Introduction

I Latin word “status” meaning “state”

I The discipline of statistics provides methods for organizingand summarizing data and for drawing conclusions based oninformation contained in the data.

I Our Focus: Drawing Conclusions or Making StatisticalInferences

Introduction

I Latin word “status” meaning “state”

I The discipline of statistics provides methods for organizingand summarizing data and for drawing conclusions based oninformation contained in the data.

I Our Focus: Drawing Conclusions or Making StatisticalInferences

Introduction

I Latin word “status” meaning “state”

I The discipline of statistics provides methods for organizingand summarizing data and for drawing conclusions based oninformation contained in the data.

I Our Focus: Drawing Conclusions or Making StatisticalInferences

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

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 todayPopulation: 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: randomlyselected 1,000 lamps; Census: impossible

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individual

usually use lowercase letters to denote variablesExamples: x = yes or no a student drove to school today

y = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of students

The collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Variable: a characteristic of the population that may differfrom individual to individualusually use lowercase letters to denote variables

Examples: x = yes or no a student drove to school todayy = maximum hours a lamp can last

I Univariate Data: observation on a single variable

I Bivariate Data: observation on each of two variables

I Multivariate Data: observations made on more than onevariable

I Examples:The collection of data about whether students drove to schooltoday and the gender of studentsThe collection of data about whether students drove to schooltoday, the gender of students and the distance from theirhome to campus

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically exist

Examples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic Studies

Enumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructed

Examples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroom

Analytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT available

Examples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Conceptual/Hypothetical Population: population whichdoes not physically existExamples: all possible values of tomorrow’s highesttemperature; all possible pH values of some unknown liquid;etc.

I Enumerative v.s. Analytic StudiesEnumerative Studies: the sample is available to aninvestigator or else can be constructedExamples: life of the GE lamps; the gender of students in thisclassroomAnalytic Studies: the sample is NOT availableExamples: tomorrow’s highest temperature; Champion of the2009 NBA

Basic Concepts

I Descriptive Statistics & Inferential StatisticsRecall: 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 andsummarizing data

I Inferential Statistics: discipline of drawing conclusions froma sample to a population

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 andsummarizing data

I Inferential Statistics: discipline of drawing conclusions froma sample to a population

Basic Concepts

I Descriptive Statistics & Inferential StatisticsRecall: 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 andsummarizing data

I Inferential Statistics: discipline of drawing conclusions froma sample to a population

Basic Concepts

I Descriptive Statistics & Inferential StatisticsRecall: 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 andsummarizing data

I Inferential Statistics: discipline of drawing conclusions froma sample to a population

Basic Concepts

I Descriptive Statistics & Inferential StatisticsRecall: 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 andsummarizing data

I Inferential Statistics: discipline of drawing conclusions froma 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 astudy of strength properties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying data on flexuralstrength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 thatcould be made in this way.

The stem-and-leaf plot:5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

The histogram graph:

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reported on astudy of strength properties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying data on flexuralstrength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 thatcould be made in this way.

The stem-and-leaf plot:

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

The histogram graph:

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reported on astudy of strength properties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying data on flexuralstrength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 thatcould be made in this way.

The stem-and-leaf plot:5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

The histogram graph:

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reported on astudy of strength properties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying data on flexuralstrength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 thatcould be made in this way.

The stem-and-leaf plot:5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

The histogram graph:

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reported on astudy of strength properties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying data on flexuralstrength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 thatcould be made in this way.

The stem-and-leaf plot:5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

The histogram graph:

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reportedon a study of strength properties of high performance concrete obtainedby using superplasticizers and certain binders. The accompanying data onflexural strength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 beamsthat 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 populationmean strength is between 7.48 MPa and 8.80 Mpa; this is called aconfidence interval or interval.Furthermore, with a high degree of confidence, the strength of a singlesuch beam will exceed 7.35 MPa; this number 7.35 is called a lowerprediction bound.

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reportedon a study of strength properties of high performance concrete obtainedby using superplasticizers and certain binders. The accompanying data onflexural strength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 beamsthat 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 populationmean strength is between 7.48 MPa and 8.80 Mpa; this is called aconfidence interval or interval.Furthermore, with a high degree of confidence, the strength of a singlesuch beam will exceed 7.35 MPa; this number 7.35 is called a lowerprediction bound.

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reportedon a study of strength properties of high performance concrete obtainedby using superplasticizers and certain binders. The accompanying data onflexural strength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 beamsthat 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 populationmean strength is between 7.48 MPa and 8.80 Mpa; this is called aconfidence interval or interval.

Furthermore, with a high degree of confidence, the strength of a singlesuch beam will exceed 7.35 MPa; this number 7.35 is called a lowerprediction bound.

Basic Concepts

I Example(Example 1.2 p5): The article ‘‘Effects of Aggregates and

Microfillers on the Flexural Properties of Concrete’’ reportedon a study of strength properties of high performance concrete obtainedby using superplasticizers and certain binders. The accompanying data onflexural strength (in MPa) appeared in the article cited:

5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 beamsthat 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 populationmean strength is between 7.48 MPa and 8.80 Mpa; this is called aconfidence interval or interval.Furthermore, with a high degree of confidence, the strength of a singlesuch beam will exceed 7.35 MPa; this number 7.35 is called a lowerprediction bound.

Probability & Statistics

I Probability: know the information of population and askquestion about sampleA probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout populationA statistic question: We have a coin and toss it 6 times. Theresults are H, T, T, H, H, H. Is this coin a fair coin?

Probability & Statistics

I Probability: know the information of population and askquestion about sampleA probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout populationA statistic question: We have a coin and toss it 6 times. Theresults are H, T, T, H, H, H. Is this coin a fair coin?

Probability & Statistics

I Probability: know the information of population and askquestion about sample

A probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout populationA statistic question: We have a coin and toss it 6 times. Theresults are H, T, T, H, H, H. Is this coin a fair coin?

Probability & Statistics

I Probability: know the information of population and askquestion about sampleA probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout populationA statistic question: We have a coin and toss it 6 times. Theresults are H, T, T, H, H, H. Is this coin a fair coin?

Probability & Statistics

I Probability: know the information of population and askquestion about sampleA probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout population

A statistic question: We have a coin and toss it 6 times. Theresults are H, T, T, H, H, H. Is this coin a fair coin?

Probability & Statistics

I Probability: know the information of population and askquestion about sampleA probability question: We have a fair coin and toss it manytimes. What’s the chance to get three consecutive heads?

I Statistics: know the information of sample and ask questionabout populationA statistic question: We have a coin and toss it 6 times. Theresults 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 strengthproperties of high performance concrete obtained by usingsuperplasticizers and certain binders. The accompanying dataon 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 6.86.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.77.4 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 forall beams that could be made in this way.

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 9

6 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 33588

7 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 00234677889

8 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 127

9 | 07710 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 7

11 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

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

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

• identification of a typicalvalue

• presence of any gaps in thedata

• extent of symmetry in thedistribution of values

• number and location ofpeaks

• presence of any outlyingvalues

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Stem-and-Leaf Displays

I 1. Select one or more leading digits for the stem values. Thetrailing digits become the leaves.

I 2. List possible stem values in a vertical column.

I 3. Record the leaf for every observation beside thecorresponding stem value.

I 4. Indicate the units for stems and leaves someplace in thedisplay.

Pictorial and Tabular Methods

I Remark:

1. Each data in the population must consist of at least twodigits.e.g. the stem-and-leaf display is not suitable for the data set1,2,1,4,1,5,2,6,1,3,2,32. Ordering the leaves from smallest to largest is not necessary

Pictorial and Tabular Methods

I Remark:1. Each data in the population must consist of at least twodigits.

e.g. the stem-and-leaf display is not suitable for the data set1,2,1,4,1,5,2,6,1,3,2,32. Ordering the leaves from smallest to largest is not necessary

Pictorial and Tabular Methods

I Remark:1. Each data in the population must consist of at least twodigits.e.g. the stem-and-leaf display is not suitable for the data set1,2,1,4,1,5,2,6,1,3,2,3

2. Ordering the leaves from smallest to largest is not necessary

Pictorial and Tabular Methods

I Remark:1. Each data in the population must consist of at least twodigits.e.g. the stem-and-leaf display is not suitable for the data set1,2,1,4,1,5,2,6,1,3,2,32. Ordering the leaves from smallest to largest is not necessary

Pictorial and Tabular Methods

The decimal point is at the |

5 | 96 | 388537 | 230609847878 | 1279 | 077

10 | 711 | 638

The decimal point is at the |

5 | 96 | 335887 | 002346778898 | 1279 | 077

10 | 711 | 368

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 thecorresponding location on a horizontal measurement scale.When a value occurs more than once, there is a dot for eachoccurrence, and these dots are stacked vertically.

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 thecorresponding location on a horizontal measurement scale.When a value occurs more than once, there is a dot for eachoccurrence, and these dots are stacked vertically.

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 thecorresponding location on a horizontal measurement scale.When a value occurs more than once, there is a dot for eachoccurrence, and these dots are stacked vertically.

Pictorial and Tabular Methods

I Histograms

e.g. The histogram for the previous example:

Pictorial and Tabular Methods

I Histograms

e.g. The histogram for the previous example:

Pictorial and Tabular Methods

I Discrete & Continuous Variables:

A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.e.g. x = number of students in this classroom who drove toschool todayUsually arising from counting

A numerical variable is continuous if its possible values consistof an entire interval on the number line.e.g y = maximum hours a GE lamp can lastUsually arising from measuring

Pictorial and Tabular Methods

I Discrete & Continuous Variables:A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.

e.g. x = number of students in this classroom who drove toschool todayUsually arising from counting

A numerical variable is continuous if its possible values consistof an entire interval on the number line.e.g y = maximum hours a GE lamp can lastUsually arising from measuring

Pictorial and Tabular Methods

I Discrete & Continuous Variables:A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.e.g. x = number of students in this classroom who drove toschool today

Usually arising from counting

A numerical variable is continuous if its possible values consistof an entire interval on the number line.e.g y = maximum hours a GE lamp can lastUsually arising from measuring

Pictorial and Tabular Methods

I Discrete & Continuous Variables:A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.e.g. x = number of students in this classroom who drove toschool todayUsually arising from counting

A numerical variable is continuous if its possible values consistof an entire interval on the number line.

e.g y = maximum hours a GE lamp can lastUsually arising from measuring

Pictorial and Tabular Methods

I Discrete & Continuous Variables:A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.e.g. x = number of students in this classroom who drove toschool todayUsually arising from counting

A numerical variable is continuous if its possible values consistof 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 Discrete & Continuous Variables:A numerical variable is discrete if its set of possible values iseither finite or can be listed in an infinite sequence.e.g. x = number of students in this classroom who drove toschool todayUsually arising from counting

A numerical variable is continuous if its possible values consistof an entire interval on the number line.e.g y = maximum hours a GE lamp can lastUsually arising from measuring

Pictorial and Tabular Methods

I Frequency: the frequency of any particular data value is thenumber of times that value occurs in the data set.

I Relative Frequency: the relative frequency of a value is thefraction 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: 2relative frequency of the value 6.8: 2

27 = 0.074

I Frequency Distribution: a tabulation of the frequenciesand/or relative frequencies.

Pictorial and Tabular Methods

I Frequency: the frequency of any particular data value is thenumber of times that value occurs in the data set.

I Relative Frequency: the relative frequency of a value is thefraction 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: 2relative frequency of the value 6.8: 2

27 = 0.074

I Frequency Distribution: a tabulation of the frequenciesand/or relative frequencies.

Pictorial and Tabular Methods

I Frequency: the frequency of any particular data value is thenumber of times that value occurs in the data set.

I Relative Frequency: the relative frequency of a value is thefraction 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: 2relative frequency of the value 6.8: 2

27 = 0.074

I Frequency Distribution: a tabulation of the frequenciesand/or relative frequencies.

Pictorial and Tabular Methods

I Frequency: the frequency of any particular data value is thenumber of times that value occurs in the data set.

I Relative Frequency: the relative frequency of a value is thefraction 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: 2relative frequency of the value 6.8: 2

27 = 0.074

I Frequency Distribution: a tabulation of the frequenciesand/or relative frequencies.

Pictorial and Tabular Methods

I Frequency: the frequency of any particular data value is thenumber of times that value occurs in the data set.

I Relative Frequency: the relative frequency of a value is thefraction 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: 2relative frequency of the value 6.8: 2

27 = 0.074

I Frequency Distribution: a tabulation of the frequenciesand/or relative frequencies.

Pictorial and Tabular Methods

Constructing a Histogram for a Data Set:

1. Divide the data set into a suitable number of class interval orclasses;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 thecorresponding relative frequency(or frequency)

Pictorial and Tabular Methods

Constructing a Histogram for a Data Set:1. Divide the data set into a suitable number of class interval orclasses;

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 thecorresponding relative frequency(or frequency)

Pictorial and Tabular Methods

Constructing a Histogram for a Data Set:1. Divide the data set into a suitable number of class interval orclasses;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 thecorresponding relative frequency(or frequency)

Pictorial and Tabular Methods

Constructing a Histogram for a Data Set:1. Divide the data set into a suitable number of class interval orclasses;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 thecorresponding relative frequency(or frequency)

Pictorial and Tabular Methods

Constructing a Histogram for a Data Set:1. Divide the data set into a suitable number of class interval orclasses;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 thecorresponding 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:

1. For discrete data, we usually don’t have to determine theclass intervals.2. There is no hard-and-fast rules for the choice of classintervals. A reasonable rule of thumb is

number of classes =√

number of observation

3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.e.g.

Pictorial and Tabular Methods

I Remark:1. For discrete data, we usually don’t have to determine theclass intervals.

2. There is no hard-and-fast rules for the choice of classintervals. A reasonable rule of thumb is

number of classes =√

number of observation

3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.e.g.

Pictorial and Tabular Methods

I Remark:1. For discrete data, we usually don’t have to determine theclass intervals.2. There is no hard-and-fast rules for the choice of classintervals. A reasonable rule of thumb is

number of classes =√

number of observation

3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.e.g.

Pictorial and Tabular Methods

I Remark:1. For discrete data, we usually don’t have to determine theclass intervals.2. There is no hard-and-fast rules for the choice of classintervals. A reasonable rule of thumb is

number of classes =√

number of observation

3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.

e.g.

Pictorial and Tabular Methods

I Remark:1. For discrete data, we usually don’t have to determine theclass intervals.2. There is no hard-and-fast rules for the choice of classintervals. A reasonable rule of thumb is

number of classes =√

number of observation

3. Equal-width classes may not be a sensible choice if a dataset “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 dataset “stretches out” to one side or the other.e.g.

Use a few wider intervals near extreme observations andnarrower intervals in the region of high concentration.

rectangle height =relative frequency of the class

class width

Pictorial and Tabular Methods

I Remark:3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.e.g.

Use a few wider intervals near extreme observations andnarrower intervals in the region of high concentration.

rectangle height =relative frequency of the class

class width

Pictorial and Tabular Methods

I Remark:3. Equal-width classes may not be a sensible choice if a dataset “stretches out” to one side or the other.e.g.

Use a few wider intervals near extreme observations andnarrower intervals in the region of high concentration.

rectangle height =relative frequency of the class

class width