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The relevant statement is as follows:

"the larger the sample size, the more the sample mean reaches that of a normal distribution EVEN if the population distribution is inherently skewed/biased. In essence, when the sample size gets to a certain threshold, the sample fails to be an accurate estimate of the trends present within the population."

I think the statement is confusing the Central Limit Theorem with the Strong Law of Large Numbers. The first statement is the CLT but the next seems to be a warped notion of how it is applied. The CLT is not a tool for finding the true distribution of the data (like if it was skewed), correct? However, the SLLN does state that the more samples I have, the closer I will come to the true distribution (and see if for example, the true distribution is likely to be skewed).

I would like to ask because the statement quoted above was made supposedly by a professional researcher but I am worried about its veracity.

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  • $\begingroup$ This is what I'm asking about in a post from today! stats.stackexchange.com/questions/473455/… Anyway, the Glivenko–Cantelli theorem says that the empirical distribution approaches the population distribution (under the usual iid conditions). $\endgroup$
    – Dave
    Commented Jun 22, 2020 at 22:28

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Is there such a thing as having too large a sample when it comes to trying to estimate the population distribution?

No. The closer the sample size is to the population size, the better. Ideally you would take a full census of the population, which occurs when the sample is the full population; in that case the sample is the population, and there is no inference problem left. (The second sentence of the quote you gave makes no sense, and does not reflect any valid rule of sampling theory.)

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  • $\begingroup$ The Glivenko–Cantelli theorem should be cited here...convergence of the empirical CDF to the population CDF. $\endgroup$
    – Dave
    Commented Jun 22, 2020 at 22:30
  • $\begingroup$ I have in mind the more realistic cases where the population in finite, such that the GC theorem is not needed to establish the result. The latter becomes important only when you want to establish uniform convergence in the limit (i.e., for an infinite superpopulation). $\endgroup$
    – Ben
    Commented Jun 22, 2020 at 22:39
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I think that the researcher used sloppy language, at best. Their second sentence appears to be incorrect.

I might summarize the relevant points as follows. As sample size increases towards infinity:

  • The distribution of the sample mean (a random variable) converges to a Normal distribution, by the Central Limit Theorem
  • The sample mean itself converges almost surely to the true population mean, by the Strong Law of Large Numbers
  • On a different point, the empirical distribution function (eCDF) converges almost surely to the true distribution function (CDF), by the Glivenko-Cantelli theorem. For instance, if the true distribution was Poisson with mean parameter 3, the empirical distribution would eventually converge to Poisson(3).
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  • $\begingroup$ That first bullet point isn't quite right. The central limit theorem concerns $\sqrt{n}(\bar{X}_n-\mu)$. $\endgroup$
    – Dave
    Commented Jun 22, 2020 at 22:29
  • $\begingroup$ @Dave: I prefer the shorthand in the answer here. It aids intuition better than the formal statement. $\endgroup$
    – Ben
    Commented Oct 15, 2020 at 22:05
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The statement is incorrect and it is obvious that it is incorrect. The first sentence is a misstatement of the central limit theorem. The second sentence is utter nonsense.

The sample mean is a number, not a distribution. What the CLT is about is the distribution of the means of a large number of samples from a population(s).

The answer to your question is "no" -- the larger the sample the more accurately it reflects the population (other things being equal).

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    $\begingroup$ I think you meant obvious that it is incorrect $\endgroup$
    – AllModelsAreWrong
    Commented Oct 28, 2019 at 16:18

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