Sample Size Calculator
With population size 100000, margin of error wanted 5 percent, expected proportion 50 percent, expected response rate 100 percent, sample size comes to 384 — responses needed. It is reached in 5 steps, the last of which is ceil(385 / (1 + (385 - 1) / 100000)), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST/SEMATECH e-Handbook of Statistical Methods, not an approximation fitted to it.
How many people a survey needs for a given margin of error, with the finite population correction that makes small populations cheaper.
Formula and sources checked · How we check
Population size 100000, Margin of error wanted 5, Confidence level 95%, Expected proportion 50
384
Responses needed for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Critical value, z
1.96- Sample for an unlimited population
ceil(pow(1.959964, 2) * 0.5 * (1 - 0.5) / pow(0.05, 2))385- After the finite population correction
ceil(385 / (1 + (385 - 1) / 100000))384- People to invite at that response rate
ceil(384 / (100 / 100))384- Share of the population sampled
384 / 100000 * 1000.384 %
Worked example
A ±5% margin at 95% confidence needs 385 responses, whether the population is a hundred thousand or a hundred million — which is why national polls sample about a thousand people and not a proportion of the country.
How to work it out yourself
- 1.Set the margin of error first: it drives the answer more than anything else, and halving it quadruples the sample.
- 2.Leave the expected proportion at 50% unless you have a real prior estimate. It is the worst case, so it is the safe one.
- 3.Use the response rate line to work out how many people to contact. A 20% response rate means inviting five times the sample you need.
Responses needed by margin of error
| Margin of error wanted (percent) | Responses needed | People to invite at that response rate |
|---|---|---|
| 1 | 8,763 | 8,763 |
| 1.5 | 4,095 | 4,095 |
| 2 | 2,345 | 2,345 |
| 2.5 | 1,514 | 1,514 |
| 3 | 1,057 | 1,057 |
| 4 | 598 | 598 |
| 5 | 384 | 384 |
| 6 | 267 | 267 |
| 8 | 151 | 151 |
| 10 | 97 | 97 |
95% confidence, population of 100,000, proportion 50%.
The formula
- Critical value, z
- Sample for an unlimited population
ceil(pow(1.959964, 2) * 0.5 * (1 - 0.5) / pow(0.05, 2)) - After the finite population correction
ceil(385 / (1 + (385 - 1) / 100000)) - People to invite at that response rate
ceil(384 / (100 / 100)) - Share of the population sampled
384 / 100000 * 100
Source: NIST/SEMATECH e-Handbook of Statistical Methods — sample sizes required
Questions people actually ask
- Why do national polls only ask about a thousand people?
- Because the population size barely matters once it is large. A ±3% margin at 95% confidence needs 1,067 responses from a town of a million and 1,068 from a country of three hundred million. What drives the number is the margin you will accept, not the size of the group you are describing.
- What is the finite population correction?
- An adjustment for sampling a large share of a small group. Surveying 400 people out of 1,000 tells you more than 400 out of a million, because you have asked 40% of everyone. The correction cuts the required sample — from 385 to 278 in a population of a thousand — and it is negligible above about 20,000.
- What does the expected proportion do?
- It sets the variance. A split near 50/50 is the most uncertain and needs the largest sample; a proportion you expect to be 5% or 95% needs far fewer. Assuming 50% when you do not know is the conservative choice, and it is what every published "sample size for a ±3% margin" figure assumes.
- Does a bigger sample fix a biased one?
- No, and this is the failure that arithmetic cannot see. If the people who respond differ systematically from those who do not, a larger sample estimates the wrong thing more precisely. The 1936 Literary Digest poll surveyed 2.4 million people and called the election wrong, because its list was drawn from car and telephone owners.
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