© 2003 Thomson/South-Western
Slide 1
Chapter 2
Descriptive Statistics:
Tabular and Graphical Methods
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Summarizing Qualitative Data
Summarizing Quantitative Data
Exploratory Data Analysis
Crosstabulations
and Scatter Diagrams
© 2003 Thomson/South-Western
Slide 2
Summarizing Qualitative Data
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Frequency Distribution
Relative Frequency
Percent Frequency Distribution
Bar Graph
Pie Chart
© 2003 Thomson/South-Western
Slide 3
Frequency Distribution
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A frequency distribution is a tabular summary of
data showing the frequency (or number) of items in
each of several nonoverlapping classes.
The objective is to provide insights about the data
that cannot be quickly obtained by looking only at
the original data.
© 2003 Thomson/South-Western
Slide 4
Example: Marada Inn
Guests staying at Marada Inn were asked to rate the
quality of their accommodations as being excellent,
above average, average, below average, or poor. The
ratings provided by a sample of 20 quests are shown
below.
Below Average Average
Above Average Above Average
Above Average Below Average
Average
Poor
Above Average Excellent
Average
Above Average
Above Average Average
© 2003 Thomson/South-Western
Above Average
Above Average
Below Average
Poor
Above Average
Average
Slide 5
Example: Marada Inn
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Frequency Distribution
Rating
Frequency
Poor
2
Below Average
3
Average
5
Above Average
9
Excellent
1
Total
20
© 2003 Thomson/South-Western
Slide 6
Relative Frequency Distribution
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The relative frequency of a class is the fraction or
proportion of the total number of data items
belonging to the class.
A relative frequency distribution is a tabular
summary of a set of data showing the relative
frequency for each class.
© 2003 Thomson/South-Western
Slide 7
Percent Frequency Distribution
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The percent frequency of a class is the relative
frequency multiplied by 100.
A percent frequency distribution is a tabular
summary of a set of data showing the percent
frequency for each class.
© 2003 Thomson/South-Western
Slide 8
Example: Marada Inn
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Relative Frequency and Percent Frequency
Distributions
Rating
Relative
Percent
Frequency Frequency
Poor
Below Average
Average
Above Average
Excellent
Total
© 2003 Thomson/South-Western
.10
.15
.25
.45
.05
1.00
10
15
25
45
5
100
Slide 9
Bar Graph
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A bar graph is a graphical device for depicting
qualitative data.
On the horizontal axis we specify the labels that are
used for each of the classes.
A frequency, relative frequency, or percent frequency
scale can be used for the vertical axis.
Using a bar of fixed width drawn above each class
label, we extend the height appropriately.
The bars are separated to emphasize the fact that
each class is a separate category.
© 2003 Thomson/South-Western
Slide 10
Example: Marada Inn
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Bar Graph
9
Frequency
8
7
6
5
4
3
2
1
Poor
Below Average Above Excellent
Average
Average
© 2003 Thomson/South-Western
Rating
Slide 11
Pie Chart
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The pie chart is a commonly used graphical device
for presenting relative frequency distributions for
qualitative data.
First draw a circle; then use the relative frequencies
to subdivide the circle into sectors that correspond to
the relative frequency for each class.
Since there are 360 degrees in a circle, a class with a
relative frequency of .25 would consume .25(360) =
90 degrees of the circle.
© 2003 Thomson/South-Western
Slide 12
Example: Marada Inn
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Pie Chart
Exc.
Poor
5%
10%
Above
Average
45%
Below
Average
15%
Average
25%
Quality Ratings
© 2003 Thomson/South-Western
Slide 13
Example: Marada Inn
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Insights Gained from the Preceding Pie Chart
• One-half of the customers surveyed gave Marada
a quality rating of “above average” or “excellent”
(looking at the left side of the pie). This might
please the manager.
• For each customer who gave an “excellent” rating,
there were two customers who gave a “poor”
rating (looking at the top of the pie). This should
displease the manager.
© 2003 Thomson/South-Western
Slide 14
Summarizing Quantitative Data
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Frequency Distribution
Relative Frequency and Percent Frequency
Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive
© 2003 Thomson/South-Western
Slide 15
Example: Hudson Auto Repair
The manager of Hudson Auto would like to get a
better picture of the distribution of costs for engine
tune-up parts. A sample of 50 customer invoices has
been taken and the costs of parts, rounded to the
nearest dollar, are listed below.
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
© 2003 Thomson/South-Western
52
75
105
68
105
99
79
77
71
79
80
75
65
69
69
97
72
80
67
62
62
76
109
74
73
Slide 16
Frequency Distribution
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Guidelines for Selecting Number of Classes
• Use between 5 and 20 classes.
• Data sets with a larger number of elements
usually require a larger number of classes.
• Smaller data sets usually require fewer classes.
© 2003 Thomson/South-Western
Slide 17
Frequency Distribution
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Guidelines for Selecting Width of Classes
• Use classes of equal width.
• Approximate Class Width =
Largest Data Value  Smallest Data Value
Number of Classes
© 2003 Thomson/South-Western
Slide 18
Example: Hudson Auto Repair
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Frequency Distribution
If we choose six classes:
Approximate Class Width = (109 - 52)/6 = 9.5 10
Cost ($)
50-59
60-69
70-79
80-89
90-99
100-109
© 2003 Thomson/South-Western
Frequency
2
13
16
7
7
5
Total
50
Slide 19
Example: Hudson Auto Repair
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Relative Frequency and Percent Frequency
Distributions
Relative
Cost ($) Frequency
50-59
.04
60-69
.26
70-79
.32
80-89
.14
90-99
.14
100-109
.10
Total 1.00
© 2003 Thomson/South-Western
Percent
Frequency
4
26
32
14
14
10
100
Slide 20
Example: Hudson Auto Repair
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Insights Gained from the Percent Frequency
Distribution
• Only 4% of the parts costs are in the $50-59 class.
• 30% of the parts costs are under $70.
• The greatest percentage (32% or almost one-third)
of the parts costs are in the $70-79 class.
• 10% of the parts costs are $100 or more.
© 2003 Thomson/South-Western
Slide 21
Dot Plot
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One of the simplest graphical summaries of data is a
dot plot.
A horizontal axis shows the range of data values.
Then each data value is represented by a dot placed
above the axis.
© 2003 Thomson/South-Western
Slide 22
Example: Hudson Auto Repair
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Dot Plot
.. .. . .
.
.
..
..
..
..
.
.
. . . ..... .......... .. . .. . . ... . .. .
50
60
70
80
90
100
110
Cost ($)
© 2003 Thomson/South-Western
Slide 23
Histogram
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Another common graphical presentation of
quantitative data is a histogram.
The variable of interest is placed on the horizontal
axis.
A rectangle is drawn above each class interval with
its height corresponding to the interval’s frequency,
relative frequency, or percent frequency.
Unlike a bar graph, a histogram has no natural
separation between rectangles of adjacent classes.
© 2003 Thomson/South-Western
Slide 24
Example: Hudson Auto Repair
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Histogram
18
Frequency
16
14
12
10
8
6
4
2
50
60
70
© 2003 Thomson/South-Western
80
90
100
110
Parts
Cost ($)
Slide 25
Cumulative Distributions
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Cumulative frequency distribution -- shows the
number of items with values less than or equal to the
upper limit of each class.
Cumulative relative frequency distribution -- shows
the proportion of items with values less than or equal
to the upper limit of each class.
Cumulative percent frequency distribution -- shows
the percentage of items with values less than or equal
to the upper limit of each class.
© 2003 Thomson/South-Western
Slide 26
Example: Hudson Auto Repair
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Cumulative Distributions
Cost ($)
< 59
< 69
< 79
< 89
< 99
< 109
Cumulative Cumulative
Cumulative
Relative
Percent
Frequency
Frequency
Frequency
2
.04
4
15
.30
30
31
.62
62
38
.76
76
45
.90
90
50
1.00
100
© 2003 Thomson/South-Western
Slide 27
Ogive
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An ogive is a graph of a cumulative distribution.
The data values are shown on the horizontal axis.
Shown on the vertical axis are the:
• cumulative frequencies, or
• cumulative relative frequencies, or
• cumulative percent frequencies
The frequency (one of the above) of each class is
plotted as a point.
The plotted points are connected by straight lines.
© 2003 Thomson/South-Western
Slide 28
Example: Hudson Auto Repair
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Ogive
• Because the class limits for the parts-cost data are
50-59, 60-69, and so on, there appear to be one-unit
gaps from 59 to 60, 69 to 70, and so on.
• These gaps are eliminated by plotting points
halfway between the class limits.
• Thus, 59.5 is used for the 50-59 class, 69.5 is used
for the 60-69 class, and so on.
© 2003 Thomson/South-Western
Slide 29
Example: Hudson Auto Repair
Ogive with Cumulative Percent Frequencies
Cumulative Percent Frequency
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100
80
60
40
20
50
60
70
© 2003 Thomson/South-Western
80
90
100
110
Parts
Cost ($)
Slide 30
Exploratory Data Analysis
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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.
© 2003 Thomson/South-Western
Slide 31
Stem-and-Leaf Display
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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.
© 2003 Thomson/South-Western
Slide 32
Example: Hudson Auto Repair
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Stem-and-Leaf Display
5
6
7
8
9
10
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
© 2003 Thomson/South-Western
Slide 33
Stretched Stem-and-Leaf Display
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If we believe the original stem-and-leaf display has
condensed the data too much, we can stretch the
display by using two more 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
values corresponds to values of 5-9.
© 2003 Thomson/South-Western
Slide 34
Example: Hudson Auto Repair
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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
© 2003 Thomson/South-Western
Slide 35
Stem-and-Leaf Display
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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.
© 2003 Thomson/South-Western
Slide 36
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
© 2003 Thomson/South-Western
Slide 37
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
© 2003 Thomson/South-Western
Slide 38
Crosstabulations and Scatter Diagrams
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Thus far we have focused on methods that are used
to summarize the data for one variable at a time.
Often a manager is interested in tabular and
graphical methods that will help understand the
relationship between two variables.
Crosstabulation and a scatter diagram are two
methods for summarizing the data for two (or more)
variables simultaneously.
© 2003 Thomson/South-Western
Slide 39
Crosstabulation
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Crosstabulation is a tabular method for summarizing
the data for two variables simultaneously.
Crosstabulation can be used when:
• One variable is qualitative and the other is
quantitative
• Both variables are qualitative
• Both variables are quantitative
The left and top margin labels define the classes for
the two variables.
© 2003 Thomson/South-Western
Slide 40
Example: Finger Lakes Homes
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Crosstabulation
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
Colonial
Home Style
Ranch Split A-Frame Total
18
12
6
14
19
16
12
3
55
45
30
20
35
15
100
© 2003 Thomson/South-Western
Slide 41
Example: Finger Lakes Homes
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Insights Gained from the Preceding Crosstabulation
• The greatest number of homes in the sample (19)
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.
© 2003 Thomson/South-Western
Slide 42
Crosstabulation: Row or Column Percentages
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Converting the entries in the table into row
percentages or column percentages can provide
additional insight about the relationship between the
two variables.
© 2003 Thomson/South-Western
Slide 43
Example: Finger Lakes Homes
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Row Percentages
Price
Range
< $99,000
> $99,000
Colonial
32.73
26.67
Home Style
Ranch Split A-Frame Total
10.91 34.55
31.11 35.56
21.82
6.67
100
100
Note: row totals are actually 100.01 due to rounding.
© 2003 Thomson/South-Western
Slide 44
Example: Finger Lakes Homes
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Column Percentages
Price
Range
< $99,000
> $99,000
Total
Colonial
Home Style
Ranch Split A-Frame
60.00
40.00
30.00
70.00
54.29
45.71
80.00
20.00
100
100
100
100
© 2003 Thomson/South-Western
Slide 45
Scatter Diagram
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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.
© 2003 Thomson/South-Western
Slide 46
Scatter Diagram
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A Positive Relationship
y
x
© 2003 Thomson/South-Western
Slide 47
Scatter Diagram
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A Negative Relationship
y
x
© 2003 Thomson/South-Western
Slide 48
Scatter Diagram
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No Apparent Relationship
y
x
© 2003 Thomson/South-Western
Slide 49
Example: Panthers Football Team
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Scatter Diagram
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
© 2003 Thomson/South-Western
y = Number of
Points Scored
14
24
18
17
27
Slide 50
Example: Panthers Football Team
Scatter Diagram
Number of Points Scored
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y
30
25
20
15
10
5
0
0
1
2
3
Number of Interceptions
© 2003 Thomson/South-Western
x
Slide 51
Example: Panthers Football Team
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The preceding scatter diagram indicates 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.
© 2003 Thomson/South-Western
Slide 52
Tabular and Graphical Procedures
Data
Qualitative Data
Tabular
Methods
•Frequency
Distribution
•Rel. Freq. Dist.
•% Freq. Dist.
•Crosstabulation
Graphical
Methods
•Bar Graph
•Pie Chart
© 2003 Thomson/South-Western
Quantitative Data
Tabular
Methods
•Frequency
Distribution
•Rel. Freq. Dist.
•Cum. Freq. Dist.
•Cum. Rel. Freq.
Distribution
•Stem-and-Leaf
Display
•Crosstabulation
Graphical
Methods
•Dot Plot
•Histogram
•Ogive
•Scatter
Diagram
Slide 53
End of Chapter 2
© 2003 Thomson/South-Western
Slide 54