Margin of Error Calculator
Estimate a survey proportion's margin of error, confidence interval precision, and standard error with an optional finite population correction.
About margin of error
A margin of error summarizes the sampling uncertainty around an estimated population proportion. Survey results are based on a sample rather than every member of a population, so another random sample would usually produce a slightly different percentage. The margin of error creates a symmetric range around the observed proportion. If 50 percent of 1,000 respondents choose an option and the 95 percent margin of error is about 3.1 percentage points, the familiar interval is approximately 46.9 to 53.1 percent. The calculation begins with the standard error of a proportion: the square root of p times one minus p divided by n. Here p is entered as a decimal and n is the sample size. A critical Z value then scales that standard error for the requested confidence level. This calculator uses 1.645 for 90 percent confidence, 1.96 for 95 percent, and 2.576 for 99 percent. Greater confidence produces a wider interval because it must cover more potential samples. When a sample is drawn without replacement from a known finite population and represents a meaningful share of that population, the finite population correction can improve precision. Entering the population size multiplies the standard error by the square root of the unsampled population divided by one less than the total population. Leave that field blank for a very large or conceptually unlimited population. The correction assumes a probability sample from the stated population. The largest conservative margin for a fixed sample size occurs at 50 percent because p times one minus p is then maximized. Researchers often use 50 percent while planning a study when the true proportion is unknown. Observed proportions near zero or one produce smaller textbook standard errors, although simple normal intervals can behave poorly at extremes or with small samples. Wilson or exact binomial intervals may be preferable in those cases. A reported margin of error covers random sampling variation under the model; it does not capture nonresponse, poor wording, coverage gaps, selection bias, weighting error, measurement error, or data processing mistakes. A huge biased sample can have a tiny calculated margin while remaining inaccurate. Always report the sampling design, confidence level, sample size, field dates, and known limitations together with the estimate.
Margin of error examples
| Survey inputs | Margin | Meaning |
|---|---|---|
| n = 1,000, p = 50%, confidence = 95% | ±3.099% | The conservative case for a large population. |
| n = 400, p = 40%, confidence = 95% | ±4.801% | An observed proportion away from 50 percent. |
| n = 1,000, p = 50%, confidence = 99% | ±4.073% | Higher confidence widens the margin. |
How to estimate survey precision
- Enter the number of completed independent observations.
- Enter the observed sample percentage between zero and one hundred.
- Choose the confidence level used for the interval.
- Optionally enter a finite population larger than the sample, then calculate.
Margin of error FAQ
Why is 50 percent the conservative proportion?
The product p times one minus p reaches its maximum at one half. That produces the largest standard error for a fixed sample size.
Does a 95 percent interval have a 95 percent chance of containing the truth?
In frequentist terms, 95 percent of intervals from repeated valid samples would cover the fixed population value. The statement concerns the long-run procedure rather than a probability assigned to one fixed interval.
When should I enter population size?
Use it for sampling without replacement when the sample is a substantial fraction of a known finite population. Otherwise leave it blank because the correction is negligible.
Does margin of error include survey bias?
No, it quantifies model-based random sampling error only. Coverage, nonresponse, wording, and selection problems require separate assessment.
How can I reduce the margin of error?
Increase the effective sample size or accept a lower confidence level. Better sampling design can also increase effective information without simply collecting more responses.