TITLE: How does accuracy of a survey depend on sample size and population size?
QUESTION [13 upvotes]: Which survey is more accurate? Assume the samples are taken perfectly randomly.

A sample of 100 people out of a population of 1000   (sample is 10% of population)
A sample of 1000 people out of a population of 1000000 (sample is 0.1% of population)


I remember my lecturer saying something like "when the sample size is small compared to the population, the accuracy depends almost all on the sample size, the population size is unimportant" . Is there a name for that result? It's quite surprising at first.
I'd love to see some graphs of these functions.
If it helps, here's a concrete example (made up by me).

An unknown proportion p of the population favour candidate Alice. The rest favour Bob. We take a random sample size k of the population (size n), and ask their preferences, to come up with an estimate p-hat.
How does the expected error $\mathbb{E}|\hat{p} - p|$ depend on k and n? And in the limit $n\to\infty$?

REPLY [9 votes]: Since you asked for graphs, I will make your example even a little more
specific. Suppose that in the sample we take, $60\%$ of the voters prefer Alice.
For a sample size of $100,$ that means we had $60$ responses for Alice.
So it looks like Alice is ahead of Bob, doesn't it?
But how sure are we?
To assess the accuracy of our poll,
we can try setting up a one-sided $95\%$ confidence interval
by finding the smallest percentage of the population
that might prefer Alice such that there is at least a $5\%$ chance that
we would have gotten at least $60$ responses favoring Alice out
of $100$ total responses.
If the population size were $100,$ we would have polled the entire
population, and the confidence interval would start at $60\%$
(zero error).
If the size of the population were just $105$,
of whom only $61$ ($58.1\%$) favored Alice, there would still be
better than a $5\%$ chance that we could get $60$ or more responses for Alice;
but if fewer people favored Alice then there would be
less than a $5\%$ chance of $60$ favorable responses.
So we set the $95\%$ confidence interval at $58.1\%$ or greater,
which is a maximum error of $1.9\%$ in the direction of overcounting
(that is, our estimate could be that much higher than the true percentage).
As the population size increases, the maximum error (in one direction)
increases rapidly at first but soon starts to level off.
The chart below shows errors for population sizes up to $3000$.

For a population of $1000$, our sample of $100$ (with $60$ favorable responses)
has a (one-sided) error of $8.2\%$ at the $95\%$ confidence level.
But the error grows very slowly after that.
In fact it will never be larger than about $8.7\%$,
even for populations of a million or more.
There is some effect of increasing the population size above
$1000$ in this example, but it is not very dramatic.
On the other hand, suppose we have a population of one million voters,
take a sample of $k$ voters from this population, and find that
$60\%$ of them prefer Alice.
If $k=100$ then we have a one-sided error of
about $8.7\%$ at the $95\%$ confidence level,
but if we make $k$ larger, the error decreases as shown in the chart below.

For $k=1000$ the error is only about $2.6\%$.
That is, while for a sample of $100$ it made little difference
whether the population was $1000$ or one million,
it makes quite a lot of difference whether the sample size is
$100$ or $1000$.
We could do a similar analysis with two-sided confidence intervals,
where we look at  possible errors in either direction.
The results would tell much the same story:
little degradation in accuracy for the $100$-voter sample as the
population grows above $1000,$ but great improvement in accuracy
when we increase the sample size.
That's what your lecturer meant by saying the accuracy depends
almost entirely on sample size
when the sample size is small relative to the population.
The only time the population size made much difference in our example
(changing the bounds of the confidence interval by
more than one percentage point)
was when the sample was considerably more than $10\%$ of the entire population.