Slides by
JOHN
LOUCKS
St. Edward’s
University
© 2009 Thomson South-Western. All Rights Reserved
Slide 1
Chapter 2, Part B
Descriptive Statistics:
Tabular and Graphical Presentations


Exploratory Data Analysis: Stem-and-Leaf Display
Crosstabulations and
y
Scatter Diagrams
x
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Slide 2
Exploratory Data Analysis
 The techniques of exploratory data analysis consist of
simple arithmetic and easy-to-draw pictures that can
be used to summarize data quickly.
 One such technique is the stem-and-leaf display.
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Slide 3
Stem-and-Leaf Display
 A stem-and-leaf display shows both the rank order
and shape of the distribution of the data.
 It is similar to a histogram on its side, but it has the
advantage of showing the actual data values.
 The first digits of each data item are arranged to the
left of a vertical line.
 To the right of the vertical line we record the last
digit for each item in rank order.
 Each line in the display is referred to as a stem.
 Each digit on a stem is a leaf.
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Slide 4
Example: Hudson Auto Repair
The manager of Hudson Auto would like to have a
better understanding of the cost of parts used in the
engine tune-ups performed in the shop. She examines
50 customer invoices for tune-ups. The costs of parts,
rounded to the nearest dollar, are listed on the next
slide.
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Slide 5
Example: Hudson Auto Repair

Sample of Parts Cost ($) for 50 Tune-ups
91
71
104
85
62
78
69
74
97
82
93
72
62
88
98
57
89
68
68
101
75
66
97
83
79
52
75
105
68
105
99
79
77
71
79
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80
75
65
69
69
97
72
80
67
62
62
76
109
74
73
Slide 6
Stem-and-Leaf Display
5
6
7
8
9
10
a stem
2
2
1
0
1
1
7
2
1
0
3
4
2
2
2
7
5
2
2
3
7
5
5
3
5
7
9
6
4
8
8
7 8 8 8 9 9 9
4 5 5 5 6 7 8 9 9 9
9
9
a leaf
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Slide 7
Stretched Stem-and-Leaf Display
 If we believe the original stem-and-leaf display has
condensed the data too much, we can stretch the
display by using two stems for each leading digit(s).
 Whenever a stem value is stated twice, the first value
corresponds to leaf values of 0 - 4, and the second
value corresponds to leaf values of 5 - 9.
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Slide 8
Stretched Stem-and-Leaf Display
5
5
6
6
7
7
8
8
9
9
10
10
2
7
2
5
1
5
0
5
1
7
1
5
2
6
1
5
0
8
3
7
4
5
2
7
2
5
2
9
2
8 8 8 9 9 9
2 3 4 4
6 7 8 9 9 9
3
7 8 9
9
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Slide 9
Stem-and-Leaf Display

Leaf Units
• A single digit is used to define each leaf.
• In the preceding example, the leaf unit was 1.
• Leaf units may be 100, 10, 1, 0.1, and so on.
• Where the leaf unit is not shown, it is assumed
to equal 1.
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Slide 10
Example: Leaf Unit = 0.1
If we have data with values such as
8.6
11.7
9.4
9.1
10.2
11.0
8.8
a stem-and-leaf display of these data will be
Leaf Unit = 0.1
8 6 8
9 1 4
10 2
11 0 7
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Slide 11
Example: Leaf Unit = 10
If we have data with values such as
1806 1717
1974
1791
1682
1910
1838
a stem-and-leaf display of these data will be
Leaf Unit = 10
16 8
17 1 9
18 0 3
19 1 7
The 82 in 1682
is rounded down
to 80 and is
represented as an 8.
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Slide 12
Crosstabulations and Scatter Diagrams
 Thus far we have focused on presentations that are
used to summarize the data for one variable at a time.
 Often a manager is interested in presentations that
will help understand the relationship between two
variables.
 Crosstabulation and a scatter diagram are two
methods for summarizing the data for two variables
simultaneously.
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Slide 13
Crosstabulation
 A crosstabulation is a tabular summary of data for
two variables.

Crosstabulation can be used when:
• one variable is qualitative and the other is
quantitative,
• both variables are qualitative, or
• both variables are quantitative.
 The left and top margin labels define the classes for
the two variables.
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Slide 14
Crosstabulation

Example: Finger Lakes Homes
The number of Finger Lakes homes sold for each
style and price for the past two years is shown below.
Price
Range
< $99,000
> $99,000
Total
quantitative
qualitative
variable
variable
Home Style
Colonial Log Split A-Frame Total
18
12
6
14
19
16
12
3
55
30
20
35
15
100
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45
Slide 15
Crosstabulation

Insights Gained from Preceding Crosstabulation
•
The greatest number of homes (19) in the sample
are a split-level style and priced at less than or
equal to $99,000.
•
Only three homes in the sample are an A-Frame
style and priced at more than $99,000.
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Slide 16
Crosstabulation
Frequency distribution
for the price variable
Home Style
Log Split A-Frame
Price
Range
Colonial
< $99,000
> $99,000
18
12
6
14
19
16
12
3
55
30
20
35
15
100
Total
Total
45
Frequency distribution
for the home style variable
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Slide 17
Crosstabulation: Row or Column Percentages

Converting the entries in the table into row
percentages or column percentages can provide
additional insight about the relationship between
the two variables.
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Slide 18
Crosstabulation: Row Percentages
Price
Range
Colonial
< $99,000
> $99,000
32.73
26.67
Home Style
Log Split A-Frame
10.91
31.11
34.55
35.56
21.82
6.67
Total
100
100
Note: row totals are actually 100.01 due to rounding.
(Colonial and > $99K)/(All >$99K) x 100 = (12/45) x 100
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Slide 19
Crosstabulation: Column Percentages
Price
Range
Colonial
< $99,000
> $99,000
60.00
40.00
30.00
70.00
54.29
45.71
80.00
20.00
Total
100
100
100
100
Home Style
Log Split A-Frame
(Colonial and > $99K)/(All Colonial) x 100 = (12/30) x 100
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Slide 20
Crosstabulation: Simpson’s Paradox
 Data in two or more crosstabulations are often
aggregated to produce a summary crosstabulation.
 We must be careful in drawing conclusions about the
relationship between the two variables in the
aggregated crosstabulation.
 Simpson’ Paradox: In some cases the conclusions
based upon an aggregated crosstabulation can be
completely reversed if we look at the unaggregated
data.
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Slide 21
Scatter Diagram and Trendline
 A scatter diagram is a graphical presentation of the
relationship between two quantitative variables.
 One variable is shown on the horizontal axis and the
other variable is shown on the vertical axis.
 The general pattern of the plotted points suggests the
overall relationship between the variables.
 A trendline is an approximation of the relationship.
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Slide 22
Scatter Diagram and Trendline

A Positive Relationship
y
x
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Slide 23
Scatter Diagram and Trendline

A Negative Relationship
y
x
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Slide 24
Scatter Diagram and Trendline

No Apparent Relationship
y
x
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Slide 25
Example: Panthers Football Team

Scatter Diagram and Trendline
The Panthers football team is interested in
investigating the relationship, if any, between
interceptions made and points scored.
x = Number of
Interceptions
1
3
2
1
3
y = Number of
Points Scored
14
24
18
17
30
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Slide 26
Scatter Diagram and Trendline
Number of Points Scored
y
35
30
25
20
15
10
5
0
0
1
x
2
3
4
Number of Interceptions
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Slide 27
Example: Panthers Football Team

Insights Gained from the Preceding Scatter Diagram
•
The scatter diagram and trendline indicate a
positive relationship between the number of
interceptions and the number of points scored.
•
Higher points scored are associated with a higher
number of interceptions.
•
The relationship is not perfect; all plotted points in
the scatter diagram are not on a straight line.
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Slide 28
Tabular and Graphical Procedures
Data
Qualitative Data
Quantitative Data
Tabular
Methods
Graphical
Methods
Tabular
Methods
• Frequency
Distribution
• Relative Freq.
Distribution
• Percent Freq.
Distribution
• Crosstabulation
• Bar Graph
• Pie Chart
• Frequency Dist.
• Rel. Freq. Dist.
• % Freq. Dist.
• Cum. Freq. Dist.
• Cum. Rel. Freq.
Distribution
• Cum. % Freq.
Distribution
• Crosstabulation
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Graphical
Methods
• Dot Plot
• Histogram
• Ogive
• Stem-andLeaf Display
• Scatter
Diagram
Slide 29
End of Chapter 2, Part B
© 2009 Thomson South-Western. All Rights Reserved
Slide 30