Below the calculator sits the responses needed for a fifty-fifty split table: 5 rows — 1 at one end, 10 at the other — which is usually faster than entering a value you have seen before. Nothing has to be entered to see it work: the default for population size is 50,000, and the output recomputes the moment that changes. The result is also read against 4 bands, from small sample at the low end to very large survey at the high end, each with a note on what that range means in practice. Margin of Error Calculator and Confidence Interval Calculator are the nearest neighbours to this one.
The formula
z- Critical value for the confidence level
p- Expected proportion, 0.5 for the worst case
e- Margin of error as a decimal
N- Population size
How it works, step by step
- Enter the population you are sampling from.
- Set the margin of error you can live with.
- Choose the confidence level and expected response split.
- The gauge shows the sample size required.
Worked examples
A population of 50,000 at ±5%
The uncorrected requirement is 385 responses, and the finite population correction brings it to 382 — barely any difference at this population size.
Why a tighter margin costs so much
Tightening from ±5% to ±2% raises the uncorrected requirement from 385 to 2401 responses — the sample scales with the inverse square of the margin, so halving it quadruples the cost.
Checking the 3 row
The reference table lists Margin of error = 3, giving Population 1000: 517, Population 10000: 965, Population 100000: 1056, Population 10000000: 1067. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
Why does population size matter so little?
Because precision comes from the absolute number of responses, not the fraction sampled. The correction only bites once your sample is a meaningful share of the whole population.
What proportion should I assume?
Fifty percent unless you have a solid prior. It maximises the variance and therefore the required sample, so it is the conservative choice.
Does this account for non-response?
The core figure is completed responses. The extras show how many invitations you would need at a twenty-five percent response rate.
Is this valid for questions with many answer options?
It is derived for a yes-or-no proportion. For multi-option questions, apply it to the single option whose share you most need to measure precisely.
What do the bands under the responses needed mean?
They label the figure so it can be read without a reference point of your own: Small sample from 0 to 100; Standard survey from 100 to 400; Large survey from 400 to 1,100; Very large survey from 1,100 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Sample Size Calculator use?
It computes n = z^2p(1−p) ÷ e^2, corrected for population N, where z is critical value for the confidence level; p is expected proportion, 0.5 for the worst case; e is margin of error as a decimal; N is population size. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Sample sizes at 95% confidence
| Margin of error | Population 1000 | Population 10000 | Population 100000 | Population 10000000 |
|---|---|---|---|---|
| 1 | 906 | 4900 | 8763 | 9595 |
| 2 | 707 | 1937 | 2345 | 2401 |
| 3 | 517 | 965 | 1056 | 1067 |
| 5 | 278 | 370 | 383 | 385 |
| 10 | 88 | 96 | 96 | 97 |
Above a population of about 100,000 the requirement stops changing meaningfully.