Pictorial and Tabular Methods
I
Example(Example 1.2 p5): The article ‘‘Effects of
Aggregates and Microfillers on the Flexural
Properties of Concrete’’ reported on a study of strength
properties of high performance concrete obtained by using
superplasticizers and certain binders. The accompanying data
on flexural strength (in MPa) appeared in the article cited:
5.9 7.2 7.3 6.3 8.1
6.8
7.0
7.6
6.8
6.5 7.0 6.3 7.9 9.0
8.2
8.7
7.8
9.7
7.4 7.7 9.7 7.8 7.7 11.6 11.3 11.8 10.7
We are interested in the average value of flexural strength for
all beams that could be made in this way.
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
|
9
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
|
|
9
33588
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
7
|
|
|
9
33588
00234677889
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
7
8
|
|
|
|
9
33588
00234677889
127
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
7
8
9
|
|
|
|
|
9
33588
00234677889
127
077
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
7
8
9
10
|
|
|
|
|
|
9
33588
00234677889
127
077
7
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
• identification of a typical
value
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
• identification of a typical
value
• presence of any gaps in the
data
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
• identification of a typical
value
• presence of any gaps in the
data
• extent of symmetry in the
distribution of values
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
• identification of a typical
value
• presence of any gaps in the
data
• extent of symmetry in the
distribution of values
• number and location of
peaks
Pictorial and Tabular Methods
5.9
6.5
7.4
7.2
7.0
7.7
7.3
6.3
9.7
6.3
7.9
7.8
8.1
9.0
7.7
6.8
8.2
11.6
7.0
8.7
11.3
7.6
7.8
11.8
6.8
9.7
10.7
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
• identification of a typical
value
• presence of any gaps in the
data
• extent of symmetry in the
distribution of values
• number and location of
peaks
• presence of any outlying
values
Pictorial and Tabular Methods
Pictorial and Tabular Methods
I
Stem-and-Leaf Displays
Pictorial and Tabular Methods
I
Stem-and-Leaf Displays
I
1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
Pictorial and Tabular Methods
I
Stem-and-Leaf Displays
I
1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
I
2. List possible stem values in a vertical column.
Pictorial and Tabular Methods
I
Stem-and-Leaf Displays
I
1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
I
2. List possible stem values in a vertical column.
I
3. Record the leaf for every observation beside the
corresponding stem value.
Pictorial and Tabular Methods
I
Stem-and-Leaf Displays
I
1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
I
2. List possible stem values in a vertical column.
I
3. Record the leaf for every observation beside the
corresponding stem value.
I
4. Indicate the units for stems and leaves someplace in the
display.
Pictorial and Tabular Methods
I
Remark:
Pictorial and Tabular Methods
I
Remark:
1. Each data in the population must consist of at least two
digits.
Pictorial and Tabular Methods
I
Remark:
1. Each data in the population must consist of at least two
digits.
e.g. the stem-and-leaf display is not suitable for the data set
1,2,1,4,1,5,2,6,1,3,2,3
Pictorial and Tabular Methods
I
Remark:
1. Each data in the population must consist of at least two
digits.
e.g. the stem-and-leaf display is not suitable for the data set
1,2,1,4,1,5,2,6,1,3,2,3
2. Ordering the leaves from smallest to largest is not necessary
Pictorial and Tabular Methods
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
38853
23060984787
127
077
7
638
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
Pictorial and Tabular Methods
I
Dotplots:
Pictorial and Tabular Methods
I
Dotplots:
e.g. The dotplot for the previous example:
Pictorial and Tabular Methods
I
Dotplots:
e.g. The dotplot for the previous example:
In a dotplot, each data is represented by a dot above the
corresponding location on a horizontal measurement scale.
When a value occurs more than once, there is a dot for each
occurrence, and these dots are stacked vertically.
Pictorial and Tabular Methods
I
Histograms
Pictorial and Tabular Methods
I
Histograms
e.g. The histogram for the previous example:
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
A numerical variable is discrete if its set of possible values is
either finite or can be listed in an infinite sequence.
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
A numerical variable is discrete if its set of possible values is
either finite or can be listed in an infinite sequence.
e.g. x = number of students in this classroom who drove to
school today
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
A numerical variable is discrete if its set of possible values is
either finite or can be listed in an infinite sequence.
e.g. x = number of students in this classroom who drove to
school today
Usually arising from counting
A numerical variable is continuous if its possible values consist
of an entire interval on the number line.
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
A numerical variable is discrete if its set of possible values is
either finite or can be listed in an infinite sequence.
e.g. x = number of students in this classroom who drove to
school today
Usually arising from counting
A numerical variable is continuous if its possible values consist
of an entire interval on the number line.
e.g y = maximum hours a GE lamp can last
Pictorial and Tabular Methods
I
Discrete & Continuous Variables:
A numerical variable is discrete if its set of possible values is
either finite or can be listed in an infinite sequence.
e.g. x = number of students in this classroom who drove to
school today
Usually arising from counting
A numerical variable is continuous if its possible values consist
of an entire interval on the number line.
e.g y = maximum hours a GE lamp can last
Usually arising from measuring
Pictorial and Tabular Methods
I
Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
Pictorial and Tabular Methods
I
Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
I
Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
Pictorial and Tabular Methods
I
Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
I
Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
relative frequency =
number of times the value occur
number of observations in the data set
Pictorial and Tabular Methods
I
Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
I
Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
relative frequency =
number of times the value occur
number of observations in the data set
e.g.
frequency of value 6.8:
relative frequency of the value 6.8:
2
2
27
= 0.074
Pictorial and Tabular Methods
I
Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
I
Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
relative frequency =
I
number of times the value occur
number of observations in the data set
e.g.
frequency of value 6.8:
2
2
relative frequency of the value 6.8: 27
= 0.074
Frequency Distribution: a tabulation of the frequencies
and/or relative frequencies.
Pictorial and Tabular Methods
Constructing a Histogram for a Data Set:
Pictorial and Tabular Methods
Constructing a Histogram for a Data Set:
1. Divide the data set into a suitable number of class interval or
classes;
Pictorial and Tabular Methods
Constructing a Histogram for a Data Set:
1. Divide the data set into a suitable number of class interval or
classes;
2. Determine the frequency and relative frequency for each class;
Pictorial and Tabular Methods
Constructing a Histogram for a Data Set:
1. Divide the data set into a suitable number of class interval or
classes;
2. Determine the frequency and relative frequency for each class;
3. Mark the class boundaries on a horizontal measurement axis;
Pictorial and Tabular Methods
Constructing a Histogram for a Data Set:
1. Divide the data set into a suitable number of class interval or
classes;
2. Determine the frequency and relative frequency for each class;
3. Mark the class boundaries on a horizontal measurement axis;
4. Above each class interval, draw a rectangle whose height is the
corresponding relative frequency(or frequency)
Pictorial and Tabular Methods
Determine frequency and relative frequency for each class:
classes
frequency relative frequency
5.00 - 5.99
1
0.037
6.00 - 6.99
5
0.185
7.00 - 7.99
11
0.407
8.00 - 8.99
3
0.111
9.00 - 9.99
3
0.111
10.00 - 10.99
1
0.037
11.00 - 11.99
3
0.111
Pictorial and Tabular Methods
Pictorial and Tabular Methods
I
Remark:
Pictorial and Tabular Methods
I
Remark:
1. For discrete data, we usually don’t have to determine the
class intervals.
Pictorial and Tabular Methods
I
Remark:
1. For discrete data, we usually don’t have to determine the
class intervals.
2. There is no hard-and-fast rules for the choice of class
intervals. A reasonable rule of thumb is
√
number of classes = number of observation
Pictorial and Tabular Methods
I
Remark:
1. For discrete data, we usually don’t have to determine the
class intervals.
2. There is no hard-and-fast rules for the choice of class
intervals. A reasonable rule of thumb is
√
number of classes = number of observation
3. Equal-width classes may not be a sensible choice if a data
set “stretches out” to one side or the other.
Pictorial and Tabular Methods
I
Remark:
1. For discrete data, we usually don’t have to determine the
class intervals.
2. There is no hard-and-fast rules for the choice of class
intervals. A reasonable rule of thumb is
√
number of classes = number of observation
3. Equal-width classes may not be a sensible choice if a data
set “stretches out” to one side or the other.
e.g.
Pictorial and Tabular Methods
I
Remark:
3. Equal-width classes may not be a sensible choice if a data
set “stretches out” to one side or the other.
e.g.
Pictorial and Tabular Methods
I
Remark:
3. Equal-width classes may not be a sensible choice if a data
set “stretches out” to one side or the other.
e.g.
Use a few wider intervals near extreme observations and
narrower intervals in the region of high concentration.
Pictorial and Tabular Methods
I
Remark:
3. Equal-width classes may not be a sensible choice if a data
set “stretches out” to one side or the other.
e.g.
Use a few wider intervals near extreme observations and
narrower intervals in the region of high concentration.
rectangle height =
relative frequency of the class
class width
Pictorial and Tabular Methods
Shapes of Histograms: