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Sampling Distribution of the Variance Ratio

Take two samples from the same population. Calculate the variance of each of those samples using the (n - 1) divisor. Form a ratio by dividing the variance of the second sample into the variance of the first.

Here is an example of the calculation of the Variance Ratio for two samples.

Sample 1 Sample 2
83.5

102.9

102.5

99.2

98.8

91.1

99.1

110.6

101.7

101.5

99.4

96.2

103.4

85.5

92.1

104.4

n 3 13
Degrees of Freedom 2 12
Mean 96.3 98.69
Sum of Squared Deviations (SS) 245.84 495.749
Variance
(SS / DF)
122.92 41.31242

Variance Ratio = 122.92 / 41.31242 = 2.9754

For this pair of samples, the variance of the second sample was almost 3 times a large as the variance of the first sample. Because both samples were randomly sampled from the same population we would expect the variance ratio to be about 1.00.


The Sampling Distribution of the Variance Ratio is the Histogram which comes from repeating the sampling process many many times. In the animation shown below this process of drawing two samples of 3 and 13 were repeated 5,000 times and the results plotted in a histogram.

Click on the controller to view the animation. The animation is updated after every 20th pair of samples.

The mean (aka "average") value of the variance ratio of these 5,000 replications is 1.16 which isn't too far from the value of 1.00. The Mean tells us that the larger variance occurred in an almost balanced way in the two samples forming the variance ratio.

Even more interesting is that 95% (actually 95.48%) of the replications produced a variance ratio of 3.8909 or smaller. If you look up the 95th percentile of the F-Ratio for DFNum = 2 and DFDenom = 12 you find the value is 3.89.

This knowledge becomes the basis for the calculation used by ANOVA to determine whether or not the independent variable is effective.


What to remember from this web page

When two samples are drawn from the same population and the variance ratio calculated, the result is lawful--we can predict what proportion of samples will produce variance ratios smaller than a particular value. The F-Distribution table describes the lawfulness.


©2006 by Burrton Woodruff. All rights reserved. Modified 9/26/06