CHAPTER 2

ORGANIZING AND GRAPHING DATA

Definition Data recorded in the sequence in which

they are collected and before they are processed or ranked are called raw data

21 ORGANIZING AND GRAPHING QUALITATIVE DATA

Table 21 Ages of 50 Students (See Pg 29)

Table 22 Status of 50 Students (See Pg 29)

Definition Ungroup data set is a data set containing

information on each member of a sample or population individually Tables 21 and 22 are examples of ungroup data set

ORGANIZING AND GRAPHING QUANTITATIVE DATA

Qualitative data sets can be Organized into tables and Displayed using graphs

To organize and display qualitative data setFrequency Distributions tableRelative Frequency and Percentage Distributions tableGraphical Presentation of Qualitative Data

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Definition Data recorded in the sequence in which

they are collected and before they are processed or ranked are called raw data

21 ORGANIZING AND GRAPHING QUALITATIVE DATA

Table 21 Ages of 50 Students (See Pg 29)

Table 22 Status of 50 Students (See Pg 29)

Definition Ungroup data set is a data set containing

information on each member of a sample or population individually Tables 21 and 22 are examples of ungroup data set

ORGANIZING AND GRAPHING QUANTITATIVE DATA

Qualitative data sets can be Organized into tables and Displayed using graphs

To organize and display qualitative data setFrequency Distributions tableRelative Frequency and Percentage Distributions tableGraphical Presentation of Qualitative Data

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 21 Ages of 50 Students (See Pg 29)

Table 22 Status of 50 Students (See Pg 29)

Definition Ungroup data set is a data set containing

information on each member of a sample or population individually Tables 21 and 22 are examples of ungroup data set

ORGANIZING AND GRAPHING QUANTITATIVE DATA

Qualitative data sets can be Organized into tables and Displayed using graphs

To organize and display qualitative data setFrequency Distributions tableRelative Frequency and Percentage Distributions tableGraphical Presentation of Qualitative Data

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 22 Status of 50 Students (See Pg 29)

Definition Ungroup data set is a data set containing

information on each member of a sample or population individually Tables 21 and 22 are examples of ungroup data set

ORGANIZING AND GRAPHING QUANTITATIVE DATA

Qualitative data sets can be Organized into tables and Displayed using graphs

To organize and display qualitative data setFrequency Distributions tableRelative Frequency and Percentage Distributions tableGraphical Presentation of Qualitative Data

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

ORGANIZING AND GRAPHING QUANTITATIVE DATA

Qualitative data sets can be Organized into tables and Displayed using graphs

To organize and display qualitative data setFrequency Distributions tableRelative Frequency and Percentage Distributions tableGraphical Presentation of Qualitative Data

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

TABLE 23 Types of Employment Students Intend to Engage In

Definition Frequency of a category is the number of

element or member of the data set that belong to a specific category

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Frequency Distributions Definition A frequency distribution table for

qualitative data lists all categories and the number of elements that belong to each of the categories

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2A sample of 30 employees from large companies was selected and these employees were asked how stressful their jobs were The responses of these employees are recorded below where very represents very stressful somewhat means somewhat stressful and none stands for not stressful at all

somewhat none somewhat very very none

very somewhat somewhat very somewhat somewhat

very somewhat none very none somewhat

somewhat very somewhat somewhat very none

somewhat very very somewhat none somewhat

Construct a frequency distribution table for these data

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2 SolutionFrequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Relative Frequency and Percentage Distributions

Calculating Relative Frequency of a Category

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Calculating Percentage Percentage = (Relative frequency) 100

Relative frequency distribution lists the relative distribution for all categories

Percentage distribution lists the percentages for all categories

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2 Determine the relative frequency and percentage for the

data in Table 24

Example 2 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Graphical Presentation of Qualitative Data Definition A bar graph is a graph made of bars whose heights

represent the frequencies of respective categories

Bar graph for the frequency

distribution of Table

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Graphical Presentation of Qualitative Data Definition A pie chart is a circle divided into portions that represent

the relative frequencies or percentages of a population or a sample belonging to different categories

Calculating Angle Sizes for the Pie Chart

Pie chart for the percentage distribution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Case Study In or Out in 30 Minutes

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

22 ORGANIZING AND GRAPHING QUANTITATIVE DATA

Quantitative data sets can be grouped into tables and Displayed using graphs

To group and display quantitative data setFrequency Distributions tableConstructing Frequency Distribution tablesRelative and Percentage Distributions tableGraphing Grouped Data

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Frequency Distribution Table of Quantitative Data Definition A class is an interval that includes all values that

falls between two numbers lower and upper limits Note that classes are mutually exclusive meaning that no

one value in the data set belong in more than one class A frequency is the number of values in the data

set that belong to a specific class A frequency distribution for quantitative data

lists all the classes and the number of values that belong to each class Data presented in the form of a frequency distribution are called grouped data

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 27 Weekly Earnings of 100 Employees of a Company ndash Frequency Distribution of Quantitative Data

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Frequency Distribution Table ndash Conversion of Class Limits to Class Boundary

Definition A class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class

A class width or size is define as

A class midpoint or mark is define as

2

classnext theoflimit Lower class theoflimit Upper Classa of Boundary Class

boundary class Lower -boundary classUpper Width Class

2

boundaryor limit class Upper boundry or limit classLower or Mark Midpoint Class

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 28 Class Boundaries Class Widths and Class Midpoints for Table 27

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Constructing Frequency Distribution Tables

classes ofNumber

alueSmallest v - lueLargest va widthclass eApproximat

Definition Number of Classes generally varies from 5 and 20 depending on the size of the data set It can also be estimated by using

c = 1 + 33 log n c is the number of classes and n is thenumber of observations

A class width or size is define as

The lower limit of the first class or starting point is any number that is equal to or less than the smallest observation or value in the data set

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-3 ndash Frequency Distribution Table

The following data give the total number of iPodsreg sold by a mail order company on each of 30 days

Construct a frequency distribution table

8 25 11 15 29 22 10 5 17 21

22 13 26 16 18 12 9 26 20 16

23 14 19 23 20 16 27 16 21 14

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-3 Solution

Lower limit of the first class

Suppose we take 5 as the lower limit of the first class Then our classes will be

5 ndash 9 10 ndash 14 15 ndash 19 20 ndash 24 and 25 ndash 29

Class Width

5~84 5

5-29 class each of Width

Number of Classes

Suppose we decide to use 5 classes of equal width

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Relative Frequency and Percentage Distributions

Calculating Relative Frequency and Percentage

100 frequency) (Relative rcentage Pe

sfrequencie all of Sum

class that ofFrequency classa offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-4

Calculate the relative frequencies and percentages for Table 29

Example 2-4 Solution

Table 210 Relative Frequency and Percentage Distributions for Table 29

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Graphing Grouped Data

DefinitionA histogram graph is used to display frequency relative frequency and percentage distributions for quantitative dataIn a histogram graph classes are marked on the horizontal axis and the frequencies relative frequencies or percentages are marked on the vertical axis The frequencies relative frequencies or percentages are represented by the heights of the barsIn a histogram the bars are drawn adjacent to each other

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Graphs for Displaying Quantitative Data

Figure 23 Frequency histogram for Table 29

Figure 24 Relative frequency histogram for Table 210

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Graphing Grouped Data

DefinitionA polygon is a graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines

Figure 25 Frequency polygon for Table 29

Figure 26 Frequency distribution curve

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Method to Use for Quantitative Data Set

Use class limit for data sets without fractional numbers or values Use ldquoless than method for data sets with fractional numbers or values

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-5On April 1 2009 the federal tax on a pack of cigarettes was increased from 39cent to $10066 a move that not only was expected to help increase federal revenue but was also expected to save about 900000 lives (Time Magazine April 2009) Table 211 shows the total tax (state plus federal) per pack of cigarettes for all 50 states as of April 1 2009

Construct a frequency distribution table Calculate the relative frequencies and percentages for all classes

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-5 SolutionNumber of Classes Suppose we decide to use 6 classes of equal widthClass Width 50~45

6

108-376 class each of Width

Lower limit of the first classSuppose we take 10 as the lower limit of the first class Then our classes will be written as

100 to less than 15 150 to less than 200 and so on

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-6 ndash Single-Value ClassesThe administration in a large city wanted to know the distribution of vehicles owned by households in that city A sample of 40 randomly selected households from this city produced the following data on the number of vehicles owned

5 1 1 2 0 1 1 2 1 11 3 3 0 2 5 1 2 3 42 1 2 2 1 2 2 1 1 14 2 1 1 2 1 1 4 1 3

Construct a frequency distribution table for these data and draw a bar graph

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-6 SolutionTable 213 Frequency Distribution of Vehicles Owned

The observations assume only six distinct values 0 1 2 3 4 and 5 Each of these six values is used as a class in the frequency distribution in Table 213

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

SHAPES OF HISTOGRAMS

1 Symmetric2 Skewed3 Uniform or Rectangular

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Figure 28 Symmetric and Skewed histograms

Symmetric histogram has the same shape on both sides of its central points

Skewed histogram is not symmetric and the tail of one side is longer than the tail of the other side

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Uniform Distribution Histogram Symmetric frequency curves and Skewed Frequency curves

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

23 CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionA cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class

To construct a cumulative frequency distribution tableAll the classes have the same lower limit but different upper limitThe cumulative frequency of a class is obtained by adding the frequency of the particular class to the frequencies of the preceding class(es)

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-7

Using the frequency distribution of Table 29 reproduced here prepare a cumulative frequency distribution for the number of iPods sold by that company

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

CUMULATIVE FREQUENCY DISTRIBUTIONS

Calculating Cumulative Relative Frequency and Cumulative Percentage

100 frequency) relative e(Cumulativ percentage Cumulative

set data in the nsobservatio Total

class a offrequency Cumulativefrequency relative Cumulative

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 215 Cumulative Relative Frequency and Cumulative Percentage Distributions for iPods Sold

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

CUMULATIVE FREQUENCY DISTRIBUTIONS

DefinitionAn ogive is a curve for displaying cumulative

frequency cumulative relative frequency and cumulative percentage distributions

To Draw an ogive Label the vertical and horizontal axes Mark a dot at the upper boundary of the

corresponding class at a height equal to the Cumulative frequencies or Cumulative relative frequencies or Cumulative percentages of respective classes

Join the dot with straight line

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Figure 212 Ogive for the cumulative frequency distribution of Table 214

Find the number of days for which 17 or fewer iPods were sold

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

24 STEM-AND-LEAF DISPLAYS

Definition Stem-and-leaf display is a method for displaying

quantitative raw data in a concise manner It divides quantitative data into two portions which

are separated by a vertical line The left of the vertical line is the stem which

represents the first digit(s) of the values in the data set and arranged in increasing order

The right of the line is the leaf which represents the remaining digit(s) of the values in the data set and aligned with the row of the corresponding stem

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-8

The following are the scores of 30 college students on a statistics test

Construct a stem-and-leaf display

756983

527284

808177

966164

657671

798687

717972

876892

935057

959298

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Figure 213 Stem-and-leaf display

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-9

The following data are monthly rents paid by a sample of 30 households selected from a small city

Construct a stem-and-leaf display for these data

88012101151

1081 985 630

72112311175

1075 932 952

1023 8501100

775 8251140

12351000 750

750 9151140

96511911370

96010351280

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-9 Solution

Figure 216 Stem-and-leaf display of rents

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-10 The following stem-and-leaf display is prepared for the number of hours that 25 students spent working on computers during the last month

Prepare a new stem-and-leaf display by grouping the stems

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-10 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

25 DOTPLOTS

Definition Values that are very small or very large

relative to the majority of the values in a data set are called outliers or extreme values

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-11

Table 216 lists the lengths of the longest field goals (in yards) made by all kickers in the American Football Conference (AFC) of the National Football League (NFL) during the 2008 season Create a dotplot for these data

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 216 Distances of Longest Field Goals (in Yards) Made by AFC Kickers During the 2008 NFL Season

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-11 Solution

Step1

Step 2

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-12

Refer to Table 216 in Example 2-11 which gives the distances of longest completed field goals for all kickers in the AFC during the 2008 NFL season Table 217 provides the same information for the kickers in the National Football Conference (NFC) of the NFL for the 2008 season Make dotplots for both sets of data and compare these two dotplots

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Table 217 Distances of Longest Field Goals (in Yards) Made by NFC Kickers During the 2008 NFL Season

Example 2-12 Solution

Example 2-12 Solution