If I draw five simple random sample observations from a non-normal population, calculate their mean, and repeat 24 more times, will the 25 means I've computed approximate a normal distribution? The central limit theorem tells us that a sample mean of sufficient size approaches a normal distribution. My sample means have a size of 5, however. Does the central limit theorem still apply?
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3$\begingroup$ The CLT doesn't apply to any finite sample size. Here's a (real) example where the means of subsamples of size 4000 still weren't anywhere near normally distributed: stats.stackexchange.com/questions/69898 $\endgroup$– whuber ♦Aug 16, 2016 at 23:28
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Note that the number of values you have from your distribution of sample means (the $25$) isn't relevant; if we're drawing samples from some distribution, $F$, they have that distribution whether you have one observation or a million. This applies to the sampling distribution of sample means as well. The $5$ is relevant but only part of the story.
In spite of what you've likely been told, the central limit theorem doesn't discuss what happens to means at some finite sample size - even extremely large ones. The central limit theorem is about convergence of the distribution of standardized sample means in the limit as $n\to\infty$ to the standard normal distribution.
There are theorems that relate to how far the cdf of the distributions of a standardized sample mean can be from that of a standardized normal. If your idea of "approximately normal" can be expressed in terms of distance between cdfs this might perhaps be useful -- but these require more than just a specification of the sample size. However, specifying that the largest difference in cdfs is less than some constant can include distributions that tend not to fall within most people's idea of approximate.
Less formally, for any specified population distribution, we might use simulation to get an idea how far the distribution of standardized sample means might be from a standard normal distribution in whatever sense we like, but for this you'd have to specify the sense in which you mean "approximate" (i.e. a specific criterion by which to assess closeness) as well as how approximate you want by that measure (i.e. you need to identify a specific cut-off at which you would say 'close enough' or 'not close enough').
You don't specify a population distribution in your question. If that might be anything, then you can say almost nothing; it may be that sample means to approach normality in any given sense at any given sample size (indeed, not even in the limit). Or it may be that you have a distribution for which the distribution of sample means will eventually be very close to normal for sufficiently large sample sizes, but which is clearly non-normal for whichever sample size you happen to have.
[whuber mentions a real data example for which sample size 4000 was not sufficient; it's also easy to construct artificial examples for which any specified sample size will not be sufficient.]
Here's an example for which samples of size 5 are pretty good (I can't tell whether they'll be close enough for your purpose though):
But by contrast here's an example where n=5 is clearly not enough for most people:
