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)

$$m=\frac{1}{b-a}{\int }_a^b{[f(x)-\phi (x)]}^2𝑑x$$

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

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

$$\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}{(\sin x-x)}^2𝑑x\approx \mathrm{\hspace{0.17em}0.04923}.$$

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