Margin of Error Calculator

What a margin of error promises

A poll talks to a sample, and a sample is not the whole population. The margin of error states how far the sample result is likely to sit from the true population value, at a stated level of confidence. A margin of plus or minus 3 percentage points at the 95 percent confidence level means that if the same survey were fielded 100 times, the result would land within 3 points of the true value in about 95 of them. Pew Research Center's guide to poll margins walks through that reading, and it is the interpretation most pollsters use.

Two numbers drive the whole calculation: the sample size and the split. Everything else in the arithmetic is fixed. The percentage matters less than people expect, because the share p enters as p x (1 - p), and that product changes only modestly across the middle of the range. It falls off near the extremes, which is why results at 90 percent or 10 percent carry a smaller margin than a 50/50 split.

The formula for a share, worked through

For a percentage from a simple random sample, the margin of error is:

margin of error = z x sqrt(p x (1 - p) / n)

where p is the reported share as a decimal, n is the number of respondents, and z is the critical value for the confidence level: 1.645 at 90 percent, 1.96 at 95 percent, 2.576 at 99 percent. The square root term is the standard error of the share, so the margin is just the standard error scaled by z.

Work the 1,000 person poll at 48 percent. The variance term is 0.48 x 0.52 = 0.2496, and 0.2496 / 1,000 = 0.0002496. Its square root is 0.015798, which is 1.58 points. At 95 percent confidence, 1.96 x 1.58 = 3.10 points. The interval runs from 44.9 to 51.1 percent.

The 90 and 95 percent rows are the same survey read two ways. The Census Bureau's margin of error training material publishes its American Community Survey margins at 90 percent, so a figure copied from a Census table needs multiplying by 1.19 to become a 95 percent margin.

Sample size for a target margin

Run the formula backwards to plan a survey. Solving for n gives:

n = z^2 x p x (1 - p) / margin^2

For a 3 point margin at 95 percent confidence with no prior guess about the split, the work is 1.96^2 x 0.25 / 0.0009 = 0.9604 / 0.0009 = 1,067.1, which rounds up to 1,068 responses. Planning numbers follow a hard rule: each halving of the margin costs four times the sample.

The table assumes the worst case split of 50 percent. If earlier rounds of a survey put the share near 30 percent, the requirement drops, since p x (1 - p) = 0.21 instead of 0.25. The same 3 point target then needs 1.96^2 x 0.21 / 0.0009 = 896.4, which rounds up to 897.

Why 50 percent is the worst case

The term p x (1 - p) peaks at 0.25 when p is 0.50 and falls toward zero at the extremes. That is why pollsters who know nothing about the answer plan with 50 percent: it buys the largest margin, so no result can come in wider than promised. Hold n at 1,000 and 95 percent confidence, and the margin moves like this:

The spread from 50 to 40 percent is a tenth of a point, which is why published margins usually quote a single number for the whole survey. The quoted number is the 50 percent worst case, and results elsewhere in the range are slightly tighter than the label says.

Reading a lead between two candidates

The margin printed on a poll applies to one candidate's share. A comparison of two shares carries a wider margin, because both numbers wobble. For a two candidate race where the shares add to 100 percent, the margin on the difference is close to twice the individual margin. A poll with a 3.10 point margin per candidate carries about a 6.2 point margin on the lead.

Work the 48 to 52 poll. The lead is 4 points, and the margin on that difference comes to 1.96 x sqrt((0.48 + 0.52 - 0.04^2) / 1,000) = 1.96 x sqrt(0.9984 / 1,000) = 1.96 x 3.16 points, which is 6.19. The 4 point lead sits well inside it, so the poll cannot distinguish the two candidates.

That logic also guards against calling a bare majority. A result of 51 percent with a 3 point margin has an interval reaching down to 48, so it is not evidence that most people hold the view. Pew's guidance to journalists makes both points: do not report decimal places, and do not call a lead real until it clears the wider difference margin.

Design effect, weighting and subgroups

Real surveys are weighted to match the sample to the population, and weighting that pushes values away from 1 inflates the variance. Statisticians bundle that inflation into a single multiplier called the design effect, usually written deff. The margin becomes:

margin = z x sqrt(p x (1 - p) / n) x sqrt(deff)

A design effect of 1.5 on the 1,000 person poll stretches the 3.10 point margin to 3.10 x 1.2247 = 3.79 points. Pew's methodology notes disclose the design effect behind each published margin for exactly this reason.

Subgroups behave like smaller surveys. The national margin says nothing about a crosstab. Quarter the sample to 250 respondents and the margin at an even split jumps to 6.20 points; take 100 respondents in a small demographic and it reaches 9.8 points. Any table that slices results by age, state or ethnicity needs its own margin, computed with the group's own count.

Small populations and the finite population correction

When the sample takes a real bite out of a small population, the ordinary formula overstates the error. The finite population correction multiplies the margin by sqrt((N - n) / (N - 1)), where N is the population size. It only bites when n exceeds about a tenth of N, and it does nothing for the national polls that sample a few thousand out of 200 million adults.

Survey 500 of a 2,000 person workforce at an even split. Without the correction the margin is 4.38 points. With it: 4.38 x sqrt(1,500 / 1,999) = 4.38 x 0.8663 = 3.80 points. The correction says a sample covering a quarter of a small group carries more information than the plain formula credits.

What the margin does not cover

The margin of error measures one source of error: sampling. Nonresponse, question wording, interviewer effects and mode effects all move results without touching the formula, and none of them shrink when the sample grows. The Census Bureau's ACS handbook, chapter 7 separates these into sampling and nonsampling error and warns that the published margin describes only the first kind. A poll can carry a tight margin and still be wrong by a wide and unmeasured amount.

Two further limits are worth knowing before quoting a number. First, the formula above leans on the normal approximation, and NIST's engineering statistics handbook recommends Wilson or Agresti-Coull intervals when a share sits near 0 or 100 percent or when the sample is small, because the symmetric interval can reach into impossible values. Second, opt-in online panels cannot compute a margin from sample size alone; Pew calls their published figure a modeled margin, built on assumptions about how repeat samples would behave.

For planning work the calculator above covers the common case: a probability sample, a single share, and a margin quoted at 90, 95 or 99 percent. For a mean instead of a share, the standard error comes from the sample spread rather than p x (1 - p), and the standard error calculator runs that version.