Questions About Standard Deviation

Date: 9/20/95 at 16:20:1
From: Brad Morris
Subject: standard deviations

Dr. Math,

One of my colleagues here at Friends' Central School in Wynnewood 
said a question came up in his precalculus class which he could 
not answer. It's in two parts:

1.  Why does a standard deviation take the square root of the 
average of the sum of the squared differences between X and 
X mean instead of the average of the absolute value differences?

2.  What is the reasoning behind dividing by n vs. n-1 in the 
population versus sample standard deviations?

Well, I couldn't come up with a satisfactory answer, so I'm ASKING 
DR MATH.

Brad Morris
Look forward to hearing from you on this as are the students!

Date: 9/26/95 at 7:53:42
From: Doctor Steve
Subject: Re: statistics

Here's a "short" answer from Professor Grinstead. Don't hesitate 
to write back to have something explained.--Steve

1. The reason that squared values are used is so that the 
algebra is easier.  For example, the variance (second central 
moment) is equal to the expected value of the square of the 
distribution (second non-central moment) minus the square of the 
mean of the distribution.  This would not be true, in general, 
if the absolute value definition were used.

   This is not to say that the absolute value definition is 
without merit.  It is quite reasonable for use as a measure of 
the spread of the distribution.  In fact, I have heard of 
someone who used it in teaching a course in statistics.  (I 
think he used it because he thought it was a more 'natural' way 
to measure the spread.)

2. The reason that n-1 is used instead of n in the formula for 
the sample variance is as follows:  The sample variance can be 
thought of as a random variable, i.e. a function which takes on 
different values for different samples of the same distribution. 
Its use is as an estimate for the true variance of the 
distribution. In statistics, one typically does not know the 
true variance; one uses the sample variance to ESTIMATE the true 
variance.  Since the sample variance is a random variable, it 
usually has a mean, or average value. One would hope that this 
average value is close to the actual value that the sample 
variance is estimating, i.e. close to the true variance.  In 
fact, if the n-1 is used in the defining formula for the sample 
variance, then it is possible to prove that the average value of 
the sample variance EQUALS the true variance.  If we replace the 
n-1 by an n, then the average value of the sample variance is 
((n-1)/n) times as large as the true variance.

    A random variable X which is used to estimate a parameter p 
of a distribution is called an unbiased estimator if the 
expected value of X equals p.  Thus, using the n-1 gives an 
unbiased estimator of the variance of a distribution.

-Doctor Steve,  The Geometry Forum