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Let's say we have a uniform distribution and we are drawing samples of size 1 so that the mean is the drawn number itself.

If we perform this activity sufficiently large number of times we would get the distribution of the means as the uniform distribution itself and not normal.

So does the CLT not apply for drawn sample size 1?

Edit: I don't understand the bashing here. What made any of you think that I posted here without research or reading? It could be the case that I couldn't understand something basic. I am a beginner to this subject as you can see from my reputations. I think I should only post complex stuffs here.

Shivendra
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  • Well the CLT is asymptotic to begin with, it requires a large sample to take effect. – JohnK Jul 17 '16 at 11:07
  • @JohnK If we take the sample size as 2 and find out the mean and take many many samples then the means follow normal distribution. Here the sample size is 2 and the number of times this experiment has been performed is large. So the sample size is low in this example. – Shivendra Jul 17 '16 at 11:18
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    Why not read some of [our posts on the CLT](http://stats.stackexchange.com/search?tab=votes&q=Central%20Limit%20Theorem) to understand what it actually says? – whuber Jul 17 '16 at 14:05
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    _"If we perform this activity sufficiently large number of times we would get the distribution of the means as the uniform distribution itself and not normal."_ Not so. As Mark Twain wrote, "It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so." – Dilip Sarwate Jul 17 '16 at 15:58
  • @DilipSarwate Mark Twain is a genius and this quote made me smile :) I should have written - *"..could be approximated by a normal distribution"* – Shivendra Jul 17 '16 at 16:03

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The mean of a sample of size 1 from a uniform distribution will follow that same uniform distribution. Hence, yes, the CLT does not apply and the mean from a small sample size will not be normally distributed.

If you generate a lot of such (independent) means, they still follow that uniform distribution (for more than one sample per mean the means start to concentrate a bit more around the middle of the range of the uniform distribution, of course, but will still not have a normal distribution). Of course, from the process of drawing a lot of them and forming the mean of the means, you will get a random variable that approaches a normal distribution as per the CLT.

Björn
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  • I think I get it now. So basically drawing a lot of samples make the outcome more symmetrical ( but still discrete) and increasing the sample size while doing so smoothes out the outcomes (as you get more unique values) and the distribution tends towards normal. Am I right? – Shivendra Jul 17 '16 at 12:23
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    Normally we do not say that a theorem "does not hold" when its assumptions are false. That would give people the false impression that the theorem is incorrect. We simply say that the theorem *does not apply* at all. – whuber Jul 17 '16 at 13:55
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    @whuber Oops. Good point. Shivendra: not sure exactly how you mean that. The difference is between the distribution of lots of samples and three distribution of their mean. – Björn Jul 17 '16 at 14:02
  • @whuber apologies for the wrong writing. I meant 'does not apply'. Thanks for correcting. – Shivendra Jul 17 '16 at 15:08
  • The distribution of the mean converges to a _constant_ (a normal random variable with zero variance if you like). This is the usual confusion between what the CLT actually says and what most beginning (and a few more advanced) students believe that the CLT says. See, for example, _[this answer](http://stats.stackexchange.com/a/22532/6633). – Dilip Sarwate Jul 17 '16 at 15:54