# N-1 as Unbiased Estimator of the Population
Variance

The purpose of this applet is to demonstrate that when we
compute the variance or standard deviation of a sample, the use
of (*N*-1) as the divisor will give us a better (less
biased) estimate of the population variance and standard
deviation than will the use of *N* as the divisor. In this
applet we have created a population consisting of each of the
numbers between 0 and 100. Because we have the whole population,
we know that the true mean is = 50, and the variance is = 853. The
true standard deviation () is thus 29.2.

## Small Samples: *N* = 3

You can draw individual samples of size 3 from this
population, and display the results using both *N* and
*N* – 1 as the denominator by clicking on the New
Sample button. First draw individual samples, and note how they
vary. Note that sometimes one denominator produces the closer
estimate, and sometimes the other denominator does so. Now click
on the “10 samples” button. This will draw 10 samples
at once, and will give you the average values for these 10
samples.

The display shows you the estimated standard deviations for
those 10 samples, as computed by using both (*N* –
1) and *N* is the denominator. It also shows you the
averages for both of those estimates. Which one was more accurate
for your data?

Click on the “100 Samples” button until you have
accumulated about 5000 samples. What is your average estimate of
the population standard deviation using the two different
divisors?

## Larger Samples: *N* = 15

We have just drawn samples of three observations at a time and
calculated their standard deviations. With a larger sample size,
the difference between (*N*-1) and *N* becomes less
important. The following applet draws 15 observations for each
sample. Run that applet and compare the results of the two
estimators.

Comments to: Gary.McClelland@Colorado.edu

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