Another of the uses 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. In order 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 checkout times for two checkers.
In order 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 pvalues 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 ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$ be the population variances and ${s}_{1}^{2}$ and ${s}_{2}^{2}$ be the sample variances. Let the sample sizes be n_{1} and n_{2}. Since we are interested in comparing the two sample variances, we use the F ratio:
$F=\frac{\left[\frac{{({s}_{1})}^{2}}{{({\sigma}_{1})}^{2}}\right]}{\left[\frac{{({s}_{2})}^{2}}{{({\sigma}_{2})}^{2}}\right]}$
F has the distribution F ~ F(n_{1} – 1, n_{2} – 1)
where n_{1} – 1 are the degrees of freedom for the numerator and n_{2} – 1 are the degrees of freedom for the denominator.
If the null hypothesis is ${\sigma}_{1}^{2}={\sigma}_{2}^{2}$, then the F Ratio becomes $F=\frac{\left[\frac{{({s}_{1})}^{2}}{{({\sigma}_{1})}^{2}}\right]}{\left[\frac{{({s}_{2})}^{2}}{{({\sigma}_{2})}^{2}}\right]}=\frac{{({s}_{1})}^{2}}{{({s}_{2})}^{2}}$.
If the two populations have equal variances, then ${s}_{1}^{2}$ and ${s}_{2}^{2}$ are close in value and $F=\frac{{({s}_{1})}^{2}}{{({s}_{2})}^{2}}$ is close to one. But if the two population variances are very different, ${s}_{1}^{2}$ and ${s}_{2}^{2}$ tend to be very different, too. Choosing ${s}_{1}^{2}$ as the larger sample variance causes the ratio $\frac{{({s}_{1})}^{2}}{{({s}_{2})}^{2}}$ to be greater than one. If ${s}_{1}^{2}$ and ${s}_{2}^{2}$ are far apart, then $F=\frac{{({s}_{1})}^{2}}{{({s}_{2})}^{2}}$ is a large number.
Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or twotailed.
Two college instructors are interested in whether or not 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%.
Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively.
n_{1} = n_{2} = 30.
H_{0}: ${\sigma}_{1}^{2}={\sigma}_{2}^{2}$ and H_{a}: ${\sigma}_{1}^{2}\text{}{\sigma}_{2}^{2}$
Calculate the test statistic: By the null hypothesis $\text{(}{\sigma}_{\text{1}}^{\text{2}}\text{=}{\sigma}_{\text{2}}^{\text{2}}\text{)}$, the F statistic is:
$F=\frac{\left[\frac{{({s}_{1})}^{2}}{{({\sigma}_{1})}^{2}}\right]}{\left[\frac{{({s}_{2})}^{2}}{{({\sigma}_{2})}^{2}}\right]}=\frac{{({s}_{1})}^{2}}{{({s}_{2})}^{2}}=\frac{52.3}{89.9}=0.5818$
Distribution for the test: F_{29,29} where n_{1} – 1 = 29 and n_{2} – 1 = 29.
Graph: This test is left tailed.
Draw the graph labeling and shading appropriately.
Probability statement: pvalue = P(F < 0.5818) = 0.0753
Compare α and the pvalue: α = 0.10 α > pvalue.
Make a decision: Since α > pvalue, reject H_{0}.
Conclusion: With a 10% level of significance, from the data, there is sufficient evidence to conclude that the variance in grades for the first instructor is smaller.
References
“MLB Vs. Division Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012/type/vsdivision/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.
 The populations from which the two samples are drawn are normally distributed.
 The two populations are independent of each other.
Formula Review
F has the distribution F ~ F(n_{1} – 1, n_{2} – 1)
F = $\frac{\frac{{s}_{1}^{2}}{{\sigma}_{1}^{2}}}{\frac{{s}_{2}^{2}}{{\sigma}_{2}^{2}}}$
If σ_{1} = σ_{2}, then F = $\frac{{s}_{1}{}^{2}}{{s}_{2}{}^{2}}$
Use the following information to answer the next two exercises. There are two assumptions that must be true in order to perform an F test of two variances.
Name one assumption that must be true.
The populations from which the two samples are drawn are normally distributed.
What is the other assumption that must be true?
Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not 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 and that his commute time is shorter. Test the claim at the 10% level.
State the null and alternative hypotheses.
H_{0}: σ_{1} = σ_{2}
H_{a}: σ_{1} < σ_{2}
or
H_{0}: ${\sigma}_{\text{1}}^{\text{2}}\text{=}{\sigma}_{\text{2}}^{\text{2}}$
H_{a}: ${\sigma}_{1}^{2}<{\sigma}_{2}^{2}$
What is s_{1} in this problem?
What is s_{2} in this problem?
4.11
What is n?
What is the F statistic?
0.7159
What is the pvalue?
Is the claim accurate?
No, at the 10% level of significance, we do not reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker.
Use the following information to answer the next four exercises. Two students are interested in whether or not 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 lower.
State the null and alternative hypotheses.
What is the F Statistic?
2.8674
What is the pvalue?
At the 5% significance level, do we reject the null hypothesis?
Reject the null hypothesis. There is enough evidence to say that the variance of the grades for the first student is higher than the variance in the grades for the second student.
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.
State the null and alternative hypotheses.
What is the F Statistic?
0.7414
At the 5% significance level, what can we say about the cyclists’ variances?
Homework
Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.
Linda's rats  Tuan's rats  Javier's rats 
43.5  47.0  51.2 
39.4  40.5  40.9 
41.3  38.9  37.9 
46.0  46.3  45.0 
38.2  44.2  48.6 
Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of 10%.
 ${H}_{0}\text{:}{\sigma}_{1}^{2}={\sigma}_{2}^{2}$
 ${H}_{a}\text{:}{\sigma}_{1}^{2}\ne {\sigma}_{1}^{2}$
 df(num) = 4; df(denom) = 4
 F_{4, 4}
 3.00
 2(0.1563) = 0.3126. Using the TI83+/84+ function 2SampFtest, you get the test statistic as 2.9986 and pvalue directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the pvalue is the same because this is a twotailed test.
 Check student't solution.
 Decision: Do not reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different.
A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt workingclass people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily oneway commuting mileage. The results are as follows.
workingclass  professional (middle incomes)  professional (wealthy) 
17.8  16.5  8.5 
26.7  17.4  6.3 
49.4  22.0  4.6 
9.4  7.4  12.6 
65.4  9.4  11.0 
47.1  2.1  28.6 
19.5  6.4  15.4 
51.2  13.9  9.3 
Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 5% significance level.
Refer to the data from [link].
Examine practice laps 3 and 4. Determine whether or not the variance in lap time is statistically the same for those practice laps.
Use the following information to answer the next two exercises. The following table lists the number of pages in four different types of magazines.
home decorating  news  health  computer 
172  87  82  104 
286  94  153  136 
163  123  87  98 
205  106  103  207 
197  101  96  146 
 H_{0}: {\sigma}_{1}^{2} = {\sigma}_{2}^{2}
 H_{a}: {\sigma}_{1}^{2} ≠ {\sigma}_{1}^{2}
 df(n) = 19, df(d) = 19
 F_{19,19}
 1.13
 0.786
 Check student’s solution.

 Alpha:0.05
 Decision: Do not reject the null hypothesis.
 Reason for decision: pvalue > alpha
 Conclusion: There is not sufficient evidence to conclude that the variances are different.
Which two magazine types do you think have the same variance in length?
Which two magazine types do you think have different variances in length?
The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances.
Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall? Suppose that the [link] shows the results of a study.
Saturday  Sunday  Saturday  Sunday 
75  44  62  137 
18  58  0  82 
150  61  124  39 
94  19  50  127 
62  99  31  141 
73  60  118  73 
89 
Are the variances for incomes on the East Coast and the West Coast the same? Suppose that [link] shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.05.
East  West 
38  71 
47  126 
30  42 
82  51 
75  44 
52  90 
115  88 
67 
 H_{0}: = {\sigma}_{1}^{2} = {\sigma}_{2}^{2}
 H_{a}: {\sigma}_{1}^{2} ≠ {\sigma}_{1}^{2}
 df(n) = 7, df(d) = 6
 F_{7,6}
 0.8117
 0.7825
 Check student’s solution.

 Alpha: 0.05
 Decision: Do not reject the null hypothesis.
 Reason for decision: pvalue > alpha
 Conclusion: There is not sufficient evidence to conclude that the variances are different.
Thirty men in college were taught a method of finger tapping. They were randomly assigned to three groups of ten, with each receiving one of three doses of caffeine: 0 mg, 100 mg, 200 mg. This is approximately the amount in no, one, or two cups of coffee. Two hours after ingesting the caffeine, the men had the rate of finger tapping per minute recorded. The experiment was double blind, so neither the recorders nor the students knew which group they were in. Does caffeine affect the rate of tapping, and if so how?
Here are the data:
0 mg  100 mg  200 mg  0 mg  100 mg  200 mg 
242  248  246  245  246  248 
244  245  250  248  247  252 
247  248  248  248  250  250 
242  247  246  244  246  248 
246  243  245  242  244  250 
King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from a fourth. These spanned most of his reign. We have data on the silver content of the coins:
First Coinage  Second Coinage  Third Coinage  Fourth Coinage 
5.9  6.9  4.9  5.3 
6.8  9.0  5.5  5.6 
6.4  6.6  4.6  5.5 
7.0  8.1  4.5  5.1 
6.6  9.3  6.2  
7.7  9.2  5.8  
7.2  8.6  5.8  
6.9  
6.2 
Did the silver content of the coins change over the course of Manuel’s reign?
Here are the means and variances of each coinage. The data are unbalanced.
First  Second  Third  Fourth  
Mean  6.7444  8.2429  4.875  5.6143 
Variance  0.2953  1.2095  0.2025  0.1314 
Here is a strip chart of the silver content of the coins:
While there are differences in spread, it is not unreasonable to use ANOVA techniques. Here is the completed ANOVA table:
Source of Variation  Sum of Squares (SS)  Degrees of Freedom (df)  Mean Square (MS)  F 
Factor (Between)  37.748  4 – 1 = 3  12.5825  26.272 
Error (Within)  11.015  27 – 4 = 23  0.4789  
Total  48.763  27 – 1 = 26 
P(F > 26.272) = 0;
Reject the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth.
The American League and the National League of Major League Baseball are each divided into three divisions: East, Central, and West. Many years, fans talk about some divisions being stronger (having better teams) than other divisions. This may have consequences for the postseason. For instance, in 2012 Tampa Bay won 90 games and did not play in the postseason, while Detroit won only 88 and did play in the postseason. This may have been an oddity, but is there good evidence that in the 2012 season, the American League divisions were significantly different in overall records? Use the following data to test whether the mean number of wins per team in the three American League divisions were the same or not. Note that the data are not balanced, as two divisions had five teams, while one had only four.
Division  Team  Wins 
East  NY Yankees  95 
East  Baltimore  93 
East  Tampa Bay  90 
East  Toronto  73 
East  Boston  69 
Division  Team  Wins 
Central  Detroit  88 
Central  Chicago Sox  85 
Central  Kansas City  72 
Central  Cleveland  68 
Central  Minnesota  66 
Division  Team  Wins 
West  Oakland  94 
West  Texas  93 
West  LA Angels  89 
West  Seattle  75 
Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season.
While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, oneway ANOVA is robust.
Here is the ANOVA table for the data:
Source of Variation  Sum of Squares (SS)  Degrees of Freedom (df)  Mean Square (MS)  F 
Factor (Between)  344.16  3 – 1 = 2  172.08  26.272 
Error (Within)  1,219.55  14 – 3 = 11  110.87  1.5521 
Total  1,563.71  14 – 1 = 13 
P(F > 1.5521) = 0.2548
Since the pvalue is so large, there is not good evidence against the null hypothesis of equal means. We decline to reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League.
 Introductory Statistics
 Preface
 Sampling and Data
 Descriptive Statistics
 Introduction
 StemandLeaf Graphs (Stemplots), Line Graphs, and Bar Graphs
 Histograms, Frequency Polygons, and Time Series Graphs
 Measures of the Location of the Data
 Box Plots
 Measures of the Center of the Data
 Skewness and the Mean, Median, and Mode
 Measures of the Spread of the Data
 Descriptive Statistics
 Probability Topics
 Discrete Random Variables
 Introduction
 Probability Distribution Function (PDF) for a Discrete Random Variable
 Mean or Expected Value and Standard Deviation
 Binomial Distribution
 Geometric Distribution
 Hypergeometric Distribution
 Poisson Distribution
 Discrete Distribution (Playing Card Experiment)
 Discrete Distribution (Lucky Dice Experiment)
 Continuous Random Variables
 The Normal Distribution
 The Central Limit Theorem
 Confidence Intervals
 Hypothesis Testing with One Sample
 Hypothesis Testing with Two Samples
 The ChiSquare Distribution
 Linear Regression and Correlation
 F Distribution and OneWay ANOVA
 Appendix A: Review Exercises (Ch 313)
 Appendix B: Practice Tests (14) and Final Exams
 Appendix C: Data Sets
 Appendix D: Group and Partner Projects
 Appendix E: Solution Sheets
 Appendix F: Mathematical Phrases, Symbols, and Formulas
 Appendix G: Notes for the TI83, 83+, 84, 84+ Calculators
 Appendix H: Tables