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July 8th, 2012, 10:44 AM  #1 
Member Joined: Oct 2009 Posts: 50 Thanks: 0  why is standard deviation squared?
Hello, I am absolutely baffeled by the fact that standard deviation s = sqrt(1/n Sum((valuemean)^2)) Why is it not just simply: 1/n(sum((abs(valuemean)) Why is it squared and then rooted again? Why isn't just the absolute value taken of the difference between the values and the mean? That would make so much more sense. Also, according to my book, if you don't root it, it's called the sample variance. What is the significance of this sample variance? it seems to me to be a completely meaningless number, because the square seems to be completely arbitrary. Obviously, it isn't nonsence, because otherwise people wouldn't do it, I just really want to know why they do it this way! Thank you! 
July 8th, 2012, 11:39 AM  #2  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: why is standard deviation squared? Quote:
 
July 8th, 2012, 11:47 AM  #3 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: why is standard deviation squared?
Here's an article that argues FOR more widespread teaching and use of the mean absolute deviation (as they call it) http://www.leeds.ac.uk/educol/documents/00003759.htm 
July 8th, 2012, 12:38 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,806 Thanks: 716  Re: why is standard deviation squared?
It is a lot easier to calculate the variance rather than the mean absolute difference.

July 8th, 2012, 01:19 PM  #5 
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: why is standard deviation squared?
The reason why standard deviations will likely always hold sway over mean absolute differences is that standard deviations have strict probabilistic interpretations under the normal distribution. Of course, statisticians can get a little fudgy about when normal distributions truly hold, but even if things are just approximate, that's better than having a statistic from which no probabilities can be calculated. But the mean absolute difference may make at least a bit of a comeback. It too has some advantages, as discussed in the paper I linked to. 
July 8th, 2012, 01:20 PM  #6  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: why is standard deviation squared? Quote:
 
July 9th, 2012, 12:42 PM  #7  
Global Moderator Joined: May 2007 Posts: 6,806 Thanks: 716  Re: why is standard deviation squared? Quote:
For computer programming purposes, the first needs one loop, while the second needs two loops.  
July 9th, 2012, 04:28 PM  #8  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: why is standard deviation squared? Quote:
 

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