I am currently reading about the standard error. I know about central limit theorem, but I don't understand why, if my variable is normally distributed in the population, the sampling distribution (the distribution of the sample means) is also a normal distribution, no matter of sample size etc.
This is not an asymptotic result, it is a finite-sample result and has nothing to do with the Central Limit Theorem.
You start by assuming that the population is normal. The normal distribution is one of the few distributions that are closed under addition, meaning that the sum of normal random variables follows a normal distribution (although with different parameters than the components of the sum). More over, scaling a normal random variable, leaves it a normal random variable (again, affecting the parametrization of it).
The Central Limit Theorem comes into effect to lead to the same result asymptotically, when the population is not normally distributed.
|show 1 more comment|