10102016

1

21

Organizing and Presenting Data

Tabulation and Graphs

Introduction to Biostatistics

Haleema Masud

Going back to the definition of Biostatisticshellip

ndash The collection organization summarization analysis presentation and dissemination of DATA and

ndash The drawing of inferences about about POPULATION from the SAMPLE observed

2

Learning Objectives

Overall To give students a basic understanding of best way of organizing and presenting data

Specific Students will be able to

bull Understand how data can be appropriately organized and displayed

bull Draw Tables

bull Draw Graphs

bull Make Frequency distribution3 24

bull Descriptive statistics involves arranging summarizing and presenting a set of data in such a way that useful information is produced

bull Descriptive statistics make use of graphical techniques and numerical techniques (such as averages) to summarize and present the data

Data

Statistics

Information

Data

QualitativeQuantitative

ContinuousDiscrete

OrdinalNominal

26

Nominal Datahellip

bull The only allowable calculation on nominal data is to count the frequency of each value of the variable

bull We can organize amp summarize the data in a table that presents the categories and their counts called a frequency distribution

10102016

2

Frequency Distributions

A frequency distribution for qualitative data lists

ndash all categories

and

ndash the number of elements that belong to each of the categories

7

Example

A 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

ndash very represents very stressful

ndash somewhat means somewhat stressful and

ndash none stands for not stressful at all

8

Example

9

Some what 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

Solution

Stress on Job Tally Frequency (f)

Very

Somewhat

None

|||| ||||

|||| |||| ||||

|||| |

10

14

6

Sum = 30

10

Table Frequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

A relative frequency distribution lists the categories and the proportion with which each occurs

Calculating Relative Frequency of a Category

11

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Relative Frequency and Percentage Distributions cont

Calculating Percentage

Percentage = (Relative frequency) 100

12

10102016

3

Determine the relative frequency and percentage for the data in Table

Stress on Job Tally Frequency (f)

Very

Somewhat

None

|||| ||||

|||| |||| ||||

|||| |

10

14

6

Sum = 30

13

Table Frequency Distribution of Stress on Job

Solution

Stress on Job Relative Frequency Percentage

Very

Somewhat

None

1030 = 333

1430 = 467

630 = 200

333(100) = 333

467(100) = 467

200(100) = 200

Sum = 100 Sum = 100

14

Table Relative Frequency and Percentage Distributions of Stress on Job

215

Nominal Data (Tabular Summary) - Organizationclassification

bull Tabulation

bull The diagrammatic or graphical representation

16

217

Nominal Data (Frequency)

Bar Charts are often used to display frequencieshellipIs there a better way to order these Would Bar Chart look different if we plotted ldquorelative frequencyrdquo rather than ldquofrequencyrdquo 218

Nominal Data (Relative Frequency)

Pie Charts show relative frequencieshellip

10102016

4

Graphical Presentation of Qualitative Data

Definition

A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph

It is used to display and compare the number frequency or other measure (eg mean) for different discrete categories or groups

19

Figure Bar graph for the frequency distribution of

Table

0

2

4

6

8

10

12

14

16

Very Somewhat None

Strees on Job

Fre

qu

en

cy

20

Bar chartsbull The heights or lengths of different bars are

proportional to the size of the category they represent

bull Since the x-axis represents the different categories it has no scale

bull The y-axis does have a scale and this indicates the units of measurement

bull The bars can be drawn either vertically or horizontally

21

Graphical Presentation of Qualitative Data cont

Definition

A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart

Pie charts display how the total data are distributed between different categories

22

Table Calculating Angle Sizes for the Pie Chart

Stress on Job Relative Frequency Angle Size

Very

Somewhat

None

333

467

200

360(333) = 11988

360(467) = 16812 360(200) = 7200

Sum = 100 Sum = 360

23

Figure Pie chart for the percentage distribution of Job Stress

24

None 20

Somewhat

4670

Very

3330

10102016

5

25

Pie chart

Civil status of men in a community

Single

31

Married

41

Divorce

d

11

Widowe

d

1

Free

union

16

Civil status of women in a

community

Single

28

Married

44

Widowe

d

8

Free

union

9

Divorce

d

11

Exercise1 Prepare a frequency distribution of different

characteristics of your class

ndash Gender

ndash Professional background

ndash From where you have got information about this institute

(choose as many as applicable)bull Websitebull Newspaperbull SMSbull Bill boardbull Friend bull Others

2 Also make suitable graphs27 28

Bar chart

Gastrintestinal infections

0

12

3

4

56

7

Cryptos Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Freq

uen

cy

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Fre

qu

en

cy

Males

Females

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

2

Frequency Distributions

A frequency distribution for qualitative data lists

ndash all categories

and

ndash the number of elements that belong to each of the categories

7

Example

A 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

ndash very represents very stressful

ndash somewhat means somewhat stressful and

ndash none stands for not stressful at all

8

Example

9

Some what 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

Solution

Stress on Job Tally Frequency (f)

Very

Somewhat

None

|||| ||||

|||| |||| ||||

|||| |

10

14

6

Sum = 30

10

Table Frequency Distribution of Stress on Job

Relative Frequency and Percentage Distributions

A relative frequency distribution lists the categories and the proportion with which each occurs

Calculating Relative Frequency of a Category

11

sfrequencie all of Sum

category that ofFrequency category a offrequency lativeRe

Relative Frequency and Percentage Distributions cont

Calculating Percentage

Percentage = (Relative frequency) 100

12

10102016

3

Determine the relative frequency and percentage for the data in Table

Stress on Job Tally Frequency (f)

Very

Somewhat

None

|||| ||||

|||| |||| ||||

|||| |

10

14

6

Sum = 30

13

Table Frequency Distribution of Stress on Job

Solution

Stress on Job Relative Frequency Percentage

Very

Somewhat

None

1030 = 333

1430 = 467

630 = 200

333(100) = 333

467(100) = 467

200(100) = 200

Sum = 100 Sum = 100

14

Table Relative Frequency and Percentage Distributions of Stress on Job

215

Nominal Data (Tabular Summary) - Organizationclassification

bull Tabulation

bull The diagrammatic or graphical representation

16

217

Nominal Data (Frequency)

Bar Charts are often used to display frequencieshellipIs there a better way to order these Would Bar Chart look different if we plotted ldquorelative frequencyrdquo rather than ldquofrequencyrdquo 218

Nominal Data (Relative Frequency)

Pie Charts show relative frequencieshellip

10102016

4

Graphical Presentation of Qualitative Data

Definition

A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph

It is used to display and compare the number frequency or other measure (eg mean) for different discrete categories or groups

19

Figure Bar graph for the frequency distribution of

Table

0

2

4

6

8

10

12

14

16

Very Somewhat None

Strees on Job

Fre

qu

en

cy

20

Bar chartsbull The heights or lengths of different bars are

proportional to the size of the category they represent

bull Since the x-axis represents the different categories it has no scale

bull The y-axis does have a scale and this indicates the units of measurement

bull The bars can be drawn either vertically or horizontally

21

Graphical Presentation of Qualitative Data cont

Definition

A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart

Pie charts display how the total data are distributed between different categories

22

Table Calculating Angle Sizes for the Pie Chart

Stress on Job Relative Frequency Angle Size

Very

Somewhat

None

333

467

200

360(333) = 11988

360(467) = 16812 360(200) = 7200

Sum = 100 Sum = 360

23

Figure Pie chart for the percentage distribution of Job Stress

24

None 20

Somewhat

4670

Very

3330

10102016

5

25

Pie chart

Civil status of men in a community

Single

31

Married

41

Divorce

d

11

Widowe

d

1

Free

union

16

Civil status of women in a

community

Single

28

Married

44

Widowe

d

8

Free

union

9

Divorce

d

11

Exercise1 Prepare a frequency distribution of different

characteristics of your class

ndash Gender

ndash Professional background

ndash From where you have got information about this institute

(choose as many as applicable)bull Websitebull Newspaperbull SMSbull Bill boardbull Friend bull Others

2 Also make suitable graphs27 28

Bar chart

Gastrintestinal infections

0

12

3

4

56

7

Cryptos Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Freq

uen

cy

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Fre

qu

en

cy

Males

Females

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

3

Determine the relative frequency and percentage for the data in Table

Stress on Job Tally Frequency (f)

Very

Somewhat

None

|||| ||||

|||| |||| ||||

|||| |

10

14

6

Sum = 30

13

Table Frequency Distribution of Stress on Job

Solution

Stress on Job Relative Frequency Percentage

Very

Somewhat

None

1030 = 333

1430 = 467

630 = 200

333(100) = 333

467(100) = 467

200(100) = 200

Sum = 100 Sum = 100

14

Table Relative Frequency and Percentage Distributions of Stress on Job

215

Nominal Data (Tabular Summary) - Organizationclassification

bull Tabulation

bull The diagrammatic or graphical representation

16

217

Nominal Data (Frequency)

Bar Charts are often used to display frequencieshellipIs there a better way to order these Would Bar Chart look different if we plotted ldquorelative frequencyrdquo rather than ldquofrequencyrdquo 218

Nominal Data (Relative Frequency)

Pie Charts show relative frequencieshellip

10102016

4

Graphical Presentation of Qualitative Data

Definition

A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph

It is used to display and compare the number frequency or other measure (eg mean) for different discrete categories or groups

19

Figure Bar graph for the frequency distribution of

Table

0

2

4

6

8

10

12

14

16

Very Somewhat None

Strees on Job

Fre

qu

en

cy

20

Bar chartsbull The heights or lengths of different bars are

proportional to the size of the category they represent

bull Since the x-axis represents the different categories it has no scale

bull The y-axis does have a scale and this indicates the units of measurement

bull The bars can be drawn either vertically or horizontally

21

Graphical Presentation of Qualitative Data cont

Definition

A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart

Pie charts display how the total data are distributed between different categories

22

Table Calculating Angle Sizes for the Pie Chart

Stress on Job Relative Frequency Angle Size

Very

Somewhat

None

333

467

200

360(333) = 11988

360(467) = 16812 360(200) = 7200

Sum = 100 Sum = 360

23

Figure Pie chart for the percentage distribution of Job Stress

24

None 20

Somewhat

4670

Very

3330

10102016

5

25

Pie chart

Civil status of men in a community

Single

31

Married

41

Divorce

d

11

Widowe

d

1

Free

union

16

Civil status of women in a

community

Single

28

Married

44

Widowe

d

8

Free

union

9

Divorce

d

11

Exercise1 Prepare a frequency distribution of different

characteristics of your class

ndash Gender

ndash Professional background

ndash From where you have got information about this institute

(choose as many as applicable)bull Websitebull Newspaperbull SMSbull Bill boardbull Friend bull Others

2 Also make suitable graphs27 28

Bar chart

Gastrintestinal infections

0

12

3

4

56

7

Cryptos Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Freq

uen

cy

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Fre

qu

en

cy

Males

Females

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

4

Graphical Presentation of Qualitative Data

Definition

A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph

It is used to display and compare the number frequency or other measure (eg mean) for different discrete categories or groups

19

Figure Bar graph for the frequency distribution of

Table

0

2

4

6

8

10

12

14

16

Very Somewhat None

Strees on Job

Fre

qu

en

cy

20

Bar chartsbull The heights or lengths of different bars are

proportional to the size of the category they represent

bull Since the x-axis represents the different categories it has no scale

bull The y-axis does have a scale and this indicates the units of measurement

bull The bars can be drawn either vertically or horizontally

21

Graphical Presentation of Qualitative Data cont

Definition

A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart

Pie charts display how the total data are distributed between different categories

22

Table Calculating Angle Sizes for the Pie Chart

Stress on Job Relative Frequency Angle Size

Very

Somewhat

None

333

467

200

360(333) = 11988

360(467) = 16812 360(200) = 7200

Sum = 100 Sum = 360

23

Figure Pie chart for the percentage distribution of Job Stress

24

None 20

Somewhat

4670

Very

3330

10102016

5

25

Pie chart

Civil status of men in a community

Single

31

Married

41

Divorce

d

11

Widowe

d

1

Free

union

16

Civil status of women in a

community

Single

28

Married

44

Widowe

d

8

Free

union

9

Divorce

d

11

Exercise1 Prepare a frequency distribution of different

characteristics of your class

ndash Gender

ndash Professional background

ndash From where you have got information about this institute

(choose as many as applicable)bull Websitebull Newspaperbull SMSbull Bill boardbull Friend bull Others

2 Also make suitable graphs27 28

Bar chart

Gastrintestinal infections

0

12

3

4

56

7

Cryptos Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Freq

uen

cy

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Fre

qu

en

cy

Males

Females

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

5

25

Pie chart

Civil status of men in a community

Single

31

Married

41

Divorce

d

11

Widowe

d

1

Free

union

16

Civil status of women in a

community

Single

28

Married

44

Widowe

d

8

Free

union

9

Divorce

d

11

Exercise1 Prepare a frequency distribution of different

characteristics of your class

ndash Gender

ndash Professional background

ndash From where you have got information about this institute

(choose as many as applicable)bull Websitebull Newspaperbull SMSbull Bill boardbull Friend bull Others

2 Also make suitable graphs27 28

Bar chart

Gastrintestinal infections

0

12

3

4

56

7

Cryptos Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Freq

uen

cy

Grouped bar chart

Gastrointestinal infections

0

1

2

3

4

5

Crypt Ehistolyt Ecoli Giardia Rotavirus Shigella

Agents

Fre

qu

en

cy

Males

Females

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

6

Bar Chart

Source Quarterly Country Summaries 2008

56

77

6670

3845

57

46

0

20

40

60

80

100

Country 1 Country 3 Country 4 Country 5

Perc

en

t

Household Ownership of at Least 1 Net or ITN 2008

Any net

LLIN

32

Stacked bar chart

36

26

9

9

11

20

0 20 40 60 80 100

2008

2007

Percent

Year

ACT Quinine

Amodiaquine Sulfadoxine-Pyrimethamine

Chloroquine Other

Children lt5 with Fever who Took Specific Antimalarial 2007-2008

34

35

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

36

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

7

ORGANIZING AND GRAPHING QUANTITATIVE DATA

bull Ordered array

bull Frequency Distributions

ndash Constructing Frequency Distribution Tables

ndash Relative and Percentage Distributions

bull Graphing Grouped Data

ndash Histograms

ndash Polygons

ndash Stem and leaf plots37

Organizing amp Grouping Data

bull To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers)

bull The main purpose in grouping data now is summarization

bull Summarization is a way of making it easier to understand the information in data

38

Ordered array

bull A first step in organizing data

bull An ordered array is a

listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value

bull If the number of measurements to be ordered is of any appreciable size the use of a computer is highly desirable

40

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

8

Frequency Distributions

43

Frequency Distributions

bull A frequency distribution for quantitative data lists

ndashall the classes

and

ndashthe number of values that belong to each class

bull Data presented in the form of a frequency distribution are called grouped data

44

45

Frequency Distributions

46

Weekly Earnings

(dollars)

Number of Employees

f

401 to 600

601 to 800

801 to 1000

1001 to 1200

1201 to 1400

1401 to 1600

9

22

39

15

9

6

Table 27 Weekly Earnings of 100 Employees of a Company

Variable

Third class

Lower limit of the sixth class

Upper limit of the sixth class

Frequency of the third class

Frequency column

Class width

Essential Question

How do we construct a frequency distribution table

Process of Constructing a Frequency Table

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

9

STEP 1 Determine the tentative number of classes (k)

k = 1 + 3322 log N

Always round ndash off

Note The number of classes should be between 5 and 15 The actual number of classes may be affected by convenience or other subjective factors

Process of Constructing a Frequency Table

STEP 2 Determine the range (R)

R = Highest Value ndash Lowest Value

STEP 3 Find the class width by dividing the range by the number of classes

(Always round ndash off )

k

Rc

classesofnumber

Rangewidthclass

STEP 4 Write the classes or categories starting with the lowest score Stop when the class already includes the highest score

Add the class width to the starting point to get the second lower class limit Add the class width to the second lower class limit to get the third and so on List the lower class limits in a vertical column and enter the upper class limits which can be easily identified at this stage

STEP 5 Determine the frequency for each class by referring to the tally columns and present the results in a table

When constructing frequency tables the following guidelines should be followed

The classes must be mutually exclusive That is each score must belong to exactly one classInclude all classes even if the frequency might be zero

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

10

All classes should have the same width although it is sometimes impossible to avoid open ndashended intervals such as ldquo65 years or olderrdquo

The number of classes should be between 5 and 15

Letrsquos Try

bull Time magazine collected information on all 464 people who died from gunfire in the Philippines during one week Here are the ages of 50 men randomly selected from that population Construct a frequency distribution table

19 18 30 40 41 33 73 25

23 25 21 33 65 17 20 76

47 69 20 31 18 24 35 24

17 36 65 70 22 25 65 16

24 29 42 37 26 46 27 63

21 27 23 25 71 37 75 25

27 23

Determine the tentative number of classes (K)

K = 1 + 3 322 log N

= 1 + 3322 log 50

= 1 + 3322 (169897)

= 664

Round ndash off the result to the next integer if the decimal part exceeds 0

K = 7

Determine the range

R = Highest Value ndash Lowest Value

R = 76 ndash 16 = 60

Find the class width (c)

Round ndash off the quotient if the decimal part exceeds 0

k

Rc

classesofnumber

Rangewidthclass

95787

60c

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

11

Write the classes starting with lowest score

Classes Tally Marks Freq

70 ndash 78

61 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 33

16 ndash 24

---

---

5

5027

14

17

Using Table

bull What is the lower class limit of the highest class

bull Upper class limit of the lowest class

bull Find the class mark of the class 43 ndash 51

bull What is the frequency of the class 16 ndash 24

Classes True Class boundaries

Tally Marks Freq x

70 ndash 7861 ndash 6952 ndash 6043 ndash 5134 ndash 4225 ndash 3316 ndash 24

695 ndash 785605 ndash 695515 ndash 605 425 ndash 515335 ndash 425245 ndash 335155 ndash 245

------

550

2714 17

74655647382920

Example

Table 29 gives the total home runs hit by all players of each of the 30 Major League Baseball teams during the 2002 season Construct a frequency distribution table

64

Table 29 Home Runs Hit by Major League Baseball

Teams During the 2002 Season

Team Home Runs Team Home Runs

Anaheim

Arizona

Atlanta

Baltimore

Boston

Chicago Cubs

Chicago White Sox

Cincinnati

Cleveland

Colorado

Detroit

Florida

Houston

Kansas City

Los Angeles

152

165

164

165

177

200

217

169

192

152

124

146

167

140

155

Milwaukee

Minnesota

Montreal

New York Mets

New York Yankees

Oakland

Philadelphia

Pittsburgh

St Louis

San Diego

San Francisco

Seattle

Tampa Bay

Texas

Toronto

139

167

162

160

223

205

165

142

175

136

198

152

133

230

187

65

Solution 2-3

2215

124230classeach of width eApproximat

66

Now we round this approximate width to a convenient number ndash say 22

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

12

Solution 2-3

The lower limit of the first class can be taken as 124 or any number less than 124 Suppose we take 124 as the lower limit of the first class Then our classes will be

124 ndash 145 146 ndash 167 168 ndash 189 190 ndash 211

and 212 - 233

67

Table 210 Frequency Distribution for the Data of

Table 29

68

Total Home Runs Tally f

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

|||| |

|||| |||| |||

||||

||||

|||

6

13

4

4

3

sumf = 30

Relative Frequency and Percentage Distributions

Relative Frequency and Percentage Distributions

69

100 frequency) (Relative Percentage

sfrequencie all of Sum

class that ofFrequency class a offrequency Relative

f

f

Example 2-4

Calculate the relative frequencies and percentages for Table 210

70

Solution 2-4

71

Total Home

RunsClass Boundaries

Relative Frequency

Percentage

124 ndash 145

146 ndash 167

168 ndash 189

190 ndash 211

212 - 233

1235 to less than 1455

1455 to less than 1675

1675 to less than 1895

1895 to less than 2115

2115 to less than 2335

200

433

133

133

100

200

433

133

133

100

Sum = 999 Sum = 999

Table 211 Relative Frequency and Percentage Distributions for

Table 210

Graphing Grouped Data

Definition

A histogram is a graph in which 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 bars In a histogram the bars are drawn adjacent to each other

72

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

13

Figure 23 Frequency histogram for Table 210

73

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

15

12

9

6

3

0

Fre

qu

en

cy

Figure 24 Relative frequency histogram for Table

210

74

124 -145

146 -167

168 -

189

190 -

211

212 -

233Total home runs

50

40

30

20

10

0

Re

lati

ve

Fre

qu

en

cy

Graphing Grouped Data cont

Definition

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

75

Figure 25 Frequency polygon for Table 210

76

124 -145

146 -167

168 -

189

190 -

211

212 -

233

15

12

9

6

3

0

Fre

qu

en

cy

Figure 26 Frequency Distribution curve

77

Fre

qu

en

cy

x

Example 2-5

The following data give the average travel time from home to work (in minutes) for 50 states The data are based on a sample survey of 700000 households conducted by the Census Bureau (USA TODAY August 6 2001)

78

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

14

Example 2-5

79

224

197

216

154

211

182

270

219

221

254

237

217

232

196

249

198

176

160

214

255

267

177

161

238

201

234

225

223

219

171

235

237

244

219

225

212

287

156

243

292

199

227

267

261

312

236

242

227

226

208

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

Solution 2-5

6326

415231classeach of width eApproximat

80

Solution 2-5

Class Boundaries fRelative

Frequency Percentage

15 to less than 18

18 to less than 21

21 to less than 24

24 to less than 27

27 to less than 30

30 to less than 33

7

7

23

9

3

1

14

14

46

18

06

02

14

14

46

18

6

2

Σf = 50 Sum = 100 Sum = 100

81

Table 212 Frequency Relative Frequency and Percentage

Distributions of Average Travel Time to Work

Example 2-6

The 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 1

1 3 3 0 2 5 1 2 3 4

2 1 2 2 1 2 2 1 1 1

4 2 1 1 2 1 1 4 1 3

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

82

Solution 2-6

Vehicles OwnedNumber of

Households (f)

0

1

2

3

4

5

2

18

11

4

3

2

Σf = 4083

Table 213 Frequency Distribution of Vehicles Owned

Figure 27 Bar graph for Table 213

0

2

4

6

8

10

12

14

16

18

20

No Car 1 Car 2 Cars 3 Cars 4 Cars 5 Cars

Vehicles owned

Fre

qu

en

cy

84

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

15

STEM-AND-LEAF DISPLAYS

Definition

In a stem-and-leaf display of quantitative data each value is divided into two portions ndash a stem and a leaf The leaves for each stem are shown separately in a display

85

Example 2-8

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

Construct a stem-and-leaf display

86

75

69

83

52

72

84

80

81

77

96

61

64

65

76

71

79

86

87

71

79

72

87

68

92

93

50

57

95

92

98

Solution 2-8

To construct a stem-and-leaf display for these scores we split each score into two parts The first part contains the first digit which is called the stem The second part contains the second digit which is called the leaf

87

Solution 2-8

We observe from the data that the stems for all scores are 5 6 7 8 and 9 because all the scores lie in the range 50 to 98

88

Figure 213 Stem-and-leaf display

89

5

6

7

8

9

2

5

Leaf for 75

Leaf for 52

Stems

Solution 2-8

After we have listed the stems we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line

90

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

16

Figure 214 Stem-and-leaf display of test scores

5

6

7

8

9

2 0 7

5 9 1 8 4

5 9 1 2 6 9 7 1 2

0 7 1 6 3 4 7

6 3 5 2 2 8

91

Figure 215 Ranked stem-and-leaf display of test

scores

5

6

7

8

9

0 2 7

1 4 5 8 9

1 1 2 2 5 6 7 9 9

0 1 3 4 6 7 7

2 2 3 5 6 8

92

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

93

880

1210

1151

1081

985

630

721

1231

1175

1075

932

952

1023

850

1100

775

825

1140

1235

1000

750

750

915

1140

965

1191

1370

960

1035

1280

Solution 2-9

6

7

8

9

10

11

12

13

30

75 50 21 50

80 25 50

32 52 15 60 85 65

23 81 35 75 00

91 51 40 75 40 00

10 31 35 80

70

94

Figure 216Stem-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

95

Example 2-10

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

96

0

1

2

3

4

5

6

7

8

6

1 7 9

2 6

2 4 7 8

1 5 6 9 9

3 6 8

2 4 4 5 7

5 6

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship

10102016

17

Solution 2-10

97

0 ndash 2 3 ndash 5

6 ndash 8

6 1 7 9 2 6

2 4 7 8 1 5 6 9 9 3 6 8

2 4 4 5 7 5 6

Figure 217 Grouped stem-and-leaf display

298

Scatter Diagramhellip

bull Example 29 A real estate agent wanted to know to what extent the selling price of a home is related to its sizehellip

bull Collect the data

1) Determine the independent variable (X ndashhouse size) and the dependent variable (Y ndashselling price)

Relationship between peoplersquos weight and height

Relationship between of calories eaten and weight gainloss

299

Scatter Diagramhellip

bull It appears that in fact there is a relationship that is the greater the house size the greater the selling pricehellip

2100

Patterns of Scatter Diagramshellip

bull Linearity and Direction are two concepts we are interested in

Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship