I'm currently reading "Concepts in Thermal Physics", and in the chapter on independent variables it has the following example:
If we have $n$ independent variables $X_i$, each with a mean $\left<X\right>$, and a variance $\sigma_X^2$, we can sum them to get the following:
$$\begin{split} Y & = \sum^n_iX_i \\ \left<Y\right> & = \sum^n_i\left<X_i\right> = n\left<X\right> \\ \sigma_Y^2 & = n\sigma_X^2 \end{split}$$
I understand the derivation of all this fine, however the following is then stated:
The results proved in this last example have some interesting applications. The first concerns experimental measurements. Imagine that a quantity $X$ is measured $n$ times, each time with an independent error, which we call $\sigma_X$. If you add up the results of the measurements to make $Y = \sum_iX_i$, then the rms error in $Y$ is only $\sqrt{n}$ times the rms error of a single $X$. Hence if you try and get a good estimate of $X$ by calculating $(\sum_iX_i)/n$, the error in this quantity is equal to $\sigma_X/ \sqrt{n}$.
I'm not entirely sure what they mean here by the root mean square error. Is that just another way of saying the standard deviation? If it is, in what sense can the above example lead to the statement that follows?
The only way I can personally see this making sense, is if they are modelling the error in a single measurement as the standard deviation of a probability distribution. This doesn't seem correct to me, is this actually what they are doing?