Chapter 2: Charts and Graphs 1
Chapter 2
Charts and Graphs
LEARNING OBJECTIVES
The overall objective of chapter 2 is for you to master several techniques for
summarizing and depicting data, thereby enabling you to:
1.
Recognize the difference between grouped and ungrouped data.
2.
Construct a frequency distribution.
3.
Construct a histogram, a frequency polygon, an ogive, a pie chart, a stem and leaf
plot, a Pareto chart, and a scatter plot.
CHAPTER TEACHING STRATEGY
Chapter 1 brought to the attention of students the wide variety and amount of data
available in the world of business. In chapter 2, we confront the problem of trying to
begin to summarize and present the data in a meaningful manner. One mechanism for
data summarization is the frequency distribution which is essentially a way of organizing
ungrouped or raw data into grouped data. It is important to realize that there is
considerable art involved in constructing a frequency distribution. There are nearly as
many possible frequency distributions for a problem as there are students in a class.
Students should begin to think about the receiver or user of their statistical product. For
example, what class widths and class endpoints would be most familiar and meaningful
to the end user of the distribution? How can the data best be communicated and
summarized using the frequency distribution?
The second part of chapter 2 presents various ways to depict data using graphs.
The student should view these graphical techniques as tools for use in communicating
characteristics of the data in an effective manner. Most business students will have some
type of management opportunity in their field before their career ends. The ability to
make effective presentations and communicate their ideas in succinct, clear ways is an
asset. Through the use of graphics packages and such techniques as frequency polygons,
ogives, histograms, and pie charts, the manager can enhance his/her personal image as a
Chapter 2: Charts and Graphs 2
communicator and decision-maker. In addition, emphasize that the final product (the
frequency polygon, etc.) is just the beginning. Students should be encouraged to study
the graphical output to recognize business trends, highs, lows, etc. and realize that the
ultimate goal for these tools is their usage in decision making.
CHAPTER OUTLINE
2.1
Frequency Distributions
Class Midpoint
Relative Frequency
Cumulative Frequency
2.2
Graphic Depiction of Data
Histograms
Frequency Polygons
Ogives
Pie Charts
Stem and Leaf Plots
Pareto Charts
2.3
Graphical Depiction of Two-Variable Numerical Data: Scatter Plots
KEY TERMS
Class Mark
Class Midpoint
Cumulative Frequency
Frequency Distribution
Frequency Polygon
Grouped Data
Histogram
Ogive
Pareto Chart
Pie Chart
Range
Relative Frequency
Scatter Plot
Stem and Leaf Plot
Ungrouped Data
SOLUTIONS TO PROBLEMS IN CHAPTER 2
2.1
a)
One possible 5 class frequency distribution:
Class Interval
10 - under 25
25 - under 40
40 - under 55
55 - under 70
Frequency
9
13
11
9
Chapter 2: Charts and Graphs 3
70 - under 85
b)
8
50
One possible 10 class frequency distribution:
Class Interval
10 - under 18
18 - under 26
26 - under 34
34 - under 42
42 - under 50
50 - under 58
58 - under 66
66 - under 74
74 - under 82
82 - under 90
c)
2.2
Frequency
7
3
5
9
7
3
6
4
4
2
The ten class frequency distribution gives a more detailed breakdown of
temperatures, pointing out the smaller frequencies for the higher temperature
intervals. The five class distribution collapses the intervals into broader
classes making it appear that there are nearly equal frequencies in each class.
One possible frequency distribution is the one below with 11 classes and class
intervals of 2.
Class Interval
39 - under 41
41 - under 43
43 - under 45
45 - under 47
47 - under 49
49 - under 51
51 - under 53
53 - under 55
55 - under 57
57 - under 59
59 - under 61
Frequency
2
1
5
10
18
13
15
15
7
9
2
The distribution reveals that only 13 of the 100 boxes of raisins contain 50 ± 1
raisin (49 -under 51). However, 71 of the 100 boxes of raisins contain between
45 and 55 raisins. It shows that there are a few boxes (5) that have 9 or more
extra raisins (59-61) and two boxes that have 9-11 less raisins (39-41) than the
boxes are supposed to contain.
2.3
Class
Interval
Frequency
Class
Midpoint
Relative
Frequency
Cumulative
Frequency
Chapter 2: Charts and Graphs 4
0-5
5 - 10
10 - 15
15 - 20
20 - 25
25 - 30
30 - 35
TOTAL
6
8
17
23
18
10
4
86
2.5
7.5
12.5
17.5
22.5
27.5
32.5
6/86 = .0698
.0930
.1977
.2674
.2093
.1163
.0465
1.0000
6
14
31
54
72
82
86
The relative frequency tells us that it is most probable that a customer is in the
15 - 20 category (.2674). Over two thirds (.6744) of the customers are between 10
and 25 years of age.
2.4
Class
Interval
0-2
2-4
4-6
6-8
8-10
TOTAL
Frequency
218
207
56
11
8
500
Class
Midpoint
1
3
5
7
9
Relative
Frequency
.436
.414
.112
.022
.016
1.0000
2.5 Some examples of cumulative frequencies in business:
sales for the fiscal year,
costs for the fiscal year,
spending for the fiscal year,
inventory build-up,
accumulation of workers during a hiring buildup,
production output over a time period.
2.6 Histogram
Cumulative
Frequency
218
425
481
492
500
Chapter 2: Charts and Graphs 5
Frequency Polygon
2.7 Histogram
Chapter 2: Charts and Graphs 6
Frequency Polygon
2.8 Ogive
Chapter 2: Charts and Graphs 7
2.9
STEM
21
22
23
24
25
26
27
2.10
LEAF
2, 8, 8, 9
0, 1, 2, 4, 6, 6, 7, 9, 9
0, 0, 4, 5, 8, 8, 9, 9, 9, 9
0, 0, 3, 6, 9, 9, 9
0, 3, 4, 5, 5, 7, 7, 8, 9
0, 1, 1, 2, 3, 3, 5, 6
0, 1,
Company
Proportion
Degrees
Andersen Worldwide
Ernst & Young
Deloitte & Touche
KPMG Peat Marwick
Coopers & Lybrand
Price Waterhouse
Grant Thornton
McGladrey & Pullen
BDO Seidman
.25
.20
.17
.12
.11
.11
.01
.01
.01
90
72
61
43
40
40
4
4
4
TOTAL
.99
358
Pie Chart
Chapter 2: Charts and Graphs 8
2. 11
Company
Delta
United
American
US Airways
Southwest
TOTAL
Proportion
Degrees
.27
.22
.21
.15
.15
97
79
76
54
54
1.00
360
2.12
Brand
Proportion
Degrees
Chapter 2: Charts and Graphs 9
Huggies
Pampers
Luvs
Drypers
Fitti
Private Labels
.413
.256
.121
.033
.009
.158
149
92
44
12
3
57
TOTAL
.990
357
Pie Chart
2.13
STEM
1
2
3
4
5
6
7
8
LEAF
3, 6, 7, 7, 7, 9, 9, 9
0, 3, 3, 5, 7, 8, 9, 9
2, 3, 4, 5, 7, 8, 8
1, 4, 5, 6, 6, 7, 7, 8, 8, 9
0, 1, 2, 2, 7, 8, 9
0, 1, 4, 5, 6, 7, 9
0, 7
0
The stem and leaf plot shows that the number of passengers per flight were
relatively evenly distributed between the high teens through the sixties. Rarely
was there a flight with at least 70 passengers. The category of 40's contained the
most flights (10).
Chapter 2: Charts and Graphs 10
2.14
Complaint
Number
% of Total
420
184
85
37
10
8
744
56.45
24.73
11.42
4.97
1.34
1.08
99.99
Busy Signal
Too long a Wait
Could not get through
Get Disconnected
Transferred to the Wrong Person
Poor Connection
Total
2.15
3500
3000
Industrial Products
2500
2000
1500
1000
500
0
0
1000
2000
3000
4000
Human Food
5000
6000
7000
Chapter 2: Charts and Graphs 11
2.16
180
160
140
Sales
120
100
80
60
40
20
0
0
2
4
6
8
Advertising
2.17
Class Interval
Frequencies
16 - under 23
23 - under 30
30 - under 37
37 - under 44
44 - under 51
51 - under 58
6
9
4
4
4
3
TOTAL
30
2.18
Class Interval Frequency Midpoint Rel.Freq. Cum.Freq.
20 - under 25
25 - under 30
30 - under 35
35 - under 40
40 - under 45
45 - under 50
17
20
16
15
8
6
22.5
27.5
32.5
37.5
42.5
47.5
.207
.244
.195
.183
.098
.073
.207
.451
.646
.829
.927
1.000
10
12
Chapter 2: Charts and Graphs 12
2.19
Class Interval
Frequencies
50 - under 60
60 - under 70
70 - under 80
80 - under 90
90 - under 100
13
27
43
31
9
TOTAL
Histogram
Frequency Polygon
123
Chapter 2: Charts and Graphs 13
Ogive
2.20
Label
A
B
C
D
TOTAL
Pie Chart
Value
Proportion
55
121
83
46
305
.180
.397
.272
.151
1.000
Degrees
65
143
98
54
360
Chapter 2: Charts and Graphs 14
2.21
STEM
28
29
30
31
32
33
2.22
LEAF
4, 6, 9
0, 4, 8
1, 6, 8, 9
1, 2, 4, 6, 7, 7
4, 4, 6
5
Problem
1
2
3
4
5
6
7
8
9
10
Frequency
673
29
108
379
73
564
12
402
54
202
2496
Pareto Chart:
Percent of Total
26.96
1.16
4.33
15.18
2.92
22.60
0.48
16.11
2.16
8.09
Chapter 2: Charts and Graphs 15
2.23
16
14
12
Y
10
8
6
4
2
0
0
2
4
6
8
10
X
2.24 Olson Company
Frequency distribution
Class Interval
32 - under 37
37 - under 42
42 - under 47
47 - under 52
52 - under 57
57 - under 62
62 - under 67
67 - under 72
TOTAL
Frequency
1
4
12
11
14
5
2
1
50
12
14
16
18
Chapter 2: Charts and Graphs 16
2.25
Class
Interval
20 – 25
25 – 30
30 – 35
35 – 40
40 – 45
45 – 50
TOTAL
Frequency
8
6
5
12
15
7
53
Class
Midpoint
22.5
27.5
32.5
37.5
42.5
47.5
Relative
Frequency
8/53 = .1509
.1132
.0943
.2264
.2830
.1321
.9999
2.26 Frequency Distribution:
Class Interval
10 - under 20
20 - under 30
30 - under 40
40 - under 50
50 - under 60
60 - under 70
70 - under 80
80 - under 90
Histogram
Frequency
2
3
9
7
12
9
6
2
50
Cumulative
Frequency
8
14
19
3
46
53
Chapter 2: Charts and Graphs 17
Frequency Polygon
The normal distribution appears to peak near the center and diminish towards the
end intervals.
2.27
a) Histogram and a Frequency Polygon for 2.25
Class
Interval
20 - 25
25 - 30
30 - 35
35 - 40
40 - 45
45 - 50
TOTAL
Frequency
8
6
5
12
15
7
53
Cumulative
Frequency
8
14
19
31
46
53
Chapter 2: Charts and Graphs 18
Histogram
Frequency Polygon
Chapter 2: Charts and Graphs 19
b) Ogive
2.28
Asking Price
$ 60,000 - under $ 70,000
$ 70,000 - under $ 80,000
$ 80,000 - under $ 90,000
$ 90,000 - under $100,000
$100,000 - under $110,000
$110,000 - under $120,000
Frequency
21
27
18
11
6
3
86
Cumulative
Frequency
21
48
66
77
83
86
Chapter 2: Charts and Graphs 20
Histogram
Frequency Polygon
Chapter 2: Charts and Graphs 21
Ogive
2.29
Amount Spent
on Prenatal Care
$ 0 - under $100
$100 - under $200
$200 - under $300
$300 - under $400
$400 - under $500
$500 - under $600
Frequency
3
6
12
19
11
6
57
Cumulative
Frequency
3
9
21
40
51
57
Chapter 2: Charts and Graphs 22
Histogram
Frequency Polygon
Chapter 2: Charts and Graphs 23
Ogive
2.30
Price
Frequency
$1.75 - under $1.90
$1.90 - under $2.05
$2.05 - under $2.20
$2.20 - under $2.35
$2.35 - under $2.50
$2.50 - under $2.65
$2.65 - under $2.80
9
14
17
16
18
8
5
87
Cumulative
Frequency
9
23
40
56
74
82
87
Chapter 2: Charts and Graphs 24
Histogram
Frequency Polygon
Chapter 2: Charts and Graphs 25
Ogive
2.31
Genre
R&B
Alternative
Rap
Country
Soundtrack
Metal
Classical
Latin
TOTAL
Albums Sold
146.4
102.6
73.7
64.5
56.4
26.6
14.8
14.5
Proportion
Degrees
.29
.21
.15
.13
.11
.05
.03
.03
104
76
54
47
40
18
11
11
1.00
361
Chapter 2: Charts and Graphs 26
Pie Chart
2.32
700
600
Manufactured Goods
500
400
300
200
100
0
0
5
10
15
20
Agricultural Products
25
30
35
Chapter 2: Charts and Graphs 27
2.33
Industry
Total Release
Proportion
Degrees
Chemicals
Primary metals
Paper
Plastics & Rubber
Transportation
Equipment
Food
Fabricated Metals
Petroleum
Electrical
Equipment
737,100,000
566,400,000
229,900,000
109,700,000
.37
.28
.11
.05
133
101
40
18
102,500,000
89,300,000
85,900,000
63,300,000
.05
.04
.04
.03
18
14
14
11
29,100,000
.01
4
0.98
353
TOTAL
Pie Chart
Elec. Equip.
Petroleum
Fab. Metals 3% 1%
4%
Food
4%
Transportation Equipment
5%
Chemicals
38%
Plas. & Rubber
5%
Paper
11%
Primary Metals
29%
Chapter 2: Charts and Graphs 28
2.34
500
100
400
80
300
60
200
40
100
20
0
Defect
Count
Percent
Cum %
0
in
ult
Fa
c
s ti
Pla
221
44.2
44.2
ss
ne
ic k
Th
le
nd
Ha
en
k
Br o
117
23.4
67.6
2.35
STEM
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
LEAF
12, 16, 24, 32, 99, 99
04, 28, 39, 46, 61, 88
20, 40, 59
12
53, 54
30, 34, 58
22, 34, 66, 78
63
48, 49, 90
66
21, 54, 57, 63, 91
38, 66, 66
31, 78
56
69
37, 50
31, 32, 58, 73
19, 23
86
17.2
84.8
ling
be
La
44
8.8
93.6
D is
n
atio
lor
co
32
6.4
100.0
Percent
Count
Problem 2.34
Chapter 2: Charts and Graphs 29
2.36
STEM
22
23
24
25
26
27
28
29
30
LEAF
00, 68
01, 37, 44, 75
05, 37, 48, 60, 68
24, 55
02, 56, 70, 77
42, 60, 64
14, 30
22, 61, 75, 76, 90, 96
02, 10
2.37 The distribution of household income is bell-shaped with an average of about $
90,000 and a range of from $ 30,000 to $ 140,000.
2.38 Family practice is most prevalent with about 20% with pediatrics next at slightly
less. A virtual tie exists between ob/gyn, general surgery, anesthesiology, and
psychiatry at about 14% each.
2.39 The fewest number of audits is 12 and the most is 42. More companies (8)
performed 27 audits than any other number. Thirty-five companies performed
between 12 and 19 audits. Only 7 companies performed 40 or more audits.
2.40 There were relatively constant sales from January through August ($4 to 6 million).
Each month from September through December sales increased with December
having the sharpest increase ($15 million in sales in December).