Difference in Standard Deviation Formulas

Date: 01/31/2006 at 17:01:11
From: Stefan
Subject: Standard Deviation

In regards to sample vs. population standard deviation, why is (n-1)
used in the denominator rather than n?

Date: 02/01/2006 at 00:52:14
From: Doctor Wilko
Subject: Re: Standard Deviation

Hi Stefan,

Thanks for writing to Dr. Math!

The answer would be found in any graduate-level book on probability. 
You would have to look up topics on biased/unbiased estimators.

The first question for us to answer is "What's an estimator?".

Definition:  An estimator is a rule/formula that tells us how to 
calculate the value of an estimate based on the measurements 
contained in the sample.  

Let's think about this using means (averages).

For example, if we take a random sample of 20 test scores from a 
class of 35 students, we can try to estimate the entire class 
average (population mean) by calculating the sample mean from these 
20 scores.  This sample mean is an estimator of the true population 
mean.  The rule/formula in this example is to take the 20 scores, 
add them up, and divide by 20.  The answer we get, the sample mean, 
is an estimator--our best inference about the true population mean 
(which we don't know, that's why we're trying to estimate it).

Now that we know what an estimator is, the short answer to your 
question is that sample variance (an estimator) COULD'VE been 
defined as "the sum of the squares of each value minus the sample 
mean divided by n", but it turns out that this is a BIASED estimator.

So, what's a biased estimator?  

A biased estimator in this case means that the sample variance 
calculated this way (divided by n) either over/underestimates the 
true population variance.  If we are trying to make an inference 
about the true population variance, we don't want to 
over/underestimate, we want to be as accurate as we can!

So, using some theory from probability, it can be shown that this 
biased sample variance (as defined by dividing by n) can be made 
UNBIASED by dividing by (n-1) instead.

So, that's the reason we divide by (n-1)--to get an unbiased 
estimator.  Now we have a good estimator that we can use when making 
inferences about a population!

Does this help?  Please write back if you have further questions.


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