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Nov 30, 2021 at 21:22 comment added John Darby Thank you @Semoi. You may be interested in my rambling answer to physics.stackexchange.com/questions/675632/…, particularly related to belief/plausibility.
Nov 30, 2021 at 19:53 comment added NotMe Great answer! It helps to realise that that the average is a weighted average with weights $w_i \propto 1/\sigma_i^2$. In addition, the formula from the standard deviation can be motivated by reading about Baysian statistics. In fact, this is the standard formula which is used to weight the "prior knowledge" and "new data" to obtain the "new state of knowledge".
Nov 30, 2021 at 13:01 comment added John Darby You are welcome.
Nov 30, 2021 at 5:19 comment added mqzhang Thank you for your answer!
Nov 30, 2021 at 3:46 history answered John Darby CC BY-SA 4.0