Margin of Error for a Proportion

This is the number in the fine print at the bottom of every poll. Take the standard error of the proportion and stretch it by the critical value for your confidence level: a survey of 1,000 people splitting 50/50 has E = 1.96 × √(0.25/1000) ≈ 0.031, so the result is reported as 50% ± 3.1 points at 95% confidence. The interval runs from about 46.9% to 53.1%, which is why a candidate "leading 51 to 49" in such a poll is not leading at all in any statistical sense.

The most consequential trap is comparing two percentages inside the same poll. The margin of error on a difference is roughly twice the margin on either figure, so a 4-point lead in a poll with ±3 points is genuinely inside the noise. A second trap: repeated polling guarantees outliers. Run twenty independent 95% polls and one is expected to land outside its interval by chance alone — the "shock result" that moves markets is often just that one. Reversed, the formula sizes a survey: to get ±4 points at 95% confidence near p = 0.5 you need n = 1.96² × 0.25/0.04² ≈ 600.25, so 601 respondents.