To estimate a proportion within three percentage points at 95 per cent confidence you need about 1,068 responses. For five points, 385. For one point, 9,604.
None of those figures depend on how many people are in the population, as long as it is reasonably large. A sample of 1,068 describes a city and a country about equally well.
Why the population size drops out
What determines precision is how much a random sample of a given size varies from draw to draw, and that variation is governed by the size of the sample rather than the size of the pot it came from. Once the population is much larger than the sample, adding more people to the population changes almost nothing.
The intuition that a bigger population needs a bigger sample comes from thinking about proportions of the population, which is not what the arithmetic depends on.
Precision gets expensive quickly
Halving the margin of error costs four times the sample, because the error falls with the square root of the sample size. That is the whole reason polls cluster around a thousand responses: it is where cost and precision meet for a three point margin, and going much finer means multiplying the fieldwork budget for a modest improvement.
It also means the difference between a poll of 1,000 and one of 1,200 is negligible, while the difference between 100 and 1,000 is large.
The margin only covers one kind of error
A margin of error describes sampling variation and nothing else. It assumes everyone in the population had a known chance of being selected and that everyone selected responded honestly.
Real surveys miss on both counts. People who answer unknown numbers differ from those who do not. Question wording moves answers. Whole groups are systematically harder to reach. None of that appears in the margin, which is why two polls with three point margins can sit six points apart and both be competently run.
Reading two polls that disagree
Before concluding that opinion has moved, check whether the gap is within what sampling variation alone would produce. Two independent samples each with a three point margin can differ by more than three points without anything having changed in the population.
The sample size calculator gives the responses needed for a target margin, and the z-score calculator is the tool for asking how unusual a particular gap is.