Chapter 2: Organizing Data STP 226: Elements of Statistics Jenifer Boshes Arizona State University.

Chapter 2:Chapter 2:Organizing DataOrganizing Data

STP 226: Elements of StatisticsSTP 226: Elements of Statistics

Jenifer BoshesJenifer Boshes

Arizona State UniversityArizona State University

2.1: Variables and Data2.1: Variables and Data

VariableVariable

HeightHeight

AgeAge

Number of siblingsNumber of siblings

SexSex

Marital statusMarital status

Blood TypeBlood Type

A A variablevariable is a characteristic that varies from one is a characteristic that varies from one person or thing to another.person or thing to another.

Example 1:Example 1:

Qualitative VariableQualitative Variable

A A qualitative variablequalitative variable is a non-numerically valued is a non-numerically valued variable. (categorical variable)variable. (categorical variable)

Example 2:Example 2:

Quantitative VariableQuantitative Variable

A A quantitative variablequantitative variable is a numerically valued is a numerically valued variable. variable.

Example 3:Example 3:

Types of Quantitative VariablesTypes of Quantitative Variables

• A A discrete variablediscrete variable is a quantitative variable whose is a quantitative variable whose possible values form a finite (or countably infinite) set of possible values form a finite (or countably infinite) set of numbers. numbers.

A A continuous variablecontinuous variable is a quantitative variable whose is a quantitative variable whose possible values form some interval of numbers. possible values form some interval of numbers.

Classify the following as qualitative, discrete, or continuous:Classify the following as qualitative, discrete, or continuous:

Height Height Age Age Number of siblings Number of siblings Place of birth Place of birth Number of credit hours Number of credit hours Eye color Eye color Ounces of coffee drank per day Ounces of coffee drank per day Number of times visited the Grand Canyon Number of times visited the Grand Canyon

Example 4:Example 4:

VariablesVariables

DataDataDataData:: Information obtained by observing values of a Information obtained by observing values of a variable. Data is classified as variable. Data is classified as qualitative dataqualitative data, , quantitative dataquantitative data, , discretediscrete or or continuous datacontinuous data depending on how it was obtained.depending on how it was obtained.

Determine whether the following examples of data are Determine whether the following examples of data are quantitative or qualitative.quantitative or qualitative.

You ask a sample of students how many hours of You ask a sample of students how many hours of sleep they get. sleep they get.

A census is taken of number of cars in a household in A census is taken of number of cars in a household in a city. a city.

First graders are asked about their favorite color.First graders are asked about their favorite color.

Example 5:Example 5:

2.2: Grouping Data2.2: Grouping Data

Why do we group data?Why do we group data?

To simplify large/complicated data sets.To simplify large/complicated data sets.

To further organize data.To further organize data.

To study a particular variable of interest.To study a particular variable of interest.

Example 1 - Cholesterol:Example 1 - Cholesterol:

The total cholesterol level The total cholesterol level for 30 twenty year old for 30 twenty year old males is given. males is given. Construct a grouped-Construct a grouped-data table for the data. data table for the data. Use a class width of 20 Use a class width of 20 and a first cutpoint of and a first cutpoint of 160.160.

180180 186186 200200 210210 190190 210210

230230 200200 198198 240240 220220 200200

210210 160160 180180 196196 200200 250250

200200 210210 250250 260260 170170 190190

210210 180180 180180 190190 200200 260260

Grouped-Data TableGrouped-Data Table

ClassesClassesFrequencyFrequencyRelative FrequencyRelative FrequencyMidpointMidpoint

Be sure to include the following columns:

180180 186186 200200 210210 190190 210210

230230 200200 198198 240240 220220 200200

210210 160160 180180 196196 200200 250250

200200 210210 250250 260260 170170 190190

210210 180180 180180 190190 200200 260260

Example 1 - Cholesterol:Example 1 - Cholesterol:

Cholesterol LevelCholesterol Level FrequencyFrequency Relative Relative FrequencyFrequency

MidpointMidpoint

22 0.0670.067 170170180160 x

Guidelines for Guidelines for Grouping Quantitative DataGrouping Quantitative Data

(1)(1) The number of classes should be small The number of classes should be small enough to effectively describe the data, but enough to effectively describe the data, but large enough to display the relevant large enough to display the relevant characteristics. (Usually 5-20.)characteristics. (Usually 5-20.)

(2)(2) Each observation must belong to one, and Each observation must belong to one, and only one, class.only one, class.

(3)(3) Whenever possible, all classes should have Whenever possible, all classes should have the same width.the same width.

Terminology in Grouping DataTerminology in Grouping Data

ClassesClasses:: Categories for grouping data. Categories for grouping data.

FrequencyFrequency:: The number of The number of observations that fall into a class.observations that fall into a class.

Frequency distributionFrequency distribution:: A table that A table that provides all classes and their provides all classes and their frequencies.frequencies.

Relative frequencyRelative frequency:: The ratio of the The ratio of the frequency of a class to the total number frequency of a class to the total number of observations.of observations.

Lower cutpointLower cutpoint:: The smallest value in The smallest value in a class.a class.

Upper cutpointUpper cutpoint:: The smallest value The smallest value that could go into the next higher class.that could go into the next higher class.

MidpointMidpoint:: The middle of a class; found The middle of a class; found by taking the average the upper and by taking the average the upper and lower cutpoints.lower cutpoints.

WidthWidth:: The difference between the The difference between the upper and lower cutpoints.upper and lower cutpoints.

Terminology in Grouping DataTerminology in Grouping Data

Example 2 – Maple Trees:Example 2 – Maple Trees:

The heights of 10 maple The heights of 10 maple trees were recorded as trees were recorded as follows. Determine the follows. Determine the frequency and relative-frequency and relative-frequency distributions frequency distributions for these data. (Use for these data. (Use classes of size 5.)classes of size 5.)

6464 6565 7272 8080 8282

6565 8080 6565 6565 7070

2.3: Graphs and Charts2.3: Graphs and Charts

Graphical Displays for Graphical Displays for Quantitative DataQuantitative DataFrequency histogramFrequency histogram:: A graph that A graph that displays the classes on the horizontal axis displays the classes on the horizontal axis and the frequencies on the vertical axis.and the frequencies on the vertical axis.Relative frequency histogramRelative frequency histogram:: A graph A graph that displays the classes on the horizontal that displays the classes on the horizontal axis and the relative frequencies on the axis and the relative frequencies on the vertical axis.vertical axis.Bar graphBar graph:: Similar to a Similar to a relative frequencyrelative frequency histogramhistogram, but the bars do not touch. This , but the bars do not touch. This is used for qualitative data.is used for qualitative data.

Example 1 - Cholesterol:Example 1 - Cholesterol:

Cholesterol LevelCholesterol Level FrequencyFrequency Relative Relative FrequencyFrequency

MidpointMidpoint

22 0.0670.067 170170

1010 0.3330.333 190190

1111 0.3660.366 210210

22 0.0670.067 230230

33 0.10.1 250250

22 0.0670.067 270270

3030 1.0001.000

180160 x200180 x220200 x240220 x260240 x280260 x

Construct a histogram for the cholesterol data, showing Construct a histogram for the cholesterol data, showing both frequencies and relative frequencies.both frequencies and relative frequencies.

Example 1-Cholesterol:Example 1-Cholesterol:

DotplotsDotplots

Dotplots are particularly useful for showing Dotplots are particularly useful for showing the relative positions of the data in a data the relative positions of the data in a data set or for comparing two or more data set or for comparing two or more data sets. They are most useful for a small sets. They are most useful for a small data set with a moderate range in values.data set with a moderate range in values.To construct a dotplot:To construct a dotplot: Draw a horizontal axis.Draw a horizontal axis. Record each data point by placing a dot over Record each data point by placing a dot over

the appropriate value.the appropriate value.

Example 2 -Cholesterol:Example 2 -Cholesterol:Construct a dotplot for the data.Construct a dotplot for the data.

180180 186186 200200 210210 190190 210210

230230 200200 198198 240240 220220 200200

210210 160160 180180 196196 200200 250250

200200 210210 250250 260260 170170 190190

210210 180180 180180 190190 200200 260260

Stem-and-Leaf DiagramStem-and-Leaf Diagram

(1)(1) Select the leading digit(s) from the Select the leading digit(s) from the data and list in a vertical column. data and list in a vertical column. (STEM) (STEM)

(2)(2) Write the final digit of each number to Write the final digit of each number to the right of the appropriate leading the right of the appropriate leading digit. (LEAVES)digit. (LEAVES)

Stem-and-Leaf DiagramStem-and-Leaf Diagram1. Select the leading digit(s) from the data and list in a vertical column. (STEM)1. Select the leading digit(s) from the data and list in a vertical column. (STEM)2. Write the final digit of each number to the right of the appropriate leading digit. (LEAVES)2. Write the final digit of each number to the right of the appropriate leading digit. (LEAVES)

2020 2222 1818 1010 2020

3030 3232 2222 5050 3030

2424 2626 2828 3030 3232

2020 1818 2424 2222 2020

A group of women who had just given birth were asked how many pre-A group of women who had just given birth were asked how many pre-natal visits they had made to a doctor. The following information natal visits they had made to a doctor. The following information was recorded. was recorded.

Example 3a – Pre-natal Visits:Example 3a – Pre-natal Visits:

Stem-and-Leaf DiagramsStem-and-Leaf Diagrams

2020 2222 1818 1010 2020

3030 3232 2222 5050 3030

2424 2626 2828 3030 3232

2020 1818 2424 2222 2020

Example 3a – Pre-natal Visits:Example 3a – Pre-natal Visits:The stem-and-leaf diagram with The stem-and-leaf diagram with twotwo lines per stem lines per stem

would be:would be:11 00

11 88 88

22 00 00 00 00 22 22 22 44 44

22 66 88

33 00 00 00 22 22

33

44

44

55 00

55

Example 4 - CholesterolExample 4 - Cholesterol

180180 186186 200200 210210 190190 210210

230230 200200 198198 240240 220220 200200

210210 160160 180180 196196 200200 250250

200200 210210 250250 260260 170170 190190

210210 180180 180180 190190 200200 260260

Create a stem-and-leaf plot for the cholesterol data using the first two Create a stem-and-leaf plot for the cholesterol data using the first two digits as the stem. digits as the stem.

1616

1717

1818

1919

2020

2121

2222

2323

2424

2525

2626

Example 4 - CholesterolExample 4 - CholesterolCompare this stem-and-leaf to the histogram for the same data. What Compare this stem-and-leaf to the histogram for the same data. What

similarities do we see? similarities do we see?

1616 00

1717 00

1818 00 00 00 00 66

1919 00 00 00 66 88

2020 00 00 00 00 00 00

2121 00 00 00 00 00

2222 00

2323 00

2424 00

2525 00 00

2626 00 00

Example 4 - CholesterolExample 4 - CholesterolCompare this stem-and-leaf to the histogram for the same data. What Compare this stem-and-leaf to the histogram for the same data. What

similarities do we see? (Note: histogram grouped by classes of similarities do we see? (Note: histogram grouped by classes of size 20, so not exactly the same.)size 20, so not exactly the same.)

1616 00

1717 00

1818 00 00 00 00 66

1919 00 00 00 66 88

2020 00 00 00 00 00 00

2121 00 00 00 00 00

2222 00

2323 00

2424 00

2525 00 00

2626 00 00

Graphing Qualitative DataGraphing Qualitative Data

A A pie chartpie chart is a circle divided into wedge- is a circle divided into wedge-shaped pieces that are proportional to the shaped pieces that are proportional to the relative frequencies.relative frequencies.

A A bar graphbar graph is like a histogram, but the is like a histogram, but the bars do not touch. bars do not touch.

Example 5 – Blood Type: Example 5 – Blood Type: A sample of 105 blood donors at A sample of 105 blood donors at

a clinic can be described as a clinic can be described as follows:follows:

TypeType FrequencyFrequency

AA 4747

BB 2222

ABAB 2020

OO 1616

TotalTotal 105105

Type, Type, xx FrequencyFrequency Relative Relative FrequencyFrequency

AA 4747 0.4480.448

BB 2222 0.2100.210

ABAB 2020 0.1900.190

OO 1616 0.1520.152

105105 11

Bar Graph for Blood Type Ex

05

101520253035404550

A B AB O

Blood Type

Fre

qu

en

cy

Pie Chart for Blood Type Ex

A

B

AB

O

A

B

AB

O

2.4: Distribution Shapes2.4: Distribution Shapes

Distribution ShapesDistribution Shapes

Group Assignment/PracticeGroup Assignment/Practice

Here are the ages for the CEOs of the 30 top-ranked Here are the ages for the CEOs of the 30 top-ranked small companies in America from small companies in America from ForbesForbes. Using . Using this data, produce the following:this data, produce the following:

(a) A grouped-data table (a) A grouped-data table using classes of size fiveusing classes of size fiveand starting with age 35.and starting with age 35.

(b) Construct a histogram(b) Construct a histogramfor the data.for the data.

(c) A dotplot.(c) A dotplot.(d) A stem-and-leaf diagram(d) A stem-and-leaf diagram

with one line per stem.with one line per stem.(e) A stem-and-leaf diagram(e) A stem-and-leaf diagram

with two lines per stem.with two lines per stem.

59593838474753536060

69694444505056566363

40404848535361614141

44444949555562624343

55556161616153534848

48485555626243434848

BibliographyBibliography

Some of the textbook images embedded in Some of the textbook images embedded in the slides were taken from:the slides were taken from:

Elementary StatisticsElementary Statistics, Sixth Edition; by , Sixth Edition; by Weiss; Addison Wesley Publishing Weiss; Addison Wesley Publishing Company Company

Copyright © 2005, Pearson Education, Inc.Copyright © 2005, Pearson Education, Inc.