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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?

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