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I have a quick question about central limit theorem. In math, we have a theorem that:

$Z = \lim\limits_{n \rightarrow \infty} \frac{\bar X_n - \mu}{\sigma/\sqrt n}$ is a random variable with normal distribution.

As I as studying statistics, I read things like "if sample size is greater than equal to 30, then distribution starts looking like normal distribution." In other words, if $n \geq 30$, then the distribution of $n$-many samples converge to the normal distribution. However, is there any restriction to each sample? (i.e. would each sample be required to have certain minimum number of data?)

Thanks in advance.

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  • $\begingroup$ Many posts & comments on site discuss the n=30 thing; counterexamples abound. For all that you can find mention of it in many locations, is demonstrably far too weak in many cases and far too strong in others - and among other things - depends on what purposes you're using the approximation for and what properties of that purpose you care about. Because the 'rule' itself contains not a hint of any discussion of the situations in which it could reasonably be claimed to work (nor any reliable way to figure out if you're even in those situations) I think we have to regard it as next-to-useless. $\endgroup$
    – Glen_b
    Jul 10, 2022 at 23:40
  • $\begingroup$ When you do see discussion of these issues, they often boil down to something like "it works when it works" ... which is undeniably true but useless in practice. Have you seen any of the rules of thumb for the normal approximation to the binomial? When $p$ is on the small side, say $1/36$ (chance of snake eyes on two dice), what does such a rule imply for $n$? Assuming that the binomial rule is adequate (it, too, is sometimes is too small), what does that tell you about a blanket $n=30$ rule? $\endgroup$
    – Glen_b
    Jul 10, 2022 at 23:42
  • $\begingroup$ read things like: where did you read this? Could you provide references? $\endgroup$ Jul 10, 2022 at 23:49
  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Jul 10, 2022 at 23:49
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    $\begingroup$ However, it looks like there's an error in phrasing in your post (which is nevertheless presumably an accurate representation of what you read, so not your fault); it seems to be saying that the distribution of observations (rather than of means or sums) becomes more nearly normal. This is not the case. It is however the case that standardized means or sums approach normality in the limit, as long as the conditions hold. This disturbingly common error is also discussed in many posts on site; a few searches turn up relevant posts. $\endgroup$
    – Glen_b
    Jul 10, 2022 at 23:50

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