# root-mean-square

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 $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$.

 Title root-mean-square Canonical name Rootmeansquare Date of creation 2013-03-22 13:10:24 Last modified on 2013-03-22 13:10:24 Owner pbruin (1001) Last modified by pbruin (1001) Numerical id 4 Author pbruin (1001) Entry type Definition Classification msc 26-00 Classification msc 26D15 Synonym root mean square Synonym rms Synonym quadratic mean Related topic ArithmeticMean Related topic GeometricMean Related topic HarmonicMean Related topic PowerMean Related topic WeightedPowerMean Related topic ArithmeticGeometricMeansInequality Related topic GeneralMeansInequality Related topic ProofOfGeneralMeansInequality