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 link carries your figures, so it reopens on exactly these numbers.

What each margin of error costs in responses

At the confidence level and expected proportion you entered. Precision is quadratic: halving the margin quadruples the sample, which is why ±1% surveys are rare and expensive.

Margin of error	Responses needed	Before the population correction	Multiplier used
±10%	97	97	1.96
±5%The figure most published polls quote.	385	385	1.96
±3%Nearly three times the responses of ±5%.	1,068	1,068	1.96
±2%	2,401	2,401	1.96
±1%A hundred times the sample of ±10%.	9,604	9,604	1.96

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.