How to Calculate Margin of Error for a Proportion
The familiar survey margin-of-error formula depends on sample size, estimated proportion and confidence level. It represents modelled sampling uncertainty, not every source of survey error.
Method
- Enter the sample size.
- Enter the estimated proportion as a percentage.
- Choose 90%, 95% or 99% confidence.
- Calculate the proportion standard error and multiply by the matching z critical value.
Worked example
n = 1000 and p = 50% at 95% gives about ±3.10 percentage points.
Using p = 50% is conservative when no prior proportion estimate is available.
Notes
- The normal approximation is not appropriate for every small or extreme-proportion sample.
- Weighting, clustering, nonresponse and measurement error are not included.
- This is a classroom/statistical planning calculation, not a guarantee about a real survey.
- The normal approximation becomes weak when expected successes or failures are too small. Use the displayed condition warning as part of the result, and switch to an exact or course-required method when the approximation assumptions are not met.
A margin of error needs a sampling design, not just a percentage
Under an illustrative simple random-sampling approximation, a sample proportion of 0.50 with n = 400 has standard error sqrt(0.50 × 0.50 ÷ 400) = 0.025. Multiplying by 1.96 gives about 0.049, or a 4.9-percentage-point 95% normal-approximation margin of error. That is not a guarantee that any voluntary online poll represents a population.
Nonresponse, selection bias, weighting, clustering and small or extreme samples can make that textbook calculation inappropriate. State the population, how people were selected and what uncertainty is being modeled before publishing a survey claim. A calculator cannot repair a biased sampling process.
Margin of error depends on the estimated proportion and sample size
For a normal approximation, the standard error of a proportion uses p(1-p) divided by sample size under a square root, then multiplies by the critical value for the selected confidence level. Proportions near 0.5 produce the largest variance for a fixed sample size.
A larger sample reduces margin of error, but again the relationship follows the square root. Cutting the margin of error in half generally requires about four times the sample size under similar assumptions.
Check whether the normal approximation and sampling design are reasonable
Very small samples or proportions close to 0 or 1 can make the simple normal approximation poor. Survey design, clustering, weighting and finite populations can also require adjustments beyond the basic formula.
The numerical margin of error addresses sampling variability under the model. It does not include nonresponse bias, measurement error, question wording effects or an unrepresentative sampling frame.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Mean Confidence Interval Calculator — Calculate a normal-approximation confidence interval for a mean from the mean, standard deviation, sample size and confidence level.
- Standard Error Calculator — Calculate the standard error of a sample mean from standard deviation and sample size.
- Z-Score & Normal Percentile Calculator — Calculate a z-score from value, mean and standard deviation, plus normal-model percentile and right-tail probability.