Introduction
I
Latin word “status” meaning “state”
Introduction
I
Latin word “status” meaning “state”
I
The discipline of statistics provides methods for organizing
and summarizing data and for drawing conclusions based on
information contained in the data.
Introduction
I
Latin word “status” meaning “state”
I
The discipline of statistics provides methods for organizing
and summarizing data and for drawing conclusions based on
information contained in the data.
I
Our Focus: Drawing Conclusions or Making Statistical
Inferences
Basic Concepts
Basic Concepts
I
Population: total collection of objects we are interested in
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
I
Examples:
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
I
Examples:
Number of students in this classroom who drove here today
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
I
Examples:
Number of students in this classroom who drove here today
Population: all the students in the class room;
Sample: All the boy; Census: possible
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
I
Examples:
Number of students in this classroom who drove here today
Population: all the students in the class room;
Sample: All the boy; Census: possible
GE manufactured 100,000,000 lamps. What’s life range?
Basic Concepts
I
Population: total collection of objects we are interested in
I
Sample: a subset of the population
I
Census: information for all objects in the population
I
Examples:
Number of students in this classroom who drove here today
Population: all the students in the class room;
Sample: All the boy; Census: possible
GE manufactured 100,000,000 lamps. What’s life range?
Population: 100,000,000 lamps; Sample: randomly
selected 1,000 lamps; Census: impossible
Basic Concepts
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Basic Concepts
I
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Univariate Data: observation on a single variable
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Univariate Data: observation on a single variable
I
Bivariate Data: observation on each of two variables
I
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Univariate Data: observation on a single variable
I
Bivariate Data: observation on each of two variables
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Multivariate Data: observations made on more than one
variable
I
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Univariate Data: observation on a single variable
I
Bivariate Data: observation on each of two variables
I
Multivariate Data: observations made on more than one
variable
I
Examples:
The collection of data about whether students drove to school
today and the gender of students
I
Basic Concepts
I
Variable: a characteristic of the population that may differ
from individual to individual
usually use lowercase letters to denote variables
Examples: x = yes or no a student drove to school today
y = maximum hours a lamp can last
Univariate Data: observation on a single variable
I
Bivariate Data: observation on each of two variables
I
Multivariate Data: observations made on more than one
variable
I
Examples:
The collection of data about whether students drove to school
today and the gender of students
The collection of data about whether students drove to school
today, the gender of students and the distance from their
home to campus
I
Basic Concepts
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
I
Enumerative v.s. Analytic Studies
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
I
Enumerative v.s. Analytic Studies
Enumerative Studies: the sample is available to an
investigator or else can be constructed
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
I
Enumerative v.s. Analytic Studies
Enumerative Studies: the sample is available to an
investigator or else can be constructed
Examples: life of the GE lamps; the gender of students in this
classroom
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
I
Enumerative v.s. Analytic Studies
Enumerative Studies: the sample is available to an
investigator or else can be constructed
Examples: life of the GE lamps; the gender of students in this
classroom
Analytic Studies: the sample is NOT available
Basic Concepts
I
Conceptual/Hypothetical Population: population which
does not physically exist
Examples: all possible values of tomorrow’s highest
temperature; all possible pH values of some unknown liquid;
etc.
I
Enumerative v.s. Analytic Studies
Enumerative Studies: the sample is available to an
investigator or else can be constructed
Examples: life of the GE lamps; the gender of students in this
classroom
Analytic Studies: the sample is NOT available
Examples: tomorrow’s highest temperature; Champion of the
2009 NBA
Basic Concepts
Basic Concepts
I
Descriptive Statistics & Inferential Statistics
Basic Concepts
I
Descriptive Statistics & Inferential Statistics
Recall: The discipline of statistics provides methods for organizing
and summarizing data and for drawing conclusions based on
information contained in the data.
Basic Concepts
I
Descriptive Statistics & Inferential Statistics
Recall: The discipline of statistics provides methods for organizing
and summarizing data and for drawing conclusions based on
information contained in the data.
I
Descriptive Statistics: discipline of organizing and
summarizing data
Basic Concepts
I
Descriptive Statistics & Inferential Statistics
Recall: The discipline of statistics provides methods for organizing
and summarizing data and for drawing conclusions based on
information contained in the data.
I
Descriptive Statistics: discipline of organizing and
summarizing data
I
Inferential Statistics: discipline of drawing conclusions from
a sample to a population
Basic Concepts
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.
Basic Concepts
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.
The stem-and-leaf plot:
Basic Concepts
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.
The stem-and-leaf plot:
5 |
9
6 |
33588
7 |
00234677889
8 |
127
9 |
077
10 |
7
11 |
368
Basic Concepts
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.
The stem-and-leaf plot:
The histogram graph:
5 |
9
6 |
33588
7 |
00234677889
8 |
127
9 |
077
10 |
7
11 |
368
Basic Concepts
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.
The stem-and-leaf plot:
The histogram graph:
5 |
9
6 |
33588
7 |
00234677889
8 |
127
9 |
077
10 |
7
11 |
368
Basic Concepts
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.
Basic Concepts
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.
Moreover, we can make statistical inferences from this data set.
Basic Concepts
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.
Moreover, we can make statistical inferences from this data set.
It can be shown that, with a high degree of confidence, the population
mean strength is between 7.48 MPa and 8.80 Mpa; this is called a
confidence interval or interval.
Basic Concepts
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.
Moreover, we can make statistical inferences from this data set.
It can be shown that, with a high degree of confidence, the population
mean strength is between 7.48 MPa and 8.80 Mpa; this is called a
confidence interval or interval.
Furthermore, with a high degree of confidence, the strength of a single
such beam will exceed 7.35 MPa; this number 7.35 is called a lower
prediction bound.
Probability & Statistics
Probability & Statistics
Probability & Statistics
I
Probability: know the information of population and ask
question about sample
Probability & Statistics
I
Probability: know the information of population and ask
question about sample
A probability question: We have a fair coin and toss it many
times. What’s the chance to get three consecutive heads?
Probability & Statistics
I
Probability: know the information of population and ask
question about sample
A probability question: We have a fair coin and toss it many
times. What’s the chance to get three consecutive heads?
I
Statistics: know the information of sample and ask question
about population
Probability & Statistics
I
Probability: know the information of population and ask
question about sample
A probability question: We have a fair coin and toss it many
times. What’s the chance to get three consecutive heads?
I
Statistics: know the information of sample and ask question
about population
A statistic question: We have a coin and toss it 6 times. The
results are H, T, T, H, H, H. Is this coin a fair coin?
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
The decimal point is at the |
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
The decimal point is at the |
5
|
9
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
The decimal point is at the |
5
6
|
|
9
33588
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
The decimal point is at the |
5
6
7
|
|
|
9
33588
00234677889
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
The decimal point is at the |
5
6
7
8
|
|
|
|
9
33588
00234677889
127
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
The decimal point is at the |
5
6
7
8
9
|
|
|
|
|
9
33588
00234677889
127
077
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
The decimal point is at the |
5
6
7
8
9
10
|
|
|
|
|
|
9
33588
00234677889
127
077
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
The decimal point is at the |
5
6
7
8
9
10
11
|
|
|
|
|
|
|
9
33588
00234677889
127
077
7
368
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
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
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Stem-and-Leaf Displays
Pictorial and Tabular Methods
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Stem-and-Leaf Displays
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1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
Pictorial and Tabular Methods
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Stem-and-Leaf Displays
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1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
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2. List possible stem values in a vertical column.
Pictorial and Tabular Methods
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Stem-and-Leaf Displays
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1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
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2. List possible stem values in a vertical column.
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3. Record the leaf for every observation beside the
corresponding stem value.
Pictorial and Tabular Methods
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Stem-and-Leaf Displays
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1. Select one or more leading digits for the stem values. The
trailing digits become the leaves.
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2. List possible stem values in a vertical column.
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3. Record the leaf for every observation beside the
corresponding stem value.
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4. Indicate the units for stems and leaves someplace in the
display.
Pictorial and Tabular Methods
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Remark:
Pictorial and Tabular Methods
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Remark:
1. Each data in the population must consist of at least two
digits.
Pictorial and Tabular Methods
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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
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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
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Dotplots:
Pictorial and Tabular Methods
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Dotplots:
e.g. The dotplot for the previous example:
Pictorial and Tabular Methods
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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
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Histograms
Pictorial and Tabular Methods
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Histograms
e.g. The histogram for the previous example:
Pictorial and Tabular Methods
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Discrete & Continuous Variables:
Pictorial and Tabular Methods
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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
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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
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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
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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
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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
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Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
Pictorial and Tabular Methods
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Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
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Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
Pictorial and Tabular Methods
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Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
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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
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Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
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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
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Frequency: the frequency of any particular data value is the
number of times that value occurs in the data set.
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Relative Frequency: the relative frequency of a value is the
fraction of proportion of times the value occurs
relative frequency =
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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
5.00 - 5.99
6.00 - 6.99
7.00 - 7.99
8.00 - 8.99
9.00 - 9.99
10.00 - 10.99
11.00 - 11.99
frequency
1
5
11
3
3
1
3
relative frequency
0.037
0.185
0.407
0.111
0.111
0.037
0.111
Pictorial and Tabular Methods
Pictorial and Tabular Methods
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Remark:
Pictorial and Tabular Methods
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Remark:
1. For discrete data, we usually don’t have to determine the
class intervals.
Pictorial and Tabular Methods
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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
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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
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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
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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
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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
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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