Margin of Error Calculator

The square-root law, and what it costs

One relationship explains almost everything people find surprising about survey precision.

The margin of error is proportional to 1/√n. Not 1/n. That distinction is the whole story.

Doubling the sample does not halve the margin — it multiplies it by 1/√2, which is a reduction of exactly 29.3%. Every doubling, at every scale, gives the same 29.3% and never more. The ladder in the tool shows the figure repeating identically from 100 to 200 and from 2,000 to 4,000.

To halve the margin you need four times the data. To get a tenth of it you need a hundred times. Since cost usually scales roughly with n, precision gets expensive fast: the last percentage point costs far more than the first ten.

That is why 1,000 is the near-universal polling number. At n = 1,000 the margin is about ±3.1 percentage points, which is precise enough to be useful. Getting to ±1.5 would need about 4,000 interviews at four times the cost, and few questions are worth that.

It also explains why sub-group results are so much shakier than headline ones. A poll of 1,000 that reports on 150 young voters has a margin of about ±8 points for that group — and those are the numbers most likely to be quoted as a finding.

Why polls quote the figure for 50%

A published margin of error is almost always the worst case, and knowing that makes the number easier to read.

For a percentage the standard error contains p(1−p). That product is largest at p = 0.5, where it reaches 0.25, and falls away symmetrically: at p = 0.9 it is 0.09, barely a third as large.

So a result near 50% has the widest margin, and a result near 5% or 95% has a much narrower one on exactly the same sample. A poll quoting “±3 points” is quoting the figure for a 50/50 split, and its 8% result is more precise than that.

Two reasons this is the right convention. Before the survey runs you do not know p, so you must plan for the worst case. And one margin is easier to publish than a different one for each question.

The practical consequence: for lopsided results the published margin is conservative, sometimes by a lot. Set the proportion field in the tool to 0.1 and watch the margin fall by about 40% against the 0.5 case.

Why the population size barely matters

The most persistent misunderstanding about surveys: that polling a large country needs a bigger sample than polling a small one.

It does not. The margin of error depends on the sample size, not the population size. A sample of 1,000 gives about ±3.1 points whether the population is 50,000 or 300 million.

The intuition people expect — that you need a fixed fraction — is simply wrong. A well-stirred pot needs one spoonful to taste, whether it is a saucepan or a vat.

There is a correction, and it works the other way round from what people expect: the finite-population correction multiplies the margin by √((N−n)/(N−1)), which reduces it when the sample is a large share of the population. Sample everyone and it goes to zero, correctly.

But it only bites when n is a substantial fraction of N. At 1,000 from 300 million the factor is 0.9999983 — invisible. At 1,000 from 5,000 it is 0.8945, a genuine 10.6% improvement worth claiming. Enter a small population in the tool to see it appear.

So: ignore population size for national polling; apply the correction when surveying a small, closed group such as one company’s staff or one school’s pupils.

The errors this number cannot see

This is the most important section on the page, and the one usually left off.

A margin of error quantifies sampling error and nothing else: the variation you would get from drawing a different random sample of the same size from the same population. It assumes the sample is genuinely random and that everything else was done perfectly.

Non-response bias. Typical response rates for telephone polling are now in the low single digits. If the few percent who answer differ systematically from those who do not — and they generally do — the result is biased by an amount no sample size reduces.

Coverage error. If your sampling frame misses part of the population — landline-only in a mobile-first country, an online panel in a place with patchy internet — those people have zero probability of selection and the margin never mentions it.

Question wording and order. Small changes in phrasing routinely move results by more than the margin of error. So does the order of the options, and what was asked immediately before.

Weighting and turnout models. Raw survey data is almost always adjusted to match known demographics, and election polls further adjust for who is likely to vote. Those models embody assumptions that can be wrong, and their contribution is not in the margin.

Post-mortems of polling misses usually find these effects were larger than the sampling error. A ±3 point margin on a survey with a 4% response rate is a precise statement about the wrong quantity, and this is the reason to treat a narrow margin as necessary rather than sufficient.

Comparing two numbers inside one poll

Candidate A is on 48% and candidate B on 45%, and the margin is ±3. Is A ahead?

The common answer — the gap is 3 points and the margin is 3, so it is within the margin — is not the right calculation.

The published margin applies to each individual number, not to the difference. The difference has its own, larger uncertainty, because both estimates are uncertain and both contribute.

Worse, in a poll where respondents choose one option the two figures are negatively correlated — a respondent counted for A is not counted for B. The correct standard error for the lead is roughly √((p₁+p₂) − (p₁−p₂)²)/√n, and the resulting margin on the difference is commonly about 1.7 times the margin on either share.

So with a ±3 point margin on each share, the margin on the lead is closer to ±5, and a 3-point lead is comfortably inside it. The rule of thumb people use is not conservative enough — it understates the uncertainty on exactly the quantity everyone cares about.

The same logic applies to comparing two separate polls, where the two samples are independent and the margins add in quadrature instead.

Using it backwards, to plan a survey

The formula runs in reverse: fix the margin you need and solve for n.

For a percentage at 95% confidence with the conservative p = 0.5, n = (1.96²)(0.25)/m², which gives the numbers survey teams quote from memory: 385 for ±5 points, 1,068 for ±3, 2,401 for ±2, and 9,604 for ±1.

Look at that sequence. Going from ±5 to ±3 costs you roughly triple. Going from ±3 to ±1 costs another factor of nine. Precision is priced by the square, and this is where the trade-off has to be made deliberately rather than by aiming at a round number.

Two adjustments in practice. If you know the proportion will be far from 50% you can use that value and need materially fewer responses. And if you intend to report on sub-groups, size for the sub-group you care about, not the whole sample — this is the most common planning mistake, and it is only discovered after the data is collected.

The Sample Size Calculator does this direction properly, including the finite-population correction.

Where the 1.96 comes from

The multiplier is the only part of the formula that encodes how confident you have chosen to be.

For 95% confidence with a normal sampling distribution it is 1.959964 — the point beyond which 2.5% of the standard normal lies in each tail. The familiar “two standard errors” rule of thumb rounds it, and the rounding is why hand-computed margins are slightly wide.

For 90% it is 1.645, and for 99% it is 2.576. Moving from 95% to 99% widens the margin by about 31% on identical data: greater certainty always costs precision, and there is no arrangement in which it does not.

For an average rather than a percentage the multiplier comes from Student-t rather than the normal, because the standard deviation was estimated from the same sample. The tool uses t for the mean case at every sample size, which is why its multiplier is a little larger than 1.96 and converges on it as n grows. That choice, and what the common shortcut costs, is worked through on the Confidence Interval Calculator.

Sources and methodology

The polling-standards and methodological references behind this page.

• AAPOR disclosure standards — what a published survey is expected to report alongside its margin of error
• Pew Research Center on survey methodology — on non-response and coverage error dominating sampling error
• NIST/SEMATECH e-Handbook — sample sizes required for a given margin
• Sample Size Calculator — the same formula solved for n instead
• Confidence Interval Calculator — the full interval this is half the width of
• How we calculate — methodology

Method. The percentage margin uses the normal multiplier and the observed proportion, defaulting to p = 0.5 because that maximises p(1−p) and is the figure published surveys quote. The average margin uses Student-t with n − 1 degrees of freedom at every sample size rather than switching to z, for the reason set out on the Confidence Interval Calculator. The finite-population correction is applied only when a population is entered and is larger than the sample. The 29.3%-per-doubling figure is not an approximation — it is exactly 1 − 1/√2, and the suite asserts that quadrupling n halves the margin to within 1e-9 across three sample sizes, which is the same statement. That engine is verified on every change against 155 assertions. The count and the per-case breakdown are published on the formula verification page.

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