Margin of Error Calculator
Calculate a survey’s margin of error and confidence interval, or find the sample size needed to meet a desired margin of error.
For proportions under simple random sampling. Use a finite population size only for sampling without replacement.
Your results
Enter values and click Calculate.
These are normal-approximation estimates, not a measure of nonresponse, coverage or measurement bias. Sample-size planning does not generate an observed confidence interval.
How to Use the Margin of Error Calculator
Choose the margin-of-error mode to analyze a completed survey. Enter its sample size, observed percentage and confidence level. Optionally enter the total population size if you sampled without replacement.
For survey planning, select Required sample size and enter your target margin of error. If the expected percentage is unknown, 50% produces the largest conventional sample-size estimate.
Margin of error formula
Margin of error = Z × √[p(1 − p) / n]
Here, Z is the confidence critical value, p is the sample proportion as a decimal, and n is sample size. Multiply the result by 100 to express percentage points. For a finite population, multiply by √[(N − n)/(N − 1)].
Required sample size formula
n₀ = Z² × p(1 − p) / e²
The target e is expressed as a decimal. With a finite population, use n = N × n₀ / (N − 1 + n₀). Round up to the next whole respondent. At 95% confidence, p = 50% and ±5 percentage points, the required sample size is 385 without population correction.
Example
With 1,000 respondents, a 50% observed proportion, and 95% confidence, the approximate margin of error is ±3.10 percentage points. The normal-approximation interval is 46.90% to 53.10%.
Important limitations
These formulas assume simple random sampling. They do not account for weighting, cluster sampling, nonresponse bias or other survey errors. For proportions very close to 0% or 100%, consider a Wilson or exact binomial interval instead.
Frequently Asked Questions
What is margin of error?
It expresses sampling uncertainty around an estimate at a chosen confidence level.
What is the margin of error for 1,000 respondents?
About ±3.10 percentage points at 95% confidence with a 50% observed proportion and no population correction.
How do I calculate a confidence interval?
Under this normal approximation, subtract and add the margin of error to the observed percentage.
Why use 50% for sample planning?
It maximizes p(1 − p) and gives a conservative sample-size estimate under this formula.
What does 95% confidence mean?
Under the method’s assumptions, about 95% of similarly constructed intervals from repeated random samples would contain the population proportion.
What is finite population correction?
It adjusts the standard error for sampling without replacement from a known finite population.
How many respondents are needed for ±5 points?
About 385 at 95% confidence and a 50% expected proportion, without finite population correction.
Does margin of error include nonresponse bias?
No. It covers random sampling uncertainty under the stated assumptions, not all possible survey errors.
Can I use it for a sample mean?
No. It is designed for percentages and proportions, not averages of continuous measurements.
Does a larger sample reduce margin of error?
Generally yes, assuming comparable sampling methods and other inputs.