A Country of 68 Million Needs Fifteen More People Than a Town of 10,000

Why the population drops out

The base figure is computed without any population at all: it depends only on the confidence level, the expected proportion and the margin. The population then enters through a single correction that shrinks the answer when the sample would be a large share of the whole — sampling 218 people out of 500 really is different from sampling 218 out of a million, because you have already met almost half of them. That correction is doing real work at 500 and 1,000, where the sample falls to 218 and 278. By 100,000 it has taken off two people, and past a few hundred thousand it has stopped mattering entirely. A survey does not become less certain because the country behind it is bigger; it becomes less certain when you talk to fewer people.

The interval that reaches below zero

The same tool reports an interval around a result you already have, and it deliberately reports two. The textbook one — the estimate plus or minus z times the standard error — breaks near the edges: 2 successes out of 50 gives 4 %, and the interval it produces runs from −1.43 % to 9.43 %. A negative share of people is not a wide answer, it is a wrong one. The second interval, Wilson's, gives 1.10 % to 13.46 % on the same data: it stays inside the possible range and it is not symmetric, because the truth about a rare event is not symmetric either. When a result sits near 0 % or near 100 %, read the second line.

Frequently asked questions

Why do national polls use around a thousand people?

Because a thousand buys a margin of about ±3 points, and ±3 is the point where the cost curve turns. Going from ±5 to ±3 nearly triples the survey — 385 to 1,068 — and going from ±3 to ±1 multiplies it by nine again, to 9,604. A thousand interviews is the last place on that curve where a small newsroom budget still buys a usable answer.

What should I enter for the expected proportion?

50 % unless you have a reason not to. It is the value that demands the largest sample, so it is the safe assumption: plan for 385 and you are covered whatever the answer turns out to be. Entering a lower figure is legitimate when previous work or a pilot survey supports it — an expected 10 % brings the same margin down to 139 people — but if the real answer then lands nearer 50 %, the margin you actually achieved is wider than the one you designed for.

Does reaching the sample size make the result representative?

No. The formula assumes every member of the population had an equal chance of being picked, and that assumption is the fragile part. A questionnaire answered by whoever clicked it has a sample size but not a random sample, and no number of respondents repairs that — a badly drawn 10,000 is worse than a well drawn 385, because it is confidently wrong. The figure the calculator gives is a floor for a survey that is already sampling properly, not a certificate that it is.