This document provides an overview of methods for visualizing data, including: - Constructing frequency distributions to summarize ungrouped data by organizing it into class intervals and frequencies. - Calculating class midpoints, relative frequencies, and cumulative frequencies for frequency distributions. - Common statistical graphs like histograms, frequency polygons, ogives, pie charts, stem-and-leaf plots, Pareto charts, and scatter plots that can be used to visualize grouped or ungrouped data.

1. Copyright 2010 John Wiley & Sons, Inc. 1
Copyright 2010 John Wiley & Sons, Inc.
Business Statistics, 6th ed.
by Ken Black
Chapter 2
Visualizing the Data

2. Copyright 2010 John Wiley & Sons, Inc. 2
Learning Objectives
Recognize the difference between grouped and
ungrouped data
Construct a frequency distribution
Construct a histogram, a frequency polygon, an ogive,
a pie chart, a stem and leaf plot, a Pareto chart, and a
scatter plot

3. Copyright 2010 John Wiley & Sons, Inc. 3
Ungrouped data
have not been summarized in any way
are also called raw data
Grouped data
have been organized into a frequency distribution
Ungrouped Versus Grouped Data

4. Copyright 2010 John Wiley & Sons, Inc. 4
42
30
53
50
52
30
55
49
61
74
26
58
40
40
28
36
30
33
31
37
32
37
30
32
23
32
58
43
30
29
34
50
47
31
35
26
64
46
40
43
57
30
49
40
25
50
52
32
60
54
Ages of a Sample of
Managers from
Urban Child Care
Centers in the
United States
Example of Ungrouped Data

5. Copyright 2010 John Wiley & Sons, Inc. 5
Frequency Distribution – summary of data presented
in the form of class intervals and frequencies
Vary in shape and design
Constructed according to the individual researcher's
preferences
Frequency Distribution

6. Copyright 2010 John Wiley & Sons, Inc. 6
Steps in Frequency Distribution
Step 1 - Determine range of frequency distribution
Range is the difference between the high and the lowest
numbers
Step 2 – determine the number of classes
Don’t use too many, or two few classes
Step 3 – Determine the width of the class interval
Approx class width can be calculated by dividing the range
by the number of classes
Values fit into only one class
Frequency Distribution

7. Copyright 2010 John Wiley & Sons, Inc. 7
Class Interval Frequency
20-under 30 6
30-under 40 18
40-under 50 11
50-under 60 11
60-under 70 3
70-under 80 1
Frequency Distribution of Child
Care Manager’s Ages

8. Copyright 2010 John Wiley & Sons, Inc. 8
42
30
53
50
52
30
55
49
61
74
26
58
40
40
28
36
30
33
31
37
32
37
30
32
23
32
58
43
30
29
34
50
47
31
35
26
64
46
40
43
57
30
49
40
25
50
52
32
60
54
Smallest
Largest
Range = Largest - Smallest
= 74 - 23
= 51
Data Range

9. Copyright 2010 John Wiley & Sons, Inc. 9
10=WidthClass
8.5=
6
51
=WidthClasseApproximat
Number of Classes and Class Width
The number of classes should be between 5 and 15.
Fewer than 5 classes cause excessive summarization.
More than 15 classes leave too much detail.
Class Width
Divide the range by the number of classes for an
approximate class width
Round up to a convenient number

10. Copyright 2010 John Wiley & Sons, Inc. 10
Class Midpoint =
beginning class endpoint + ending class endpoint
2
=
30 + 40
2
= 35
( )
Class Midpoint = class beginning point +
1
2
class width
= 30 +
1
2
10
= 35
The midpoint of each class interval is called the
class midpoint or the class mark.
Class Midpoint

11. Copyright 2010 John Wiley & Sons, Inc. 11
Relative
Class Interval Frequency Frequency
20-under 30 6 .12
30-under 40 18 .36
40-under 50 11 .22
50-under 60 11 .22
60-under 70 3 .06
70-under 80 1 .02
Total 50 1.00
6
50
=
18
50
=
The relative frequency is the proportion of the total frequency
that is any given class interval in a frequency distribution.
Relative Frequency

12. Copyright 2010 John Wiley & Sons, Inc. 12
The cumulative frequency is a running total of frequencies
through the classes of a frequency distribution.
Cumulative Frequency
Cumulative
Class Interval Frequency Frequency
20-under 30 6 6
30-under 40 18 24
40-under 50 11 35
50-under 60 11 46
60-under 70 3 49
70-under 80 1 50
Total 50
18 + 6
11 + 24

13. Copyright 2010 John Wiley & Sons, Inc. 13
Class midpoint – The midpoint of each class interval
Midpoint is half way across the class interval
Midpoint is the average of the class end points
Class Midpoint

14. Copyright 2010 John Wiley & Sons, Inc. 14
Relative frequency – is the proportion of the total
frequency that is in any given class interval in a
frequency distribution
Relative frequency = individual class frequency divided
by the total frequency
Example => Frequency/Total = 16/40 = .40
Probability of occurrence
Relative Frequency

15. Copyright 2010 John Wiley & Sons, Inc. 15
Cumulative frequency – the running total of
frequencies through the classes of a frequency
distribution
Cumulative frequency for each class is the frequency for
that class interval added to the preceding cumulative total
At the last interval, the cumulative total equals the sum of
the frequencies
Cumulative Frequency

16. Copyright 2010 John Wiley & Sons, Inc. 16
Relative Cumulative
Class Interval Frequency Midpoint Frequency Frequency
20-under 30 6 25 .12 6
30-under 40 18 35 .36 24
40-under 50 11 45 .22 35
50-under 60 11 55 .22 46
60-under 70 3 65 .06 49
70-under 80 1 75 .02 50
Total 50 1.00
Class Midpoints, Relative Frequencies,
and Cumulative Frequencies

17. Copyright 2010 John Wiley & Sons, Inc. 17
Cumulative
Relative Cumulative Relative
Class Interval Frequency Frequency Frequency Frequency
20-under 30 6 .12 6 .12
30-under 40 18 .36 24 .48
40-under 50 11 .22 35 .70
50-under 60 11 .22 46 .92
60-under 70 3 .06 49 .98
70-under 80 1 .02 50 1.00
Total 50 1.00
The cumulative relative frequency is a running total of the
relative frequencies through the classes of a frequency
distribution.
Cumulative Relative Frequencies

18. Copyright 2010 John Wiley & Sons, Inc. 18
Common Statistical Graphs
Histogram -- vertical bar chart of frequencies
Frequency Polygon -- line graph of frequencies
Ogive -- line graph of cumulative frequencies
Pie Chart -- proportional representation for
categories of a whole

19. Copyright 2010 John Wiley & Sons, Inc. 19
Stem and Leaf Plot -- display is a graphical method
of displaying data. It is particularly useful when your
data are not too numerous.
Pareto Chart -- type of chart which contains both bars
and a line graph.
The bars display the values in descending order, and the
line graph shows the cumulative totals of each category,
left to right.
The purpose is to highlight the most important among a
(typically large) set of factors.
Common Statistical Graphs

20. Copyright 2010 John Wiley & Sons, Inc. 20
Scatter Plot -- type of display using Cartesian
coordinates to display values for two variables for a
set of data.
The data is displayed as a collection of points, each having
the value of one variable determining the position on the
horizontal axis and the value of the other variable
determining the position on the vertical axis.
A scatter plot is also called a scatter chart, scatter diagram
and scatter graph.
Common Statistical Graphs

21. Copyright 2010 John Wiley & Sons, Inc. 21
Class IntervalFrequency
20-under 30 6
30-under 40 18
40-under 50 11
50-under 60 11
60-under 70 3
70-under 80 1
Histogram
01020
0 10 20 30 40 50 60 70 80
Years
Frequency

22. Copyright 2010 John Wiley & Sons, Inc. 22
Class IntervalFrequency
20-under 30 6
30-under 40 18
40-under 50 11
50-under 60 11
60-under 70 3
70-under 80 1
01020
0 10 20 30 40 50 60 70 80
Years
Frequency
Histogram Construction

23. Copyright 2010 John Wiley & Sons, Inc. 23
Class IntervalFrequency
20-under 30 6
30-under 40 18
40-under 50 11
50-under 60 11
60-under 70 3
70-under 80 1 01020
0 10 20 30 40 50 60 70 80
Years
Frequency
Frequency Polygon

24. Copyright 2010 John Wiley & Sons, Inc. 24
Cumulative
Class Interval Frequency
20-under 30 6
30-under 40 24
40-under 50 35
50-under 60 46
60-under 70 49
70-under 80 50
0204060
0 10 20 30 40 50 60 70 80
Years
Frequency
Ogive

25. Copyright 2010 John Wiley & Sons, Inc. 25
Cumulative
Relative
Class Interval Frequency
20-under 30 .12
30-under 40 .48
40-under 50 .70
50-under 60 .92
60-under 70 .98
70-under 80 1.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 10 20 30 40 50 60 70 80
Years
CumulativeRelativeFrequency
Relative Frequency Ogive

26. Copyright 2010 John Wiley & Sons, Inc. 26
COMPLAINT NUMBER PROPORTION DEGREES
Stations, etc. 28,000 .40 144.0
Train
Performance
14,700 .21 75.6
Equipment 10,500 .15 50.4
Personnel 9,800 .14 50.6
Schedules,
etc.
7,000 .10 36.0
Total 70,000 1.00 360.0
Complaints by Amtrak Passengers

27. Copyright 2010 John Wiley & Sons, Inc. 27
Complaints by Amtrak Passengers

28. Copyright 2010 John Wiley & Sons, Inc. 28
Second Quarter U.S. Truck Production
Second Quarter Truck
Production in the U.S.
(Hypothetical values)
2d Quarter
Truck
Production
Company
A
B
C
D
E
Totals
357,411
354,936
160,997
34,099
12,747
920,190

29. Copyright 2010 John Wiley & Sons, Inc. 29
Second Quarter U.S. Truck Production
39%
17%
4%
1%
39%
A B C D E

30. Copyright 2010 John Wiley & Sons, Inc. 30
2d Quarter
Truck
Production
Proportion DegreesCompany
A
B
C
D
E
Totals
357,411
354,936
160,997
34,099
12,747
920,190
.388
.386
.175
.037
.014
1.000
140
139
63
13
5
360
357,411
920,190
=
.388 360=
Pie Chart Calculations for Company A

31. Copyright 2010 John Wiley & Sons, Inc. 31
Safety Examination Scores
for Plant Trainees
86
76
23
77
81
79
68
77
92
59
68
75
83
49
91
47
72
82
74
70
56
60
88
75
97
39
78
94
55
67
83
89
67
91
81
Raw Data Stem
2
3
4
5
6
7
8
9
Leaf
3
9
7 9
5 6 9
0 7 7 8 8
0 2 4 5 5 6 7 7 8 9
1 1 2 3 3 6 8 9
1 1 2 4 7

32. Copyright 2010 John Wiley & Sons, Inc. 32
Construction of Stem and Leaf Plot
86
76
23
77
81
79
68
77
92
59
68
75
83
49
91
47
72
82
74
70
56
60
88
75
97
39
78
94
55
67
83
89
67
91
81
Raw Data Stem
2
3
4
5
6
7
8
9
Leaf
3
9
7 9
5 6 9
0 7 7 8 8
0 2 4 5 5 6 7 7 8 9
1 1 2 3 3 6 8 9
1 1 2 4 7
Stem
Leaf
Stem
Leaf

33. Copyright 2010 John Wiley & Sons, Inc. 33
Pareto Chart
0
10
20
30
40
50
60
70
80
90
100
Poor
Wiring
Short in
Coil
Defective
Plug
Other
Frequency
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%

34. Copyright 2010 John Wiley & Sons, Inc. 34
Registered
Vehicles
(1000's)
Gasoline Sales
(1000's of
Gallons)
5 60
15 120
9 90
15 140
7 60
0
100
200
0 5 10 15 20
RegisteredVehicles
GasolineSales
Scatter Plot