root-mean-squareRootMeanSquare32002-11-23 11:55:392002-11-23 11:55:39Definitionmeanexpectation% this is the default PlanetMath preamble. as your knowledge
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% define commands hereIf $x_1,x_2,\ldots,x_n$ are real numbers, we define their
\emph{root-mean-square} or \emph{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 $\sigma_1,\sigma_2,\ldots,\sigma_n$.