Transparent method
The formula
Worked example
Example inputs
How to interpret the result
Cutting a margin of error in half doesn't double the sample size needed: it roughly quadruples it, because required sample size scales with the square of how tight the margin needs to be. A survey aiming for a very precise result can need dramatically more respondents than one accepting a looser estimate, out of proportion to how much tighter the target actually is.
Before you rely on it
What to check
Check the relationship between your target margin and the resulting sample size before committing to a survey budget: small precision gains get expensive fast.
The common error
Where people go wrong with sample size calculator
Assuming sample size scales in a straight line with precision. Wanting a margin of error half as wide isn't a modest ask; it's close to a fourfold increase in the respondents needed.
Sensitivity evidence
How desired margin of error changes the required sample size
| Desired margin of error | Required sample size | Z-value used |
|---|---|---|
| 3.0% | 1,068 | 2 |
| 4.0% | 601 | 2 |
| 5.0%worked example | 385 | 2 |
| 6.0% | 267 | 2 |
| 7.0% | 196 | 2 |
Every input, tested
Which input moves the required sample size most
| Input | Tested from | To | Required sample size at each end | Swing |
|---|---|---|---|---|
| Confidence level | 90% | 99% | 271 to 664 | 393 (102%) |
| Desired margin of error | 4.0% | 6.0% | 601 to 267 | 334 (87%) |
| Estimated population proportion | 48.0% | 52.0% | 384 to 384 | none |
Two variables at once
Required sample size by desired margin of error and estimated population proportion
| Desired margin of error \ Estimated population proportion | 46.0% | 50.0% | 54.0% |
|---|---|---|---|
| 3.0% | 1,061 | 1,068 | 1,061 |
| 4.0% | 597 | 601 | 597 |
| 5.0% | 382 | 385 | 382 |
| 6.0% | 266 | 267 | 266 |
| 7.0% | 195 | 196 | 195 |
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Desired margin of error | 5.0% | |
| Confidence level | 95% | How confident the result should be. |
| Estimated population proportion | 50.0% | |
| Required sample size | 385 | |
| Z-value used | 2 | |
Inputs, definitions and assumptions
Desired margin of error
Confidence level
How confident the result should be.
Estimated population proportion
Where the inputs come from
Enter the desired margin of error as a percentage, the z value for your confidence level, 1.96 for 95%, and the expected proportion, which is 50% when you have no idea, since that gives the largest sample. Use this for surveys where you are estimating a percentage. A smaller margin of error or higher confidence increases the required sample. The population size matters only when the sample is a large share of it.
When the result stops applying
The calculation assumes a random sample and ignores bias from who responds. If the population is small, a correction reduces the needed sample. Surveys of means rather than proportions need a different formula. Statistical power for comparing groups requires another calculation.
What to compare it with
See the interval that results on the confidence interval calculator, the normal distribution on the z-score and normal probability pages, and testing on the p-value calculator. The binomial probability page covers success counts, and the percentage calculator helps with reporting.
People also ask
Frequently asked questions
Why does the sample size stop growing with the population?
Because the precision of a random sample depends mainly on its size, not on the population size, unless the sample is a large fraction of it.
What proportion should I use if I do not know it?
Use 50%, which gives the largest and therefore most conservative sample size, so your survey will be precise enough whatever the true proportion turns out to be.
How do I reduce the margin of error?
Increase the sample size. Cutting the margin of error in half requires four times as many responses, so precision gets expensive quickly.