What is the uncertainty of a discrete sum given the uncertainty of an individual element?

Question

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $X_i$. It is also known that $\sigma_i = \sigma_j$ for for all $i, j = 1, 2, \dotsc, n$.

Answer 1

If $X_i$s are independent and you have already their standard deviations, say $\sigma_i$, for $X_i$, which seem to be the same, then the variance of each $X_i$, for $i=1,\cdots, n$ is $\sigma^2_i$, and so the variance of $X$ is equal to $\sum_{i=1}^n\sigma^2_i=n\times \sigma^2_i$. As a result, the standard deviation of $X$ is $\sqrt{n\times \sigma^2_i}=\sigma_i\sqrt{n}.$