# How does population size impact the precision of the results?

Suppose we want to conduct a poll among a sample from the population. I know that increasing a sample size generally improves the precision of the results, but I was wondering whether given fixed sample size, population also affects the precision.

For example, there are two polls conducted:

• Poll 1, a sample size of 1000 people from a population of 1 000 000.
• Poll 2, a sample size of 1000 people from a population of 10 000 000.

Can we say those polls differ by any way? That poll 1 has larger variance than poll 2, or poll 2 has larger variance than poll 1? For simplicity let's assume that in both cases the sampling fraction is small enough for no finite sampling correction to be required. With no additional assumptions, can we say anything about the relationship of poll 1 results and poll 2 results?

Intuitively, the same sample from a smaller population should have less variance, because there are less ways to pick extremely biased samples, but I cannot find a formal proof for this statement.

As a real life example - suppose we know that conducting polls on a group of 1000 people gives satisfactory results in France. Is the same result valid for the US, which has a population almost fivefold the size?

• A formula about correcting for finite populations is given here. You might want to think about what happens as you vary the $N$ and $n$ variables.
– Dave
Commented Jun 23, 2023 at 0:09

"For simplicity let's assume that in both cases the sampling fraction is small enough for no finite sampling/population correction (FPC) to be required." But the finite population correction is what actually makes the difference!

Without FPC we're in the situation of sampling with replacement. Now imagine estimating a proportion, and in both populations we have 60% yes and 40% no. With replacement it simply doesn't matter whether we sample from a population of 10,000,000 or 1,000,000, as we can just emulate any sample from the bigger population by sampling elements of the smaller population several times. This is only different if the larger population has some values (maybe even extreme outliers in a non-binary situation) that the smaller one doesn't have, but if the overall distribution in the populations is the same, there is no difference.

This changes with sampling without replacement, which is where the FPC comes into play. The difference is not large but you'll have a smaller variance indeed with a smaller population. This should become clear if you think of the extreme case that the sample size is equal to the population size, in which case you have perfect precision, zero variance, as you have the whole population in your sample. The bigger the population size is, the more variation is then possible. If the populations are much bigger than the samples, however, the difference is very small (hence the FPC is often ignored).

Intuitively, the same sample from a smaller population should have less variance...

That is correct. Suppose you have a population of size $$N$$ and a sample of size $$n$$. If the underlying (infinite) superpopulation has variance parameter $$\sigma^2$$ then the variance of the difference between the sample mean and the population mean is:

$$\mathbb{V}(\bar{X}_n - \bar{X}_N) = \frac{N-n}{N} \cdot \frac{\sigma^2}{n}.$$

The first term in this formula is the "finite population correction" term which arises when we have a finite population. It is easy to see that this variance is strictly increasing in $$N$$, so the variance in estimating the population mean is larger for a larger population. In particular, it is worth noting that a full census with $$n=N$$ gives $$\mathbb{V}(\bar{X}_n - \bar{X}_N) = 0$$ which allows perfect estimation of the population mean. A consequence of this result is that confidence intervals for the population mean will be more accurate (i.e., narrower) for smaller population sizes. It is also noteable that if you look at confidence intervals for the unsampled part of the population, the opposite is true --- in this case the accuracy is better for larger populations.

If you would like to learn more about the moment results in sampling problems and consequent variability of estimators, you might find it useful to read O'Neill (2014). This paper sets out a range of useful quantities in sampling problems involving a finite population and shows the moments of various quantities of interest. It also derives relevant confidence intervals for problems involving finite populations. This paper examines inferences for a finite population mean but also inferences for the mean of the unsampled part of a finite population.

If you play around with the Finite Population Correction formula in a spreadsheet, then you'll find that using the FPC rarely makes any practical difference to your conclusions. Differences become noticeable when a sample size is more than about 10% of the population size.

Sampling variation is more often a function of sampling frame and sampling method rather than sample size. This is especially the case in social research.

To quote D'Alessandro et al., (2020) Marketing Research 5ed. Cengage Australia.: "In a poorly conducted study, a larger sample size just gives you more confidence in a wrong answer."