Copyright ©2011 Nelson Education Limited Describing Bivariate Data CHAPTER 3.

Copyright ©2011 Nelson Education Limited

Describing Bivariate Data

CHAPTER 3CHAPTER 3CHAPTER 3CHAPTER 3


Bivariate DataBivariate Data• When two variables are measured on a single

experimental unit, the resulting data are called bivariate data.bivariate data.

• You can describe each variable individually, and you can also explore the relationshiprelationship between the two variables.

• Bivariate data can be described with– GraphsGraphs– Numerical MeasuresNumerical Measures


Graphs for Qualitative VariablesGraphs for Qualitative Variables• When at least one of the variables is qualitative, you

can use comparative pie charts or bar charts.comparative pie charts or bar charts.

Variable #1 =

Variable #2 =

Do you think that men and women are treated equally in the workplace?

Do you think that men and women are treated equally in the workplace?

Opinion

Gender

WomenMen


Comparative Bar ChartsComparative Bar Charts

• Stacked Bar ChartStacked Bar Chart

Percent

OpinionGender

No OpinionDisagreeAgreeWomenMenWomenMenWomenMen

70

60

50

40

30

20

10

0

Percent

OpinionGender

No OpinionDisagreeAgreeWomenMenWomenMenWomenMen

70

60

50

40

30

20

10

0

• Side-by-Side Bar ChartSide-by-Side Bar ChartDescribe the relationship between opinion and gender:

More women than men feel that they are not treated equally in the workplace.

More women than men feel that they are not treated equally in the workplace.

Percent

Opinion No OpinionDisagreeAgree

120

100

80

60

40

20

0

GenderMenWomen

Percent

Opinion No OpinionDisagreeAgree

120

100

80

60

40

20

0

GenderMenWomen


Two Quantitative VariablesTwo Quantitative Variables

When both of the variables are quantitative, call one variable x and the other y. A single measurement is a pair of numbers (x, y) that can be plotted using a two-dimensional graph called a scatterplot.scatterplot.

y

x

(2, 5)

x = 2

y = 5


• What pattern pattern or form form do you see? • Straight line upward or downward • Curve or no pattern at all

• How strongstrong is the pattern?• Strong, moderate, or weak

• Are there any unusual observationsunusual observations?• Clusters or outliers

Describing the ScatterplotDescribing the Scatterplot


Positive linear - strong Negative linear -weak

Curvilinear No relationship

ExamplesExamples


Numerical Measures for Numerical Measures for Two Quantitative VariablesTwo Quantitative Variables

• Assume that the two variables x and y exhibit a linear pattern linear pattern or formform.

• There are two numerical measures to describe– The strength strength and direction direction of the relationship

between x and y.

– The form form of the relationship.


The Correlation CoefficientThe Correlation Coefficient• The strength and direction of the relationship between

x and y are measured using the correlation correlation coefficient, coefficient, rr..

where

sx = standard deviation of the x’s

sy = standard deviation of the y’s

sx = standard deviation of the x’s

sy = standard deviation of the y’s

yx

xy

ss

sr

yx

xy

ss

sr

1

))((

nn

yxyx

s

iiii

xy 1

))((

nn

yxyx

s

iiii

xy


ExampleExample

Residence 1 2 3 4 5

x (sq meters) 126.3 134.5 137.5 144.0 148.6y ($000) 178.5 188.6 168.8 229.8 205.2

•The scatterplot indicates a positive linear relationship.

•The scatterplot indicates a positive linear relationship.

• Living area x and selling price y of 5 homes.


ExampleExample x y xy

126.3 178.5 22544.55134.7 188.6 25404.42137.5 168.8 23210.00144.0 229.8 33091.20148.6 205.2 30492.72691.1 970.9 134742.89

038.24 18.194

604.8 22.138

Calculate

y

x

sy

sx

yx

xy

ss

sr

0.659)038.24(604.8

27.136

1

))((

nn

yxyx

s

iiii

xy

27.1364

5)9.970)(1.691(

134742.89


Interpreting Interpreting rr

All points fall exactly on a straight line.

Strong relationship; either positive or negative

Weak relationship; random scatter of points

•-1 r 1

•r 0

•r 1 or –1

•r = 1 or –1

Sign of r indicates direction of the linear relationship.

APPLETAPPLETMY


The Regression LineThe Regression Line• Sometimes x and y are related in a particular way—the

value of y dependsdepends on the value of x.

– y = response variable (dependent)– x = explanatory variable (independent)

• The form of the linear relationship between x and y can be described by fitting a line as best we can through the points. This is the regression line, regression line,

yy = = aa + + bxbx..

– a = y-intercept of the line– b = slope of the line

APPLETAPPLETMY


The Regression LineThe Regression Line• To find the slope and y-intercept of the

best fitting line, use:

b rsy

sx

a y bx

b rsy

sx

a y bx

• The least squares

• regression line is y = a + bx


x

y

s

srb

xby a

ExampleExample

84.1604.8

038.24)659(.

194.18 1.84(138.22)

-60.23

xy 84.114.60 :Line Regression

x y xy126.3 178.5 22544.55134.7 188.6 25404.42137.5 168.8 23210.00144.0 229.8 33091.20148.6 205.2 30492.72691.1 970.9 134742.89

659.

038.24 18.194

604.8 22.138

Calculate

r

sy

sx

y

x


Predict:

ExampleExample• Predict the selling price for another residence

with 140 square meter of living area.

)140(84.114.60 197.471 or $197,471xy 84.114.60


Key ConceptsKey ConceptsI. Bivariate DataI. Bivariate Data

1. Both qualitative and quantitative variables

2. Describing each variable separately3. Describing the relationship between the variables

II. Describing Two Qualitative VariablesII. Describing Two Qualitative Variables1. Side-by-Side pie charts2. Comparative line charts3. Comparative bar chartsSide-by-SideStacked

4. Relative frequencies to describe the relationship between the two variables


Key ConceptsKey ConceptsIII. Describing Two Quantitative VariablesIII. Describing Two Quantitative Variables

1. Scatterplots

Linear or nonlinear pattern

Strength of relationship

Unusual observations; clusters and outliers2. Covariance and correlation coefficient (resistant to outliers?)

3. The best fitting line

Calculating the slope and y-intercept

Graphing the line

Using the line for prediction

Copyright ©2011 Nelson Education Limited Describing Bivariate Data CHAPTER 3.

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