Sample Size for a Mean

Every study design eventually reduces to this line: decide how much error you can live with, decide how confident you want to be, guess the standard deviation, and the sample size falls out. Estimating a mean to within 2 units at 95% confidence when σ ≈ 15 needs n = (1.96 × 15/2)² = 216.09: always round up, so 217. Rounding down is the classic slip; it leaves you fractionally short of the precision you promised. The σ you plug in is usually borrowed from a pilot study, published literature, or the crude rule that a range spans about four standard deviations.

The square in the formula is the brutal part of research economics. Precision is bought at quadratic cost: tightening E from 2 to 1 in the example above takes the requirement from 217 to 865. Note too that the population size never appears: a well-drawn sample of 1,000 estimates a mean about as well in a city of 100,000 as in a country of 100 million, which is the single most counter-intuitive fact in survey work. Only when the sample is a sizeable fraction of a finite population (over about 5%) does a correction factor start to help you.