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Mean Deviation


The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size N, the mean deviation is defined by

 MD=1/Nsum_(i=1)^N|x_i-x^_|,
(1)

where x^_ is the mean of the distribution. The mean deviation of a list of numbers is implemented in the Wolfram Language as MeanDeviation[data].

The mean deviation for a discrete distribution P_i defined for i=1, 2, ..., N is given by

 MD=sum_(i=1)^NP_i|x_i-x^_|.
(2)

Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation

 sigma=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2).
(3)

As a result, least squares fitting and other standard statistical techniques rely on minimizing the sum of square residuals instead of the sum of absolute residuals.

For example, consider the discrete uniform distribution consisting of n possible outcomes with P_i=1/N for i=1, 2, ..., N. The mean is given by

 x^_=sum_(i=1)^NiP_i=1/Nsum_(i=1)^Ni=1/2(N+1).
(4)

The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as

 sigma^2=sum_(i=1)^N(i-x^_)^2P_i=1/(12)(N-1)(N+1).
(5)

On the other hand, the mean deviation is given by

 MD=sum_(i=1)^N|i-x^_|P_i=1/Nsum_(i=1)^N|i-x^_|.
(6)

This can be obtained in closed form, but is much more unwieldy since it requires breaking up the summand into two pieces and treating the cases of n even and odd separately.

The following table summarizes the mean absolute deviations for some named continuous distributions, where B(z;a,b) is an incomplete beta function, B(a,b) is a beta function, Gamma(z) is a gamma function, gamma is the Euler-Mascheroni constant, G_(2,1)^(0,2)(a,b|m) is a Meijer G-function, Ei(z) is the exponential integral function, erf(z) is erf, and erfc(z) is erfc.

The following table summarizes the mean absolute deviations for some named discrete distributions, where mu=zeta(rho)/zeta(rho+1).


See also

Absolute Deviation, Standard Deviation, Variance

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References

Havil, J. "Ways of Means." §13.1 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962.

Referenced on Wolfram|Alpha

Mean Deviation

Cite this as:

Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MeanDeviation.html

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