Margin of Error for a Mean

The margin of error is the "plus or minus" you see quoted next to every poll: take the standard error σ/√n and stretch it by the critical value for your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). With σ = 10 and n = 100, a 95% interval reaches E = 1.96 × 10/10 = 1.96 units either side of the sample mean. The interval is the estimate ± E. George Gallup made this arithmetic famous in 1936, when his sample of a few thousand correctly called Roosevelt's landslide while the Literary Digest's two-million-response mail-in poll called it catastrophically wrong: proof that how you sample matters more than how much.

The trap hidden in that story is the one this formula cannot fix: E measures only sampling error. It says nothing about a biased frame, leading questions, or people who refuse to answer, which is exactly what sank the Digest. Note also that n appears under a square root, so halving your margin of error costs four times the data: going from E = 3 to E = 1.5 with σ = 12 and 95% confidence takes you from 62 observations to 246.