I have come up with an estimator for mu in a population of prices which I call Pbar.

I want to show that this estimate is consistent. This is what I've done:

$\displaystyle E [(\frac{1}{n}\sum P_i)^2]$ - $\displaystyle E([\frac{1}{n}\sum P_i] )^2$

= $\displaystyle \frac{1}{n}P^2 bar - \frac{1}{n}Pbar^2$

= $\displaystyle \frac{P^2 bar -P bar^2} {n}$

are there any glaring mistakes here?