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.