# F Distribution and One-way Anova

## Test of Two Variances

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:


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 
  and 
  be the population variances and 
  and 
  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 
 , then the F ratio becomes 
 .
If the two populations have equal variances, then 
  and 
  are close in value and 
  is close to 1. But if the two population variances are very different, 
  and 
  tend to be very different, too. Choosing 
  as the larger sample variance causes the ratio 
  to be greater than 1. If 
  and 
  are far apart, then 
  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.

### References

ESPN. (2012). MLB standings  2012. Retrieved from http://espn.go.com/mlb/standings/_/year/2012/type/vs-division/order/true.

### Chapter Review

The F test for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as an F statistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. A test of two variances hypothesis test determines if two variances are the same. The distribution for the hypothesis test is the F distribution with two different degrees of freedom.
2. The populations from which the two samples are drawn are normally distributed.
3. The two populations are independent of each other.

### Formula Review

F has the distribution F ~ F(n1 – 1, n2 – 1)
F = 
 
If σ1 = σ2, then F = 
 
Use the following information to answer the next two exercises. There are two assumptions that must be true to perform an F test of two variances.
Use the following information to answer the next seven exercises. Two coworkers commute from the same building. They are interested in whether there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first worker’s times have a variance of 12.1. The second worker’s times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10 percent level. Assume that commute times are normally distributed.
Use the following information to answer the next four exercises. Two students are interested in whether there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first student’s grades have a standard deviation of 38.1. The second student’s grades have a standard deviation of 22.5. The second student thinks his scores are more consistent.
Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up 35 hills. The first cyclist has a variance of 23.8, and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different. Assume that speeds are normally distributed.

### Homework

Use the following information to answer the next two exercises. The following table lists the number of pages in four different types of magazines.