Standard Error of a Proportion

Counting successes out of n trials is a binomial problem, and the standard deviation of a single yes/no outcome is √(p(1 − p)). Average n of them and, as always, the spread shrinks by √n — so a sample proportion of 0.50 from 100 people carries a standard error of √(0.25/100) = 0.05, five percentage points. Multiply by 1.96 and you have the familiar ±10 points that makes small polls nearly useless. Abraham de Moivre worked out this normal approximation to the binomial in 1733, decades before anyone was polling anything, while analysing games of chance.

Two traps. First, the p(1 − p) term is remarkably flat near the middle: the standard error at p = 0.5 is only about 2% larger than at p = 0.4, but it collapses toward zero as p approaches 0 or 1 — which is exactly where the normal approximation stops working. The usual rule is to require np ≥ 10 and n(1 − p) ≥ 10 before trusting it. Second, note that p enters as a decimal, not a percentage; enter 25% as 0.25 or pick the % unit. Reversed, the formula sizes a study: to pin a proportion near 0.25 down to a standard error of 0.05 you need n = 0.25 × 0.75/0.05² = 75 observations.