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If I want to average two values with errors, say

x1 = 10 +/- 2

x2 = 12 +/- 4

I would do a regular weighted average that would reflect the fact that x1 is a more precise measurement.

But what if my errors are asymmetric? So if I had

x1 = 10 +2/-1

x2 = 12 +3/-4

How would I take the average of those values? My first thought was to somehow "normalize" x1 and x2 such that they have symmetric errors, but I am not sure whether that is valid or how I would do that. Anyone have any ideas? thanks...