Where plus or minus 3.1 comes from
Draw 1,000 people at random and ask a yes or no question. Draw a different 1,000 and the answer moves a little. Draw a thousand different samples and the answers scatter around the true value in a predictable pattern, and the width of that scatter is what the margin of error describes. At 95 percent confidence it is 1.96 times the square root of p(1 minus p) divided by n. With p at 0.5 and n at 1,000 that is 1.96 times the square root of 0.00025, which is 0.031 — the ±3.1 points printed under the poll.
Because n sits under a square root, precision improves slowly. Tripling the sample from 1,000 to 3,000 does not triple the precision; it improves it by the square root of three, from ±3.1 to ±1.8 points. That relationship is why around a thousand became the industry standard: it is where the next increment of precision starts costing more than it returns.
Percent and percentage points
A margin of error is always in percentage points. If support moves from 40 percent to 43 percent, that is a rise of 3 percentage points and a rise of 7.5 percent relative to where it started. Both statements are true and they are not interchangeable. Reporting the relative figure when the reader expects points, or the reverse, is a common and effective way to make a small movement sound large.
The gap between two options needs a different test
A poll puts A on 45 and B on 42, margin of error ±3.1 points. The gap is 3 points. It is tempting to say that because 3 is under 3.1 the race is tied, but the comparison is not like for like: the ±3.1 applies to each figure separately, while the gap inherits uncertainty from both. The rough convention, and what this page applies, is to compare the gap against roughly twice the margin — about 6.2 points here — so a 3 point lead is not resolved by this poll.
The convention is deliberately conservative. Two shares measured in the same sample are negatively correlated, since a respondent counted for A cannot also be counted for B, and a proper test of the difference exploits that and sets a slightly lower bar. The honest summary: a gap well under the doubled margin is unresolved, a gap well over it is real, and a gap near the line should be reported as unresolved rather than pushed either way.
What 95 percent confidence actually claims
It is a statement about the method, not about this poll. If the same sampling procedure were repeated many times, about 95 percent of the intervals it produced would contain the true value. This particular interval either contains it or it does not; no probability remains in it once it has been calculated. The everyday phrasing — there is a 95 percent chance the true figure lies in this range — is a shorthand that is not quite right, and the distinction matters because the correct version also tells you the flip side: roughly one poll in twenty lands outside its own margin with everything done correctly. Occasional badly wrong polls are a feature of the arithmetic, not evidence of manipulation.
Sampling error is the smallest of the errors
The margin of error prices exactly one thing: the luck of the draw in a genuine random sample. It has nothing to say about the people who never answer, and response rates for telephone polling now sit in the single digits in many markets. It has nothing to say about a question that primes one answer, about respondents who tell an interviewer what they think is acceptable, or about the weighting model that stretches an unrepresentative raw sample back toward the census. Firms adjust for these as best they can and the adjustments carry their own uncertainty, none of which is inside the ±3.1. When a whole set of polls misses in the same direction, sampling error is almost never the reason — that pattern is the signature of a systematic problem shared across the industry, and no amount of interviewing fixes it.
Questions people ask
Why is the margin of error largest at 50 percent?
The variance of a proportion is p(1 minus p), a downward parabola peaking at 0.25 when p is 0.5 and shrinking toward zero at both extremes. A result near 50 percent is the noisiest thing a survey can measure; a result near 3 percent or 97 percent is measured much more precisely. That is why published margins are usually calculated at 50 percent — it is the worst case, so quoting it guarantees no figure in the survey is less precise than advertised. It also means the ±3.1 attached to a poll is genuinely too wide for the minor options in it.
My sample is nearly the whole population. Does the margin still apply?
Not without a correction. When you have surveyed a large fraction of a closed group the uncertainty shrinks, and in the limit where you ask everyone it disappears entirely — a census has no sampling error. The adjustment is the finite population correction, multiplying the margin by the square root of (N minus n) over (N minus 1). At 200 responses from a population of 500 it cuts the margin by about 22 percent. The sample size calculator applies it in both directions, so use that one when you know N.
Does the margin of error apply to subgroups in the poll?
No, and this is where a lot of reporting goes wrong. The headline margin is computed from the full sample. A crosstab covering 150 of the 1,000 respondents carries a margin of about ±8 points at 95 percent, not ±3.1. Differences between subgroups are shakier still, because both figures are noisy. If a story about a poll rests on a movement inside a demographic slice, put that slice size into this calculator before you believe it.
Two polls disagree by more than their margins. Is one of them wrong?
Not necessarily, and there are three separate reasons. Different polls are taken at different times and opinion moves. They use different sampling frames, weighting schemes and question wordings, and those choices produce systematic house differences entirely outside the reported margin. And a difference between two independent estimates is subject to both of their errors, so the threshold for calling them inconsistent is wider than either margin alone. Disagreement between individual polls is normal, which is why aggregates exist.