I'm looking through my notes and something my lecture said seems off, just want to clarify.
"Let $Z_1,\ldots, Z_n$ be iid N(0,1) random variables and let $\overline{Z}$ be their average. Then $\overline{Z}$ is normal with mean 0 and deviation $\frac{1}{n}$. So
$$\frac{\overline{Z}-0}{\frac{1}{\sqrt{n}}}=\sqrt{n}\overline{Z}$$ is N(0,1). Is this true? would it not be that the variance of $\overline{Z}$ would be $\frac{1}{n}$?