KS, your question is interesting, why not the absolute value and why the square. I guess, the answer below should be helpful.
A fundamental task in many statistical analyses is to characterize the spread, or variability, of a data set. Measures of scale are simply attempts to estimate this variability.
When assessing the variability of a data set, there are two key components:
How spread out are the data values near the center?
How spread out are the tails?
The variance is roughly the arithmetic average of the squared distance from the mean. Squaring the distance from the mean has the effect of giving greater weight to values that are further from the mean. For example, a point 2 units from the mean adds 4 to the above sum while a point 10 units from the mean adds 100 to the sum. Although the variance is intended to be an overall measure of spread, it can be greatly affected by the tail behavior.
The standard deviation is the square root of the variance. The standard deviation restores the units of the spread to the original data units (the variance squares the units).
the average absolute deviation measure does not square the distance from the mean, so it is less affected by extreme observations than are the variance and standard deviation. Hope this clears your dobut.
Rajesh Mohandas.
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