# Why is the sampling distribution of normal distributed variable automatically also normal distributed

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.

The sample mean of a sample from a normal population, is a sum of normal random variables, scaled by $1/n$. Hence it too follows a normal distribution.