# Is the sampling distribution for small samples of a normal population normal or t distributed? [closed]

If I know that the population is normally distributed, and then take small samples from this population, is it more correct to claim that the sampling distribution is normal or instead follows the t distribution?

I understand that small samples tend to be t distributed, but does this only apply when the underlying population distribution is unknown?

Thanks!

1) a set of random observations from a population with distribution $F$ are samples from that distribution. So even single values sampled from a normal population are normally distributed. (Well, speaking slightly more strictly, the random variable that represents the single draw is the thing that’s normally distributed.)

2) If the observations are independent draws from a normal distribution, the sample means are normal. (If they’re dependent, it matters what the dependence structure is.)

3) Here’s something that will be t-distributed, if the data are i.i.d draws from a normal population: t-statistics. (We get something other than normal because there’s a numerator and a denominator)

I understand that small samples tend to be t distributed

This is a mistaken understanding. On what is this understanding based?

[This seems to be such a common misunderstanding that I can only assume it’s in some popular or once-popular book somewhere. If you do find such a book, post details in your question or in a comment, because I’d love to know where it comes from.]