TITLE: If $x_i$ is from a random sample is $Var(\bar x \mid x_i)=0$?
QUESTION [0 upvotes]: If $x_i$ is from a random sample, is the conditional variance of the mean (or the sum of squares, really any statistic based on $x$) just treated as a constant? I saw this in a OLS variance of a parameter proof. I'm just looking for some rational as to why it is true.
I think this would also work for expected values So for example:
$$E(\bar x \mid x_i)=\bar x$$ is true as well. Whats the reasoning being these things?
I believe it can also be assumed that $x_i$ is identically distributed random sample.

REPLY [1 votes]: Whenever you see $$\bar{X}\mid \mathbf{x}$$
or some other form of it, it means to fix the value of $\mathbf{x}$.
So, in your case, we seek $$\text{Var}\left[\bar{X}\mid X_1, \cdots, X_n\right]\text{.}$$
But here's the thing. By definition, $$\bar{X} = \dfrac{1}{n}\sum\limits_{i=1}^{n}X_i$$
so thus $\bar{X}$ is constant because $X_1, \cdots, X_n$ are fixed (or constant). Since the variance of a constant is $0$, the result follows.
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