Frequency Distributions and Graphs

Frequency Distributions and Graphs

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CHAPTER 2
Frequency Distributions and Graphs
(Inset) Copyright 2005 Nexus Energy Software Inc. All Rights Reserved. Used with Permission.
Objectives
After completing this chapter, you should be able to
1Organize data using a frequency distribution.
2Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.
3Represent data using bar graphs, Pareto charts, time series graphs, and pie graphs.
4Draw and interpret a stem and leaf plot.
5Draw and interpret a scatter plot for a set of paired data.
Outline
Introduction
2–1Organizing Data
2–2Histograms, Frequency Polygons, and Ogives
2–3Other Types of Graphs
2–4Paired Data and Scatter Plots
Summary
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Statistics Today
How Your Identity Can Be Stolen
Identity fraud is a big business today. The total amount of the fraud in 2006 was $56.6 billion. The average amount of the fraud for a victim is $6383, and the average
time to correct the problem is 40 hours. The ways in which a person’s identity can be stolen are presented in the following table:
Lost or stolen wallet, checkbook, or credit card 38%
Friends, acquaintances 15
Corrupt business employees 15
Computer viruses and hackers 9
Stolen mail or fraudulent change of address 8
Online purchases or transactions 4
Other methods 11
Source: Javelin Strategy & Research; Council of Better Business Bureau, Inc.
Looking at the numbers presented in a table does not have the same impact as presenting numbers in a well-drawn chart or graph. The article did not include any graphs.
This chapter will show you how to construct appropriate graphs to represent data and help you to get your point across to your audience.
See Statistics Today—Revisited at the end of the chapter for some suggestions on how to represent the data graphically.
Introduction
When conducting a statistical study, the researcher must gather data for the particular variable under study. For example, if a researcher wishes to study the number
of people who were bitten by poisonous snakes in a specific geographic area over the past several years, he or she has to gather the data from various doctors,
hospitals, or health departments.
To describe situations, draw conclusions, or make inferences about events, the researcher must organize the data in some meaningful way. The most convenient method of
organizing data is to construct a frequency distribution.
After organizing the data, the researcher must present them so they can be understood by those who will benefit from reading the study. The most useful method of
presenting the data is by constructing statistical charts and graphs. There are many different types of charts and graphs, and each one has a specific purpose.
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This chapter explains how to organize data by constructing frequency distributions and how to present the data by constructing charts and graphs. The charts and graphs
illustrated here are histograms, frequency polygons, ogives, pie graphs, Pareto charts, and time series graphs. A graph that combines the characteristics of a
frequency distribution and a histogram, called a stem and leaf plot, is also explained. Two other graphs, the stem and leaf plot and the scatter plot, are also
included in this chapter.
Objective 1
Organize data using a frequency distribution.
2–1Organizing Data
Wealthy People
Suppose a researcher wished to do a study on the ages of the top 50 wealthiest people in the world. The researcher first would have to get the data on the ages of the
people. In this case, these ages are listed in Forbes Magazine. When the data are in original form, they are called raw data and are listed next.

Since little information can be obtained from looking at raw data, the researcher organizes the data into what is called a frequency distribution. A frequency
distribution consists of classes and their corresponding frequencies. Each raw data value is placed into a quantitative or qualitative category called a class. The
frequency of a class then is the number of data values contained in a specific class. A frequency distribution is shown for the preceding data set.

Now some general observations can be made from looking at the frequency distribution. For example, it can be stated that the majority of the wealthy people in the
study are over 55 years old.
A frequency distribution is the organization of raw data in table form, using classes and frequencies.
Unusual Stat
Of Americans 50 years old and over, 23% think their greatest achievements are still ahead of them.
The classes in this distribution are 35–41, 42–48, etc. These values are called class limits. The data values 35, 36, 37, 38, 39, 40, 41 can be tallied in the first
class; 42, 43, 44, 45, 46, 47, 48 in the second class; and so on.
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Two types of frequency distributions that are most often used are the categorical frequency distribution and the grouped frequency distribution. The procedures for
constructing these distributions are shown now.
Categorical Frequency Distributions
The categorical frequency distribution is used for data that can be placed in specific categories, such as nominal-or ordinal-level data. For example, data such as
political affiliation, religious affiliation, or major field of study would use categorical frequency distributions.
Example 2–1
Distribution of Blood Types
Twenty-five army inductees were given a blood test to determine their blood type. The data set is

Construct a frequency distribution for the data.
Solution
Since the data are categorical, discrete classes can be used. There are four blood types: A, B, O, and AB. These types will be used as the classes for the
distribution.
The procedure for constructing a frequency distribution for categorical data is given next.
Step 1Make a table as shown.

Step 2Tally the data and place the results in column B.
Step 3Count the tallies and place the results in column C.
Step 4Find the percentage of values in each class by using the formula

where f = frequency of the class and n = total number of values. For example, in the class of type A blood, the percentage is

Percentages are not normally part of a frequency distribution, but they can be added since they are used in certain types of graphs such as pie graphs. Also, the
decimal equivalent of a percent is called a relative frequency.
Step 5Find the totals for columns C (frequency) and D (percent). The completed table is shown.
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For the sample, more people have type O blood than any other type.
Grouped Frequency Distributions
When the range of the data is large, the data must be grouped into classes that are more than one unit in width, in what is called a grouped frequency distribution.
For example, a distribution of the number of hours that boat batteries lasted is the following.
Unusual Stat
Six percent of Americans say they find life dull.

The procedure for constructing the preceding frequency distribution is given in Example 2–2; however, several things should be noted. In this distribution, the values
24 and 30 of the first class are called class limits. The lower class limit is 24; it represents the smallest data value that can be included in the class. The upper
class limit is 30; it represents the largest data value that can be included in the class. The numbers in the second column are called class boundaries. These numbers
are used to separate the classes so that there are no gaps in the frequency distribution. The gaps are due to the limits; for example, there is a gap between 30 and
31.
Students sometimes have difficulty finding class boundaries when given the class limits. The basic rule of thumb is that the class limits should have the same decimal
place value as the data, but the class boundaries should have one additional place value and end in a 5. For example, if the values in the data set are whole numbers,
such as 24, 32, and 18, the limits for a class might be 31–37, and the boundaries are 30.5–37.5. Find the boundaries by subtracting 0.5 from 31 (the lower class limit)
and adding 0.5 to 37 (the upper class limit).
Lower limit – 0.5 = 31 – 0.5 = 30.5 = lower boundary
Upper limit + 0.5 = 37 + 0.5 = 37.5 = upper boundary
If the data are in tenths, such as 6.2, 7.8, and 12.6, the limits for a class hypothetically might be 7.8–8.8, and the boundaries for that class would be 7.75–8.85.
Find these values by subtracting 0.05 from 7.8 and adding 0.05 to 8.8.
Unusual Stat
One out of every hundred people in the United States is color-blind.
Finally, the class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class
limit of the next class. For example, the class width in the preceding distribution on the duration of boat batteries is 7, found from 31 – 24 = 7.
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The class width can also be found by subtracting the lower boundary from the upper boundary for any given class. In this case, 30.5 – 23.5 = 7.
Note: Do not subtract the limits of a single class. It will result in an incorrect answer.
The researcher must decide how many classes to use and the width of each class. To construct a frequency distribution, follow these rules:
1.There should be between 5 and 20 classes. Although there is no hard-and-fast rule for the number of classes contained in a frequency distribution, it is of the
utmost importance to have enough classes to present a clear description of the collected data.
2.It is preferable but not absolutely necessary that the class width be an odd number. This ensures that the midpoint of each class has the same place value as the
data. The class midpoint Xm is obtained by adding the lower and upper boundaries and dividing by 2, or adding the lower and upper limits and dividing by 2:

or

For example, the midpoint of the first class in the example with boat batteries is

The midpoint is the numeric location of the center of the class. Midpoints are necessary for graphing (see Section 2–2). If the class width is an even number, the
midpoint is in tenths. For example, if the class width is 6 and the boundaries are 5.5 and 11.5, the midpoint is

Rule 2 is only a suggestion, and it is not rigorously followed, especially when a computer is used to group data.
3.The classes must be mutually exclusive. Mutually exclusive classes have nonoverlapping class limits so that data cannot be placed into two classes. Many times,
frequency distributions such as
Age
10–20
20–30
30–40
40–50
are found in the literature or in surveys. If a person is 40 years old, into which class should she or he be placed? A better way to construct a frequency distribution
is to use classes such as
Age
10–20
21–31
32–42
43–53
4.The classes must be continuous. Even if there are no values in a class, the class must be included in the frequency distribution. There should be no gaps in a
frequency distribution. The only exception occurs when the class with a zero frequency is the first or last class. A class with a zero frequency at either end can be
omitted without affecting the distribution.
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5.The classes must be exhaustive. There should be enough classes to accommodate all the data.
6.The classes must be equal in width. This avoids a distorted view of the data.
One exception occurs when a distribution has a class that is open-ended. That is, the class has no specific beginning value or no specific ending value. A frequency
distribution with an open-ended class is called an open-ended distribution. Here are two examples of distributions with open-ended classes.
Age Frequency
10–20 3
21–31 6
32–42 4
43–53 10
54 and above 8
Minutes Frequency
Below 110 16
110–114 24
115–119 83
120–124 14
125–129 5
The frequency distribution for age is open-ended for the last class, which means that anybody who is 54 years or older will be tallied in the last class. The
distribution for minutes is open-ended for the first class, meaning that any minute values below 110 will be tallied in that class.
Example 2–2 shows the procedure for constructing a grouped frequency distribution, i.e., when the classes contain more than one data value.
Example 2–2
Record High Temperatures
These data represent the record high temperatures in degrees Fahrenheit (°F) for each of the 50 states. Construct a grouped frequency distribution for the data using
7 classes.

Source: The World Almanac and Book of Facts.
Unusual Stats
America’s most popular beverages are soft drinks. It is estimated that, on average, each person drinks about 52 gallons of soft drinks per year, compared to 22 gallons
of beer.
Solution
The procedure for constructing a grouped frequency distribution for numerical data follows.
Step 1Determine the classes.
Find the highest value and lowest value: H = 134 and L = 100.
Find the range: R = highest value – lowest value = H – L, so R = 134 – 100 = 34
Select the number of classes desired (usually between 5 and 20). In this case, 7 is arbitrarily chosen.
Find the class width by dividing the range by the number of classes.

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Round the answer up to the nearest whole number if there is a remainder: 4.9 ˜ 5. (Rounding up is different from rounding off. A number is rounded up if there is any
decimal remainder when dividing. For example, 85 ÷ 6 = 14.167 and is rounded up to 15. Also, 53 ÷ 4 = 13.25 and is rounded up to 14. Also, after dividing, if there is
no remainder, you will need to add an extra class to accommodate all the data.)
Select a starting point for the lowest class limit. This can be the smallest data value or any convenient number less than the smallest data value. In this case, 100
is used. Add the width to the lowest score taken as the starting point to get the lower limit of the next class. Keep adding until there are 7 classes, as shown, 100,
105, 110, etc.
Subtract one unit from the lower limit of the second class to get the upper limit of the first class. Then add the width to each upper limit to get all the upper
limits.
105 – 1 = 104
The first class is 100–104, the second class is 105–109, etc.
Find the class boundaries by subtracting 0.5 from each lower class limit and adding 0.5 to each upper class limit:
99.5–104.5, 104.5–109.5, etc.
Step 2Tally the data.
Step 3Find the numerical frequencies from the tallies.
The completed frequency distribution is

The frequency distribution shows that the class 109.5–114.5 contains the largest number of temperatures (18) followed by the class 114.5–119.5 with 13 temperatures.
Hence, most of the temperatures (31) fall between 109.5 and 119.5°F.
Sometimes it is necessary to use a cumulative frequency distribution. A cumulative frequency distribution is a distribution that shows the number of data values less
than or equal to a specific value (usually an upper boundary). The values are found by adding the frequencies of the classes less than or equal to the upper class
boundary of a specific class. This gives an ascending cumulative frequency. In this example, the cumulative frequency for the first class is 0 + 2 = 2; for the second
class it is 0 + 2 + 8 = 10; for the third class it is 0 + 2 + 8 + 18 = 28. Naturally, a shorter way to do this would be to just add the cumulative frequency of the
class below to the frequency of the given class. For example, the cumulative frequency for the number of data values less than 114.5 can be found by adding 10 + 18 =
28. The cumulative frequency distribution for the data in this example is as follows:
Cumulative frequency
Less than 99.5 0
Less than 104.5 2
Less than 109.5 10
Less than 114.5 28
Less than 119.5 41
Less than 124.5 48
Less than 129.5 49
Less than 134.5 50
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Cumulative frequencies are used to show how many data values are accumulated up to and including a specific class. In Example 2–2, 28 of the total record high
temperatures are less than or equal to 114°F. Forty-eight of the total record high temperatures are less than or equal to 124°F.
After the raw data have been organized into a frequency distribution, it will be analyzed by looking for peaks and extreme values. The peaks show which class or
classes have the most data values compared to the other classes. Extreme values, called outliers, show large or small data values that are relative to other data
values.
When the range of the data values is relatively small, a frequency distribution can be constructed using single data values for each class. This type of distribution
is called an ungrouped frequency distribution and is shown next.
Example 2–3
MPGs for SUVs
The data shown here represent the number of miles per gallon (mpg) that 30 selected four-wheel-drive sports utility vehicles obtained in city driving. Construct a
frequency distribution, and analyze the distribution.

Source: Model Year Fuel Economy Guide. United States Environmental Protection Agency.
Solution
Step 1Determine the classes. Since the range of the data set is small (19 – 12 = 7), classes consisting of a single data value can be used. They are 12, 13, 14, 15,
16, 17, 18, 19.
Note: If the data are continuous, class boundaries can be used. Subtract 0.5 from each class value to get the lower class boundary, and add 0.5 to each class value to
get the upper class boundary.
Step 2Tally the data.
Step 3Find the numerical frequencies from the tallies, and find the cumulative frequencies.
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The completed ungrouped frequency distribution is

In this case, almost one-half (14) of the vehicles get 15 or 16 miles per gallon. The cumulative frequencies are
Cumulative frequency
Less than 11.5 0
Less than 12.5 6
Less than 13.5 7
Less than 14.5 10
Less than 15.5 16
Less than 16.5 24
Less than 17.5 26
Less than 18.5 29
Less than 19.5 30
The steps for constructing a grouped frequency distribution are summarized in the following Procedure Table.
Procedure Table
Constructing a Grouped Frequency Distribution
Step 1Determine the classes.
Find the highest and lowest values.
Find the range.
Select the number of classes desired.
Find the width by dividing the range by the number of classes and rounding up.
Select a starting point (usually the lowest value or any convenient number less than the lowest value); add the width to get the lower limits.
Find the upper class limits.
Find the boundaries.
Step 2Tally the data.
Step 3Find the numerical frequencies from the tallies, and find the cumulative frequencies.
When you are constructing a frequency distribution, the guidelines presented in this section should be followed. However, you can construct several different but
correct frequency distributions for the same data by using a different class width, a different number of classes, or a different starting point.
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Interesting Fact
Male dogs bite children more often than female dogs do; however, female cats bite children more often than male cats do.
Furthermore, the method shown here for constructing a frequency distribution is not unique, and there are other ways of constructing one. Slight variations exist,
especially in computer packages. But regardless of what methods are used, classes should be mutually exclusive, continuous, exhaustive, and of equal width.
In summary, the different types of frequency distributions were shown in this section. The first type, shown in Example 2–1, is used when the data are categorical
(nominal), such as blood type or political affiliation. This type is called a categorical frequency distribution. The second type of distribution is used when the
range is large and classes several units in width are needed. This type is called a grouped frequency distribution and is shown in Example 2–2. Another type of
distribution is used for numerical data and when the range of data is small, as shown in Example 2–3. Since each class is only one unit, this distribution is called an
ungrouped frequency distribution.
All the different types of distributions are used in statistics and are helpful when one is organizing and presenting data.
The reasons for constructing a frequency distribution are as follows:
1.To organize the data in a meaningful, intelligible way.
2.To enable the reader to determine the nature or shape of the distribution.
3.To facilitate computational procedures for measures of average and spread (shown in Sections 3–1 and 3–2).
4.To enable the researcher to draw charts and graphs for the presentation of data (shown in Section 2–2).
5.To enable the reader to make comparisons among different data sets.
The factors used to analyze a frequency distribution are essentially the same as those used to analyze histograms and frequency polygons, which are shown in Section
2–2.
Applying the Concepts 2–1
Ages of Presidents at Inauguration
The data represent the ages of our Presidents at the time they were first inaugurated.

1.Were the data obtained from a population or a sample? Explain your answer.
2.What was the age of the oldest President?
3.What was the age of the youngest President?
4.Construct a frequency distribution for the data. (Use your own judgment as to the number of classes and class size.)
5.Are there any peaks in the distribution?
6.ldentify any possible outliers.
7.Write a brief summary of the nature of the data as shown in the frequency distribution.
See page 108 for the answers.
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Exercises 2–1
1.List five reasons for organizing data into a frequency distribution.
2.Name the three types of frequency distributions, and explain when each should be used.
3.Find the class boundaries, midpoints, and widths for each class.
a.12–18
b.56–74
c.695–705
d.13.6–14.7
e.2.15–3.93
4.How many classes should frequency distributions have? Why should the class width be an odd number?
5.Shown here are four frequency distributions. Each is incorrectly constructed. State the reason why.
a.
Class Frequency
27–32 1
33–38 0
39–44 6
45–49 4
50–55 2
b.
Class Frequency
5–9 1
9–13 2
13–17 5
17–20 6
20–24 3
c.
Class Frequency
123–127 3
128–132 7
138–142 2
143–147 19
d.
Class Frequency
9–13 1
14–19 6
20–25 2
26–28 5
29–32 9
6.What are open-ended frequency distributions? Why are they necessary?
7.Trust in Internet Information A survey was taken on how much trust people place in the information they read on the Internet. Construct a categorical frequency
distribution for the data. A = trust in everything they read, M = trust in most of what they read, H = trust in about one-half of what they read, S = trust in a small
portion of what they read. (Based on information from the UCLA Internet Report.)

8. State Gasoline Tax The state gas tax in cents per gallon for 25 states is given below. Construct a grouped frequency distribution and a cumulative frequency
distribution with 5 classes.

Source: The World Almanac and Book of Facts.
9. Weights of the NBA’s Top 50 Players Listed are the weights of the NBA’s top 50 players. Construct a grouped frequency distribution and a cumulative frequency
distribution with 8 classes. Analyze the results in terms of peaks, extreme values, etc.

Source: www.msn.foxsports.com
10. Stories in the World’s Tallest Buildings The number of stories in each of the world’s 30 tallest buildings is listed below. Construct a grouped frequency
distribution and a cumulative frequency distribution with 5 classes.

Source: New York Times Almanac.
11. GRE Scores at Top-Ranked Engineering Schools The average quantitative GRE scores for the top 30 graduate schools of engineering are listed. Construct a grouped
frequency distribution and a cumulative frequency distribution with 5 classes.

Source: U.S. News & World Report Best Graduate Schools.
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12. Airline Passengers The number of passengers (in thousands) for the leading U.S. passenger airlines in 2004 is indicated below. Use the data to construct a
grouped frequency distribution and a cumulative frequency distribution with a reasonable number of classes and comment on the shape of the distribution.

Source: The World Almanac and Book of Facts.
13. Ages of Declaration of Independence Signers The ages of the signers of the Declaration of Independence are shown. (Age is approximate since only the birth year
appeared in the source, and one has been omitted since his birth year is unknown.) Construct a grouped frequency distribution and a cumulative frequency distribution
for the data using 7 classes. (The data for this exercise will be used for Exercise 5 in Section 2–2 and Exercise 23 in Section 3–1.)

Source: The Universal Almanac.
14. Online Gambling Online computer gaming has become a popular leisure time activity. Fifty-six percent of the 117 million active gamers play games online. Below
are listed the numbers of players playing a free online game at various times of the day. Construct a grouped frequency distribution and a cumulative frequency
distribution with 6 classes.

Source: www.msn.tech.com
15. Presidential Vetoes The number of total vetoes exercised by the past 20 Presidents is listed below. Use the data to construct a grouped frequency distribution
and a cumulative frequency distribution with 5 classes. What is challenging about this set of data?

Source: World Almanac and Book of Facts.
16. U.S. National Park Acreage The acreage of the 39 U.S. National Parks under 900,000 acres (in thousands of acres) is shown here. Construct a grouped frequency
distribution and a cumulative frequency distribution for the data using 8 classes. (The data in this exercise will be used in Exercise 11 in Section 2–2.)

Source: The Universal Almanac.
17. Heights of Alaskan Volcanoes The heights (in feet above sea level) of the major active volcanoes in Alaska are given here. Construct a grouped frequency
distribution and a cumulative frequency distribution for the data using 10 classes. (The data in this exercise will be used in Exercise 9 in Section 3–1 and Exercise
17 in Section 3–2.)

Source: The Universal Almanac.
18. Home Run Record Breakers During the 1998 baseball season, Mark McGwire and Sammy Sosa both broke Roger Maris’s home run record of 61. The distances (in feet) for
each home run follow. Construct a grouped frequency distribution and a cumulative frequency distribution for each player, using 8 classes. (The information in this
exercise will be used for Exercise 12 in Section 2–2, Exercise 10 in Section 3–1, and Exercise 14 in Section 3–2.)

Source: USA TODAY.
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Extending the Concepts
19.JFK Assassination A researcher conducted a survey asking people if they believed more than one person was involved in the assassination of John F. Kennedy. The
results were as follows: 73% said yes, 19% said no, and 9% had no opinion. Is there anything suspicious about the results?
Technology Step by Step
MINITAB
Step by Step
Make a Categorical Frequency Table (Qualitative or Discrete Data)
1.Type in all the blood types from Example 2–1 down C1 of the worksheet.
A B B AB O O O B AB B B B O A O A O O O AB AB A O B A
2.Click above row 1 and name the column BloodType.
3.Select Stat>Tables>Tally Individual Values.
The cursor should be blinking in the Variables dialog box. If not, click inside the dialog box.
4.Double-click C1 in the Variables list.
5.Check the boxes for the statistics: Counts, Percents, and Cumulative percents.
6.Click [OK]. The results will be displayed in the Session Window as shown.

Make a Grouped Frequency Distribution (Quantitative Variable)
1.Select File>New>New Worksheet. A new worksheet will be added to the project.
2.Type the data used in Example 2–2 into C1. Name the column TEMPERATURES.
3.Use the instructions in the textbook to determine the class limits.
In the next step you will create a new column of data, converting the numeric variable to text categories that can be tallied.
4.Select Data>Code>Numeric to Text.
a)The cursor should be blinking in Code data from columns. If not, click inside the box, then double-click C1 Temperatures in the list. Only quantitative variables
will be shown in this list.
b)Click in the Into columns: then type the name of the new column, TempCodes.
c)Press [Tab] to move to the next dialog box.
d)Type in the first interval 100:104.
Use a colon to indicate the interval from 100 to 104 with no spaces before or after the colon.
e)Press [Tab] to move to the New: column, and type the text category 100–104.
f)Continue to tab to each dialog box, typing the interval and then the category until the last category has been entered.
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The dialog box should look like the one shown.

5.Click [OK]. In the worksheet, a new column of data will be created in the first empty column, C2. This new variable will contain the category for each value in C1.
The column C2-T contains alphanumeric data.
6.Click Stat>Tables>Tally Individual Values, then double-click TempCodes in the Variables list.
a)Check the boxes for the desired statistics, such as Counts, Percents, and Cumulative percents.
b)Click [OK].
The table will be displayed in the Session Window. Eighteen states have high temperatures between 110 and 114°F. Eighty-two percent of the states have record high
temperatures less than or equal to 119°F.

7.Click File>Save Project As … , and type the name of the project file, Ch2-2. This will save the two worksheets and the Session Window.
Excel
Step by Step
Categorical Frequency Table (Qualitative or Discrete Data)
1.In an open workbook select cell A1 and type in all the blood types from Example 2–1 down column A.
2.Type in the variable name Blood Type in cell B1.
3.Select cell B2 and type in the four different blood types down the column.
4.Type in the name Count in cell C1.
5.Select cell C2. From the toolbar, select the Formulas tab on the toolbar.
6.Select the Insert Function icon , then select the Statistical category in the Insert Function dialog box.
7.Select the Countif function from the function name list.
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8.In the dialog box, type A1:A25 in the Range box. Type in the blood type “A” in quotes in the Criteria box. The count or frequency of the number of data corresponding
to the blood type should appear below the input. Repeat for the remaining blood types.
9.After all the data have been counted, select cell C6 in the worksheet.
10.From the toolbar select Formulas, then AutoSum and type in C2:C5 to insert the total frequency into cell C6.

After entering data or a heading into a worksheet, you can change the width of a column to fit the input. To automatically change the width of a column to fit the
data:
1.Select the column or columns that you want to change.
2.On the Home tab, in the Cells group, select Format.
3.Under Cell Size, click Autofit Column Width.
Making a Grouped Frequency Distribution (Quantitative Data)
1.Press [Ctrl]-N for a new workbook.
2.Enter the raw data from Example 2–2 in column A, one number per cell.
3.Enter the upper class boundaries in column B.
4.From the toolbar select the Data tab, then click Data Analysis.
5.In the Analysis Tools, select Histogram and click [OK].
6.In the Histogram dialog box, type A1:A50 in the Input Range box and type B1:B7 in the Bin Range box.
7.Select New Worksheet Ply, and check the Cumulative Percentage option. Click [OK].
8.You can change the label for the column containing the upper class boundaries and expand the width of the columns automatically after relabeling:
Select the Home tab from the toolbar.
Highlight the columns that you want to change.
Select Format, then AutoFit Column Width.

Note: By leaving the Chart Output unchecked, a new worksheet will display the table only.
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Objective 2
Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.
2–2Histograms, Frequency Polygons, and Ogives
After you have organized the data into a frequency distribution, you can present them in graphical form. The purpose of graphs in statistics is to convey the data to
the viewers in pictorial form. It is easier for most people to comprehend the meaning of data presented graphically than data presented numerically in tables or
frequency distributions. This is especially true if the users have little or no statistical knowledge.
Statistical graphs can be used to describe the data set or to analyze it. Graphs are also useful in getting the audience’s attention in a publication or a speaking
presentation. They can be used to discuss an issue, reinforce a critical point, or summarize a data set. They can also be used to discover a trend or pattern in a
situation over a period of time.
The three most commonly used graphs in research are
1.The histogram.
2.The frequency polygon.
3.The cumulative frequency graph, or ogive (pronounced o-jive).
An example of each type of graph is shown in Figure 2–1. The data for each graph are the distribution of the miles that 20 randomly selected runners ran during a given
week.
The Histogram
The histogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the
frequencies of the classes.
Example 2–4
Record High Temperatures
Construct a histogram to represent the data shown for the record high temperatures for each of the 50 states (see Example 2–2).
Class boundaries Frequency
99.5–104.5 2
104.5–109.5 8
109.5–114.5 18
114.5–119.5 13
119.5–124.5 7
124.5–129.5 1
129.5–134.5 1
Solution
Step 1Draw and label the x and y axes. The x axis is always the horizontal axis, and the y axis is always the vertical axis.
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Figure 2–1
Examples of Commonly Used Graphs

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Figure 2–2
Histogram for Example 2–4

Step 2Represent the frequency on the y axis and the class boundaries on the x axis.
Step 3Using the frequencies as the heights, draw vertical bars for each class. See Figure 2–2.
Historical Note
Graphs originated when ancient astronomers drew the position of the stars in the heavens. Roman surveyors also used coordinates to locate landmarks on their maps.
The development of statistical graphs can be traced to William Playfair (1748–1819), an engineer and drafter who used graphs to present economic data pictorially.
As the histogram shows, the class with the greatest number of data values (18) is 109.5–114.5, followed by 13 for 114.5–119.5. The graph also has one peak with the
data clustering around it.
The Frequency Polygon
Another way to represent the same data set is by using a frequency polygon.
The frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies
are represented by the heights of the points.
Example 2–5 shows the procedure for constructing a frequency polygon.
Example 2–5
Record High Temperatures
Using the frequency distribution given in Example 2–4, construct a frequency polygon.
Solution
Step 1Find the midpoints of each class. Recall that midpoints are found by adding the upper and lower boundaries and dividing by 2:

and so on. The midpoints are
Class boundaries Midpoints Frequency
99.5–104.5 102 2
104.5–109.5 107 8
109.5–114.5 112 18
114.5–119.5 117 13
119.5–124.5 122 7
124.5–129.5 127 1
129.5–134.5 132 1
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Figure 2–3
Frequency Polygon for Example 2–5

Step 2Draw the x and y axes. Label the x axis with the midpoint of each class, and then use a suitable scale on the y axis for the frequencies.
Step 3Using the midpoints for the x values and the frequencies as the y values, plot the points.
Step 4Connect adjacent points with line segments. Draw a line back to the x axis at the beginning and end of the graph, at the same distance that the previous and next
midpoints would be located, as shown in Figure 2–3.
The frequency polygon and the histogram are two different ways to represent the same data set. The choice of which one to use is left to the discretion of the
researcher.
The Ogive
The third type of graph that can be used represents the cumulative frequencies for the classes. This type of graph is called the cumulative frequency graph, or ogive.
The cumulative frequency is the sum of the frequencies accumulated up to the upper boundary of a class in the distribution.
The ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.
Example 2–6 shows the procedure for constructing an ogive.
Example 2–6
Record High Temperatures
Construct an ogive for the frequency distribution described in Example 2–4.
Solution
Step 1Find the cumulative frequency for each class.
Cumulative frequency
Less than 99.5 0
Less than 104.5 2
Less than 109.5 10
Less than 114.5 28
Less than 119.5 41
Less than 124.5 48
Less than 129.5 49
Less than 134.5 50
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Figure 2–4
Plotting the Cumulative Frequency for Example 2–6

Figure 2–5
Ogive for Example 2–6

Step 2Draw the x and y axes. Label the x axis with the class boundaries. Use an appropriate scale for the y axis to represent the cumulative frequencies. (Depending on
the numbers in the cumulative frequency columns, scales such as 0, 1, 2, 3, … , or 5, 10, 15, 20, … , or 1000, 2000, 3000, … can be used. Do not label the y axis with
the numbers in the cumulative frequency column.) In this example, a scale of 0, 5, 10, 15, … will be used.
Step 3Plot the cumulative frequency at each upper class boundary, as shown in Figure 2–4. Upper boundaries are used since the cumulative frequencies represent the
number of data values accumulated up to the upper boundary of each class.
Step 4Starting with the first upper class boundary, 104.5, connect adjacent points with line segments, as shown in Figure 2–5. Then extend the graph to the first lower
class boundary, 99.5, on the x axis.
Cumulative frequency graphs are used to visually represent how many values are below a certain upper class boundary. For example, to find out how many record high
temperatures are less than 114.5°F, locate 114.5°F on the x axis, draw a vertical line up until it intersects the graph, and then draw a horizontal line at that point
to the y axis. The y axis value is 28, as shown in Figure 2–6.
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Figure 2–6
Finding a Specific Cumulative Frequency

The steps for drawing these three types of graphs are shown in the following Procedure Table.
Unusual Stat
Twenty-two percent of Americans sleep 6 hours a day or fewer.
Procedure Table
Constructing Statistical Graphs
Step 1Draw and label the x and y axes.
Step 2Choose a suitable scale for the frequencies or cumulative frequencies, and label it on the y axis.
Step 3Represent the class boundaries for the histogram or ogive, or the midpoint for the frequency polygon, on the x axis.
Step 4Plot the points and then draw the bars or lines.
Relative Frequency Graphs
The histogram, the frequency polygon, and the ogive shown previously were constructed by using frequencies in terms of the raw data. These distributions can be
converted to distributions using proportions instead of raw data as frequencies. These types of graphs are called relative frequency graphs.
Graphs of relative frequencies instead of frequencies are used when the proportion of data values that fall into a given class is more important than the actual number
of data values that fall into that class. For example, if you wanted to compare the age distribution of adults in Philadelphia, Pennsylvania, with the age distribution
of adults of Erie, Pennsylvania, you would use relative frequency distributions. The reason is that since the population of Philadelphia is 1,478,002 and the
population of Erie is 105,270, the bars using the actual data values for Philadelphia would be much taller than those for the same classes for Erie.
To convert a frequency into a proportion or relative frequency, divide the frequency for each class by the total of the frequencies. The sum of the relative
frequencies will always be 1. These graphs are similar to the ones that use raw data as frequencies, but the values on the y axis are in terms of proportions. Example
2–7 shows the three types of relative frequency graphs.
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Example 2–7
Miles Run per Week
Construct a histogram, frequency polygon, and ogive using relative frequencies for the distribution (shown here) of the miles that 20 randomly selected runners ran
during a given week.
Class boundaries Frequency
5.5–10.5 1
10.5–15.5 2
15.5–20.5 3
20.5–25.5 5
25.5–30.5 4
30.5–35.5 3
35.5–40.5 2
20
Solution
Step 1Convert each frequency to a proportion or relative frequency by dividing the frequency for each class by the total number of observations.
For class 5.5–10.5, the relative frequency is = 0.05; for class 10.5–15.5, the relative frequency is = 0.10; for class 15.5–20.5, the relative frequency is = 0.15;
and so on.
Place these values in the column labeled Relative frequency.
Class boundaries Midpoints Relative frequency
5.5–10.5 8 0.05
10.5–15.5 13 0.10
15.5–20.5 18 0.15
20.5–25.5 23 0.25
25.5–30.5 28 0.20
30.5–35.5 33 0.15
35.5–40.5 38 0.10
1.00
Step 2Find the cumulative relative frequencies. To do this, add the frequency in each class to the total frequency of the preceding class. In this case, 0 + 0.05 =
0.05, 0.05 + 0.10 = 0.15, 0.15 + 0.15 = 0.30, 0.30 + 0.25 = 0.55, etc. Place these values in the column labeled Cumulative relative frequency.
An alternative method would be to find the cumulative frequencies and then convert each one to a relative frequency.
Cumulative frequency Cumulative relative frequency
Less than 5.5 0 0.00
Less than 10.5 1 0.05
Less than 15.5 3 0.15
Less than 20.5 6 0.30
Less than 25.5 11 0.55
Less than 30.5 15 0.75
Less than 35.5 18 0.90
Less than 40.5 20 1.00
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Step 3Draw each graph as shown in Figure 2–7. For the histogram and ogive, use the class boundaries along the x axis. For the frequency polygon, use the midpoints on
the x axis. The scale on the y axis uses proportions.
Figure 2–7
Graphs for Example 2–7

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Distribution Shapes
When one is describing data, it is important to be able to recognize the shapes of the distribution values. In later chapters you will see that the shape of a
distribution also determines the appropriate statistical methods used to analyze the data.
A distribution can have many shapes, and one method of analyzing a distribution is to draw a histogram or frequency polygon for the distribution. Several of the most
common shapes are shown in Figure 2–8: the bell-shaped or mound-shaped, the uniform-shaped, the J-shaped, the reverse J-shaped, the positively or right-skewed shape,
the negatively or left-skewed shape, the bimodal-shaped, and the U-shaped.
Distributions are most often not perfectly shaped, so it is not necessary to have an exact shape but rather to identify an overall pattern.
A bell-shaped distribution shown in Figure 2–8(a) has a single peak and tapers off at either end. It is approximately symmetric; i.e., it is roughly the same on both
sides of a line running through the center.
Figure 2–8
Distribution Shapes

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A uniform distribution is basically flat or rectangular. See Figure 2–8(b).
A J-shaped distribution is shown in Figure 2–8(c), and it has a few data values on the left side and increases as one moves to the right. A reverse J-shaped
distribution is the opposite of the J-shaped distribution. See Figure 2–8(d).
When the peak of a distribution is to the left and the data values taper off to the right, a distribution is said to be positively or right-skewed. See Figure 2–8(e).
When the data values are clustered to the right and taper off to the left, a distribution is said to be negatively or left-skewed. See Figure 2–8(f). Skewness will be
explained in detail in Chapter 3. Distributions with one peak, such as those shown in Figure 2–8(a), (e), and (f), are said to be unimodal. (The highest peak of a
distribution indicates where the mode of the data values is. The mode is the data value that occurs more often than any other data value. Modes are explained in
Chapter 3.) When a distribution has two peaks of the same height, it is said to be bimodal. See Figure 2–8(g). Finally, the graph shown in Figure 2–8(h) is a U-shaped
distribution.
Distributions can have other shapes in addition to the ones shown here; however, these are some of the more common ones that you will encounter in analyzing data.
When you are analyzing histograms and frequency polygons, look at the shape of the curve. For example, does it have one peak or two peaks? Is it relatively flat, or is
it U-shaped? Are the data values spread out on the graph, or are they clustered around the center? Are there data values in the extreme ends? These may be outliers.
(See Section 3–3 for an explanation of outliers.) Are there any gaps in the histogram, or does the frequency polygon touch the x axis somewhere other than at the ends?
Finally, are the data clustered at one end or the other, indicating a skewed distribution?
For example, the histogram for the record high temperatures shown in Figure 2–2 shows a single peaked distribution, with the class 109.5–114.5 containing the largest
number of temperatures. The distribution has no gaps, and there are fewer temperatures in the highest class than in the lowest class.
Applying the Concepts 2–2
Selling Real Estate
Assume you are a realtor in Bradenton, Florida. You have recently obtained a listing of the selling prices of the homes that have sold in that area in the last 6
months. You wish to organize that data so you will be able to provide potential buyers with useful information. Use the following data to create a histogram, frequency
polygon, and cumulative frequency polygon.

1.What questions could be answered more easily by looking at the histogram rather than the listing of home prices?
2.What different questions could be answered more easily by looking at the frequency polygon rather than the listing of home prices?
3.What different questions could be answered more easily by looking at the cumulative frequency polygon rather than the listing of home prices?
4.Are there any extremely large or extremely small data values compared to the other data values?
5.Which graph displays these extremes the best?
6.Is the distribution skewed?
See page 108 for the answers.
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Exercises 2–2
1.Do Students Need Summer Development? For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained.
Construct a histogram, frequency polygon, and ogive for the data. (The data for this exercise will be used for Exercise 13 in this section.)
Class limits Frequency
90–98 6
99–107 22
108–116 43
117–125 28
126–134 9
Applicants who score above 107 need not enroll in a summer developmental program. In this group, how many students do not have to enroll in the developmental program?
2.Number of College Faculty The number of faculty listed for a variety of private colleges which offer only bachelor’s degrees is listed below. Use these data to
construct a frequency distribution with 7 classes, a histogram, a frequency polygon, and an ogive. Discuss the shape of this distribution. What proportion of schools
have 180 or more faculty?

Source: World Almanac and Book of Facts.
3.Counties, Divisions, or Parishes for 50 States The number of counties, divisions, or parishes for each of the 50 states is given below. Use the data to construct a
grouped frequency distribution with 6 classes, a histogram, a frequency polygon, and an ogive. Analyze the distribution.

Source: World Almanac and Book of Facts.
4.NFL Salaries The salaries (in millions of dollars) for 31 NFL teams for a specific season are given in this frequency distribution.
Class limits Frequency
39.9–42.8 2
42.9–45.8 2
45.9–48.8 5
48.9–51.8 5
51.9–54.8 12
54.9–57.8 5
Source: NFL.com
Construct a histogram, a frequency polygon, and an ogive for the data; and comment on the shape of the distribution.
5.Automobile Fuel Efficiency Thirty automobiles were tested for fuel efficiency, in miles per gallon (mpg). The following frequency distribution was obtained.
Construct a histogram, a frequency polygon, and an ogive for the data.
Class boundaries Frequency
7.5–12.5 3
12.5–17.5 5
17.5–22.5 15
22.5–27.5 5
27.5–32.5 2
6.Construct a frequency histogram, a frequency polygon, and an ogive for the data in Exercise 9 in Section 2–1. Analyze the results.
7.Air Quality Standards The number of days that selected U.S. metropolitan areas failed to meet acceptable air quality standards is shown below for 1998 and 2003.
Construct grouped frequency distributions and a histogram for each set of data, and compare your results.

Source: World Almanac.
8.How Quick Are Dogs? In a study of reaction times of dogs to a specific stimulus, an animal trainer obtained the following data, given in seconds. Construct a
histogram, a frequency polygon, and an ogive for the data; analyze the results. (The histogram in this exercise will be used for Exercise 18 in this section, Exercise
16 in Section 3–1, and Exercise 26 in Section 3–2.)
Class limits Frequency
2.3–2.9 10
3.0–3.6 12
3.7–4.3 6
4.4–5.0 8
5.1–5.7 4
5.8–6.4 2
9.Quality of Health Care The scores of health care quality as calculated by a professional risk management company are listed on the next page for selected states. Use
the data to construct a frequency distribution, a histogram, a frequency polygon, and an ogive.

Source: New York Times Almanac.
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10.Making the Grade The frequency distributions shown indicate the percentages of public school students in fourth-grade reading and mathematics who performed at or
above the required proficiency levels for the 50 states in the United States. Draw histograms for each, and decide if there is any difference in the performance of the
students in the subjects.
Class Reading frequency Math frequency
17.5–22.5 7 5
22.5–27.5 6 9
27.5–32.5 14 11
32.5–37.5 19 16
37.5–42.5 3 8
42.5–47.5 1 1
Source: National Center for Educational Statistics.
11.Construct a histogram, a frequency polygon, and an ogive for the data in Exercise 16 in Section 2–1, and analyze the results.
12.For the data in Exercise 18 in Section 2–1, construct a histogram for the home run distances for each player and compare them. Are they basically the same, or are
there any noticeable differences? Explain your answer.
13.For the data in Exercise 1 in this section, construct a histogram, a frequency polygon, and an ogive, using relative frequencies. What proportion of the applicants
needs to enroll in the summer development program?
14.For the data for 2003 in Exercise 4 in this section, construct a histogram, a frequency polygon, and an ogive, using relative frequencies.
15. Cereal Calories The number of calories per serving for selected ready-to-eat cereals is listed here. Construct a frequency distribution using 7 classes. Draw a
histogram, a frequency polygon, and an ogive for the data, using relative frequencies. Describe the shape of the histogram.

Source: The Doctor’s Pocket Calorie, Fat, and Carbohydrate Counter.
16. Protein Grams in Fast Food The amount of protein (in grams) for a variety of fast-food sandwiches is reported here. Construct a frequency distribution using 6
classes. Draw a histogram, a frequency polygon, and an ogive for the data, using relative frequencies. Describe the shape of the histogram.

Source: The Doctor’s Pocket Calorie, Fat, and Carbohydrate Counter.
17.For the data for year 2003 in Exercise 7 in this section, construct a histogram, a frequency polygon, and an ogive, using relative frequencies.
18.How Quick Are Older Dogs? The animal trainer in Exercise 8 in this section selected another group of dogs who were much older than the first group and measured
their reaction times to the same stimulus. Construct a histogram, a frequency polygon, and an ogive for the data.
Class limits Frequency
2.3–2.9 1
3.0–3.6 3
3.7–4.3 4
4.4–5.0 16
5.1–5.7 14
5.8–6.4 4
Analyze the results and compare the histogram for this group with the one obtained in Exercise 8 in this section. Are there any differences in the histograms? (The
data in this exercise will be used for Exercise 16 in Section 3–1 and Exercise 26 in Section 3–2.)
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Extending the Concepts
19.Using the histogram shown here, do the following.

a.Construct a frequency distribution; include class limits, class frequencies, midpoints, and cumulative frequencies.
b.Construct a frequency polygon.
c.Construct an ogive.
20.Using the results from Exercise 19, answer these questions.
a.How many values are in the class 27.5–30.5?
b.How many values fall between 24.5 and 36.5?
c.How many values are below 33.5?
d.How many values are above 30.5?
Technology Step by Step
MINITAB
Step by Step
Construct a Histogram
1.Enter the data from Example 2–2, the high temperatures for the 50 states.
2.Select Graph>dHistogram.
3.Select [Simple], then click [OK].
4.Click C1 TEMPERATURES in the Graph variables dialog box.
5.Click [Labels]. There are two tabs, Title/Footnote and Data Labels.
a)Click in the box for Title, and type in Your Name and Course Section.
b)Click [OK]. The Histogram dialog box is still open.
6.Click [OK]. A new graph window containing the histogram will open.

7.Click the File menu to print or save the graph.

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8.Click File>Exit.
9.Save the project as Ch2-3.mpj.
TI-83 Plus or TI-84 Plus
Step by Step
Constructing a Histogram
To display the graphs on the screen, enter the appropriate values in the calculator, using the WINDOW menu. The default values are Xmin = –10, Xmax = +10, Ymin = –10,
and Ymax = +10.
The Xscl changes the distance between the tick marks on the x axis and can be used to change the class width for the histogram.
To change the values in the WINDOW:
1.Press WINDOW.
2.Move the cursor to the value that needs to be changed. Then type in the desired value and press ENTER.
3.Continue until all values are appropriate.
4.Press [2nd] [QUIT] to leave the WINDOW menu.
To plot the histogram from raw data:
1.Enter the data in L1.
2.Make sure WINDOW values are appropriate for the histogram.
3.Press [2nd] [STAT PLOT] ENTER.
4.Press ENTER to turn the plot on, if necessary.
5.Move cursor to the Histogram symbol and press ENTER, if necessary.
6.Make sure Xlist is L1.
7.Make sure Freq is 1.
8.Press GRAPH to display the histogram.
9.To obtain the number of data values in each class, press the TRACE key, followed by or keys.

Example TI2–1
Plot a histogram for the following data from Examples 2–2 and 2–4.

Press TRACE and use the arrow keys to determine the number of values in each group.
To graph a histogram from grouped data:
1.Enter the midpoints into L1.
2.Enter the frequencies into L2.
3.Make sure WINDOW values are appropriate for the histogram.
4.Press [2nd] [STAT PLOT] ENTER.
5.Press ENTER to turn the plot on, if necessary.
6.Move cursor to the histogram symbol, and press ENTER, if necessary.
7.Make sure Xlist is L1.
8.Make sure Freq is L2.
9.Press GRAPH to display the histogram.
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Example TI2–2
Plot a histogram for the data from Examples 2–4 and 2–5.
Class boundaries Midpoints Frequency
99.5–104.5 102 2
104.5–109.5 107 8
109.5–114.5 112 18
114.5–119.5 117 13
119.5–124.5 122 7
124.5–129.5 127 1
129.5–134.5 132 1

To graph a frequency polygon from grouped data, follow the same steps as for the histogram except change the graph type from histogram (third graph) to a line graph
(second graph).
To graph an ogive from grouped data, modify the procedure for the histogram as follows:
1.Enter the upper class boundaries into L1.
2.Enter the cumulative frequencies into L2.
3.Change the graph type from histogram (third graph) to line (second graph).
4.Change the Ymax from the WINDOW menu to the sample size.

Excel
Step by Step
Constructing a Histogram
1.Press [Ctrl]-N for a new workbook.
2.Enter the data from Example 2–2 in column A, one number per cell.
3.Enter the upper boundaries into column B.
4.From the toolbar, select the Data tab, then select Data Analysis.
5.In Data Analysis, select Histogram and click [OK].
6.In the Histogram dialog box, type A1:A50 in the Input Range box and type B1:B7 in the Bin Range box.

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7.Select New Worksheet Ply and Chart Output. Click [OK].

Editing the Histogram
To move the vertical bars of the histogram closer together:
1.Right-click one of the bars of the histogram, and select Format Data Series.
2.Move the Gap Width bar to the left to narrow the distance between bars.
To change the label for the horizontal axis:
1.Left-click the mouse over any region of the histogram.
2.Select the Chart Tools tab from the toolbar.
3.Select the Layout tab, Axis Titles and Primary Horizontal Axis Title.

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Once the Axis Title text box is selected, you can type in the name of the variable represented on the horizontal axis.
Constructing a Frequency Polygon
1.Press [Ctrl]-N for a new workbook.
2.Enter the midpoints of the data from Example 2–2 into column A. Enter the frequencies into column B.

3.Highlight the Frequencies (including the label) from column B.
4.Select the Insert tab from the toolbar and the Line Chart option.
5.Select the 2-D line chart type.

We will need to edit the graph so that the midpoints are on the horizontal axis and the frequencies are on the vertical axis.
1.Right-click the mouse on any region of the graph.
2.Select the Select Data option.
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3.Select Edit from the Horizontal Axis Labels and highlight the midpoints from column A, then click [OK].
4.Click [OK] on the Select Data Source box.
Inserting Labels on the Axes
1.Click the mouse on any region of the graph.
2.Select Chart Tools and then Layout on the toolbar.
3.Select Axis Titles to open the horizontal and vertical axis text boxes. Then manually type in labels for the axes.
Changing the Title
1.Select Chart Tools, Layout from the toolbar.
2.Select Chart Title.
3.Choose one of the options from the Chart Title menu and edit.

Constructing an Ogive
To create an ogive, you can use the upper class boundaries (horizontal axis) and cumulative frequencies (vertical axis) from the frequency distribution.
1.Type the upper class boundaries and cumulative frequencies into adjacent columns of an Excel worksheet.
2.Highlight the cumulative frequencies (including the label) and select the Insert tab from the toolbar.
3.Select Line Chart, then the 2-D Line option.
As with the frequency polygon, you can insert labels on the axes and a chart title for the ogive.

2–3Other Types of Graphs
In addition to the histogram, the frequency polygon, and the ogive, several other types of graphs are often used in statistics. They are the bar graph, Pareto chart,
time series graph, and pie graph. Figure 2–9 shows an example of each type of graph.
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Figure 2–9
Other Types of Graphs Used in Statistics

Objective 3
Represent data using bar graphs, Pareto charts, time series graphs, and pie graphs.
Bar Graphs
When the data are qualitative or categorical, bar graphs can be used to represent the data. A bar graph can be drawn using either horizontal or vertical bars.
A bar graph represents the data by using vertical or horizontal bars whose heights or lengths represent the frequencies of the data.
Example 2–8
College Spending for First-Year Students
The table shows the average money spent by first-year college students. Draw a horizontal and vertical bar graph for the data.
Electronics $728
Dorm decor 344
Clothing 141
Shoes 72
Source: The National Retail Federation.
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Solution
1.Draw and label the x and y axes. For the horizontal bar graph place the frequency scale on the x axis, and for the vertical bar graph place the frequency scale on
the y axis.
2.Draw the bars corresponding to the frequencies. See Figure 2–10.
Figure 2–10
Bar Graphs for Example 2–8

The graphs show that first-year college students spend the most on electronic equipment including computers.
Pareto Charts
When the variable displayed on the horizontal axis is qualitative or categorical, a Pareto chart can also be used to represent the data.
A Pareto chart is used to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are
arranged in order from highest to lowest.
Example 2–9
Turnpike Costs
The table shown here is the average cost per mile for passenger vehicles on state turnpikes. Construct and analyze a Pareto chart for the data.
State Number
Indiana 2.9¢
Oklahoma 4.3
Florida 6.0
Maine 3.8
Pennsylvania 5.8
Source: Pittsburgh Tribune Review.
Historical Note
Vilfredo Pareto (1848–1923) was an Italian scholar who developed theories in economics, statistics, and the social sciences. His contributions to statistics include
the development of a mathematical function used in economics. This function has many statistical applications and is called the Pareto distribution. In addition, he
researched income distribution, and his findings became known as Pareto’s law.
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Solution
Step 1Arrange the data from the largest to smallest according to frequency.
State Number
Florida 6.0¢
Pennsylvania 5.8
Oklahoma 4.3
Maine 3.8
Indiana 2.9
Step 2Draw and label the x and y axes.
Step 3Draw the bars corresponding to the frequencies. See Figure 2–11. The Pareto chart shows that Florida has the highest cost per mile. The cost is more than twice
as high as the cost for Indiana.
Suggestions for Drawing Pareto Charts
1.Make the bars the same width.
2.Arrange the data from largest to smallest according to frequency.
3.Make the units that are used for the frequency equal in size.
When you analyze a Pareto chart, make comparisons by looking at the heights of the bars.
The Time Series Graph
When data are collected over a period of time, they can be represented by a time series graph.
Figure 2–11
Pareto Chart for Example 2–9

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A time series graph represents data that occur over a specific period of time.
Example 2–10 shows the procedure for constructing a time series graph.
Example 2–10
Arson Damage to Churches
The arson damage to churches for the years 2001 through 2005 is shown. Construct and analyze a time series graph for the data.
Year Damage (in millions)
2001 $2.8
2002 3.3
2003 3.4
2004 5.0
2005 8.5
Source: U.S. Fire Administration.
Historical Note
Time series graphs are over 1000 years old. The first ones were used to chart the movements of the planets and the sun.
Solution
Step 1Draw and label the x and y axes.
Step 2Label the x axis for years and the y axis for the damage.
Step 3Plot each point according to the table.
Step 4Draw line segments connecting adjacent points. Do not try to fit a smooth curve through the data points. See Figure 2–12. The graph shows a steady increase over
the 5-year period.
Figure 2–12
Time Series Graph for Example 2–10

When you analyze a time series graph, look for a trend or pattern that occurs over the time period. For example, is the line ascending (indicating an increase over
time) or descending (indicating a decrease over time)? Another thing to look for is the slope, or steepness, of the line. A line that is steep over a specific time
period indicates a rapid increase or decrease over that period.
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Figure 2–13
Two Time Series Graphs for Comparison

Two data sets can be compared on the same graph (called a compound time series graph) if two lines are used, as shown in Figure 2–13. This graph shows the number of
snow shovels sold at a store for two seasons.
The Pie Graph
Pie graphs are used extensively in statistics. The purpose of the pie graph is to show the relationship of the parts to the whole by visually comparing the sizes of
the sections. Percentages or proportions can be used. The variable is nominal or categorical.
A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.
Example 2–11 shows the procedure for constructing a pie graph.
Example 2–11
Super Bowl Snack Foods
This frequency distribution shows the number of pounds of each snack food eaten during the Super Bowl. Construct a pie graph for the data.
Snack Pounds (frequency)
Potato chips 11.2 million
Tortilla chips 8.2 million
Pretzels 4.3 million
Popcorn 3.8 million
Snack nuts 2.5 million
Total n = 30.0 million
Source: USA TODAY Weekend.
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Speaking of Statistics
Cell Phone Usage
The graph shows the estimated number (in millions) of cell phone subscribers since 1995. How do you think the growth will affect our way of living? For example,
emergencies can be handled faster since people are using their cell phones to call 911.

Source: Cellular Telecommunications and Internet Association.
Solution
Step 1Since there are 360° in a circle, the frequency for each class must be converted into a proportional part of the circle. This conversion is done by using the
formula

where f = frequency for each class and n = sum of the frequencies. Hence, the following conversions are obtained. The degrees should sum to 360°.*

Step 2Each frequency must also be converted to a percentage. Recall from Example 2–1 that this conversion is done by using the formula

Hence, the following percentages are obtained. The percentages should sum to 100%.†

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Step 3Next, using a protractor and a compass, draw the graph using the appropriate degree measures found in step 1, and label each section with the name and
percentages, as shown in Figure 2–14.
Figure 2–14
Pie Graph for Example 2–11

*Note: The degrees column does not always sum to 360° due to rounding.
†Note: The percent column does not always sum to 100% due to rounding.
Example 2–12
Construct a pie graph showing the blood types of the army inductees described in Example 2–1. The frequency distribution is repeated here.
Class Frequency Percent
A 5 20
B 7 28
O 9 36
AB 4 16
25 100
Solution
Step 1Find the number of degrees for each class, using the formula

For each class, then, the following results are obtained.

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Step 2Find the percentages. (This was already done in Example 2–1.)
Step 3Using a protractor, graph each section and write its name and corresponding percentage, as shown in Figure 2–15.
Figure 2–15
Pie Graph for Example 2–12

The graph in Figure 2–15 shows that in this case the most common blood type is type O.
To analyze the nature of the data shown in the pie graph, look at the size of the sections in the pie graph. For example, are any sections relatively large compared to
the rest?
Figure 2–15 shows that among the inductees, type O blood is more prevalent than any other type. People who have type AB blood are in the minority. More than twice as
many people have type O blood as type AB.
Misleading Graphs
Graphs give a visual representation that enables readers to analyze and interpret data more easily than they could simply by looking at numbers. However,
inappropriately drawn graphs can misrepresent the data and lead the reader to false conclusions. For example, a car manufacturer’s ad stated that 98% of the vehicles
it had sold in the past 10 years were still on the road. The ad then showed a graph similar to the one in Figure 2–16. The graph shows the percentage of the
manufacturer’s automobiles still on the road and the percentage of its competitors’ automobiles still on the road. Is there a large difference? Not necessarily.
Notice the scale on the vertical axis in Figure 2–16. It has been cut off (or truncated) and starts at 95%. When the graph is redrawn using a scale that goes from 0 to
100%, as in Figure 2–17, there is hardly a noticeable difference in the percentages. Thus, changing the units at the starting point on the y axis can convey a very
different visual representation of the data.
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Figure 2–16
Graph of Automaker’s Claim Using a Scale from 95 to 100%

Figure 2–17
Graph in Figure 2–16 Redrawn Using a Scale from 0 to 100%

It is not wrong to truncate an axis of the graph; many times it is necessary to do so. However, the reader should be aware of this fact and interpret the graph
accordingly. Do not be misled if an inappropriate impression is given.
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Let us consider another example. The projected required fuel economy in miles per gallon for General Motors vehicles is shown. In this case, an increase from 21.9 to
23.2 miles per gallon is projected.

Source: National Highway Traffic Safety Administration.
When you examine the graph shown in Figure 2–18(a) using a scale of 0 to 25 miles per gallon, the graph shows a slight increase. However, when the scale is changed to
21 to 24 miles per gallon, the graph shows a much larger increase even though the data remain the same. See Figure 2–18(b). Again, by changing the units or starting
point on the y axis, one can change the visual representation.
Figure 2–18
Projected Miles per Gallon

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Another misleading graphing technique sometimes used involves exaggerating a one-dimensional increase by showing it in two dimensions. For example, the average cost of
a 30-second Super Bowl commercial has increased from $42,000 in 1967 to $2.5 million in 2006 (Source: USA TODAY).
The increase shown by the graph in Figure 2–19(a) represents the change by a comparison of the heights of the two bars in one dimension. The same data are shown two-
dimensionally with circles in Figure 2–19(b). Notice that the difference seems much larger because the eye is comparing the areas of the circles rather than the
lengths of the diameters.
Note that it is not wrong to use the graphing techniques of truncating the scales or representing data by two-dimensional pictures. But when these techniques are used,
the reader should be cautious of the conclusion drawn on the basis of the graphs.
Figure 2–19
Comparison of Costs for a 30-Second Super Bowl Commercial

Another way to misrepresent data on a graph is by omitting labels or units on the axes of the graph. The graph shown in Figure 2–20 compares the cost of living,
economic growth, population growth, etc., of four main geographic areas in the United States. However, since there are no numbers on the y axis, very little
information can be gained from this graph, except a crude ranking of each factor. There is no way to decide the actual magnitude of the differences.
Figure 2–20
A Graph with No Units on the y Axis

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Finally, all graphs should contain a source for the information presented. The inclusion of a source for the data will enable you to check the reliability of the
organization presenting the data. A summary of the types of graphs and their uses is shown in Figure 2–21.
Figure 2–21
Summary of Graphs and Uses of Each

Objective 4
Draw and interpret a stem and leaf plot.
Stem and Leaf Plots
The stem and leaf plot is a method of organizing data and is a combination of sorting and graphing. It has the advantage over a grouped frequency distribution of
retaining the actual data while showing them in graphical form.
A stem and leaf plot is a data plot that uses part of the data value as the stem and part of the data value as the leaf to form groups or classes.
Example 2–13 shows the procedure for constructing a stem and leaf plot.
Example 2–13
At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data.

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Speaking of Statistics
How Much Paper Money Is in Circulation Today?
The Federal Reserve estimated that during a recent year, there were 22 billion bills in circulation. About 35% of them were $1 bills, 3% were $2 bills, 8% were $5
bills, 7% were $10 bills, 23% were $20 bills, 5% were $50 bills, and 19% were $100 bills. It costs about 3¢ to print each $1 bill.
The average life of a $1 bill is 22 months, a $10 bill 3 years, a $20 bill 4 years, a $50 bill 9 years, and a $100 bill 9 years. What type of graph would you use to
represent the average lifetimes of the bills?

Solution
Step 1Arrange the data in order:
02, 13, 14, 20, 23, 25, 31, 32, 32, 32,
32, 33, 36, 43, 44, 44, 45, 51, 52, 57
Note: Arranging the data in order is not essential and can be cumbersome when the data set is large; however, it is helpful in constructing a stem and leaf plot. The
leaves in the final stem and leaf plot should be arranged in order.
Step 2Separate the data according to the first digit, as shown.
02 13, 14 20, 23, 25 31, 32, 32, 32, 32, 33, 36
43, 44, 44, 45 51, 52, 57
Step 3A display can be made by using the leading digit as the stem and the trailing digit as the leaf. For example, for the value 32, the leading digit, 3, is the stem
and the trailing digit, 2, is the leaf. For the value 14, the 1 is the stem and the 4 is the leaf. Now a plot can be constructed as shown in Figure 2–22.
Leading digit (stem) Trailing digit (leaf)
0 2
1 3 4
2 0 3 5
3 1 2 2 2 2 3 6
4 3 4 4 5
5 1 2 7
Figure 2–22
Stem and Leaf Plot for Example 2–13

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Figure 2–22 shows that the distribution peaks in the center and that there are no gaps in the data. For 7 of the 20 days, the number of patients receiving cardiograms
was between 31 and 36. The plot also shows that the testing center treated from a minimum of 2 patients to a maximum of 57 patients in any one day.
If there are no data values in a class, you should write the stem number and leave the leaf row blank. Do not put a zero in the leaf row.
Example 2–14
An insurance company researcher conducted a survey on the number of car thefts in a large city for a period of 30 days last summer. The raw data are shown. Construct
a stem and leaf plot by using classes 50–54, 55–59, 60–64, 65–69, 70–74, and 75–79.

Solution
Step 1Arrange the data in order.
50, 51, 51, 52, 53, 53, 55, 55, 56, 57, 57, 58, 59, 62, 63,
65, 65, 66, 66, 67, 68, 69, 69, 72, 73, 75, 75, 77, 78, 79
Step 2Separate the data according to the classes.
50, 51, 51, 52, 53, 53 55, 55, 56, 57, 57, 58, 59
62, 63 65, 65, 66, 66, 67, 68, 69, 69 72, 73
75, 75, 77, 78, 79
Step 3Plot the data as shown here.
Leading digit (stem) Trailing digit (leaf)
5 0 1 1 2 3 3
5 5 5 6 7 7 8 9
6 2 3
6 5 5 6 6 7 8 9 9
7 2 3
7 5 5 7 8 9
The graph for this plot is shown in Figure 2–23.
Figure 2–23
Stem and Leaf Plot for Example 2–14

Interesting Fact
The average number of pencils and index cards David Letterman tosses over his shoulder during one show is 4.
When the data values are in the hundreds, such as 325, the stem is 32 and the leaf is 5. For example, the stem and leaf plot for the data values 325, 327, 330, 332,
335, 341, 345, and 347 looks like this.
32 5 7
33 0 2 5
34 1 5 7
When you analyze a stem and leaf plot, look for peaks and gaps in the distribution. See if the distribution is symmetric or skewed. Check the variability of the data
by looking at the spread.
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Related distributions can be compared by using a back-to-back stem and leaf plot. The back-to-back stem and leaf plot uses the same digits for the stems of both
distributions, but the digits that are used for the leaves are arranged in order out from the stems on both sides. Example 2–15 shows a back-to-back stem and leaf
plot.
Example 2–15
The number of stories in two selected samples of tall buildings in Atlanta and Philadelphia is shown. Construct a back-to-back stem and leaf plot, and compare the
distributions.

Source: The World Almanac and Book of Facts.
Solution
Step 1Arrange the data for both data sets in order.
Step 2Construct a stem and leaf plot using the same digits as stems. Place the digits for the leaves for Atlanta on the left side of the stem and the digits for the
leaves for Philadelphia on the right side, as shown. See Figure 2–24.
Figure 2–24
Back-to-Back Stem and Leaf Plot for Example 2–15

Step 3Compare the distributions. The buildings in Atlanta have a large variation in the number of stories per building. Although both distributions are peaked in the
30- to 39-story class, Philadelphia has more buildings in this class. Atlanta has more buildings that have 40 or more stories than Philadelphia does.
Stem and leaf plots are part of the techniques called exploratory data analysis. More information on this topic is presented in Chapter 3.
Applying the Concepts 2–3
Leading Cause of Death
The following shows approximations of the leading causes of death among men ages 25–44 years. The rates are per 100,000 men. Answer the following questions about the
graph.
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1.What are the variables in the graph?
2.Are the variables qualitative or quantitative?
3.Are the variables discrete or continuous?
4.What type of graph was used to display the data?
5.Could a Pareto chart be used to display the data?
6.Could a pie chart be used to display the data?
7.List some typical uses for the Pareto chart.
8.List some typical uses for the time series chart.
See page 108 for the answers.
Exercises 2–3
1.Women’s Softball Champions The NCAA Women’s Softball Division 1 Champions since 1982 are listed below. Use the data to construct a Pareto chart and a vertical bar
graph.
’82 UCLA
’83 Texas A&M
’84 UCLA
’85 UCLA
’86 Cal St – Fullerton
’87 Texas A&M
’88 UCLA
’89 UCLA
’90 UCLA
’91 Arizona
’92 UCLA
’93 Arizona
’94 Arizona
’95 UCLA
’96 Arizona
’97 Arizona
’98 Fresno State
’99 UCLA
’00 Oklahoma
’01 Arizona
’02 California
’03 UCLA
’04 UCLA
’05 Michigan
Source: New York Times Almanac.
2.Delegates Who Signed the Declaration of Independence The state represented by each delegate who signed the Declaration of Independence is indicated. Organize the
data in a Pareto chart and a vertical bar graph and comment on the results.
MA 5 PA 9 SC 4
NH 3 RI 2 CT 4
VA 7 NY 4 DE 3
MD 4 GA 3
NJ 5 NC 3
Source: New York Times Almanac.
3.Internet Connections The following data represent the estimated number (in millions) of computers connected to the Internet worldwide. Construct a Pareto chart and a
horizontal bar graph for the data. Based on the data, suggest the best place to market appropriate Internet products.
Location Number of computers
Homes 240
Small companies 102
Large companies 148
Government agencies 33
Schools 47
Source: IDC.
4.Roller Coaster Mania The World Roller Coaster Census Report lists the following number of roller coasters on each continent. Represent the data graphically, using a
Pareto chart and a horizontal bar graph.
Africa 17
Asia 315
Australia 22
Europe 413
North America 643
South America 45
Source: www.rcdb.com
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5.World Energy Use The following percentages indicate the source of energy used worldwide. Construct a Pareto chart and a vertical bar graph for the energy used.
Petroleum 39.8%
Coal 23.2
Dry natural gas 22.4
Hydroelectric 7.0
Nuclear 6.4
Other (wind, solar, etc.) 1.2
Source: New York Times Almanac.
6.Airline Departures Draw a time series graph to represent the data for the number of airline departures (in millions) for the given years. Over the years, is the
number of departures increasing, decreasing, or about the same?

Source: The World Almanac and Book of Facts.
7.Average Global Temperatures Represent these average global temperatures in a time series graph.
1900–09 56.5
1910–19 56.6
1920–29 56.7
1930–39 57.0
1940–49 57.1
1950–59 57.1
1960–69 57.1
1970–79 57.0
1980–89 57.4
1990–99 57.6
Source: World Almanac.
8.Nuclear Power Reactors Draw a time series graph for the data shown and comment on the trend. The data represent the number of active nuclear reactors.

Source: The World Almanac and Book of Facts.
9.Percentage of Voters in Presidential Elections Listed are the percentages of voters who voted in past Presidential elections since 1964. Illustrate the data with a
time series graph. The day before the 2006 election, a website published a survey where 90% of the respondents said they voted in the 2004 election. Give possible
reasons for the discrepancy.
1964 95.83
1968 89.65
1972 79.85
1976 77.64
1980 76.53
1984 74.63
1988 72.48
1992 78.04
1996 65.97
2000 67.50
2004 64.0
Source: New York Times Almanac.
10.Reasons We Travel The following data are based on a survey from American Travel Survey on why people travel. Construct a pie graph for the data and analyze the
results.
Purpose Number
Personal business 146
Visit friends or relatives 330
Work-related 225
Leisure 299
Source: USA TODAY.
11.Characteristics of the Population 65 and Over Two characteristics of the population aged 65 and over are shown below for 2004. Illustrate each characteristic with a
pie graph.

Source: New York Times Almanac.
12.Components of the Earth’s Crust The following elements comprise the earth’s crust, the outermost solid layer. Illustrate the composition of the earth’s crust with a
pie graph.
Oxygen 45.6%
Silicon 27.3
Aluminum 8.4
Iron 6.2
Calcium 4.7
Other 7.8
Source: New York Times Almanac.
13.Workers Switch Jobs In a recent survey, 3 in 10 people indicated that they are likely to leave their jobs when the economy improves. Of those surveyed, 34%
indicated that they would make a career change, 29% want a new job in the same industry, 21% are going to start a business, and 16% are going to retire. Make a pie
chart and a Pareto chart for the data. Which chart do you think better represents the data?
Source: National Survey Institute.
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14.State which graph (Pareto chart, time series graph, or pie graph) would most appropriately represent the given situation.
a.The number of students enrolled at a local college for each year during the last 5 years.
b.The budget for the student activities department at a certain college for each year during the last 5 years.
c.The means of transportation the students use to get to school.
d.The percentage of votes each of the four candidates received in the last election.
e.The record temperatures of a city for the last 30 years.
f.The frequency of each type of crime committed in a city during the year.
15. Presidents’ Ages at Inauguration The age at inauguration for each U.S. President is shown. Construct a stem and leaf plot and analyze the data.

Source: New York Times Almanac.
16. Calories in Salad Dressings A listing of calories per one ounce of selected salad dressings (not fat-free) is given below. Construct a stem and leaf plot for the
data.

17. Twenty Days of Plant Growth The growth (in centimeters) of two varieties of plant after 20 days is shown in this table. Construct a back-to-back stem and leaf
plot for the data, and compare the distributions.

18. Math and Reading Achievement Scores The math and reading achievement scores from the National Assessment of Educational Progress for selected states are listed
below. Construct a back-to-back stem and leaf plot with the data and compare the distributions.

Source: World Almanac.
19.The sales of recorded music in 2004 by genre are listed below. Represent the data with an appropriate graph.
Rock 23.9
Country 13.0
Rap/hip-hop 12.1
R&B/urban 11.3
Pop 10.0
Religious 6.0
Children’s 2.8
Jazz 2.7
Classical 2.0
Oldies 1.4
Soundtracks 1.1
New age 1.0
Other 8.9
Source: World Almanac.
Extending the Concepts
20.Successful Space Launches The number of successful space launches by the United States and Japan for the years 1993–1997 is shown here. Construct a compound time
series graph for the data. What comparison can be made regarding the launches?

Source: The World Almanac and Book of Facts.
21.Meat Production Meat production for veal and lamb for the years 1960–2000 is shown here. (Data are in millions of pounds.) Construct a compound time series graph
for the data. What comparison can be made regarding meat production?

Source: The World Almanac and Book of Facts.
22.Top 10 Airlines The top 10 airlines with the most aircraft are listed. Represent these data with an appropriate graph.
American 714
United 603
Delta 600
Northwest 424
U.S. Airways 384
Continental 364
Southwest 327
British Airways 268
American Eagle 245
Lufthansa (Ger.) 233
Source: Top 10 of Everything.
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23.Nobel Prizes in Physiology or Medicine The top prize-winning countries for Nobel Prizes in Physiology or Medicine are listed here. Represent the data with an
appropriate graph.
United States 80
United Kingdom 24
Germany 16
Sweden 8
France 7
Switzerland 6
Denmark 5
Austria 4
Belgium 4
Italy 3
Australiaata.
5.No, a Pareto chart could not be used to display the data, since we can only have one quantitative variable and one categorical variable in a Pareto chart.
6.We cannot use a pie chart for the same reasons as given for the Pareto chart.
7.A Pareto chart is typically used to show a categorical variable listed from the highest-frequency category to the category with the lowest frequency.
8.A time series chart is used to see trends in the data. It can also be used for forecasting and predicting.
Page 109
Section 2–4 Absences and Final Grades
1.The number of absences can be considered to be the independent variable.
2.The final grade can be considered to be the dependent variable.
3.The scatter plot shows that there is a somewhat linear negative relationship between the variables.
4.It can be concluded in general that more classes missed are associated with lower grades, and fewer classes missed are associated with higher grades.