The margin of error, both directions
Every survey headline, "accurate to within 3 points", is this number. The margin of error is the half-width of a confidence interval, and it obeys one unforgiving law: it shrinks with the square root of the sample size. This calculator works both directions: given a target margin, the sample size you must collect; given a sample size, the margin you will actually get.
Estimating a mean with σ ≈ 15 to within E = 3 at 95%: n = (1.96 × 15 ÷ 3)² = 96.04, rounded up to 97. Always round up, or the delivered margin is wider than the one you promised. For a proportion within ±3 points with no prior guess (p = 0.5): n = 1,068, which is why national polls cluster around a thousand respondents. And a sample of 200 with σ = 15 delivers a margin of ±2.08, whatever you hoped for.
The square-root tax
Halving the margin costs four times the sample. Going from ±3 to ±1.5 on the proportion example takes the count from 1,068 to 4,272 interviews. Precision is bought at quadratic prices, and honest planning says so before the fieldwork starts, not after.
Guessing a rosy p to shrink the required n. p(1−p) peaks at p = 0.5; planning with p = 0.2 when the truth is 0.45 leaves you under-sampled and over-margined. No prior knowledge means p = 0.5: pay for the worst case.
Frequently asked questions
Why do polls always seem to use about 1,000 people?
Because ±3 points at 95% confidence for a proportion needs n = 1,068 at worst-case p = 0.5, and ±2 points would need 2,401. A thousand is where cost and precision shake hands.
Does the population size matter?
Almost never, until the sample is a visible fraction (say over 5%) of the population. Then a finite population correction narrows the margin slightly. A sample of 1,000 measures a city or a country with the same margin.