If $x_1,x_2,\ldots,x_n$ are real numbers, we define their root-mean-square or quadratic mean as $$ R(x_1,x_2,\ldots,x_n)=\sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}. $$

The root-mean-square of a random variable X is defined as the square root of the expectation of $X^2$ $$ R(X)=\sqrt{E(X^2)} $$

If $X_1,X_2,\ldots,X_n$ are random variables with standard deviations $\sigma_1,\sigma_2,\ldots,\sigma_n$ then the standard deviation of their arithmetic mean, $\frac{X_1+X_2+\cdots+X_n}{n}$ is the root-mean-square of $\sigma_1,\sigma_2,\ldots,\sigma_n$