Required Statistics:

Standard Deviation

Consider a population consisting of the following eight values:

    2,\  4,\  4,\  4,\  5,\  5,\  7,\  9

These eight data points have the mean (average) of 5:

    \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5

To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each:

    \begin{array}{lll}     (2-5)^2 = (-3)^2 = 9   &&  (5-5)^2 = 0^2 = 0 \\     (4-5)^2 = (-1)^2 = 1  &&  (5-5)^2 = 0^2 = 0 \\     (4-5)^2 = (-1)^2 = 1  &&  (7-5)^2 = 2^2 = 4 \\     (4-5)^2 = (-1)^2 = 1  &&  (9-5)^2 = 4^2 = 16     \end{array}

Next compute the average of these values, and take the square root:

    \sqrt{ \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} } = 2


Mean Squared Error
See Mean Squared Error on Wikipedia


Absolute Error
Given some value v and its approximation vapprox, the absolute error is

\epsilon = |v-v_{\text{approx}}|\,

where the vertical bars denote the absolute value. If v\ne 0, the relative error is

 

Relative Error

\eta = \frac{|v-v_{\text{approx}}|}{|v|} = \left| \frac{v-v_{\text{approx}}}{v} \right|,

and the percent error is

\delta = \frac{|v-v_{\text{approx}}|}{|v|}\times 100 = \left| \frac{v-v_{\text{approx}}}{v} \right|\times 100.

These definitions can be extended to the case when v and vapprox are n-dimensional vectors, by replacing the absolute value with an n-norm.[1]

 

References:
Approximation Error
Standard Deviation
Mean Squared Error on Wikipedia

Last edited Apr 17, 2011 at 11:33 PM by sesmar, version 1

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