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### 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!

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.

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) underestimates the true
population variance.  If we are trying to make an inference about the
true population variance, we don't want to 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

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

- Doctor Wilko, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Statistics
High School Statistics

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