Student’s Solutions Manual and Study Guide: Chapter 2
Page 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. Explain the difference between grouped and un- grouped data and construct a
frequency distribution from a set of data and explain what the distribution
represents.
2. Describe and construct different types of quantitative data graphs, including
histograms, frequency polygons, ogives, and stem and leaf plots. Explain when
these graphs should be used.
3. Describe and construct different types of qualitative data graphs, including pie
charts, bar charts, and Pareto charts. Explain when these graphs should be used.
4. Display and analyze two variables simultaneously using cross tabulation
and scatter plots.
CHAPTER OUTLINE
2.1
Frequency Distributions
Class Midpoint
Relative Frequency
Cumulative Frequency
2.2
Quantitative Data Graphs
Histograms
Frequency Polygons
Ogives
Stem and Leaf Plots
2.3
Qualitative Data Graphs
Pie Charts
Bar Charts
Pareto Charts
2.4
Charts and Graphs for Two Variables
Cross Tabulation
Scatter Plot
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 2
KEY TERMS
Bar Chart or Graph
Class Midpoint or Mark
Cross Tabulation
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
STUDY QUESTIONS
1. The following data represents the number of printer ribbons used annually in a company by
twenty-eight departments. This is an example of ______________ data.
8 4 5 10 6 5 4 6 3 4 4 6 1 12
2 11 2 5 3 2 6 7 6 12 7 1 8 9
2. Below is a frequency distribution of ages of managers with a large retail firm. This is an
example of _______________ data.
Age
20-29
30-39
40-49
50-59
over 60
f
11
32
57
43
18
3. For best results, a frequency distribution should have between _____ and _____ classes.
4. The difference between the largest and smallest numbers is called the _______________.
5. Consider the values below. In constructing a frequency distribution, the beginning point
of the lowest class should be at least as small as _____ and the endpoint of the highest
class should be at least as large as _____.
27 21 8 10 9 16 11 12 21 11 29 19 17 22 28 28 29 19 18 26 17 34 19 16 20
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
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6. The class midpoint can be determined by _______________.
7-9 Examine the frequency distribution below:
class
5-under 10
10-under 15
15-under 20
20-under 25
25-under 30
30-under 35
frequency
56
43
21
11
12
8
7. The relative frequency for the class 15-under 20 is _______________.
8. The cumulative frequency for the class 20-under 25 is _______________.
9. The midpoint for the class 25-under 30 is ___________.
10. The graphical depiction that is a type of vertical bar chart and is used to depict a frequency
distribution is a _______________.
11. The graphical depiction that utilizes cumulative frequencies is a _______________.
12. The graph shown below is an example of a _______________.
13. Consider the categories below and their relative amounts:
Category
A
B
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Amount
112
319
Student’s Solutions Manual and Study Guide: Chapter 2
C
D
E
Page 4
57
148
202
If you were to construct a Pie Chart to depict these categories, then you would allot
_______________ degrees to category D.
14. A graph that is especially useful for observing the overall shape of the distribution of
data points along with identifying data values or intervals for which there are
groupings and gaps in the data is called a ______________________.
15. Given the values below, construct a stem and leaf plot using two digits for the stem.
346 340 322 339 342 332 338
357 328 329 346 341 321 332
16. A vertical bar chart that displays the most common types of defects that occur with a product,
ranked in order from left to right, is called a __________________.
17. A two-dimensional plot of pairs of points often used to examine the relationship of two
numerical variables is called a _________________.
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 5
ANSWERS TO STUDY QUESTIONS
1. Raw or Ungrouped
10. Histogram
2. Grouped
11. Ogive
3. 5, 15
12. Frequency Polygon
4. Range
13. 148/838 of 360o = 63.6o
5. 8, 34
14. Dot Plot
6. Averaging the two class endpoints
15. 32
33
34
35
7. 21/151 = .1391
1 2 8 9
2 2 8 9
0 1 2 6 6
7
8. 131
16. Pareto Chart
9. 27.5
17. Scatter Plot
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 6
SOLUTIONS TO THE ODD-NUMBERED PROBLEMS IN CHAPTER 2
2.1
a)
One possible 5 class frequency distribution:
Class Interval
Frequency
–15 - under –6
7
–6 - under 3
12
3 - under 12
13
12 - under 21
9
21 - under 30
9
Totals
50
b) One possible 10 class frequency distribution:
Class Interval
–15 - under –10
–10 - under –5
–5 - under 0
0 - under 5
5 - under 10
10 - under 15
15 - under 20
20 - under 25
25 - under 30
30 - under 35
Totals
Frequency
2
5
7
10
7
3
7
4
5
0
50
c) The ten class frequency distribution gives a more detailed breakdown of
temperatures. It allows locating more accurately the temperatures with the greatest
frequency. The temperatures with the highest frequency, 10, are in the 0 – under 5
class. The five class distribution collapses the intervals into broader classes making
it appear that there are nearly equal frequencies in each class.
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 7
2.3
Class
Interval
Frequency
0-5
6
5 - 10
8
10 - 15
17
15 - 20
23
20 - 25
18
25 - 30
10
30 - 35
4
TOTAL
86
Class
Midpoint
2.5
7.5
12.5
17.5
22.5
27.5
32.5
Relative
Frequency
6/86 = .0698
.0930
.1977
.2674
.2093
.1163
.0465
1.0000
Cumulative
Frequency
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.
2.5
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.
2.7 Histogram:
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 8
Frequency Polygon:
Comment: The histogram indicates that the number of calls per shift varies widely.
However, the heavy numbers of calls per shift fall in the 50 to 80 range. Since these
numbers occur quite frequently, staffing planning should be done with these number of
calls in mind realizing from the rest of the graph that there may be shifts with as few as
10 to 20 calls.
2.9
STEM
21
22
23
24
25
26
27
LEAF
2
0
0
0
0
0
0
8
1
0
0
3
1
1
8
2
4
3
4
1
3
9
4
5
6
5
2
6
8
9
5
3
6
8
9
7
3
7
9
9
7
5
9 9
9 9 9
8 9
6
The stem and leaf plot indicates that sales prices vary quite a bit within the range of
$212,000 and $273,000. It is evident from the stem and leaf plot that there is a
strong grouping of prices in the five price ranges from the $220’s through the $260’s.
2.11 The histogram shows that there is only one airport with more than 70 million
passengers and from the given problem information, we know that that airport is
Atlanta’s Hartsfield-Jackson International Airport which has more than 90 million
passengers. There are no airports with 70 to 90 million passengers. Nearly one-half
(14) of the top 30 airports have between 30 and 40 million passengers. The next
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 9
largest grouping is between 50 and 60 million passengers in which there are six
airports.
2.13
Airlines
Delta
United
Southwest
American
China Southern
Ryanair
Lufthansa
Totals
Number of passengers
(in millions)
164.6
140.4
134.0
107.9
86.5
79.6
74.7
787.7
Proportion
164.6/787.7= 0.209
0.178
0.170
0.137
0.110
0.101
0.095
1.000
Pie chart:
Lufthansa, 74.7,
9%
Delta , 164.6, 21%
Ryanair, 79.6, 10%
China Southern, 86.5,
11%
United , 140.4, 18%
American, 107.9, 14%
Southwest , 134.0,
17%
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Degrees
75
64
61
49
40
36
34
360
Student’s Solutions Manual and Study Guide: Chapter 2
Page 10
Bar chart:
180.0
164.6
160.0
140.4
Number of passengers (in millions)
140.0
134.0
120.0
107.9
100.0
86.5
79.6
80.0
74.7
60.0
40.0
20.0
0.0
Delta
United
Southwest
American
China
Southern
Ryanair
Lufthansa
Airlines
2.15
Underwriting Firm
BMO Capital Markets
RBC Capital Markets
TD Securities
Scotia Capital
National Bank
Financial
Totals
Bar chart:
Gross Proceeds
Proportion
($ millions)
1,668/3,646=0.457
1,668
0.187
682
0.137
500
0.118
431
365
3,646
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Degrees
165
67
49
43
0.100
36
1.000
360
Student’s Solutions Manual and Study Guide: Chapter 2
Page 11
Pie chart:
The proportion, sizes and color of the pie slices clearly shows that BMO Capital Markets
has the highest revenue ($1,668 million, 45.7%) and National Bank Financial has the
lowest revenue ($365 million, 10.0%).
2.17
Complaint
Busy Signal
Too long a Wait
Could not get through
Got Disconnected
Transferred to the Wrong Person
Poor Connection
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Number
% of Total
420
184
85
37
10
8
56.45
24.73
11.42
4.97
1.34
1.08
Student’s Solutions Manual and Study Guide: Chapter 2
Page 12
Total
744
99.99
800
700
600
500
400
300
200
100
0
100
80
60
40
20
C1
sy
Bu
l
na
is g
o
To
Co
Count
Percent
Cum %
d
ul
420
56.5
56.5
ng
lo
a
e
tg
o
n
w
t
ai
hr
tt
gh
ou
to
an
t
en
g
a
t
Go
T
184
24.7
81.2
2.19
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
e
nn
o
sc
di
rr
fe
s
n
ra
85
11.4
92.6
ed
ed
ct
to
e
th
w
37
5.0
97.6
ng
ro
n
so
r
pe
n
tio
c
ne
on
c
or
Po
10
1.3
98.9
8
1.1
100.0
0
Percent
Count
Customer Complaints
Student’s Solutions Manual and Study Guide: Chapter 2
Page 13
Generally speaking, the tendency is that sales are higher when more money is spent on
advertising.
2.21
Number of Annual
Non-vacation -Day
Absences
0-2
3-5
More than 5
Total
One-Way Commute Distance (in km)
0-3
4 - 10
More than 10
95
184
117
21
40
53
3
7
12
119
231
182
Total
396
114
22
532
There is a slight tendency for there to be a few more absences as plant workers commute
further distances. Say, 6.6% of those who commute more than 10 km had more than 5 nonvacation absent days, as compared to 2.5% and 3% for those who commute 0-3 km and 410 km respectively. Comparing workers who travel 4-10 km to those who travel 0-3 km,
there is about a 2:1 ratio in all three cells (0-2, 3-5, more than 5 non-vacation day absences)
indicating that for these two categories (0-3 and 4-10), number of absences is essentially
independent of commute distance.
2.23
2.25
Class Interval
Frequencies
16 - under 23
23 - under 30
30 - under 37
37 - under 44
44 - under 51
51 - under 58
TOTAL
6
9
4
4
4
3
30
Class Interval
Frequencies
50 - under 60
60 - under 70
70 - under 80
80 - under 90
90 - under 100
TOTAL
13
27
43
31
9
123
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Histogram:
Frequency Polygon:
Ogive:
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Page 14
Student’s Solutions Manual and Study Guide: Chapter 2
2.27
Label
A
B
C
D
TOTAL
Value
55
121
83
46
305
Page 15
Proportion
.180
.397
.272
.151
1.000
Pie Chart:
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Degrees
65
143
98
54
360
Student’s Solutions Manual and Study Guide: Chapter 2
2.29
Problem
1
2
3
4
5
6
7
8
9
10
Page 16
Frequency
673
29
108
379
73
564
12
402
54
202
2496
Pareto Chart:
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Percent of Total
26.96
1.16
4.33
15.18
2.92
22.60
0.48
16.11
2.16
8.09
Student’s Solutions Manual and Study Guide: Chapter 2
Page 17
2.31 Yellowknife Steel Company
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
The highest frequencies are between 42 and 57.
2.33
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
Frequency
2
3
9
7
12
9
6
2
50
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 18
Histogram:
Frequency Polygon:
The normal distribution appears to peak near the center and diminish towards the
end intervals.
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 19
2.35
Asking Price
$ 80,000 - under $ 100,000
$ 100,000 - under $ 120,000
$ 120,000 - under $ 140,000
$ 140,000 - under $ 160,000
$ 160,000 - under $ 180,000
$ 180,000 - under $ 200,000
Frequency
21
27
18
11
6
3
86
Cumulative
Frequency
21
48
66
77
83
86
Histogram:
Frequency Polygon:
30
25
20
y
c
n
e 15
u
q
e
r
F
10
5
0
90,000
110,000
130,000
Class Midpoints
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
150,000
170,000
190,000
Student’s Solutions Manual and Study Guide: Chapter 2
Page 20
Ogive:
100
90
80
70
Frequencies
60
50
40
30
20
10
0
80,000
100,000
120,000
140,000
160,000
180,000
Class Endpoints
2.37
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
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Cumulative
Frequency
9
23
40
56
74
82
87
200,000
Student’s Solutions Manual and Study Guide: Chapter 2
Histogram:
Frequency Polygon:
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Student’s Solutions Manual and Study Guide: Chapter 2
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Ogive:
2.39
90
Canadian Travellers to the U.S.
80
70
60
50
40
30
20
10
0
0.000
0.100
0.200
0.300
0.400
0.500
0.600
Exchange Rate
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
0.700
0.800
0.900
1.000
Student’s Solutions Manual and Study Guide: Chapter 2
Page 23
50
Travellers to Canada from U.S.
45
40
35
30
25
20
15
10
5
0
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
Exchange Rate
The higher is the exchange rate more Canadian travellers go to the U.S. and less the U.S.
travellers do to Canada.
2.41
The Pareto chart indicates that faulty plastic causes 44.2% of the defects and becomes the
major problem. According to the chart, 23.4% of the plastic bottles were rejected because
of incorrect thickness which can be identified as the second severe problem. The steepest
slopes correspond to “fault in plastic”, “thickness”, and “broken handle” categories. They
represent 84.8% causes of poor-quality bottles.
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 24
500
100
400
80
300
60
200
40
100
20
0
C1
Count
Percent
Cum %
Fault in plastic
221
44.2
44.2
Thickness Broken handle
117
86
23.4
17.2
67.6
84.8
Labeling
44
8.8
93.6
Discoloration
32
6.4
100.0
0
2.43
STEM
92
93
94
95
96
97
98
99
100
2.45
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
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.
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Percent
Count
Causes of Poor Quality Bottles
Student’s Solutions Manual and Study Guide: Chapter 2
2.47
Page 25
There were relatively constant sales from January through October (about $6 million on
average). In November and December sales dramatically increased with December
having the sharpest increase ($30 million in sales).
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition
Student’s Solutions Manual and Study Guide: Chapter 2
Page 26
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