chapter 2 organizing and graphing data. definition data recorded in the sequence in which they are...
TRANSCRIPT
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