Mean Deviation,Stander Deviation of the Data and Some More Concepts about statistic Data
 

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 Attachment 6148, the mean deviation is defined by

Attachment 6142

(1)

where Attachment 6149 is the mean of the distribution. The mean deviation of a list of numbers is implemented in Mathematica as MeanDeviation[data].
The mean deviation for a discrete distribution Attachment 6150 defined for Attachment 6151, 2, ..., Attachment 6152 is given by

Attachment 6143

(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

Attachment 6144

(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 Attachment 6153 possible outcomes with Attachment 6154 for Attachment 6155, 2, ..., Attachment 6156. The mean is given by

Attachment 6145

(4)

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

Attachment 6146

(5)

On the other hand, the mean deviation is given by

Attachment 6147

(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 Attachment 6157 even and odd separately.
The following table summarizes the mean absolute deviations for some named continuous distributions, where Attachment 6158 is an incomplete beta function, Attachment 6159 is a beta function, Attachment 6160 is a gamma function, Attachment 6161 is the Euler-Mascheroni constant, Attachment 6162 is a Meijer G-function, Attachment 6163 is the exponential integral function, Attachment 6164 is erf, and Attachment 6165 is erfc.