Margin of Error Calculator
Find the margin of error for a survey proportion at a chosen confidence level. Tells you how far a poll result might be from the true population value.
Last updated: May 2026
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About this calculator
The margin of error quantifies the random sampling uncertainty in a survey estimate — how far a measured proportion is likely to sit from the true value in the whole population. For a proportion it is MOE = z × √(p(1 − p) / n), where p is the Sample Proportion (as a decimal), n is the Sample Size, and z is the Z-Score corresponding to your confidence level (1.96 for 95%, 1.645 for 90%, 2.576 for 99%). The result is usually expressed as a percentage and reported as 'plus or minus' that amount. Two forces drive it: larger samples shrink the margin (via the √n in the denominator), and the proportion itself matters because p(1 − p) is largest at p = 0.5, which is why pollsters often quote the conservative 'maximum' margin assuming a 50/50 split. A crucial feature is diminishing returns — because of the square root, cutting the margin in half requires quadrupling the sample, so going from 1,000 to 4,000 respondents only halves the error. Edge cases: this standard formula assumes simple random sampling from a large population and a roughly normal sampling distribution, which holds well when both np and n(1 − p) exceed about 10. For small populations a finite-population correction reduces the margin; for very small samples or extreme proportions, exact methods are more accurate. The margin captures only sampling error — not bias from poor question wording, non-response, or unrepresentative samples.
How to use
Example 1 — a poll of 1,000 people where 50% favor a candidate, at 95% confidence. Enter Sample Size = 1000, Sample Proportion = 50, Z-Score = 1.96. MOE = 1.96 × √(0.5 × 0.5 / 1000) × 100 = 1.96 × 0.01581 × 100 ≈ 3.10%. Verify: this is the familiar '±3 points' you hear in election coverage, meaning the true support is likely between 47% and 53%. Example 2 — a smaller survey of 400 people where 30% chose an option, at 95% confidence. Enter 400, 30, 1.96. MOE = 1.96 × √(0.3 × 0.7 / 400) × 100 = 1.96 × 0.02291 × 100 ≈ 4.49%. Verify: the smaller sample and the calculation give a wider ±4.5% band, so the true value plausibly ranges from about 25.5% to 34.5%.
Frequently asked questions
What z-score should I use for my confidence level?
The z-score is set by how confident you want to be that the interval contains the true value. The three standard choices are 1.645 for 90% confidence, 1.96 for 95% confidence (by far the most common in polling), and 2.576 for 99% confidence. A higher confidence level uses a larger z-score and therefore produces a wider margin of error — you trade precision for certainty. If you need a confidence level between the standard ones, you can look up the corresponding z-value from a normal distribution table. Always report which confidence level you used, because a margin of error is meaningless without it.
Why does the margin of error use a 50% proportion as the worst case?
The term p(1 − p) inside the square root is maximized when p = 0.5, giving the largest possible margin for a given sample size. Pollsters often quote this 'maximum margin of error' so they can report a single conservative figure regardless of the actual result. As the proportion moves toward 0% or 100%, p(1 − p) shrinks and the true margin narrows. This is why a result like 90%–10% actually has a tighter margin than a 50%–50% split at the same sample size. Using the 50% assumption is safe but can overstate the uncertainty for lopsided results.
How much does increasing the sample size help?
It helps, but with sharply diminishing returns, because the margin of error scales with 1/√n. To cut the margin in half you must quadruple the sample: going from 1,000 to 2,000 respondents only narrows a ±3.1% margin to about ±2.2%, and you would need 4,000 to reach roughly ±1.5%. This square-root law is why most national polls settle around 1,000–1,500 respondents — beyond that, the extra precision rarely justifies the cost. It also means a poll of 600 is not dramatically worse than one of 1,200. Understanding this relationship helps you judge whether a larger, more expensive survey is actually worth it.
What error does the margin of error NOT include?
The margin of error captures only random sampling error — the luck of which people happened to be selected. It completely ignores systematic errors, which are often larger and more dangerous: biased question wording, non-response bias when certain groups are harder to reach, coverage gaps from the sampling frame, and respondents who answer untruthfully. A poll can have a tiny ±2% sampling margin yet be wildly wrong because its sample was not representative. This is the most common misuse of the statistic — treating ±3% as the total uncertainty when it is only one component. Always weigh methodology and potential bias alongside the reported margin.
When should I NOT use this margin-of-error formula?
This formula assumes simple random sampling from a large population and a normal approximation, so it breaks down in several cases. For small samples or proportions near 0% or 100% (when np or n(1−p) is below about 10), the normal approximation is poor and you should use an exact method such as the Wilson or Clopper–Pearson interval. If your sample is a large fraction of a small population, apply a finite-population correction, which reduces the margin. Complex survey designs with stratification or clustering need design-adjusted standard errors, not this simple formula. And it does not apply to means or other statistics — those use a different margin-of-error expression based on the standard deviation.