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What intuitive explanation is there for the central limit theorem?

I am in an introductory statistics course, and I am having trouble understanding the Central Limit Theorem. The way I conceptually think about the theorem is that samples of size n will approach normality as n increases to infinity. I learned that the sample standard deviation equals the population standard deviation divided by the square root of n. But doesn't that mean that the sample standard deviation will be zero as n goes to infinity, since you are dividing by the square root on n? If the standard deviation is zero then how can data be normally distributed? Can someone please explain the Central Limit Theorem in plain English without complicated equations.

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  • $\begingroup$ This question has appeared in many guises here. Please consider searching out site. Some good answers are at stats.stackexchange.com/questions/643, stats.stackexchange.com/questions/3734, and stats.stackexchange.com/a/11969. $\endgroup$
    – whuber ♦
    Nov 15, 2012 at 21:48
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    $\begingroup$ There is a difference between what you said, "samples of size $n$ will approach normality," and the more correct statement, "the sum or average of a sample of size $n$ will approach normality." $\endgroup$
    – Douglas Zare
    Nov 15, 2012 at 23:25
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    $\begingroup$ "The way I conceptually think about the theorem is that samples of size n will approach normality as n increases to infinity" is plain wrong. It's a theorem about a standardized mean. $\endgroup$
    – Glen_b
    Nov 16, 2012 at 6:08
  • $\begingroup$ it could also be the case that if n is infinity, then you will always get the same exact mean value. so the standard deviation should in theory be 0 since there's no variability when you've sampled an infinite number of values? $\endgroup$
    – Matt
    Jun 28, 2021 at 15:40

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