Suppose you have a data set of experimental measurements were for each measurement you have the uncertainty associated to it.
What is the correct way to calculate the uncertainty associated with the average of all values in the data set?
2026-03-24 22:51:54.1774392714
How to calculate uncertanty based on the uncertanty of a data set?
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If the samples are independent and identically distributed from a population with variance $\sigma^2$, i.e. standard deviation $\sigma$, then
the sum of $n$ samples has variance $n\sigma^2$ and standard deviation $\sqrt{n} \sigma$
the average of $n$ samples has variance $\frac1n\sigma^2$ and standard deviation $\frac1{\sqrt{n}} \sigma$
Whether the samples here are your measurements or are the errors in your measurements will depend on what precisely you are measuring. The i.i.d. assumption is important and if not true then you may need another approach