A proportion is the easiest parameter to estimate and the easiest one to get quietly wrong. You count the successes, divide by n, and you have p̂. The interval has the same shape as every other one in this chapter — estimate ± multiplier × standard error — except that the standard error is now built out of p̂ itself. What makes proportions their own case is that the normal curve underneath the multiplier can fail. When successes or failures are scarce, the interval still prints, still looks respectable, and captures p far less often than the label on it promises. Draw real samples below, check the conditions as you go, and watch the 95% come apart the moment the conditions do.
One sample → one interval
successesfailuresp̂ — the point estimatep — the true proportionimpossible values
The same recipe, run again and again
Every horizontal line is one whole study: draw n, count, compute, print an interval. Green captured p, coral missed it. In real work you see exactly one of these lines and never learn its color — which is why the promise has to be about the recipe rather than about your one interval.
Intervals built0
Captured p0
Simulated coverage—
Exact coverage—promised —
Ballot measure supportpress “Draw a new sample” to build an interval
What you are counting — p is known here only because this is a simulation
Confidence level — sets the multiplier z*
100
0.52
Estimate p̂ = x / n——
Multiplier z*1.960
Std. error √(p̂q̂/n)—
Margin of error—
Check the conditions — before you trust a single digit of it
Random / independent—
n is under 10% of N—
At least 10 successes—
At least 10 failures—
Why the success–failure condition exists
The bars are the exact distribution of p̂ over every sample you could possibly draw — real binomial probabilities, not a simulation. The smooth curve is the normal approximation the formula quietly assumes. Push p toward 0 or 1, or pull n down, until the two stop agreeing.
Where the margin is widest
The standard error is built from p̂ itself, so the width of the interval depends on the answer you got. The curve below is the margin of error at every possible p̂, holding n and the confidence level fixed.
p̂ ± z* √(p̂(1−p̂)/n). Three ingredients, same as always. The estimate is the count over the sample size. The standard error says how much an estimate like yours typically wobbles from sample to sample, and for a proportion it can be computed from p̂ alone — there is no separate spread to measure, because a proportion's spread is its center. The multiplier is set by the confidence level and nothing else. One detail worth carrying: np̂ is literally the number of successes you observed and n(1−p̂) is the number of failures, so the success–failure condition is not an extra calculation. Look at your two counts. If either is under 10, stop.
Five things this interval will not tell you.(1) It is not a probability statement about the interval in front of you. Once the numbers are computed, p is either inside it or it is not; nothing is left to chance. The 95% belongs to the procedure — repeat the whole study endlessly and 95% of the intervals it produces contain p. The stack of lines above is that sentence drawn out. (2) It is not a statement about individuals. An interval of (0.48, 0.56) does not mean 48% to 56% of anything; it is a claim about one fixed number, the population proportion. (3) It does not cover bias. The whole margin is built from √(p̂q̂/n), which describes random sampling wobble and nothing else. Volunteers only, one neighborhood, whoever answers the phone, a question worded to lead — none of that is in the formula. A biased study returns a tight interval centered on the wrong number, and raising n only tightens it. (4) It does not repair a broken condition. When successes or failures are scarce, the arithmetic still runs and still prints an answer — watch the exact coverage tile fall to the 70s and 80s while the label still says 95%. It can even hand you a lower bound below 0 or an upper bound above 1, which is the approximation telling on itself. In that situation use a method built for it, such as the Wilson or Clopper–Pearson interval, rather than the formula on this page. (5) It does not settle on 10. Books variously require 5, 10, or 15 successes and failures, and none of the three is a cliff — the approximation degrades smoothly, which is exactly what the exact-coverage number lets you watch it do. Treat 10 as a working floor, not a law.