Why does the expected proportion change the answer?

Because variance peaks at 50-50. If you already know a result will land near 90-10, you need fewer responses to pin it down. Using 50% when you are unsure is the conservative choice — it never underestimates.

When does the population size start to matter?

When your sample would be a meaningful share of it. Surveying 385 people out of 1,000 is a third of them, and the correction cuts the requirement to 278. Out of a million, the correction changes nothing.

Does this account for people not responding?

No. This is the number of completed responses you need. If a third of people you approach reply, you have to contact three times as many — and non-response is rarely random, which is a bigger problem than the arithmetic.

Math

Sample Size Calculator

Enter the margin of error you want and the size of the group you are studying.

The formula

n = z² × p × (1 − p) ÷ margin²

with a finite population: n ÷ (1 + (n − 1) ÷ population)

Worked example

±5% at 95% confidence, on a population large enough to ignore

1.95996² × 0.5 × 0.5 = 0.9604
0.9604 ÷ 0.05² = 384.1
Round up: 385 responses

Where this goes wrong

Sizing the sample to the population

A national survey and a city survey need almost the same number of responses. Above roughly 20,000 people the population barely enters the arithmetic: 385 responses give ±5% whether you are studying a town of 50,000 or a country of 50 million. The instinct to sample a fixed percentage is what makes surveys needlessly expensive.

Questions

Why does the expected proportion change the answer?: Because variance peaks at 50-50. If you already know a result will land near 90-10, you need fewer responses to pin it down. Using 50% when you are unsure is the conservative choice — it never underestimates.
When does the population size start to matter?: When your sample would be a meaningful share of it. Surveying 385 people out of 1,000 is a third of them, and the correction cuts the requirement to 278. Out of a million, the correction changes nothing.
Does this account for people not responding?: No. This is the number of completed responses you need. If a third of people you approach reply, you have to contact three times as many — and non-response is rarely random, which is a bigger problem than the arithmetic.