Another use of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. For a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.

To perform a F test of two variances, it is important that the following are true:

  • The populations from which the two samples are drawn are normally distributed.
  • The two populations are independent of each other.

Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p-values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here.

Suppose we sample randomly from two independent normal populations. Let σ12σ12 and σ22σ22 be the population variances and s12s12 and s22s22 be the sample variances. Let the sample sizes be n1 and n2. Since we are interested in comparing the two sample variances, we use the F ratio


F has the distribution F ~ F(n1 – 1, n2 – 1),

where n1 – 1 are the degrees of freedom for the numerator and n2 – 1 are the degrees of freedom for the denominator.

If the null hypothesis is σ12=σ22σ12=σ22, then the F ratio becomes F=[(s1)2(σ1)2][(s2)2(σ2)2]=(s1)2(s2)2F=[(s1)2(σ1)2][(s2)2(σ2)2]=(s1)2(s2)2.


The F ratio could also be (s2)2(s1)2(s2)2(s1)2. It depends on Ha and on which sample variance is larger.

If the two populations have equal variances, then s12s12 and s22s22 are close in value and F=(s1)2(s2)2F=(s1)2(s2)2 is close to 1. But if the two population variances are very different, s12s12 and s22s22 tend to be very different, too. Choosing s12s12 as the larger sample variance causes the ratio (s1)2(s2)2(s1)2(s2)2 to be greater than 1. If s12s12 and s22s22 are far apart, then F=(s1)2(s2)2F=(s1)2(s2)2 is a large number.

Therefore, if F is close to 1, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than 1, then the evidence is against the null hypothesis. A test of two variances may be left-tailed, right-tailed, or two-tailed.

Example 13.5

Two college instructors are interested in whether or there is any variation in the way they grade math exams. They each grade the same set of 30 exams. The first instructor’s grades have a variance of 52.3. The second instructor’s grades have a variance of 89.9. Test the claim that the first instructor’s variance is smaller. In most colleges, it is desirable for the variances of exam grades to be nearly the same among instructors. The level of significance is 10 percent.

Solution 13.5

Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively.

n1 = n2 = 30.

H0: σ12=σ22σ12=σ22 and Ha: σ12 σ22σ12 σ22.

Calculate the test statistic: By the null hypothesis (σ12 = σ22)(σ12 = σ22), the F statistic is


Distribution for the test: F29,29 where n1 – 1 = 29 and n2 – 1 = 29.

Graph: This test is left-tailed.

Draw the graph, labeling and shading appropriately.

This graph shows a nonsymmetrical F distribution curve. The curve is slightly skewed to the right, but is approximately normal. The value 0.5818 is marked on the vertical axis to the right of the curve's peak. A vertical upward line extends from 0.5818 to the curve and the area to the left of this line is shaded to represent the p-value.
Figure 13.7

Probability statement: p-value = P(F

Compare α and the p-value: α = 0.10 α > p-value.

Make a decision: Since α > p-value, reject H0.

Conclusion: With a 10 percent level of significance from the data, there is sufficient evidence to conclude that the variance in grades for the first instructor is smaller.

Using the TI-83, 83+, 84, 84+ Calculator

Press STAT and arrow over to TESTS. Arrow down to D:2-SampFTest. Press ENTER. Arrow to Stats and press ENTER. For Sx1, n1, Sx2, and n2, enter (52.3)(52.3), 30, (89.9)(89.9), and 30. Press ENTER after each. Arrow to σ1: and σ2. Press ENTER. Arrow down to Calculate and press ENTER. F = 0.5818 and p-value = 0.0753. Do the procedure again and try Draw instead of Calculate.

Try It 13.5

The New York Choral Society divides male singers into four categories from highest voices to lowest: Tenor1, Tenor2, Bass1, and Bass2. In the table are heights of the men in the Tenor1 and Bass2 groups. One suspects that taller men will have lower voices, and that the variance of height may go up with the lower voices as well. Do we have good evidence that the variance of the heights of singers in each of these two groups (Tenor1 and Bass2) are different?

Tenor1 Bass2 Tenor1 Bass2 Tenor1 Bass2
69 72 67 72 68 67
72 75 70 74 67 70
71 67 65 70 64 70
66 75 72 66   69
76 74 70 68   72
74 72 68 75   71
71 72 64 68   74
66 74 73 70   75
68 72 66 72    
Table 13.11