So I got to start working on some statistics notes and at first everything seemed intuitive, the concepts were pretty basic. But then they got a little more complicated. So I do know the neat idea behind the mean deviation which is basically that it is a measure of dispersion from the mean. But why it is given by this formula? $$ \frac{1}{n}\sum_{i=1}^n |x_i-m(X)|. $$ That is, I want to see a derivation of that formula. Thanks in advance.
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I think you're missing a square root over the $n$. – Jun 04 '14 at 17:00
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@Bryan The only square root I know is for the standard deviation $$\sigma=\sqrt{\frac{1}{n}\sum_{i=1}^n (x_i-m(X))^2.}$$ – user155181 Jun 04 '14 at 17:07
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Oh, I'm sorry. I read standard deviation instead of mean deviation. – Jun 04 '14 at 17:08
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I hadn't heard of this before, but the wiki article explains it pretty well I think. http://en.wikipedia.org/wiki/Absolute_deviation#Average_absolute_deviation_about_median. The section on minimization helped me the most. Basically, standard deviation corresponds to $L^2$ distance while mean deviation corresponds to $L^1$ distance, if you have some background in real analysis. Why it isn't used so much, I found a good answer here: http://forums.udacity.com/questions/10006597/standard-deviation-vs-mean-absolute-deviation – Jun 04 '14 at 17:23
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I wish I learned real analysis before... – user155181 Jun 04 '14 at 17:32
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Mean deviation is classically calculated as the arithmetic mean of the absolute values of the differences between observations and the mean or median. This corresponds to the formula that you posted.
Albeit potentially important and rather intuitive, this measure of dispersion is uncommonly used in statistics. This is because the presence of the absolute value would make various types of other statistical calculations more difficult as compared to those obtainable using the SD.
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