CHAPTER TWO

DESCRIPTIVE STATISTICS:
TABULAR AND GRAPHICAL METHODS
CHAPTER OUTLINE AND REVIEW
In Chapter 1, you were introduced to the concept of statistics and in exercise *6 of that chapter you were given a frequency distribution of the ages of 180 students at a local college, but you were not told how this frequency distribution was formulated. In
Chapter 2 of your text, you were informed how such frequency distributions could be formulated and were introduced to several tabular and graphical procedures for summarizing data. Furthermore, you were shown how crosstabulations and scatter diagrams can be used to summarize data for two variables simultaneously. The terms that you should have learned from this chapter include:
A.

Qualitative Data:

Data that are measured by either nominal or ordinal scales of measurement. Each value serves as a name or label for identifying an item.

B.

Quantitative Data:

Data that are measured by interval or ratio scales of measurement. Quantitative data are numerical values on which mathematical operations can be performed.

C.

Bar Graph:

A graphical method of presenting qualitative data that have been summarized in a frequency distribution or a relative frequency distribution.

D.

Pie Chart:

A graphical device for presenting qualitative data by subdividing a circle into sectors that correspond to the relative frequency of each class.

23

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Chapter Two

E.

Frequency
Distribution:

A tabular presentation of data, which shows the frequency of the appearance of data elements in several nonoverlapping classes. The purpose of the frequency distribution is to organize masses of data elements into smaller and more manageable groups. The frequency distribution can present both qualitative and quantitative data. F.

Relative Frequency
Distribution:

A tabular presentation of a set of data which shows the frequency of each class as a fraction of the total frequency. The relative frequency distribution can present both qualitative and quantitative data.

G.

Percent Frequency
Distribution:

A tabular presentation of a set of data which shows the percentage of the total number of items in each class.
The percent frequency of a class is simply the relative frequency multiplied by 100.

H.

Class:

A grouping of data elements in order to develop a frequency distribution.

I.

Class Width:

The length of the class interval. Each class has two limits. The lowest value is referred to as the lower class limit, and the highest value is the upper class limit. The difference between the upper and the lower class limits represents the class width.

J.

Class Midpoint:

The point in each class that is halfway between the lower and the upper class limits.

K.

Cumulative
Frequency
Distribution:

A tabular presentation of a set of quantitative data which shows for each class the total number of data elements with values less than the upper class limit.

L.

Cumulative Relative
Frequency
Distribution:

A tabular presentation of a set of quantitative data which shows for each class the fraction of the total frequency with values less than the upper class limit.

M.

Cumulative Percent
Frequency
Distribution:

A tabular presentation of a set of quantitative data which shows for each class the fraction of the total frequency with values less than the upper class limit.

N.

Dot Plot:

A graphical presentation of data, where the horizontal axis shows the range of data values and each observation is plotted as a dot above the axis.

Descriptive Statistics: Tabular and Graphical Methods

25

O.

Histogram:

A graphical method of presenting a frequency or a relative frequency distribution.

P.

Ogive:

A graphical method of presenting a cumulative frequency distribution or a cumulative relative frequency distribution. Q.

Exploratory Data
Analysis:

The use of simple arithmetic and easy-to-draw pictures to look at data more effectively.

R.

Stem-and-Leaf
Display:

An exploratory data analysis technique that simultaneously rank orders quantitative data and provides insight into the shape of the underlying distribution.

S.

Crosstabulation:

A tabular presentation of data for two variables. Rows and columns show the classes of categories for the two variables. T.

Scatter Diagram:

A graphical method of presenting the relationship between two quantitative variables. One variable is shown on the horizontal and the other on the vertical axis.

In this chapter, you have also been informed about the role that computers play in statistical analysis. You have been shown how to use Microsoft Excel and Minitab for your data analysis. At this point, I recommend that you ask your instructor or the staff at your computer center whether or not Excel and/or Minitab are available at your institution; and if they are available, how you can access them. Find out what other statistical software packages are available to you, and how you can use them.

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Chapter Two
CHAPTER FORMULAS

Relative Frequency of a Class =

Frequency of the Class n (2.1)

where n = total number of observations

Approximate Class Width =

Largest Data Value - Smallest Data Value
Number of Classes
(2.2)

Descriptive Statistics: Tabular and Graphical Methods

27

EXERCISES
*1. A student has completed 20 courses in the School of Business Administration. Her grades in the 20 courses are shown below.
A
C
B
C
(a)

B
C
A
B

A
B
B
C

B
B
B
B

C
B
B
A

Develop a frequency distribution for her grades.

Answer: To develop a frequency distribution we simply count her grades in each category. Thus, the frequency distribution of her grades can be presented as
Grade
A
B
C
(b)

Frequency
4
11
5
20

Develop a relative frequency distribution for her grades.

Answer: The relative frequency distribution is a distribution that shows the fraction or proportion of data items that fall in each category. The relative frequencies of each category can be computed by equation 2.1. Thus, the relative frequency distribution can be shown as follows:
Grade
A
B
C
(c)

Relative Frequency
4/20 = 0.20
11/20 = 0.55
5/20 = 0.25

Develop a percent frequency distribution for her grades.

Answer: A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. The percent frequency of a class is simply the relative frequency multiplied by 100. Thus, we can multiply the relative frequencies that we found in Part b to arrive at the percent frequency distribution. Hence, the percent frequency distribution can be shown as follows.

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Chapter Two
Grade
A
B
C

(d)

Percent Frequency
20
55
25

Develop a bar graph.

Answer: A bar graph is a graphical device for presenting the information of a frequency distribution for qualitative data. Bars of equal width are drawn to represent various classes (in this case, grades). The height of each bar represents the frequencies of various classes. Figure 2.1 shows the bar graph for the above data.

BAR GRAPH OF GRADES
12

Frequency

10
8
6
4
2
0
A

B
Grades

Figure 2.1

C

Descriptive Statistics: Tabular and Graphical Methods
(e)

29

Construct a pie chart

Answer: A pie chart is a pictorial device for presenting a relative frequency distribution of qualitative data. The relative frequency distribution is used to subdivide a circle into sections, where each section's size corresponds to the relative frequency of each class.
Figure 2.2 shows the pie chart for the student's grades.
PIE CHART FOR GRADES
20.00%
25.00%

A
B
C

55.00%

Figure 2.2
2.
There are 800 students in the School of Business Administration at UTC. There are four majors in the school: Accounting, Finance, Management and Marketing. The following shows the number of students in each major:
Major

Number of Students

Accounting
Finance
Management
Marketing
(a)

240
160
320
80

Develop a relative and a percent frequency distribution.

30

Chapter Two

(b)

Construct a bar chart.

(c)

Construct a pie chart.

3. Thirty students in the School of Business were asked what their majors were. The following represents their responses (M = Management;
A = Accounting; E = Economics; O = Others).
A
E
M

M
E
A

M
M
O

A
A
A

M
O
M

M
E
E

E
E
E

M
A
M

O
M
A

A
A
M

(a)

Construct a frequency distribution.

(b)

Construct a relative frequency and a percent frequency distribution.

Descriptive Statistics: Tabular and Graphical Methods

31

4.
Twenty employees of ABC corporation were asked if they liked or disliked the new district manager. Below you are given their responses. Let L represent liked and D represent disliked.
L
D
D
D

L
D
L
D

D
L
D
D

L
L
D
D

D
D
L
L

(a)

Construct a frequency distribution.

(b)

Construct a relative frequency and a percent frequency distribution.

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Chapter Two

5.

Five hundred recent graduates indicated their majors as follows.
Major
Accounting
Finance
Economics
Management
Marketing
Engineering
Computer Science
Total

Frequency
60
100
40
120
80
60
40
500

(a)

Construct a relative frequency distribution.

(b)

Construct a percent frequency distribution.

Descriptive Statistics: Tabular and Graphical Methods

33

*6.
In a recent campaign, many airlines reduced their summer fares in order to gain a larger share of the market. The following data represent the prices of round-trip tickets from Atlanta to Boston for a sample of nine airlines.
120
160
160

140
160
180

140
160
180

Construct a dot plot for the above data.
Answer: The dot plot is one of the simplest graphical presentations of data. The horizontal axis shows the range of data values, and each observation is plotted as a dot above the axis. Figure 2.3 shows the dot plot for the above data. The four dots shown at the value of 160 indicate that four airlines were charging $160 for the round-trip ticket from Atlanta to Boston.
DOT PLOT FOR TICKET PRICES

100

•
120

•
•
•
•
160

•
•
140

•
•
180

200

Ticket Prices
Figure 2.3
7.

A sample of the ages of 10 employees of a company is shown below.
20
30

30
20

40
30

30
20

Construct a dot plot for the above data.

50
40

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Chapter Two

*8. The following data elements represent the amount of time (rounded to the nearest second) that 30 randomly selected customers spent in line before being served at a branch of First County Bank.
183
90
320
235
263
165
(a)

121
62
110
250
295
179

140
135
185
242
146
359

198
60
85
193
160
220

199
175
172
75
210
170

Develop a frequency distribution for the above data.

Answer: The first step for developing a frequency distribution is to decide how many classes are needed. There are no "hard" rules for determining the number of classes; but generally, using anywhere from five to twenty classes is recommended, depending on the number of observations. Fewer classes are used when there are fewer observations, and more classes are used when there are numerous observations. In our case, there are only
30 observations. With such a limited number of observations, let us use 5 classes. The second step is to determine the width of each class. By using equation 2.2 which states
Approximate Class Width =

Largest Data Value - Smallest Data Value
Number of Classes

we can determine the class width. In the above data set, the highest value is 359, and the lowest value is 60. Therefore,
Approximate Class Width =

359 - 60
5

= 59.8

We can adjust the above class width (59.8) and use a more convenient value of 60 for the development of the frequency distribution. Note that I decided to use five classes. If you had used 6 or 7 or any other reasonable number of classes, you would not have been wrong and would have had a frequency distribution with a different class width than the one shown above.
Now that we have decided on the number of classes and have determined the class width, we are ready to prepare a frequency distribution by simply counting the number of data items belonging to each class. For example, let us count the

Descriptive Statistics: Tabular and Graphical Methods

35

number of observations belonging to the 60 - 119 class. Six values of 60, 62, 75, 85, 980, and 110 belong to the class of 60 - 119. Thus, the frequency of this class is 6. Since we want to develop classes of equal width, the last class width is from 300 to 359.
THE FREQUENCY DISTRIBUTION OF WAITING TIMES
AT FIRST COUNTY BANK
Waiting Times
(Seconds)

Frequency

60 - 119
120 - 179
180 - 239
240 - 299
300 - 359
Total

6
10
8
4
2
30

(b) What are the lower and the upper class limits for the first class of the above frequency distribution?
Answer: The lower class limit shows the smallest value that is included in a class.
Therefore, the lower limit of the first class is 60. The upper class limit identifies the largest value included in a class. Thus, the upper limit of the first class is 119. (Note:
The difference between the lower limits of adjacent classes provides the class width.
Consider the lower class limits of the first two classes, which are 60 and 120. We note that the class width is 120 - 60 = 60.)
(c) Develop a relative frequency distribution and a percent frequency distribution for the above.
Answer: The relative frequency for each class is determined by the use of equation 2.1.
Relative Frequency of a Class =

Frequency of the Class n where n is the total number of observations. The percent frequency distribution is simply the relative frequencies multiplied by 100. Hence, the relative frequency distribution and the percent frequency distribution are developed as shown on the next page.

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Chapter Two

RELATIVE FREQUENCY AND PERCENT FREQUENCY DISTRIBUTIONS OF
WAITING TIMES AT FIRST COUNTY BANK
Waiting Times
(Seconds)

Relative
Frequency

Percent
Frequency

60 - 119
120 - 179
180 - 239
240 - 299
300 - 359
Total
(d)

Frequency
6
10
8
4
2
30

6/30 = 0.2000
10/30 = 0.3333
8/30 = 0.2667
4/30 = 0.1333
2/30 = 0.0667
1.0000

20.00
33.33
26.67
13.33
6.67
100.00

Develop a cumulative frequency distribution.

Answer: The cumulative frequency distribution shows the number of data elements with values less than or equal to the upper limit of each class. For instance, the number of people who waited less than or equal to 179 seconds is 16 (6 + 10), and the number of people who waited less than or equal to 239 seconds is 24 (6 + 10 + 8). Therefore, the frequency and the cumulative frequency distributions for the above data will be as follows. FREQUENCY AND CUMULATIVE FREQUENCY DISTRIBUTIONS
FOR THE WAITING TIMES AT FIRST COUNTY BANK
Waiting Times
(Seconds)

Cumulative
Frequency

60 - 119
120 - 179
180 - 239
240 - 299
300 - 359
(e)

Frequency
6
10
8
4
2

6
16
24
28
30

How many people waited less than or equal to 239 seconds?

Answer: The answer to this question is given in the table of the cumulative frequency.
You can see that 24 people waited less than or equal to 239 seconds.

Descriptive Statistics: Tabular and Graphical Methods

37

(f) Develop a cumulative relative frequency distribution and a cumulative percent frequency distribution.
Answer: The cumulative relative frequency distribution can be developed from the relative frequency distribution. It is a table that shows the fraction of data elements with values less than or equal to the upper limit of each class. Using the table of relative frequency, we can develop the cumulative relative and the cumulative percent frequency distributions as follows:
RELATIVE FREQUENCY AND CUMULATIVE RELATIVE FREQUENCY AND
CUMULATIVE PERCENT FREQUENCY DISTRIBUTIONS OF WAITING
TIMES AT FIRST COUNTY BANK
Waiting Times
(Seconds)

Relative
Frequency

Cumulative
Relative
Frequency

60 - 119
120 - 179
180 - 239
240 - 299
300 - 359

0.2000
0.3333
0.2667
0.1333
0.0667

0.2000
0.5333
0.8000
0.9333
1.0000

Cumulative
Percent
Frequency
20.00
53.33
80.00
93.33
100.00

NOTE: To develop the cumulative relative frequency distribution, we could have used the cumulative frequency distribution and divided all the cumulative frequencies by the total number of observations, that is, 30.
(g)

Construct a histogram for the waiting times in the above example.

Answer: One of the most common graphical presentations of data sets is a histogram.
We can construct a histogram by measuring the class intervals on the horizontal axis and the frequencies on the vertical axis. Then we can plot bars with the widths equal to the class intervals and the height equivalent to the frequency of the class that they represent.
In Figure 2.4, the histogram of the waiting times is presented. As you note, the width of each bar is equal to the width of the various classes (60 seconds), and the height represents the frequency of the various classes. Note that the first class ends at 119; the next class begins at 120, and one unit exists between these two classes (and all other classes). To eliminate these spaces, the vertical lines are drawn halfway between the class limits. Thus, the vertical lines are drawn at 59.5, 119.5, 179.5, 239.5, 299.5, and
359.5.

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Chapter Two
HISTOGRAM OF THE WAITING TIMES
AT FIRST COUNTY BANK

10
9
8
7
6
5
4
3
2
1
0

0

60

120

180

240

300

360

Waiting Times (in seconds)
Figure 2.4
(h)

Construct an ogive for the above example.

Answer: An ogive is a graphical representation of the cumulative frequency distribution or cumulative relative frequency distribution. It is constructed by measuring the class intervals on the horizontal axis and the cumulative frequencies (or cumulative relative frequencies) on the vertical axis. Then points are plotted halfway between class limits
(i.e., 119.5, 179.5, 239.5, etc.) at a height equal to the cumulative frequency (or cumulative relative frequency). One additional point is plotted at 59.5 on the horizontal axis and 0 on the vertical axis. This point shows that there are no data values below the
60 - 119 class. Finally, these points are connected by straight lines. The result is an ogive that is shown in Figure 2.5.

Descriptive Statistics: Tabular and Graphical Methods

39

OGIVE FOR THE CUMULATIVE FREQUENCY DISTRIBUTION
OF THE WAITING TIMES AT FIRST COUNTY BANK
Waiting Times (in seconds)

Waiting Times
Figure 2.5
9.
The following data set shows the number of hours of sick leave that some of the employees of Bastien's, Inc. have taken during the first quarter of the year (rounded to the nearest hour).
19
23
36
12
59

22
47
25
20
39

27
11
34
28
48

24
55
16
29
32

28
25
45
21
40

12
42
49
10
31

(a) Develop a frequency distribution for the above data. (Let the width of your classes be 10 units and start your first class as 10 - 19.)

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Chapter Two

(b) Develop a relative frequency distribution and a percent frequency distribution for the data.

(c)

Develop a cumulative frequency distribution.

(d)

How many employees have taken less than 40 hours of sick leave?

10.

The grades of 20 students on their first statistics test are shown below.
71
71
72
82

52
68
75
85

66
55
78
87

76
77
62
98

78
91
93
65

(a) Develop a frequency distribution for the grades. (Let your first class be
50 - 59.)

Descriptive Statistics: Tabular and Graphical Methods
(b)

41

Develop a percent frequency distribution.

11. The sales record of a real estate company for the month of May shows the following house prices (rounded to the nearest $1,000). Values are in thousands of dollars.
105
30

55
60

45
75

85
79

75
95

(a) Develop a frequency distribution and a percent frequency distribution for the house prices. (Use 5 classes and have your first class be 20 - 39.)

(b) Develop a cumulative frequency and a cumulative percent frequency distribution for the above data.

(c)

What percentage of the houses sold at a price below $80,000?

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Chapter Two

12.

The hourly wages of 12 employees are shown below.
7
17
24

8
15
10

14
10
21

17
20
25

(a)

Develop a frequency distribution. (Let your first class be 7 - 11.)

(b)

Develop a cumulative frequency distribution.

Descriptive Statistics: Tabular and Graphical Methods

43

13. A group of freshmen at an area university decided to sell magazine subscriptions to help pay for their Christmas party. Below is a list of the 18 students who participated and the number of subscriptions they sold.
Student

# of Subscriptions Sold

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

30
79
59
65
40
64
52
53
57
39
61
47
50
60
48
50
58
67

(a) Develop a frequency distribution and a percent frequency distribution. (Let the width of your classes be 10 units.)

(b)

Develop a cumulative frequency and a cumulative percent frequency distribution.

(c)

How many students sold less than 60 subscriptions?

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Chapter Two

14. The temperatures for the month of June in a southern city were recorded as follows.
(Rounded to the nearest degree)
70
82
91
97
105

75
89
92
95
104

79
88
93
98
108

80
87
95
100
111

78
90
94
107
109

82
92
95
107
116

(a) Develop a frequency distribution and a relative frequency distribution for the above data. (Let the width of your classes be 10 units.)

(b)

Develop a cumulative frequency and a cumulative relative frequency distribution.

(c)

How many days was the temperature at least 90 degrees?

Descriptive Statistics: Tabular and Graphical Methods

45

15. The frequency distribution below shows the yearly income distribution of a sample of 160 Kern County residents.
Yearly Income
(in thousands of dollars)
10 - 14
15 - 19
20 - 24
25 - 29
30 - 34
35 - 39
Total

Frequency
10
25
30
40
35
20
160

(a)

What percentage of the individuals in the sample had incomes of less than $25,000?

(b)

How many individuals had incomes of at least $30,000?

16. The Alex Food Company bakes quiches and sells their products in the greater Los
Angeles area. Their records over the past 60 days are shown below.
Sales Volume
(Number of Quiches)

Number of Days

100 - 199
200 - 299
300 - 399
400 - 499
500 - 599
600 - 699
Total

6
10
20
12
8
4
60

(a)

Develop a cumulative frequency distribution and a percent frequency distribution.

(b)

What percentage of the days did the company sell at least 400 quiches?

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Chapter Two

17.

The ages of 16 employees are shown below.
22
35
39
45

40
27
46
36

34
30
32
41

36
32
48
41

(a)

Develop a frequency distribution. Let your first class be 20 - 25.

(b)

Develop a cumulative frequency distribution.

18.

For the following distribution, develop a percent frequency distribution.
Class
40 - 59
60 - 79
80 - 99
100 - 119
120 - 139
140 - 159
160 - 179

Frequency
48
110
82
70
58
20
12

Percent
Frequency

Descriptive Statistics: Tabular and Graphical Methods

47

19. The frequency distribution below was constructed from data collected from a group of 250 students.
Height in Inches

Frequency

58 - 63
64 - 69
70 - 75
76 - 81
82 - 87
88 - 93
94 - 99
Total

30
50
20
60
40
30
20
250

(a)

Construct a percent frequency distribution.

(b)

Construct a cumulative frequency distribution.

(c)

Construct a cumulative percent frequency distribution.

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Chapter Two

*20. The test scores of 14 individuals on their first statistics examination are shown below. 95
75
a)

87
63

52
92

43
81

77
83

84
91

78
88

Construct a stem-and-leaf display for these data.

Answer: To construct a stem-and-leaf display, the first digit of each data item is arranged in an ascending order and written to the left of a vertical line. Then, the second digit of each data item is written to the right of the vertical line next to its corresponding first digit as follows.
4
5
6
7
8
9

3
2
3
7 8 5
7 4 1 3 8
5 2 1

Now, the second digits are rank ordered horizontally, thus leading to the following stemand-leaf display.
4
5
6
7
8
9
b)

3
2
3
5 7 8
1 3 4 7 8
1 2 5

What does the above stem-and-leaf show?

Answer: Each line in the above display is called a stem, and each piece of information on a stem is a leaf. For instance, let us consider the fourth line:
7 5 7 8
The stem indicates that there are 3 scores in the seventies. These values are
75

77

78

Similarly, we can look at line five (where the first digit is 8) and see
8 1 3 4 7 8

Descriptive Statistics: Tabular and Graphical Methods

49

This stem indicates that there are 5 scores in the eighties, and they are
81

83

84

87

88

At a glance, one can see the overall distribution for the grades. There is one score in the forties (43), one score in the fifties (52), one score in the sixties (63), three scores in the seventies (75, 77, 78), five scores in the eighties
(81, 83, 84, 87, 88), and three scores in the nineties (91, 92, 95).
21.

Construct a stem-and-leaf display for the following data.
22
18

44
48

36
32

45
19

49
43

57
31

38
26

47
40

51
37

12
52

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Chapter Two

*22. The following is a crosstabulation of starting salaries (in $1,000s) of a sample of business school graduates by their gender.
Starting Salary
Gender

Less than 20

20 up to 25

25 and more

Total

Female

12

84

24

120

Male

20

48

12

80

Total

32

132

36

200

(a) What general comments can be made about the distribution of starting salaries and the gender of the individuals in the sample?
Answer: Using the frequency distribution at the bottom margin of the above table it is noted that majority of the individuals in the sample (132) have starting salaries in the range of $20,000 up to $25,000, followed by 36 individuals whose salaries are at least
$25,000, and only 32 individuals had starting salaries of under $20,000. Now considering the right-hand margin it is noted that the majority of the individuals in the sample (120) are female while 80 are male.
(b) Compute row percentages and comment on the relationship between starting salaries and gender.
Answer: To compute the row percentages we divide the values of each cell by the row total and express the results as percentages. Let us consider the row representing females. The row percentages (across) are computed as
(12/120)(100)=10%; (84/120)(100)=70%; (24/120)(100)=20%
Continuing in the same manner and computing the row percentages for the other row we determine the following row percentages table:
Starting Salary
Gender

Less than 20

20 up to 25

25 and more

Total

Female

10%

70%

20%

100%

Male

25%

60%

15%

100%

From the above percentages it can be noted that the largest percentage of both genders' starting salaries are in the $20,000 to $25,000 range. However, 70% of females and only
60% of the males have starting salaries in this range. Also it can be noted that 10% of females' starting salaries are under $20,000, whereas, 25% of the males' starting salaries fall in this category.

Descriptive Statistics: Tabular and Graphical Methods

51

(c) Compute column percentages and comment on the relationship between gender and starting salaries.
Column percentages are computed by dividing the values in each cell by column total and expressing the results as percentages. For instance for the category of "Less than 20" the column percentages are computed as (12/32)(100)=37.5 and (20/32)(100)=62.5
(rounded). Continuing in the same manner the column percentages will be as follows.
Gender

Starting Salary
Less than 20
20 up to 25

25 and more

Female

37.5%

63.6%

66.7%

Male

62.5%

36.4%

33.3%

Total

100%

100%

100%

Considering the "Less than 20" category it is noted that the majority (62.5%) are male. In the next category of "20 up to 25" the majority (63.6%) are female. Finally in the last category of "25 and more" the majority (66.7%) are female.
23. A survey of 400 college seniors resulted in the following crosstabulation regarding their undergraduate major and whether or not they plan to go to graduate school.
Undergraduate Major
Graduate School

Business

Engineering

Others

Total

Yes

35

42

63

140

No

91

104

65

260

Total

126

146

128

400

(a)

Are majority of seniors in the survey planning to attend graduate school?

(b)

Which discipline constitutes the majority of the individuals in the survey?

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Chapter Two

(c) Compute row percentages and comment on the relationship between the students' undergraduate major and their intention of attending graduate school.

(d) Compute the column percentages and comment on the relationship between the students' intention of going to graduate school and their undergraduate major.

Descriptive Statistics: Tabular and Graphical Methods

53

*24. The average grades of 8 students in statistics and the number of absences they had during the semester are shown below.

Student

Number of
Absences
(x)

Average
Grade
(y)

1
2
3
4
5
6
7
8

1
2
2
1
3
4
8
3

94
78
70
88
68
40
30
60

Develop a scatter diagram for the relationship between the number of absences (x) and their average grade (y).
Answer: A scatter diagram is a graphical method of presenting the relationship between two variables. The scatter diagram is shown in Figure 2.6. The number of absences (x) is shown in the horizontal axis and the average grade (y) on the vertical axis. The first student has one absence (x=1) and an average grade of 94 (y=94). Therefore, a point with coordinates of x=1 and y=94 is plotted on the scatter diagram. In a similar manner all other points for all 8 students are plotted.

Figure 2.6
The scatter diagram shows that there is a negative relationship between the number of absences and the average grade. That is, the higher the number of absences, the lower the average grade appears to be.

54

Chapter Two

25.

You are given the following ten observations on two variables, x and y. x y

1
5
6
4
2
8
9
1
6
8

8
15
20
12
10
20
26
5
18
26

(a)

Develop a scatter diagram for the relationship between x and y.

(b)

What relationship, if any, appears to exist between x and y?

Descriptive Statistics: Tabular and Graphical Methods

55

SELF-TESTING QUESTIONS
In the following multiple choice questions, circle the correct answer. An answer key is provided following the questions.
1.
A tabular summary of a set of data, which shows the frequency of the appearance of data elements in several nonoverlapping classes is termed
a)
b)
c)
d)

the class width a frequency polygon a frequency distribution a histogram

2.
A tabular summary of a set of data showing classes of the data and the fraction of the items belonging to each class is called
a)
b)
c)
d)

the class width a relative frequency distribution a cumulative relative frequency distribution an ogive

3.

A histogram is

a)
b)
c)
d)

a graphical presentation of a frequency or relative frequency distribution a graphical method of presenting a cumulative frequency or a cumulative relative frequency distribution the history of data elements the same as a pie chart

4.

The length of the interval forming a class is called

a)
b)
c)
d)

the class midpoint the lower class limit the upper class limit the class width

5.
A graphical method of presenting qualitative data by frequency distribution is termed a)
b)
c)
d)

a frequency polygon an ogive a bar graph a qualitative chart

56

Chapter Two

Exhibit 2-1
Michael's Rent-A-Car, a national car rental company, has kept a record of the number of cars they have rented for a period of 80 days. Their rental records are shown below:
Number of Cars Rented

Number of Days

0 - 19
20 - 39
40 - 59
60 - 79
80 - 99
Total

5
15
30
20
10
80

6.

Refer to Exhibit 2-1. The class width of the above distribution is

a)
b)
c)
d)

0 to 100
20
80
5

7.

Refer to Exhibit 2-1. The lower limit of the first class is

a)
b)
c)
d)

5
80
0
20

8. Refer to Exhibit 2-1. If one develops a cumulative frequency distribution for the above data, the last class will have a frequency of
a)
b)
c)
d)

10
100
0 to 100
80

9. Refer to Exhibit 2-1. The percentage of days in which the company rented at least
40 cars is
a)
b)
c)
d)

37.5%
62.5%
90.0%
75.0%

Descriptive Statistics: Tabular and Graphical Methods

57

10. Refer to Exhibit 2-1. The number of days in which the company rented less than 60 cars is
a)
b)
c)
d)

20
30
50
60

11.

The sum of frequencies for all classes will always equal

a)
b)
c)
d)

1 the number of elements in a data set the number of classes a value between 0 to 1

12.

The relative frequency of a class is computed by

a)
b)
c)
d)

dividing the midpoint of the class by the sample size dividing the frequency of the class by the midpoint dividing the sample size by the frequency of the class dividing the frequency of the class by the sample size

13. If several frequency distributions are constructed from the same data set, the distribution with the narrowest class width will have
a)
b)
c)
d)

the fewest classes the most classes the same number of classes as the other distributions since all are constructed from the same data at least 10 classes

14.

The sum of the relative frequencies for all classes will always equal

a)
b)
c)
d)

the sample size the number of classes one any value larger than one

15. In a cumulative relative frequency distribution, the last class will have a cumulative relative frequency equal to
a)
b)
c)
d)

one zero the total number of elements in the data set any positive value

58

Chapter Two

16. In a cumulative percent frequency distribution, the last class will have a cumulative percent frequency equal to
a)
b)
c)
d)

one zero the total number of elements in the data set
100

17. A tabular method that can be used to summarize the data on two variables simultaneously is called
a)
b)
c)
d)

simultaneous equations an ogive a histogram crosstabulation Descriptive Statistics: Tabular and Graphical Methods
ANSWERS TO THE SELF-TESTING QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

c b a d c b c d d c b d b c a d d

59

60

Chapter Two
ANSWERS TO CHAPTER TWO EXERCISES

2.

(a)
Major
Accounting
Finance
Management
Marketing

Relative Frequency
0.3
0.2
0.4
0.1

Percent Frequency
30
20
40
10

(b)
350
300
250
200
150
100
50
0
Accounting

Finance

Management

Marketing

(c)
10.00%
30.00%
Accounting
Finance
Management
40.00%
Marketing
20.00%

Descriptive Statistics: Tabular and Graphical Methods
3.

(a) and (b)
Major

Frequency

Relative
Frequency

M
A
E
O

11
9
7
3

0.37
0.30
0.23
0.10

37
30
23
10

Preferences

Frequency

Relative
Frequency

Percent
Frequency

L
D

8
12
20

0.4
0.6
1.0

40
60
100

4.

Percent
Frequency

(a) and (b)

Total
5.
Major
Accounting
Finance
Economics
Management
Marketing
Engineering
Computer Science
Total

Frequency

(a)
Relative
Frequency

(b)
Percent
Frequency

60
100
40
120
80
60
40
500

0.12
0.20
0.08
0.24
0.16
0.12
0.08
1.00

12
20
8
24
16
12
8
100

7.

10

•
•
•
20

•
•
•
•
30

•
•
40

•
50

60

61

62

Chapter Two

9.

(a)
Hours of
Sick Leave Taken
10 - 19
20 - 29
30 - 39
40 - 49
50 - 59
(d) 22

(b)
Relative
Freq.
0.20
0.37
0.16
0.20
0.07

Freq.
6
11
5
6
2

(b)
Percent
Freq.
20
37
16
20
7

(c)
Cum.
Freq.
6
17
22
28
30

10.
Class

(a)
Frequency

(b)
Percent Frequency

2
4
8
3
3
20

10
20
40
15
15
100

50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
11.

(a)
Sales Price
(In Thousands of Dollars)
20 - 39
40 - 59
60 - 79
80 - 99
100 - 119
(c)

(a)

(b)

Freq.
1
2
4
2
1

Percent
Freq.
10
20
40
20
10

Cum.
Freq.
1
3
7
9
10

(b)
Cum.
Percent
Freq.
10
30
70
90
100

70%

12.
Class
7 - 11
12 - 16
17 - 21
22 - 26

(a)
Frequency

(b)
Cumulative Frequency

4
2
4
2
12

4
6
10
12

Descriptive Statistics: Tabular and Graphical Methods
13.

(a)

(c)

(a)

Freq.
4
6
11
7
2

Rel.
Freq.
0.13
0.20
0.37
0.23
0.07

Temperature
70 - 79
80 - 89
90 - 99
100 - 109
110 - 119
(c)

20

(a)
(b)

40.625%
55

16.
Sales Volume
100 - 199
200 - 299
300 - 399
400 - 499
500 - 599
600 - 699
(b)

(b)
Cum.
Freq.
2
5
12
17
18

(b)
Cum.
Percent
Freq.
11
28
67
95
100

12

14.

15.

Freq.
2
3
7
5
1

Percent
Freq.
11
17
39
28
5

(a)

No. of Subscriptions
Sold
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79

(a)

(a)
Cumulative
Frequency
6
16
36
48
56
60

(a)
Percent
Frequency
10
17
33
20
13
7

40%

(b)
Cum.
Freq.
4
10
21
28
30

(b)
Cum.
Rel.
Freq.
0.13
0.33
0.70
0.93
1.00

63

64

Chapter Two

17.
Class

(a)
Frequency

(b)
Cumulative Frequency

20 - 25
26 - 31
32 - 37
38 - 43
44 - 49

1
2
6
4
3

1
3
9
13
16

18.
Class

Percent
Frequency

40 - 59
60 - 79
80 - 99
100 - 119
120 - 139
140 - 159
160 - 179

0.120
0.275
0.205
0.175
0.145
0.050
0.030

19.
(a)
Height
(In Inches)

Frequency

58 - 63
64 - 69
70 - 75
76 - 81
82 - 87
88 - 93
94 - 99

30
50
20
60
40
30
20

21.

1
2
3
4
5

2
2
1
0
1

8
6
2
3
2

9
6 7 8
4 5 7 8 9
7

(b)

Percent
Frequency

Cumulative
Frequency

(c)
Cumulative
Percent
Frequency

12
20
8
24
16
12
8
100

30
80
100
160
200
230
250

12
32
40
64
80
92
100

Descriptive Statistics: Tabular and Graphical Methods
23.

(a)
(b)
(c)

65

No, majority (260) will not attend graduate school
Majority (146) are engineering major
Undergraduate Major

Graduate School

Business

Engineering

Others

Total

Yes

25%

30%

45%

100%

No

35%

40%

25%

100%

Majority who plan to go to graduate school are from "Other" majors. Majority of those who will not go to graduate school are engineering majors.
(d)
Undergraduate Major
Graduate School

Business

Engineering

Others

Yes

27.8%

28.8%

49.2%

No

72.2%

71.2%

50.8%

Total

100%

100%

100%

Approximately the same percentages of Business and engineering majors plan to attend graduate school (27.8% and 28.8% respectively). Of the "Other" majors approximately half (49.2%) plan to go to graduate school.

66

Chapter Two

25.

(a)
30
25
20
15

Y values

10
5
0

0

1

2

3

4

5

6

7

X values
(b)

A positive relationship between x and y appears to exist.

8

9