If I have a distribution of data, X, representing N samples taken during one measurement, then the mean square of X is $\bar{X^2} = \langle X^2 \rangle$, the variance of $X^2$, $\mathrm{var}(X^2)$ is $\langle\langle X^4 \rangle - \langle X^2 \rangle \rangle$, and the standard error of the mean square is $\sqrt{\frac{\mathrm{var}(X^2)}{N}}$.
Thus, the standard error of the mean square represents one standard deviation of the distribution that would be produced by repeating the measurement (taking N samples each time), assuming that $X^2$ is normally distributed with the variance of the original measurement.
What is the standard error of the quantity $\sqrt{\langle X^2 \rangle}$ (the standard error of the root mean square?)?
A rephrasing: assume Y is normally distributed with mean $\mu$ and variance $\sigma^2$. Define Z = $\sqrt{\frac{1}{N}\sum_1^N Y_i}$ What is the variance of Z?