# root-mean-square

If ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ are real numbers, we define their
*root-mean-square ^{}* or

*quadratic mean*as

$$R({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})=\sqrt{\frac{{x}_{1}^{2}+{x}_{2}^{2}+\mathrm{\cdots}+{x}_{n}^{2}}{n}}.$$ |

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 ${X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}$ are random variables with standard deviations^{}
${\sigma}_{1},{\sigma}_{2},\mathrm{\dots},{\sigma}_{n}$, then the standard deviation of
their arithmetic mean^{}, $\frac{{X}_{1}+{X}_{2}+\mathrm{\cdots}+{X}_{n}}{n}$, is the
root-mean-square of ${\sigma}_{1},{\sigma}_{2},\mathrm{\dots},{\sigma}_{n}$.

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 |