# Why independence assumption is reasonable if one uses a random sample of less than 10% of a population?

The title is self-explanatory. I was reading the openIntro statistics book by Diez et al., where this sort of rule of thumb is stated frequently. Where all this comes from?

This has been a well-known approximation in survey analysis since at least the 1940s where samples of populations are usually taken without replacement.

The usual variance of the mean of a sample of size n is $\frac{s^2}{n}$.

For a sample without replacement the correct formula is $(1-\frac{n}{N} ) \frac{s^2}{n}$.

if $n$ is much smaller than $N$ (the population size) then the finite population correction (fpc), $1-\frac{n}{N}$, is close to $1$ and both variances are more similar. Clearly, the smaller $\frac{n}{N}$ is, the more similar are the two estimates.

There has been general agreement, from the days of mechanical calculators, that smaller than 10% was close enough. For a long time using such an approximation for calculation has not been necessary.

I should have noted originally that when sampling from a finite population without replacement there is a small negative correlation between individual observations. The modified formula accounts for this.

## Exhausting the population

Without further context, I believe this statement is based on the possibility of 'exhausting' the population to such an extent, the first (large) lot of individuals you sample noticeably changes the probability of sampling specific individuals at a further point in time, but during the same 'sampling'.

## An example

Think of taking colored marbles (sampling) out of a very large jar (the population). Notice that the jar is so large, taking a few would not really change the distribution of marble colors. So, especially if you sample at random and only a small amount of marbles, exhausting a color has a low probability of occurring. However, at a certain sample size some colors might run out, changing the probability of drawing the exhausted color(s) and those still in the population ==> independence assumption could be violated.

## 10%?

Finally, if I understand correctly, the '10%' cut-off is (a possibly arbitrary) point for which Diez et al found that independence can reasonably be assumed without large errors in the models or tests depending on this assumption.

• Yep, context is correct. And now that I think about it, the answer was pretty trivial. Marbles always help. Thank you!
– mp85
Commented Jun 1, 2017 at 10:07
• If you like it, you could also accept the answer (so future readers know your problem was solved). :)
– IWS
Commented Jun 1, 2017 at 13:06