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I have a few values which are results of measurement. Every of these values has certain uncertainty, given as absolute portion of the value (eg. $y_i = 16\pm 0.5$, uncertainty $u_i = 0.5$, $x_i = 16$). I need to calculate the final result from this set of values.

I was thinking about it, and I think that I just need to make a sort of inverse weighted mean of the values where the values $u_i$ are the lowest. I've been playing around and for $n$ values I produced this equation, which is wrong:

$$result = \frac{\sum\limits_{i=1}^n \frac{x_i}{u_i} * \sum\limits_{i=1}^n u_i}{n} $$

And actually, even if it worked it would tell me nothing about the resulting uncertainty, which is, I think, a value smaller than largest uncertainty and bigger than the smallest one.

How can I calculate such inverse weighted mean? Is it the right way to solve the problem? Would it be wrong to pick the value with smallest uncertainty right away (I assume it would, but that's an irrational guess)?

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