# mean square deviation

If $f$ is a Riemann integrable^{} real function on the interval $[a,b]$ which is wished to be approximated by another function^{} $\phi $ with the same property, then the mean (http://planetmath.org/MeanValueTheorem)

$$m=\frac{1}{b-a}{\int}_{a}^{b}{[f(x)-\phi (x)]}^{2}\mathit{d}x$$ |

is called the mean square deviation of $\phi $ from $f$.

For example, if $\mathrm{sin}x$ is approximated by $x$ on $[0,\frac{\pi}{2}]$, the mean square deviation is

$$\frac{2}{\pi}{\int}_{0}^{\frac{\pi}{2}}{(\mathrm{sin}x-x)}^{2}\mathit{d}x\approx \mathrm{\hspace{0.17em}0.04923}.$$ |

Title | mean square deviation |

Canonical name | MeanSquareDeviation |

Date of creation | 2013-03-22 18:21:57 |

Last modified on | 2013-03-22 18:21:57 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26A06 |

Classification | msc 41A99 |

Classification | msc 26A42 |

Synonym | mean squared error |

Related topic | Variance |

Related topic | RmsError |

Related topic | AverageValueOfFunction |