# 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 (http://planetmath.org/MeanValueTheorem)

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

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

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