# central limit theorem and sample size

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?)

• 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? Jul 10, 2022 at 23:42
• read things like: where did you read this? Could you provide references? Jul 10, 2022 at 23:49