Margin of Error Calculator
Compute the margin of error for surveys and polls (MOE = z* × √(p(1−p)/n)), solve the sample size for a target MOE, or build a confidence interval — with optional finite population correction and a z-score table.
Margin of error from sample size
Sample size for a target margin of error
Confidence interval for a proportion
Critical z* values (two-sided)
| Confidence | z* (4 dp) | z* (6 dp) |
|---|---|---|
| 80% | 1.2816 | 1.281552 |
| 85% | 1.4395 | 1.439531 |
| 90% | 1.6449 | 1.644854 |
| 95% | 1.9600 | 1.959964 |
| 97% | 2.1701 | 2.170090 |
| 98% | 2.3263 | 2.326348 |
| 99% | 2.5758 | 2.575829 |
| 99.5% | 2.8070 | 2.807034 |
| 99.9% | 3.2905 | 3.290527 |
z* is the standard-normal value with the stated probability in the middle: for 95%, Φ−1(0.975) = 1.96. Wider confidence needs a larger z*, hence a larger margin of error.
Worked example
A classic poll: n = 1,068, p = 50%, 95% confidence → MOE = 1.96 × √(0.25/1068) = ±2.9987% ≈ ±3.0 percentage points.
Observed 620 of 1,000 (95%): 62% ± 3.0084% → 58.99% to 65.01%.
Need ±3 points at 95%? Sample-size mode gives 1,068 respondents; for ±5 points it is only 385 — precision is expensive (halving the MOE quadruples n).
Scope & related tools
This calculator covers the one-sample survey margin of error for proportions. It does not compute z-scores from raw data, plan A/B tests by minimum detectable effect, or build intervals for a continuous mean:
- Z-Score Calculator — z-values from raw scores, p-values, and the normal distribution.
- AB Test Calculator — significance testing and sample-size planning by MDE and power.
- Standard Deviation of the Sample Mean — standard error and confidence intervals for a mean.
Frequently asked questions
What is margin of error?
The margin of error is the largest likely gap between a survey result and the true value in the whole population, at the stated confidence level. A poll reporting “52% ± 3 points (95% confidence)” means the true support is plausibly between 49% and 55%.
What is the margin of error formula?
MOE = z* × √(p(1−p)/n), where z* is the critical value for the confidence level (1.96 for 95%), p is the estimated proportion, and n is the sample size. The square-root term is the standard error of the proportion.
Which z-score do I use for 90%, 95%, and 99% confidence?
Use z* = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence (two-sided). These are standard-normal critical values; the z-score table on this page lists more levels.
Why do pollsters often assume p = 50%?
The product p(1−p) — and therefore the margin of error — is largest at p = 0.5. Using 50% gives the conservative, worst-case margin of error when the true proportion is unknown.
How do I find the sample size for a target margin of error?
Rearrange the formula: n = (z*/E)² × p(1−p), then round up. For ±3 points at 95% confidence with p = 0.5: n = (1.96/0.03)² × 0.25 ≈ 1,067.1, so you need 1,068 respondents.
What is the finite population correction (FPC)?
When the sample is a large fraction of the population (rule of thumb: more than 5%), multiply the margin of error by √((N−n)/(N−1)), where N is the population size. It shrinks the MOE because sampling without replacement from a small population carries less uncertainty.
How do I read “±3 percentage points”?
Add and subtract 3 points from the reported figure: 52% ± 3 means the 95% confidence interval is 49% to 55%. Saying “percentage points” keeps it distinct from a 3% relative change.
Does 95% confidence mean a 95% chance the true value is inside?
Not exactly. It means that if you repeated the survey many times, about 95% of the intervals computed this way would contain the true population value. One single computed interval either contains it or does not.
What is the difference between margin of error and standard error?
The standard error, √(p(1−p)/n), measures the sampling variability of the estimate; the margin of error multiplies it by z* to reach the chosen confidence level. MOE = z* × SE.
When is the normal-approximation formula valid?
A common rule of thumb is np ≥ 10 and n(1−p) ≥ 10, so the sampling distribution of the proportion is close enough to normal. For tiny samples or extreme proportions, exact methods are safer.
Why does doubling precision require quadrupling the sample?
The margin of error shrinks with 1/√n. To halve the MOE you must multiply n by four: going from ±6 points (about 267 respondents) to ±3 points (about 1,068) at 95% confidence, for example.
Can the margin of error be zero?
Only if you survey the entire population — then the finite population correction makes it exactly zero — or with an infinite sample. Any real sample leaves some sampling uncertainty.
Does the margin of error cover biased samples?
No. It quantifies sampling error only — the luck of the draw. Nonresponse, leading questions, and unrepresentative samples create bias that the margin of error does not measure.
What is a typical margin of error for public polls?
About ±3 percentage points at 95% confidence, which corresponds to roughly 1,000–1,100 respondents when p = 0.5. Larger polls report smaller margins; subgroup results have larger ones.
How should I report a confidence interval?
Give the estimate, the margin, and the interval: “52% ± 3 points; 95% CI 49–55%.” Always state the confidence level, since the same data gives a wider interval at 99% than at 90%.
Free tool by NRTechStudio