Margin of Error, Explained: What a Poll's ± Covers
A seeded simulation of polls of 1,000 people: where ±3.1 points comes from, why a 2-point lead needs ±6.2, and what the margin cannot see.
The electorate, the two candidates and the 52 percent share are an example, not a real race. Every number on this page is produced by the simulation in the page, from a fixed random seed, so a reload shows the same polls.
A poll asks a sample, not everyone: the true share exists, but the pollster cannot see it.
A poll puts its question to a sample of people instead of the whole electorate. Somewhere there is a true answer: in this made-up electorate, Candidate A holds exactly 52 percent. The pollster never sees that number. Everything that follows is about how close a sample gets to it, and how the margin of error describes that closeness. The electorate, the candidates and the share are an example chosen for the simulation.
Draw 1,000 people at random: this sample says 52.3 percent, near the truth but not on it.
The simulation draws 1,000 people at random, each with a 52 percent chance of backing Candidate A. In this draw, 523 of them do, a sample share of 52.3 percent. That is 0.3 points away from the truth: close, but not exact. Another draw of 1,000 would land somewhere else, and a pollster holding only one sample cannot tell how far off this one is.
Repeat the draw 100 times: the shares scatter, and 98 land within 3.1 points of the truth.
Run the same poll 100 times, each time with 1,000 new random people. The sample shares scatter between 48.3 and 54.6 percent and pile up around 52. The spread is predictable. Pollsters publish the worst case, a 50-50 split: 1.96 times the square root of 0.25 divided by 1,000 gives 3.1 points. In this run, 98 of the 100 shares fall within that distance of the truth.
Put ±3.1 around each share: 98 of these 100 intervals catch the true 52 percent.
Now put a band of 3.1 points on either side of every sample share. A band that crosses the dashed line has caught the true 52 percent. In this run 98 of the 100 bands do, and 2 miss. That is what the 95 percent confidence level promises: not that any single poll is right, but that about 95 in 100 such bands would catch the truth.
Smaller samples, wider margins: 400 people give ±4.9 points, 100 people give ±9.8.
The margin shrinks with the square root of the sample size, so precision is expensive. With 400 people the published margin grows to 4.9 points; with 100 people it reaches 9.8. Cutting the sample to a quarter doubles the margin. In the simulation, 94 of 100 bands caught the truth at each of the smaller sizes: wider bands, the same promise.
A 51–49 poll: ±3.1 on each share, but ±6.2 on the 2-point lead.
Say a poll of 1,000 shows Candidate A at 51 and Candidate B at 49. Each share carries a margin of 3.1 points, but the lead is the difference of two numbers that move in opposite directions, so its margin is 6.2 points; Pew Research Center puts it at about twice the single margin. Rerun that poll 100 times on a race that really is 51 to 49, and the trailing candidate leads or ties in 29.
Opt-in panel, one side 1.4 times as likely to answer: the samples cluster 8.3 points low.
The margin measures only the luck of the draw. Suppose the poll uses an opt-in online panel where Candidate B's supporters are 1.4 times as likely to take part. All 100 samples of 1,000 now cluster around 43.7 percent instead of 52, and not one band catches the truth. The printed margin is still 3.1 points. It cannot see this bias, nor gaps in coverage, non-response or question wording.
Across 10,000 random samples, 94.5 percent of intervals caught the truth, against a promised 95.
Check the promise against the run. Across all 300 random-sample polls here, 286 bands caught the truth: 95.3 percent. A run of 10,000 polls of 1,000 gives 94.5 percent, and the exact binomial calculation 94.7, a hair under 95 because the formula is an approximation. The opt-in panel scored zero. AAPOR ties the margin of sampling error to probability samples; opt-in polls report a model-based credibility interval instead.
Sources: Pew Research Center, Andrew Mercer: 5 key things to know about the margin of error in election polls (2016) · AAPOR: Understanding a credibility interval and how it differs from the margin of sampling error in a public opinion poll