Margin of Error Calculator

Background.

The Quanta margin of error calculator takes the three numbers behind every reported poll or survey result — the sample size n, the sample proportion p̂ (the share who answered a certain way), and a confidence level — and returns the margin of error, the resulting confidence bounds around p̂, and the z critical value used to compute it. Enter the defaults, n = 1000 and p̂ = 0.5 at 95% confidence, and the calculator returns a margin of error of about 0.031, or ±3.1 percentage points — the exact figure quoted in the fine print of nearly every major US national poll conducted with a sample of roughly a thousand respondents.

The margin of error answers a specific, precise question: if you drew many different random samples of the same size from the same population, how far would the sample proportion p̂ typically wander from the true population proportion, purely due to the luck of who happened to get sampled? A tighter margin means your single sample result is a more reliable stand-in for the truth; a wider margin means there is more sampling noise to account for. The now-familiar newspaper phrase 'this poll has a margin of error of plus or minus 3 percentage points, 19 times out of 20' is a direct translation of a 95% confidence level: if the same polling method were repeated over and over, about 95% of the resulting margins of error would actually contain the true population value.

The formula itself, e = z·√(p̂(1−p̂)/n), traces back to William Cochran's 1977 textbook Sampling Techniques, still the standard graduate reference for survey statisticians decades later. Three facts fall directly out of the formula and are worth internalizing before reading any poll. First, the margin of error shrinks only with the square root of the sample size, not proportionally — quadrupling your sample size (say from 1,000 to 4,000 respondents) only cuts the margin of error in half, which is why campaigns and pollsters rarely commission surveys much larger than a few thousand respondents; the marginal accuracy gained per additional dollar spent drops off fast. Second, the margin of error is largest when p̂ = 0.5 and shrinks as p̂ moves toward 0 or 1 — a race polling at 50-50 has more inherent sampling noise to report than one polling at 90-10, which is precisely why this calculator (and most professional pollsters) defaults p̂ to 0.5 when the true split is not yet known: it is the conservative, worst-case choice that never understates the true uncertainty. Third, the margin of error does not depend on the size of the overall population at all, once that population is reasonably large — polling 1,000 residents of a city of 50,000 produces essentially the same margin of error as polling 1,000 residents of a nation of 330 million, which surprises most people encountering the formula for the first time but follows directly from the fact that N does not appear in the base formula.

That last point comes with one qualification: when the population itself is small relative to the sample — for example, surveying 200 employees out of a company with only 800 total staff — sampling without replacement measurably shrinks the true uncertainty below what the infinite-population formula predicts, because each additional response removes a meaningfully large share of the remaining uncertainty. Cochran's finite population correction, √((N−n)/(N−1)), captures exactly this effect and is available here as an optional input; leave the population size at 0 to skip it whenever your population is large, unknown, or effectively unlimited (the standard case for national or state-level polling).

This calculator is the direct mirror image of Quanta's sample-size calculator: this page starts from a sample size and reports the resulting margin of error, while sample-size starts from a target margin of error and reports the sample size needed to hit it. Use whichever matches the question you actually have — designing a new survey (sample-size) or interpreting one that has already been run (margin-of-error).

What is margin of error calculator?

The margin of error for a sample proportion is the half-width of the confidence interval built around a survey's observed proportion p̂, expressed as e = z·√(p̂(1−p̂)/n), where n is the sample size, p̂ is the observed proportion (as a decimal between 0 and 1), and z is the critical value of the standard normal distribution corresponding to the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). Subtracting and adding e to p̂ produces the confidence interval [p̂ − e, p̂ + e], the range that is expected to contain the true population proportion with the stated confidence level, under repeated random sampling. The formula descends from the normal approximation to the binomial distribution: for a reasonably large n, the sampling distribution of p̂ is approximately normal with standard error √(p̂(1−p̂)/n), and multiplying that standard error by the appropriate z critical value produces the margin at the desired confidence level (Cochran, 1977, Sampling Techniques, 3rd ed., Ch. 4). When the sample is drawn without replacement from a finite, fully enumerated population of size N that is not overwhelmingly larger than the sample, Cochran's finite population correction factor, √((N−n)/(N−1)), is applied as an additional multiplier, shrinking the margin of error because sampling a larger share of a small population removes proportionally more of the true uncertainty. The correction approaches 1 (no effect) as N grows much larger than n, which is why it is routinely omitted for national-scale surveys and included only for surveys of small, well-defined groups such as a single company, school, or club.

How to use this calculator.

1. 01Enter your sample size n — the number of people or units that actually responded.
2. 02Enter the sample proportion p̂ as a decimal between 0 and 1 (0.5 = 50%). If you don't know the true split in advance, leave it at 0.5, the conservative choice that maximizes the reported margin of error.
3. 03Choose your confidence level: 90%, 95% (the standard default used by most media polling), or 99% for a stricter standard of confidence.
4. 04Leave population size at 0 unless you are surveying a small, fully-known group (a single company, club, or classroom) where the sample represents a meaningful share of everyone who exists — enter the true population size there to apply the finite population correction, which will narrow the margin of error.
5. 05Read the primary result, the margin of error, and add/subtract it from your sample proportion to state the confidence interval, e.g. '52% ± 3.1 points, 95% confidence'.
6. 06Check the lower and upper bounds directly if you don't want to do the addition/subtraction yourself — they are already clamped to the valid [0, 1] proportion range.
7. 07If you are instead designing a new survey and need to know how large a sample to collect for a target margin of error, use Quanta's sample-size calculator — the inverse of this one.

The formula.

The margin of error formula comes from the normal approximation to the sampling distribution of a proportion. For a sample of size n drawn independently from a population with true proportion p, the sample proportion p̂ has an approximately normal sampling distribution (for reasonably large n) with standard error SE = √(p̂(1−p̂)/n). Multiplying that standard error by the z critical value for a chosen confidence level produces the margin of error: a 95% confidence level corresponds to z = 1.96 because 95% of the area under the standard normal curve lies within ±1.96 standard errors of the mean; 90% corresponds to z = 1.645, and 99% to z = 2.576 (Cochran, 1977, Ch. 4; NIST/SEMATECH e-Handbook of Statistical Methods).

Three structural properties fall directly out of the formula. Because n sits under a square root, the margin of error shrinks with the square root of the sample size, not proportionally — doubling the margin-of-error precision requires quadrupling the sample size, which is why pollsters rarely push far past a few thousand respondents. Because p̂(1−p̂) is maximized at p̂ = 0.5 (where it equals 0.25) and shrinks toward 0 as p̂ approaches either 0 or 1, the margin of error is largest for a 50-50 split and smaller for lopsided splits — which is exactly why 0.5 is the standard conservative default when the true proportion is unknown. And because the base formula has no N (total population size) in it at all, the margin of error for a well-designed sample of, say, 1,000 respondents is essentially the same whether the underlying population is a town of 50,000 or a nation of 330 million, a fact that regularly surprises people encountering the formula for the first time.

That last property holds only in the limit of an effectively infinite population relative to the sample. When the sample represents a non-trivial fraction of a small, fully enumerated population — surveying 300 of a company's 1,200 employees, for instance — Cochran's finite population correction (fpc) factor, √((N−n)/(N−1)), is multiplied onto the base margin of error, shrinking it because sampling without replacement from a small population removes proportionally more of the total uncertainty with each additional respondent. As N grows large relative to n, (N−n)/(N−1) approaches 1 and the correction has essentially no effect, which is why it is routinely skipped (left at N = 0 in this calculator) for national and state-level polling and included only for small, closed populations. Finally, the resulting confidence bounds p̂ − MoE and p̂ + MoE are clamped to the valid [0, 1] proportion range in this calculator's output, since a poll cannot meaningfully report 'negative 2 percent support' even when a very small p̂ combined with a large margin of error would otherwise push the raw arithmetic below zero.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Cochran, W.G. (1977). Sampling Techniques, 3rd ed. New York: John Wiley & Sons. Ch. 4 (proportion margin of error) and Ch. 9 (finite population correction).
2. [2]Israel, G.D. "Determining Sample Size." University of Florida IFAS Extension, PEOD-6 — presents the identical Cochran formulas and worked examples for both the base margin of error and the finite population correction.
3. [3]NIST/SEMATECH e-Handbook of Statistical Methods.
4. [4]Krejcie, R.V. & Morgan, D.W. (1970). "Determining Sample Size for Research Activities." Educational and Psychological Measurement, 30, 607–610.