My Math Forum Standard deviation VS Mean absolute deviation

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 April 27th, 2011, 08:43 AM #1 Newbie   Joined: Apr 2011 Posts: 14 Thanks: 0 Standard deviation VS Mean absolute deviation Hi, From my understanding STD has been the de facto deviation measure for a century now because, among other things, it is easier to compute algebraically(!). Now, I can't understand how absolute values are harder to compute than squares. However, this is what I keep reading on and on in disbelief. Can you clear that up? Also there is a whole literature that advocates the use of MAD over STD because it is more accurate in more chaotic situations (eg. society, in contrast with agriculture) that deviate more from the normal distribution as well as it is better to grasp intuitevely. What's your say on that? And, oh, how on earth are absolutes more difficult than squares?
April 27th, 2011, 04:43 PM   #2
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Re: Standard deviation VS Mean absolute deviation

Quote:
 Originally Posted by Axel Hi, From my understanding STD has been the de facto deviation measure for a century now because, among other things, it is easier to compute algebraically(!). Now, I can't understand how absolute values are harder to compute than squares. However, this is what I keep reading on and on in disbelief. Also there is a whole literature that advocates the use of MAD over STD
Whole literature? Could you provide a citation?

Standard deviation is roughly speaking an average distance from the mean of a set of numbers to the numbers themselves. Think about how you calculate distance in a coordinate setting. You use square root of sum of squares - not absolute values.

As far as which is easier ...

1. Suppose you have a set of numbers $\{x_1, x_2, \ldots, x_n\}$ and suppose you want to find the value of $m$ that minimizes

$\sum \mid x_i-m \mid$

2.Suppose you have a set of numbers $\{x_1, x_2, \ldots, x_n\}$ and suppose you want to find the value of $m$ that minimizes

$\sum (x_i-m)^2$

The first problem appears very difficult to solve. The second is trivial. It is a parabola opening up. Find the vertex. Or if you know calculus, find a derivative and set it equal to zero.

 April 28th, 2011, 03:25 AM #3 Newbie   Joined: Apr 2011 Posts: 14 Thanks: 0 Re: Standard deviation VS Mean absolute deviation Here is a paper debating the pros & cons of each: http://www.leeds.ac.uk/educol/documents/00003759.htm As for the easiness of calculation, the example you provided is conscerned with minimum values. But in finding deviation you seek average values, namely the average distance from the mean. So suppose you want to find STD and MAD. In the first case you will have to square each pair. add them up, and then take the square root. With MAD, though, all you have to do is subtract and always put the "+" sign in front of the values. You avoid squaring and finding the root. In the problems I solve for my statistics module there are no variables to find, [color=#FF4000]mrtwhs[/color], just the statistics, including STD. I can see no shortcut in those cases where STD can be easier to calculate than MAD. Am I missing something here?

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# comparing mean absolute deviation and standard deviation

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