B50 Appendix B Review of Graphs, Equations, and Inequalities

B.5 Representing Data Graphically

What you should learn● Use line plots to order and

analyze data.

● Use histograms to represent

frequency distributions.

● Use bar graphs to represent and

analyze data.

● Use line graphs to represent and

analyze data.

Why you should learn itDouble bar graphs allow you to

compare visually two sets of data

over time. For example, in Exercises

17 and 18 on page B56, you are

asked to estimate the difference in

tuition between public and private

institutions of higher education.

Line PlotsStatistics is the branch of mathematics that studies techniques for collecting, organizing,and interpreting data. In this appendix, you will study several ways to organize data. Thefirst is a line plot, which uses a portion of a real number line to order numbers. Lineplots are especially useful for ordering small sets of numbers (about 50 or less) by hand.

Many statistical measures can be obtained from a line plot. Two such measures arethe frequency and range of the data. The frequency measures the number of times avalue occurs in a data set. The range is the difference between the greatest and leastdata values. For example, consider the data values

20, 21, 21, 25, 32.

The frequency of 21 in the data set is 2 because 21 occurs twice. The range is 12because the difference between the greatest and least data values is

Example 1 Constructing a Line Plot

The scores from an economics class of 30 students are listed below. The scores are for a100-point exam.

93, 70, 76, 67, 86, 93, 82, 78, 83, 86, 64, 78, 76, 66, 8383, 96, 74, 69, 76, 64, 74, 79, 76, 88, 76, 81, 82, 74, 70

a. Use a line plot to organize the scores.

b. Which score occurs with the greatest frequency?

c. What is the range of scores?

Solutiona. Begin by scanning the data to find the least and greatest numbers. For the data, the

least number is 64 and the greatest is 96. Next, draw a portion of a real number linethat includes the interval

To create the line plot, start with the first number, 93, and record an above 93 onthe number line. Continue recording an for each number in the list until you obtainthe line plot shown in Figure B.62.

Figure B.62

b. From the line plot, you can see that 76 occurs with the greatest frequency.

c. Because the range is the difference between the greatest and least data values, therange of scores is

Now try Exercise 9.

96 � 64 � 32.

65 70 75 80 85 90 95 100

××

×× ×××

×××

×××××

××

× ×××

×××

××

× ××

×

Test scores

�

�

�64, 96�.

32 � 20 � 12.

Appendix B.5 Representing Data Graphically B51

Histograms and Frequency DistributionsWhen you want to organize large sets of data, it is useful to group the data into intervals and plot the frequency of the data in each interval. A frequency distributioncan be used to construct a histogram. A histogram uses a portion of a real number lineas its horizontal axis. The bars of a histogram are not separated by spaces.

Example 2 Constructing a Histogram

The table at the right shows the percent of the resident population of each state and the District of Columbia that was at least 65 years old in 2008. Construct a frequency distribution and a histogram for the data. (Source: U.S. Census Bureau)

SolutionTo begin constructing a frequency distribution, you must first decide on the number ofintervals. There are several ways to group the data. However, because the least numberis 7.3 and the greatest is 17.4, it seems that six intervals would be appropriate. The firstwould be the interval

the second would be

and so on. By tallying the data into the six intervals, you obtain the frequency distributionshown below.

Interval Tally

You can construct the histogram by drawing a vertical axis to represent the number ofstates and a horizontal axis to represent the percent of the population 65 and older.Then, for each interval, draw a vertical bar whose height is the total tally, as shown inFigure B.63. Note that the break in the horizontal axis indicates that the numbers 0 through 6 have been omitted.

Figure B.63

Now try Exercise 13.

��17, 19�����15, 17�

���� ������ ���� ���� ���� �13, 15����� ���� �����11, 13������9, 11���7, 9�

�9, 11�

�7, 9�

AK 7.3AL 13.8AR 14.3AZ 13.3CA 11.2CO 10.4CT 13.7DC 11.9DE 13.9FL 17.4GA 10.1HI 14.8IA 14.8ID 12.0IL 12.2IN 12.8KS 13.1KY 13.3LA 12.3MA 13.4MD 12.1ME 15.1MI 13.0MN 12.5MO 13.6MS 12.7

MT 14.2NC 12.4ND 14.7NE 13.5NH 12.9NJ 13.3NM 13.1NV 11.4NY 13.4OH 13.7OK 13.5OR 13.3PA 15.4RI 14.1SC 13.3SD 14.4TN 13.2TX 10.2UT 9.0VA 12.1VT 14.0WA 12.0WI 13.3WV 15.7WY 12.3

B52 Appendix B Review of Graphs, Equations, and Inequalities

Bar GraphsA bar graph is similar to a histogram, except that the bars can be either horizontal or vertical and the labels of the bars are not necessarily numbers. For instance, the labels of the bars can be the months of a year, names of cities, or types of vehicles. Another difference between a bar graph and a histogram is that the bars in a bar graph are usuallyseparated by spaces.

Example 3 Constructing a Bar Graph

The data below show the normal monthly precipitation (in inches) in Houston, Texas.Construct a bar graph for the data. What can you conclude? (Source: U.S. NationalOceanic and Atmospheric Administration)

SolutionTo create a bar graph, begin by drawing a vertical axis to represent the precipitation anda horizontal axis to represent the month. The bar graph is shown in Figure B.64.

Figure B.64

From the graph, you can see that Houston receives a fairly consistent amount of rainthroughout the year—the driest month tends to be February and the wettest month tendsto be June.

Now try Exercise 15.

Mon

thly

nor

mal

prec

ipita

tion

(in

inch

es)

Month

1

2

3

4

5

6

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Monthly Precipitation

Month Precipitation

January 3.7February 3.0March 3.4April 3.6May 5.2June 5.4July 3.2

August 3.8September 4.3

October 4.5November 4.2December 3.7

Appendix B.5 Representing Data Graphically B53

Example 4 Constructing a Double Bar Graph

The table shows the percents of associate degrees awarded to males and females forselected fields of study in the United States in 2007. Construct a double bar graph forthe data. (Source: U.S. National Center for Education Statistics)

SolutionFor the data, a horizontal bar graph seems to be appropriate. This makes it easier tolabel and read the bars. Such a graph is shown in Figure B.65.

Figure B.65

Now try Exercise 19.

Education

Engineering

Family and Consumer Sciences

Fiel

d of

stu

dy

Percent of associate degrees

Emergency Medical Technician

Law and Legal Studies

Liberal/General Studies

Mathematics

Physical Sciences

Precision Production Trades

Psychology

Social Sciences

10 20 30 40 50 60 70 80 90 100

FemaleMale

Agriculture and Natural Resources

Biological Sciences/Life Sciences

Accounting

Business

Communications

Associate Degrees

Field of Study % Female % Male

Agriculture and Natural Resources 37.9 62.1Biological Sciences Life Sciences� 67.2 32.8

Accounting 77.5 22.5Business 60.3 39.7

Communications 49.8 50.2Education 86.4 13.6

Engineering 14.1 85.9Family and Consumer Sciences 96.4 3.6Emergency Medical Technician 32.4 67.6

Law and Legal Studies 89.9 10.1Liberal General Studies� 62.5 37.5

Mathematics 33.7 66.3Physical Sciences 43.0 57.0

Precision Production Trades 6.3 93.7Psychology 76.6 23.4

Social Sciences 66.3 33.7

B54 Appendix B Review of Graphs, Equations, and Inequalities

Line GraphsA line graph is similar to a standard coordinate graph. Line graphs are usually used toshow trends over periods of time.

Example 5 Constructing a Line Graph

The table at the right shows the number of immigrants (in thousands) entering the United States for each decade from 1901 through 2000. Construct a line graph for the data. What can you conclude? (Source: U.S.Immigration and Naturalization Service)

SolutionBegin by drawing a vertical axis to represent the number of immigrants in thousands.Then label the horizontal axis with decades and plot the points shown in the table.Finally, connect the points with line segments, as shown in Figure B.66. From the linegraph, you can see that the number of immigrants hit a low point during the depressionof the 1930s. Since then the number has steadily increased.

Figure B.66

Now try Exercise 29.

You can use a graphing utility to check your sketch, as shown in Figure B.67.

Figure B.67

19000

2010

10,000

Decade Number(in thousands)

1901–1910 87951911–1920 57361921–1930 41071931–1940 5281941–1950 10351951–1960 25151961–1970 33221971–1980 44931981–1990 73381991–2000 9095

Technology TipYou can use the statistical plottingfeature of a graphing

utility to create different typesof graphs, such as line graphs.For instructions on how to usethe statistical plotting feature,see Appendix A; for specifickeystrokes, go to this textbook’sCompanion Website.

Appendix B.5 Representing Data Graphically B55

B.5 Exercises For instructions on how to use a graphing utility, see Appendix A.

9. Economics The line plot shows a sample of prices ofunleaded regular gasoline in 25 different cities.

(a) What price occurred with the greatest frequency?

(b) What is the range of prices?

10. Waste Management The line plot shows the weights(to the nearest hundred pounds) of municipal wastehauled by a garbage truck in 30 trips to a landfill.

(a) What weight occurred with the greatest frequency?

(b) What is the range of weights?

Education In Exercises 11 and 12, use the followingscores from an algebra class of 30 students. The scoresare for one 25-point quiz and one 100-point exam.

Quiz 20, 15, 14, 20, 16, 19, 10, 21, 24, 15, 15, 14, 15, 21,19, 15, 20, 18, 18, 22, 18, 16, 18, 19, 21, 19, 16, 20, 14, 12

Exam 77, 100, 77, 70, 83, 89, 87, 85, 81, 84, 81, 78, 89, 78,88, 85, 90, 92, 75, 81, 85, 100, 98, 81, 78, 75, 85, 89, 82, 75

11. Construct a line plot for the quiz. Which score(s)occurred with the greatest frequency?

12. Construct a line plot for the exam. Which score(s)occurred with the greatest frequency?

13. Demographics The list shows the percents of individuals living below the poverty level in the 50 statesin 2008. Use a frequency distribution and a histogram toorganize the data. (Source: U.S. Census Bureau)

AK 9.5 AL 16.3 AR 17.6 AZ 14.3CA 12.9 CO 11.9 CT 8.5 DE 10.4FL 12.6 GA 14.5 HI 8.8 IA 11.2ID 12.4 IL 12.1 IN 12.7 KS 11.7KY 17.2 LA 18.5 MA 10.0 MD 8.0ME 12.6 MI 14.0 MN 9.7 MO 13.3MS 21.0 MT 14.3 NC 14.6 ND 12.1NE 11.1 NH 7.6 NJ 8.7 NM 17.9NV 10.8 NY 13.8 OH 13.2 OK 16.2OR 13.4 PA 11.9 RI 11.6 SC 15.5SD 13.2 TN 15.7 TX 16.3 UT 10.0VA 9.9 VT 10.5 WA 11.6 WI 10.7WV 17.2 WY 8.9

14. Education The list shows the numbers of public highschool graduates (in thousands) in the 50 states and theDistrict of Columbia in 2007. Use a frequency distributionand a histogram to organize the data. (Source: U.S.National Center for Education Statistics)

AK 7.7 AL 38.9 AR 27.2 AZ 56.0CA 356.6 CO 45.6 CT 37.5 DC 2.9DE 7.2 FL 142.3 GA 77.8 HI 11.1IA 34.1 ID 16.2 IL 130.2 IN 59.9KS 30.1 KY 39.1 LA 34.3 MA 63.9MD 57.6 ME 13.2 MI 111.8 MN 59.5MO 60.3 MS 24.2 MT 10.1 NC 76.0ND 7.2 NE 19.9 NH 14.5 NJ 93.0NM 16.1 NV 16.5 NY 168.3 OH 117.7OK 37.1 OR 33.4 PA 128.6 RI 10.4SC 35.1 SD 8.3 TN 54.5 TX 241.2UT 28.3 VA 74.0 VT 7.3 WA 62.8WI 64.0 WV 17.4 WY 5.4

××

×××

××××

×××××××××

××××

××××××

××

600 800 1000 1200 1400

2.859 2.879 2.899 2.919 2.939 2.959 2.979 2.999 3.019 3.039 3.059

× × × ×××

× ××

××××××

××

××

× ××××

×

Vocabulary and Concept CheckIn Exercises 1–4, fill in the blank.

1. _______ are useful for ordering small sets of numbers by hand.

2. A _______ uses a portion of a real number line as its horizontal axis, and the barsare not separated by spaces.

3. You can use a _______ to construct a histogram.

4. _______ show trends over periods of time.

In Exercises 5–8, match the data display with its name.

5. Line plot 6. Bar graph 7. Histogram 8. Line graph

(a) (b) (c) (d)

Procedures and Problem Solving

× × ×××××

××××

B56 Appendix B Review of Graphs, Equations, and Inequalities

15. Business The table shows the numbers of Wal-Martstores from 1995 through 2008. Construct a bar graphfor the data. Write a brief statement regarding the numberof Wal-Mart stores over time. (Source: Wal-MartStores, Inc.)

16. Business The table shows the revenues (in billions ofdollars) for Costco Wholesale stores in the years from1995 through 2009. Construct a bar graph for the data. Write a brief statement regarding the revenue ofCostco Wholesale stores over time. (Source: CostcoWholesale Corporation)

(p. B50) In Exercises 17 and18, the double bar graph shows the mean tuitions (in dollars) charged by public and private institutions ofhigher education in the United States from 2003 through2008. (Source: U.S. National Center for EducationStatistics)

17. Approximate the difference in tuition charges for publicand private schools for each year.

18. Approximate the increase in tuition charges for eachtype of institution from year to year.

19. Education The table shows the total college enrollments (in thousands) for women and men in theUnited States from 2002 through 2008. Construct adouble bar graph for the data. (Source: U.S. NationalCenter for Education Statistics)

20. Demographics The table shows the populations (inmillions) in the coastal regions of the United States in1990 and 2008. Construct a double bar graph for thedata. (Source: U.S. Census Bureau)

6,000

2,0004,000

8,000

2003 2004 2005 2006 2007 2008

10,000 12,000

16,00014,000

20,00022,000

18,000

Public Private

Tui

tion

(in

dolla

rs)

Year

Year Number of stores

1995 29431996 30541997 34061998 35991999 39852000 41892001 44142002 46882003 49062004 52892005 61412006 67792007 72622008 7720

Year Revenue(in billions of dollars)

1995 18.2471996 19.5661997 21.8741998 24.2701999 27.4562000 32.1642001 34.7972002 38.7622003 42.5462004 48.1072005 52.9352006 60.1512007 64.9092008 72.4832009 71.422

Year Women(in thousands)

Men(in thousands)

2002 8998 68752003 9258 72402004 9319 73182005 9885 73872006 10,032 74562007 10,184 75752008 10,432 7816

Region1990

population(in millions)

2008population

(in millions)

Atlantic 59.0 69.2Gulf of Mexico 15.2 20.1

Great Lakes 25.9 27.4Pacific 33.2 40.8

Appendix B.5 Representing Data Graphically B57

Economics In Exercises 21–24, use the line graph,which shows the average prices of a gallon of premiumunleaded gasoline from 2000 through 2008. (Source:U.S. Energy Information Administration)

21. Describe the trend in the price of premium unleadedgasoline from 2000 to 2002.

22. Describe the trend in the price of premium unleadedgasoline from 2002 to 2008.

23. Approximate the percent increase in the price per gallonof premium unleaded gasoline from 2005 to 2008.

24. Approximate the percent increase in the price per gallonof premium unleaded gasoline from 2002 to 2008.

Agriculture In Exercises 25–28, use the line graph,which shows the average retail price (in dollars) of onedozen Grade A large eggs in the United States for eachmonth in 2009. (Source: U.S. Bureau of Labor Statistics)

25. What is the highest price of one dozen Grade A largeeggs shown in the graph? When did this price occur?

26. What was the difference between the highest price andthe lowest price of one dozen Grade A large eggs in2009?

27. Determine when the price of eggs showed the greatestrate of decrease from one month to the next.

28. Describe any trends shown by the line graph. Then predict the average price of eggs in February of 2010.Are you confident that your prediction is within $0.10of the actual average retail price? Explain your reasoning.

29. Human Resources The table shows the total numbersof women in the workforce (in thousands) in the UnitedStates from 1998 through 2009. Construct a line graphfor the data. Write a brief statement describing what thegraph reveals. (Source: U.S. Bureau of Labor Statistics)

30. Economics The table shows the total amounts (in trillions of dollars) of private fixed assets held by noncorporate entities in the United States from 2000through 2008. Construct a line graph for the data. Writea brief statement describing what the graph reveals.(Source: U.S. Bureau of Economic Analysis)

Ret

ail p

rice

(in

dol

lars

)

Month

Jan. Mar. May July Sept. Nov.

1.40

1.50

1.60

1.70

1.80

1.90

2.00

Year

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

Ave

rage

pri

ce(i

n do

llars

per

gal

lon)

2000 2001 2002 2003 20052004 2007 20082006

Year Fixed assets (in trillions of dollars)

2000 13.042001 13.952002 14.792003 15.942004 17.772005 19.742006 21.152007 21.562008 20.96

Year Women in the workforce(in thousands)

1998 63,7141999 64,8552000 66,3032001 66,8482002 67,3632003 68,2722004 68,4212005 69,2882006 70,1732007 70,9882008 71,7672009 72,019

B58 Appendix B Review of Graphs, Equations, and Inequalities

31. Human Resources The table shows the averagehourly earnings (in dollars) of construction workers in the United States from 1994 through 2009. Use agraphing utility to construct a line graph for the data.(Source: U.S. Bureau of Labor Statistics)

32. Networking The list shows the percent of households in each of the 50 states and the District ofColumbia with Internet access in 2009. Use a graphingutility to organize the data in the graph of your choice.Explain your choice of graph. (Source: NationalTelecommunications and Information Administration)

AK 77.3 AL 56.1 AR 55.9AZ 72.1 CA 73.1 CO 72.9CT 74.8 DC 71.8 DE 71.1FL 71.5 GA 67.6 HI 73.3IA 67.8 ID 72.2 IL 68.3IN 62.5 KS 70.5 KY 59.5LA 60.6 MA 75.9 MD 73.4ME 70.2 MI 68.0 MN 72.4MO 63.7 MS 51.8 MT 64.1NC 63.9 ND 67.1 NE 70.0NH 78.7 NJ 76.8 NM 61.7NV 72.4 NY 70.7 OH 66.9OK 60.7 OR 76.0 PA 67.3RI 71.6 SC 58.3 SD 65.5TN 62.3 TX 63.9 UT 77.9VA 71.0 VT 70.8 WA 77.9WI 71.8 WV 59.7 WY 69.8

33. Athletics The table shows the numbers of participants(in thousands) in high school athletic programs in theUnited States from 1998 through 2009. Organize thedata in an appropriate display. Explain your choice ofgraph. (Source: National Federation of State HighSchool Associations)

Conclusions34. Writing Describe the differences between a bar graph

and a histogram.

35. Writing Describe the differences between a line plotand a scatter plot.

35. Think About It The graphs shown below represent thesame data points.

(a) Which of the two graphs is misleading, and why?

(b) Discuss other ways in which graphs can be misleading.

(c) Why would it be beneficial for someone to use amisleading graph?

37. Think About It How can you decide which type ofgraph to use when you are organizing data?

32.0

32.4

32.8

33.2

33.6

34.0

34.4

NSJMMJ

Month

Com

pany

pro

fits

0

10

20

30

40

50

Month

J M M J S N

Com

pany

pro

fits

Year Hourly earnings(in dollars)

1994 14.381995 14.731996 15.111997 15.671998 16.231999 16.802000 17.482001 18.002002 18.522003 18.952004 19.232005 19.462006 20.022007 20.952008 21.872009 22.67

Year Female athletes(in thousands)

Male athletes(in thousands)

1998 2570 37631999 2653 38322000 2676 38622001 2784 39212002 2807 39612003 2856 39892004 2865 40382005 2908 41102006 2953 42072007 3022 43212008 3057 43722009 3114 4423