Margin of Error Calculator
Find the margin of error of a poll or survey percentage from the sample size, the result and the confidence level.
Last updated: October 2026
Margin of Error
3.1 %
Formula below · 2 sources (pewresearch.org, Wikipedia) · Updated Oct 2026
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About this calculator
The margin of error tells you how far a survey percentage could plausibly be from the true value in the whole population, because only a sample was asked.
MOE = z × √(p × (1 − p) ÷ n)
p is the sample proportion as a decimal, n the sample size and z the critical value for your confidence level: 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. The result is in percentage points and is read as plus or minus: 52% ± 3.1 points means the true figure is plausibly between 48.9% and 55.1%.
Two things drive the margin. Larger samples shrink it, but only with the square root of n, so quadrupling the sample halves the margin. And the result itself matters: p × (1 − p) is largest at 50%, so pollsters usually quote the margin for a 50/50 split as the worst case.
The formula assumes a simple random sample from a large population and at least about 10 respondents in each answer group (n × p and n × (1 − p) both above 10). It covers sampling error only. Poorly worded questions, people who do not respond and unrepresentative samples add error that no formula captures. The difference between two candidates in the same poll has a larger margin than either share alone.
How to use
Enter the number of respondents, the percentage who gave the answer you care about and the z-score for your confidence level. The default, 1,000 respondents split 50/50 at 95% confidence, gives 1.96 × √(0.5 × 0.5 ÷ 1,000) = 0.0310, or ±3.1 points, the figure often quoted for national polls. For 400 respondents with 30% choosing an option, the margin is 1.96 × √(0.3 × 0.7 ÷ 400) = ±4.49 points, so the true share is plausibly 25.5% to 34.5%.
Frequently asked questions
Which z-score should I use?
1.96 for 95% confidence, the standard for published polls; 1.645 for 90%; and 2.576 for 99%. Higher confidence gives a wider margin.
How much does a bigger sample help?
Less than you might expect. At 50% and 95% confidence, 1,000 respondents give ±3.1 points, 2,000 give ±2.2 and 4,000 give ±1.5. Halving the margin takes four times the sample.
Why do pollsters quote the margin at 50%?
The term p × (1 − p) is largest at 50%, so the 50/50 margin is the widest possible for that sample size. A lopsided result such as 90/10 has a smaller margin.
What does the margin of error leave out?
Only random sampling error is included. Non-response, question wording, people who answer untruthfully and a sample that does not represent the population can all move results by more than the stated margin.
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