What this calculator does
Sample size determines how precise a survey can be. The relationship is governed by the square root, which has a counterintuitive consequence: halving the margin of error requires four times as many responses, and the returns diminish steeply after the first few hundred.
Population size matters far less than most people expect. Once the population is large relative to the sample, adding more people to it barely changes the requirement. A national survey of 200 million and a city survey of 200,000 need almost identical sample sizes for the same precision.
The formula
The base formula multiplies the squared critical value by the expected variance and divides by the squared margin of error. The finite population correction then reduces this when the sample is a substantial share of the population, since sampling without replacement from a small pool is more informative per response.
| Term | Meaning |
|---|---|
| Margin of error | The ± precision you want. |
| p | The expected proportion. 50% maximises variance and is the safe assumption. |
| N | The population size. Enter zero when it is very large. |
| Finite population correction | The adjustment for sampling a large share of a small population. |
The inputs explained
| Field | What to enter |
|---|---|
| Margin of error (%) | Your target margin of error as a percentage. ±5% is a common survey standard. |
| Expected proportion (%) | The proportion you expect to find. If you have no idea, use 50 per cent: it gives the largest and therefore safest sample size. |
| Population size (0 = very large) | The size of the population you are sampling from. Enter zero if it is very large or unknown. |
| Confidence level | The confidence level. 95 per cent is the usual convention in survey research. |
When to use it
Planning a customer survey
Decide the margin of error you can live with and read off the sample. ±5 per cent is common in commercial research; ±3 per cent is closer to a political polling standard.
Sampling a small population
If you are surveying a company of 500 or a school of 800, enter that population. The correction can reduce the required sample substantially.
Justifying a research budget
The comparison rows for ±3 and ±1 per cent make the cost of extra precision explicit: the jump to ±1 per cent is roughly twenty-five times the sample of ±5 per cent.
Assessing someone else’s survey
A poll of 400 people carries a margin of about ±5 per cent. If two options in that poll sit within a few points of each other, the poll cannot distinguish them.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Sample size by margin of error
The sample needed at different levels of precision, assuming a 50 per cent proportion and a large population.
| Margin of error | Sample size needed | For a ±3% margin | For a ±1% margin |
|---|---|---|---|
| ± 10% | 97 | 1,068 | 9,604 |
| ± 5% | 385 | 1,068 | 9,604 |
| ± 4% | 601 | 1,068 | 9,604 |
| ± 3% | 1,068 | 1,068 | 9,604 |
| ± 2% | 2,401 | 1,068 | 9,604 |
| ± 1% | 9,604 | 1,068 | 9,604 |
How population size affects the requirement
The same target precision against populations of very different sizes.
| Population | Sample size needed | Ignoring population size |
|---|---|---|
| 100 | 80 | 385 |
| 500 | 218 | 385 |
| 1,000 | 278 | 385 |
| 5,000 | 357 | 385 |
| 100,000 | 383 | 385 |
| Very large | 385 | 385 |
Questions
What proportion should I assume?
Use 50 per cent unless you have solid prior information. It maximises the variance term and therefore gives the largest, safest sample size. Assuming a more extreme proportion reduces the requirement but risks under-sampling if you are wrong.
Why does population size matter so little?
Because precision depends on the absolute number of responses, not the fraction of the population they represent. Once the population is large relative to the sample, the correction factor approaches one and drops out.
Is 1,000 responses enough for a national survey?
For a ±3 per cent margin at 95 per cent confidence, roughly 1,068 is needed: which is why so many national polls report around a thousand respondents. It is not a coincidence.
Does this account for non-response?
No. It gives the number of completed responses required. If you expect a 20 per cent response rate, you need to contact five times that many people.
What about sampling bias?
Sample size controls random error only. A biased sample gives a precisely wrong answer: and a larger biased sample gives a more precisely wrong one. How you select respondents matters at least as much as how many you get.
For the interval around a result, see the confidence interval calculator.