why n>=30 for central limit theorem to hold?

Question

From population choosing samples(size n=30) and calculate its mean then repeating it N times will converge to normal distribution as N->inf when mean of each sample is plotted as a histogram.

From my understanding as size of n increase normal distribution will have smaller standard deviation, this makes sense because using larger sample size will be better at estimating population mean than smaller sample.

is my understanding correct?

edit: for clarification I am referring to n,N as follows:

• n = number of data in a sample
• N = number of time sample(of size n) is sampled

Answer 1

As a historical example of a different set of folklore, one way people used to generate Normal random numbers was as a sum of $n=12$ Unif[0,1] variables (minus 6, to recenter at zero)

rclt<-function(n){
  u<-runif(12*n,0,1)
  rowSums(matrix(u, ncol=12))-6
}
qqnorm(rclt(1000))

As you can see, this is quite accurately Normal -- we now have ways of doing this that are better and faster, but the Normal approximation worked very well just adding 12 things

Peter Hall wrote a paper giving two-sided bounds on the approximation accuracy of the CLT. It depends on the tails of the distributions you're adding up.

Answer 2

N times will show normal distribution when mean of each sample mean is plotted.

That just isn't true. The sample mean of $n$ exponential random variables with rate parameter $\lambda$ has Gamma distribution with shape and rate parameters $n$ and $\lambda n$ respectively.

The central limit theorem says that in the limit as $n$ approaches infinity, the sampling distribution of the sample mean converges in distribution to a normal. As to your titular question, there is no threshold as to when the central limit theorem does or does not apply.

You are likely referencing the "use a Z test (as opposed to a t test) when there are more than 30 samples" rule of thumb. The line in the sand of 30 samples was likely chosen because beyond $n=30$ there is at most a 5% relative error when using the normal to approximate the t on the interval [-2,2] which contains almost 95% of the probability mass of each distribution.