Let's say I have two measurements
$$x_1 \pm \Delta x_1$$ $$x_2 \pm \Delta x_2$$
where $x_i$ are indirect measurements and $\Delta x_i$ come from error propagation and might differ from one another.
I can estimate $x$ with the mean
$$\bar x = \frac{x_1 + x_2}{2}$$
But how do I handle the errors? Applying simple rules of error propagation to the above formula I could say
$$ \Delta \bar x = \frac{1}{2} \left(\Delta x_1 + \Delta x_2\right)$$
So the error on the mean goes down as $1/N$... but shouldn't it be $1/\sqrt{N}$?
Also this way all the measurements have the same weight. Should I take into a bigger account more precise measurements with some kind of weighted mean?
