understanding the Central Limit Theorem with different sample size

I'm trying to understanding the standard deviation of the sampling distribution from the Central Limit Theorem.

$$\bar{X}\rightarrow (\mu ,\frac{\sigma^{2} }{n})$$

I can understand from a mathematical point of view that when n increases, it will reduce the standard deviation of the sampling distribution, but I can't explain this phenomenon from the perspective of the real world. Is there a more intuitive explanation?

• Do you have the same common misconception about the central limit theorem that I once had?
– Dave
Mar 11, 2022 at 10:39

• Thank you for your reply, from your point of view, I try to use dice to understand, when n=1, the probability of $\bar{X}=1$ or $6$ is $1/3$, and when n=2, the probability of $\bar{X}=1$ or $6$ = $2/36$. As n becomes larger, the probability of the average = 1 or 6 approaches 0, which proves that the sample average is more concentrated in the expected value of the dice. Inspiration also comes from elonen.iki.fi/articles/centrallimit/index.en.html#demo