Margin of Error Calculator
Enter your sample result, sample size and confidence level to get the margin of error and the resulting interval. Add a population size to apply the finite population correction.
Last updated November 4, 2026
From 0 to 100. Use 50 if unsure.
Whole number of people or items, 1 or more.
Leave blank for a very large population.
Result
Margin of error = ±3.1%
| Measure | Value |
|---|---|
| Margin of error | ±3.1% |
| Confidence interval | 46.9% to 53.1% |
| Critical z | 1.96 |
| Standard error | 1.581% |
Assumes a simple random sample. The margin covers sampling error only, not biased questions or non-response.
Show the work
Formula: MOE = z × √(p × (1 − p) ÷ n) × √((N − n) ÷ (N − 1)); the last factor only when N is given
- p = 50% = 0.5; z for 95% confidence = 1.96.
- Standard error = √(0.5 × 0.5 ÷ 1,000) = 0.015811.
- Margin = 1.96 × 0.015811 = 0.03099, or ±3.1%.
- Interval = 50% ± 3.1% = 46.9% to 53.1%.
Result: Margin of error = ±3.1%
Margin of error formula
MOE = z × √(p(1 − p) ÷ n) [× √((N − n) ÷ (N − 1)) when the population N is known]
Here p is the sample proportion, n the sample size and z the critical value for the confidence level: about 1.645 at 90%, 1.960 at 95% and 2.576 at 99%.
When the sample is a large share of a small population, the finite population correction shrinks the margin. If the sample is the whole population, the margin is 0.
Using 50% gives the widest margin for a given sample size, which is why surveys plan with it when the result is unknown.
The margin covers random sampling error only. It does not account for leading questions, non-response or a non-random sample.
See how we calculate for the formulas, sources and review schedule behind every calculator.
How to use the Margin of Error Calculator
- Enter the sample result as a percentage; use 50 if unknown.
- Enter the sample size and pick the confidence level.
- Optionally enter the population size.
- Read the margin of error and interval, then open Show the work.
Worked example
A poll of 1,000 people with a 50% result at 95% confidence has a margin of error of about ±3.10 points. Margins at 50% and 95%:
| Sample size | Margin of error |
|---|---|
| 100 | ±9.8% |
| 385 | ±4.99% |
| 1,000 | ±3.1% |
| 2,000 | ±2.19% |
Frequently asked questions
What does a margin of error of 3% mean?
If the survey were repeated many times, about 95% of the intervals built this way (at 95% confidence) would contain the true value, each ±3 points around its result.
How does sample size affect the margin of error?
The margin shrinks with the square root of the sample size. Quadrupling the sample cuts the margin in half.
When should I enter a population size?
When your sample is a sizeable part of a known, small population. For 400 people from 1,000 the margin falls from 4.90% to 3.80%.
Can I use this for a mean instead of a percentage?
No. This tool is for proportions. For a mean, use the Confidence Interval Calculator.