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Is a 2-point lead in a poll with a ±3 margin of error a real lead?
A poll reports 48% to 46%, "with a margin of error of plus or minus 3 points." It sounds like a lead that is inside the error but close to real.
Two numbers, two margins
A poll of about 1,000 likely voters
The reported margin of error applies to each candidate's number, so the margin on the gap between them is about twice as big. If A could be 3 points too high while B is 3 points too low, the true lead could be very different from what the poll shows. In a two-person race, the margin of error on the lead is roughly double the reported margin: about ±6 points here. A 2-point lead in this poll is a statistical tie, not a near-lead.
Uncertainty adds up when you subtract. Whenever a headline compares two estimates, the uncertainty of the difference is bigger than the uncertainty of either one.
Try it somewhere else
A survey says 52% of customers prefer the new menu this year, up from 49% last year, each with a ±3 margin of error.
Can the restaurant say that more customers prefer the new menu now?
What the margin doesn't cover
The margin of error only counts the randomness of who happened to be sampled. It does not include people who refuse to answer, voters who change their minds, or questions worded in a way that nudges answers. Pew Research Center notes that real-world poll errors are often larger than the stated margin for these reasons.
Related exhibits
Sources: Andrew Mercer, "5 key things to know about the margin of error in election polls," Pew Research Center, 2016. Margin of error for n ≈ 1,000 at 95% confidence: about ±3 points. Illustrative poll numbers.