Every poll headline that says “plus or minus 3 points” is quoting a margin of error. It is half the width of a confidence interval: the distance on either side of a sample result within which the true population value plausibly lies. This calculator finds the margin for a percentage or an average, applies the finite population correction when you sample a large share of a small group, and answers the question readers actually ask: is that lead real?
How to use the margin of error calculator
- Pick percentage for a poll or survey share, or average for a mean.
- Enter the sample size. For a percentage, enter the reported share; for an average, enter the standard deviation (and the mean, if you want the interval).
- To test a lead, enter the second option’s percentage from the same poll.
- Choose the confidence level and, for a small population, its size.
- The tape shows the margin and plausible range; the table shows how the margin changes with sample size.
Margin of error formulas
For a sample proportion p̂ from n responses:
For a sample mean with standard deviation s:
For the lead between two shares from the same poll, the two shares are negatively related (a vote for one is a vote not cast for the other), so the variance of the difference is
When the sample is a noticeable fraction of a population of size N, multiply any margin by √((N − n) ÷ (N − 1)).
Worked example
A poll of 1,000 likely voters puts Candidate A at 52% and Candidate B at 44%.
- Standard error for A: √(0.52 × 0.48 ÷ 1,000) = 0.015799.
- Margin of error at 95%: 1.96 × 0.015799 = 0.0310, so A’s support is 52% ± 3.1 points, or 48.9% to 55.1%.
- The lead is 8 points. Its standard error is √((0.52 + 0.44 − 0.08²) ÷ 1,000) = 0.03088.
- Margin on the lead: 1.96 × 0.03088 = ±6.05 points. Since 8 > 6.05, the lead is statistically significant at 95% confidence.
Notice that the margin on the lead (6.05) is close to double the single-share margin (3.1). Had the lead been 5 points, it would have looked “outside the margin of error” by the naive comparison while actually being too close to call.
How sample size drives the margin
The margin falls with the square root of the sample size, so gains slow down quickly:
| Sample size | Worst-case margin (95%) |
|---|---|
| 100 | ±9.8 points |
| 400 | ±4.9 points |
| 1,000 | ±3.1 points |
| 2,500 | ±2.0 points |
| 10,000 | ±1.0 point |
Quadrupling the sample only halves the margin. To plan a survey around a target margin, run the numbers backward with the sample size calculator.
Subgroups have bigger margins
A national poll of 1,000 might include only 150 voters under 30. Their results carry a margin of about ±8 points, so swings in subgroup numbers between polls are often noise. Always check the subgroup n before reading much into a breakdown.
Worst case versus actual share
Pollsters often publish one margin for the whole survey, computed at 50%, because that is the largest possible value. A share far from 50% has a smaller margin: at 10% with n = 1,000 it is only ±1.9 points.
Related tools
For an interval with a choice of methods (Wilson, exact) use the confidence interval calculator. To turn raw vote counts into shares first, use the vote percentage calculator.
The margin of error covers random sampling error only, assuming a probability sample. Online opt-in panels and weighted samples have larger effective margins than these formulas suggest.
Frequently asked questions
What does a margin of error of ±3 points mean?
If a poll of 1,000 people shows 52% and the margin is ±3.1 points at 95% confidence, the method used would capture the true population share in about 95 of 100 repeated polls with intervals like 48.9% to 55.1%. It measures random sampling error only.
Is a lead within the margin of error a tie?
Not exactly. The margin on the gap between two candidates is roughly double the margin on each share, so comparing the lead with the single-share margin is the wrong test. This calculator computes the margin on the lead directly. A lead smaller than that margin is too close to call, not proof the race is tied.
Why do most national polls use about 1,000 people?
Because the margin of error depends on the sample size, not the population size. At 1,000 respondents the worst-case margin is about ±3.1 points; getting to ±2 points takes about 2,400 respondents, a large extra cost for a modest gain.
What does the margin of error not cover?
It ignores every non-random error: people who refuse to answer, leading question wording, an outdated list of phone numbers, or respondents who change their minds. Those biases can be larger than the sampling error and do not shrink with a bigger sample.
Should I use z or t for the margin of error of a mean?
Use t when the standard deviation is estimated from the same sample, which is the default here. With large samples the difference is tiny; with 1,000 observations the 95% critical value is 1.962 for t versus 1.960 for z.