PlanetMath: root-mean-square

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root-mean-square

(Definition)

If $x_1,x_2,\ldots,x_n$ are real numbers, we define their root-mean-square or quadratic mean as

$\displaystyle R(x_1,x_2,\ldots,x_n)=\sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}.$

The root-mean-square of a random variable X is defined as the square root of the expectation of :

$\displaystyle R(X)=\sqrt{E(X^2)}$

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 $\sigma_1,\sigma_2,\ldots,\sigma_n$ .

"root-mean-square" is owned by pbruin.

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Other names:

root mean square, rms, quadratic mean

Keywords:

mean, expectation

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Cross-references: arithmetic mean, standard deviations, expectation, square root, random variable, real numbers
There are 3 references to this entry.

This is version 1 of root-mean-square, born on 2002-11-23.
Object id is 3618, canonical name is RootMeanSquare3.
Accessed 22562 times total.

Classification:

AMS MSC:	26-00 (Real functions :: General reference works )
	26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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