On the wikipedia page https://en.wikipedia.org/wiki/Fermi_problem, it claims that
"In continuous terms, if one makes a Fermi estimate of n steps, with standard deviation σ units on the log scale from the actual value, then the overall estimate will have standard deviation σ√n, since the standard deviation of a sum scales as √n in the number of summands."
Could someone provide a rigorous proof of this? I have been unable to.
The statement is a consequence of a well-known theorem in statistics stating that if $X$ and $Y$ are independent random variables with variances $\sigma_X^2$ and $\sigma_Y^2$ respectively, then the variance of $X+Y$ is $\sigma_x^2 + \sigma_Y^2$. Therefore if $X_1, X_2, X_3, \dots ,X_n$ are independent random variables each of which has variance $\sigma^2$, then the variance of $X_1 + X_2 + X_3+ \dots + X_n$ is $n \sigma^2$, and the standard deviation of the sum is $\sqrt{ n \sigma^2} = \sigma \sqrt n$.