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The central limit theorem states that:

for identically distributed independent samples, the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed.

But I'm confused about the precise meaning of this statement. If the original random variable is uniformally distributed between (0,1), then the sample mean will always be a real number >= 0, and the standardized sample mean will always have some cutoff point somewhere on the domain, regardless of the size of the sample N. Whereas a Normal distribution PDF is non zero across the entire domain. So there seems to me that there is a fundamental difference.

Am I wrong in this, and is the standardized sample mean for this uniform random variable truly normally distributed when N = infinity?

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  • $\begingroup$ youtube.com/watch?v=zeJD6dqJ5lo $\endgroup$ Commented Mar 15, 2023 at 7:44
  • $\begingroup$ By construction, a standardized sample or standardized distribution has a mean of zero. The original mean is irrelevant. $\endgroup$
    – whuber
    Commented Mar 15, 2023 at 16:58
  • $\begingroup$ @whuber but that doesn't remove the cutoff point somewhere along the domain, it just shifts it to the left $\endgroup$
    – WalksB
    Commented Mar 15, 2023 at 23:40
  • $\begingroup$ I don't get your problem with a "cutoff point." I think if you read some of our better posts on what the CLT means, you'll see this is irrelevant. One way to see this is to play a game: if you think there's a definite cutoff point that causes a problem for the CLT, then name it. In response, it's an easy task to compute a sample size where the distribution of the standardized mean extends far past that cutoff. $\endgroup$
    – whuber
    Commented Mar 16, 2023 at 0:06

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then the sample mean will always be a real number >= 0

Yes. And less than 1. Sure.

the standardized sample mean will always have some cutoff point somewhere on the domain, regardless of the size of the sample N.

Those hard limits grow with sample size (they grow in proportion to $\sqrt{n}$). As $n\to\infty$ those bounds pass beyond any finite value, - but they also become less important (as n increases, the bounds are more and more standard deviations from the mean). e.g. at n=1000000, those bounds are over 577 standard deviations from the mean.

Whereas a Normal distribution PDF is non zero across the entire domain. So there seems to me that there is a fundamental difference.

The key phrase in the actual CLT (a phrase that's missing from what you're quoting there, by the look) is in the limit as $n\to\infty$.

Am I wrong in this, and is the standardized sample mean for this uniform random variable truly normally distributed when N = infinity?

The sample size is never actually infinity. You have a sequence of standardized means at increasing sample size, with a corresponding sequence of distributions (in this case, standardized versions of the Irwin-Hall distributions). The standard normal is the limiting case of that sequence; you get closer and closer to it -- and ultimately, as close as you like (e.g. If you look at the biggest absolute difference between the cdf of the standardized mean $F_n$ and that of the standard normal, $\Phi$, there's an $n$ that will bring that below any $\epsilon>0$).

In other words it gets as close as you like; at some finite sample size it can be closer than any positive bound (you may need a different one for each bound, naturally).

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  • $\begingroup$ Sorry, meant to say uniformly distributed ~ [0, 1]. Fixed question. $\endgroup$
    – WalksB
    Commented Mar 15, 2023 at 19:25
  • $\begingroup$ I will edit in response $\endgroup$
    – Glen_b
    Commented Mar 15, 2023 at 23:45

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