Why Do We Calculate Standard Deviation the Way We Do?

Date: 02/05/2004 at 00:52:43
From: Greg
Subject: Standard Deviations

Why do we square the deviations and then take the square root when 
calculating standard deviation?  Why can't we just take the absolute 
value of the deviations?  I've tried similar calculations, and the
answers are close, but in most cases, squaring appears to be more correct.

Date: 02/05/2004 at 11:16:40
From: Doctor George
Subject: Re: Standard Deviations

Hi Greg,

Thanks for writing to Doctor Math.

There is more than one way to measure the spread of a distribution. 
Having different methods is a separate issue from the issue of 
accuracy.  Taking the sum of absolute deviations is a perfectly valid 
method, but it will in general lead to a different result from 
standard deviation.  The two methods characterize the spread of the 
distribution in different ways.

I can think of two reasons why the absolute method is not commonly used.

  1. When independent random variables are added their variances add. 
     Remember that variance is the square of the standard deviation.

  2. Variance has the property of being differentiable, which becomes 
     helpful in estimation theory. 

Absolute deviations do not have these two important properties.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/