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 mu (1K) = 50, and the variance is var (1K) = 853. The true standard deviation (sigma (1K)) 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

back arrow Return to index