What the margin of error measures, and what it ignores
The margin of error quantifies one thing only: the random variation that comes from surveying a sample instead of everyone. Draw a different random 1,000 people and you would get a different percentage. The margin of error says how far apart those percentages would typically land.
It says nothing about the other ways a survey goes wrong. A question worded to lead the respondent, a sampling frame that excludes people without landlines, a 4% response rate where the people who answered differ systematically from the people who did not — none of that appears in the formula. Those are non-sampling errors, they are usually larger than the sampling error in modern polling, and no amount of sample size reduces them. When a published poll says “±3 points”, that number is a floor on the uncertainty, not a bound on it.
The margin of error is also always attached to a confidence level. “±3.1 points” on its own is incomplete; “±3.1 points at 95% confidence” is the full statement. Raising the confidence level widens the margin, because you are demanding that the interval succeed more often. At the same sample size, moving from 95% to 99% multiplies the margin by 2.575829/1.959964 = 1.314.
Where the formula comes from
Every margin of error has the same shape: a critical value times a standard error. The critical value comes from the normal distribution and depends only on the confidence level. The standard error depends on how variable the thing you measured is, and on how many observations you took.
For a percentage, each respondent contributes a 0 or a 1, and the variance of that Bernoulli variable is p(1−p). Divide by n and take the square root and you have the standard error of the proportion. That product is largest at p = 0.5, where it equals 0.25, and it falls away symmetrically towards either boundary: at p = 0.1 it is 0.09, so the standard error is √(0.09/0.25) = 0.6 times its value at 50%. This is why survey planners who do not yet know the answer assume 50% — it is the worst case, and any other true value gives a narrower interval than promised.
For a mean, the variance you divide by n is the variance of the measurement itself, σ², so the standard error is σ/√n. The rest of the machinery is identical. If σ was estimated from the same sample rather than known, the strictly correct critical value comes from the t distribution rather than the normal; the confidence interval calculator handles that case, and the difference is negligible above a few hundred responses.
The finite population correction, √((N−n)/(N−1)), handles sampling without replacement from a population that is not effectively infinite. Sample everyone and it becomes zero: a census has no sampling error. Sample 5% of the population and it is √(0.95) ≈ 0.975, a 2.5% reduction that is usually not worth mentioning. Sample half and it is about 0.707, which cuts the margin of error by nearly a third — far too large to ignore.
Worked example: 1,000 respondents, 52% support
A national poll of 1,000 adults finds 52% support for a proposal, and you want the 95% margin of error. The population is 40 million, so the finite population correction is irrelevant.
- Convert the percentage. p = 0.52, so p(1−p) = 0.52 × 0.48 = 0.2496.
- Standard error. √(0.2496 / 1000) = √0.0002496 = 0.0157988, or 1.5799 percentage points.
- Critical value. At 95%, z = 1.959964.
- Margin of error. 1.959964 × 1.5799 = 3.0966 percentage points.
- Interval. 52 − 3.10 = 48.90% to 52 + 3.10 = 55.10%.
Report it as “52%, ±3.1 points at 95% confidence”. Note that the interval contains 50%, so this poll does not establish majority support.
Now change one thing: suppose the population is a professional body of 4,000 members rather than the whole country. The correction is √((4000−1000)/(4000−1)) = √(3000/3999) = √0.750188 = 0.866134, and the margin of error falls to 3.0966 × 0.866134 = 2.682 percentage points. Sampling a quarter of a population buys you a genuinely tighter estimate, and ignoring the correction here would overstate your uncertainty by 15%.
Reading a margin of error correctly
Apply it to each candidate or category separately, not to the gap between them. A 3-point margin on 52% and a 3-point margin on 48% does not mean the 4-point lead is “within the margin of error” in any simple sense. The margin of error on a difference between two shares of the same sample is roughly twice the margin on either one, because the two percentages move in opposite directions as the sample changes. Poll reporting gets this wrong constantly.
Watch for subgroup margins. A national poll of 1,000 that reports on 180 voters under 30 carries a margin of error on that subgroup of 1.959964 × √(0.25/180) × 100 = 7.30 points, not 3.10. Any crosstab shrinks n, and the margin grows as 1/√n.
Remember that halving the margin costs four times the sample. Going from ±3.1 to ±1.55 points at 95% means moving from 1,000 respondents to 4,000. That square-root relationship is the reason national polls cluster around 1,000 to 1,500 completes: it is the point where further spending buys very little precision.
Finally, a margin of error assumes a probability sample. Opt-in web panels, river samples and convenience samples do not satisfy that assumption, and reputable pollsters using them report a “credibility interval” or a modelled error estimate rather than a classical margin of error.
Margin of error at 95% confidence, by sample size and observed percentage
| Sample size | p = 50% | p = 30% | p = 10% |
|---|---|---|---|
| 100 | 9.80 | 8.98 | 5.88 |
| 250 | 6.20 | 5.68 | 3.72 |
| 384 | 5.00 | 4.58 | 3.00 |
| 500 | 4.38 | 4.02 | 2.63 |
| 1,000 | 3.10 | 2.84 | 1.86 |
| 2,000 | 2.19 | 2.01 | 1.31 |
| 5,000 | 1.39 | 1.27 | 0.83 |
| 10,000 | 0.98 | 0.90 | 0.59 |
At p = 50% the margin passes 5.00 points between n = 384 (5.0010) and n = 385 (4.9944). Cochran's formula solves to 384.15, so survey plans specify 385 respondents — the first sample size that is genuinely inside a 5-point target.
Common mistakes
- Applying the whole-sample margin to a subgroup. The margin depends on the number of people in the cell being reported, not on the size of the survey.
- Treating the margin as covering every kind of error. It covers random sampling variation only. Non-response bias, question wording and frame coverage sit entirely outside it.
- Comparing two candidates by checking whether the gap exceeds one margin. The margin on a difference is larger than the margin on either share.
- Forgetting the finite population correction on internal surveys. Surveying 200 of a company's 600 employees without correcting overstates the margin by a factor of 1/√(400/599) = 1.22.
- Using the observed percentage when planning, then reporting as if 50% had been assumed. Plan with 50% for the worst case, but report with the percentage you actually observed.
- Quoting a margin of error for a non-probability panel. The formula assumes every member of the population had a known, non-zero chance of selection.
Where this fits with sample size and interval calculators
The margin of error, the confidence interval and the required sample size are three views of one equation. This page fixes n and solves for E. The survey sample size calculator fixes E and solves for n, using Cochran's rearrangement of the same expression. The confidence interval for a proportion reports the same arithmetic as a pair of limits and offers the Wilson interval, which behaves far better than this normal approximation when p is close to 0 or 1.
For averages rather than percentages, use the confidence interval calculator for a mean, which applies the t distribution when your standard deviation is a sample estimate. If you are comparing two survey percentages — this month against last month, treatment against control — the right tool is the two-proportion z-test, which builds a standard error for the difference rather than for either share.
The finite population correction and the p(1−p) variance term both come from classical survey sampling theory as set out by Cochran; the American Association for Public Opinion Research publishes the disclosure standards that govern how a margin of error should be reported alongside a poll.