Margin of Error for a Mean
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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.
- = Margin of error
- = Critical z-value
- = Standard deviation
- = Sample size
- Margin of error — Margin of Error for a Proportion, Sample Size for a Mean
- Critical z-value — Confidence Interval Lower Limit, Confidence Interval Upper Limit
- Standard deviation — Z-Score (Standard Score), Variance and Standard Deviation
- Sample size — Standard Error of the Mean, Confidence Interval Lower Limit