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Probability and Statistics By Sir Dr. Asghar


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Old 09-03-2009, 03:43 AM
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Read Mean Deviation,Stander Deviation of the Data and Some More Concepts about statistic Data

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 Name:  Inline1.gif
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Size:  316 Bytes, the mean deviation is defined by
Name:  NumberedEquation1.gif
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Size:  1.8 KB (1)

where Name:  Inline2.gif
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Size:  283 Bytes 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 Name:  Inline3.gif
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Size:  321 Bytes defined for Name:  Inline4.gif
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Size:  351 Bytes, 2, ..., Name:  Inline5.gif
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Size:  316 Bytes is given by
Name:  NumberedEquation2.gif
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Size:  1.7 KB (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
Name:  NumberedEquation3.gif
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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 Name:  Inline6.gif
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Size:  277 Bytes possible outcomes with Name:  Inline7.gif
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Size:  1.1 KB for Name:  Inline8.gif
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Size:  351 Bytes, 2, ..., Name:  Inline9.gif
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Size:  316 Bytes. The mean is given by
Name:  NumberedEquation4.gif
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Size:  2.3 KB (4)

The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as
Name:  NumberedEquation5.gif
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Size:  2.3 KB (5)

On the other hand, the mean deviation is given by
Name:  NumberedEquation6.gif
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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 Name:  Inline10.gif
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The following table summarizes the mean absolute deviations for some named continuous distributions, where Name:  Inline11.gif
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Size:  1.2 KB is an incomplete beta function, Name:  Inline12.gif
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Size:  1.1 KB is a beta function, Name:  Inline13.gif
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Size:  1.6 KB is a gamma function, Name:  Inline14.gif
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Size:  309 Bytes is the Euler-Mascheroni constant, Name:  Inline15.gif
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Size:  1.6 KB is a Meijer G-function, Name:  Inline16.gif
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Size:  2.2 KB is the exponential integral function, Name:  Inline17.gif
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Size:  1.2 KB is erf, and Name:  Inline18.gif
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Size:  1.5 KB is erfc.
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