Fusing measurement results

I have n measurements, statistically independant with n values x_i and n standard deviations \sigma_i. How can I combine them to get one value and one standard deviation?

By “measurements” here you mean you only have the summary statistics of the measurements? (Otherwise you could just combine all the data)

Maybe this answers your question:

I personally like Barlow. On page 54, chapter 4 on errors, he writes:

Just be careful and ensure you really believe the original individual errors. Otherwise, imagine you have measurements [1±0.01, 1±0.01, 2±0.01, 2±0.01], and the result will be 1.5±0.005. When you don’t believe original errors, better to do

using StatsBase
mean(vals) ± sem(vals)