How to check for independence within a sample I see sometimes written (see below for examples) that given a random sample from less than 10% of the population then it's reasonable to assume that the data is independent. I wonder why 10% and not another percentage.
Examples:


*

*http://www.openintro.org/stat/textbook.php, see equation 4.4 on page 164

*http://apcentral.collegeboard.com/apc/members/repository/ap03_stats_assumption_31840.pdf
 A: When sampling $n$ items out of a population of $N$ without replacement, the finite sample population correction,
scales the 'independent' (i.e. sampling with replacement/infinite population) standard error by $\sqrt{\frac{N-n}{N-1}} = \sqrt{\frac{1-n/N}{1-1/N}}\approx 1-\frac{n}{2N}$. 
So we see that when $n$ is a small fraction of $N$, this will reduce the standard error by approximately half the percentage that $n$ is of $N$. So if $n$ is 10% of $N$, the standard error you get by ignoring the finite sample correction will be about 5% too large.
(Note that the above link suggests using 5% as the cut-off for ignoring the correction rather than 10% -- with arbitrary cut-offs, you should expect to find different rules.)
As often the case with arbitrary rules of thumb, it's very likely that someone decided that the 5% difference in standard error between using and ignoring the finite sample population correction when $n/N$ was around 10% would generally be sufficiently close to nothing for their purposes. 
At some point they will have said it or written it down, and since people tend to prefer arbitrary rules to thinking, it got repeated and repeated in turn. 
Just because some arbitrary rule is found in several places doesn't imply the reason is any better than that. 
Since the finite sample population correction is very simple to apply, you should not feel bound by someone else's assessment of what's close enough, but simply work out whether you care enough to use it in each situation; roughly computing the size of the effect ("halve the percentage") is a trivial mental calculation. Since actually applying the exact formula for the correction is so simple, you may consider setting your cut off a fair bit lower, though at some point you'll have a host of approximations in your model of the situation that will swamp small changes (like assuming independence if you were sampling with replacement when perfect independence isn't likely to be exactly true, or assuming equal sampling probability when some subjects actually won't be sampled); below the effect size of that sort of thing, it's not going to matter what you do.
