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Homemean square deviation

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# 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 $\varphi$ with the same property, then the mean

$m\;=\;\frac{1}{b\!-\!a}\int_{a}^{b}[f(x)\!-\!\varphi(x)]^{2}\,dx$ |

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

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

$\frac{2}{\pi}\int_{0}^{{\frac{\pi}{2}}}(\sin{x}-x)^{2}\,dx\,\approx\,0.04923.$ |

Related:

Variance, RmsError, AverageValueOfFunction

Synonym:

mean squared error

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

26A06*no label found*41A99

*no label found*26A42

*no label found*

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new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

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new question: A trascendental number. by Ron Castillo

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new question: Banach lattice valued Bochner integrals by math ias