Chapter 4 Two-Variables Analysis

09/19-20/2013

Outline

Issue: How to identify the linear relationship between two variables? Relationship: • Scatter Plot is a collection of observations on an X-Y graph

• Covariance conveys the direction of the potential relationship• Correlation coefficient measures the strength of

a linear relationship between two variables Causality and predictions: • Least squares line

Scatter Plot

Scatter Plot: Degree of Association

Covariance A measure of the strength of a linear

relationship between two variables While the magnitude changes with the units, its

sign conveys direction only. Positive covariance Positive linear relationship Negative covariance Negative linear relationship

Population Covariance

Sample Covariance

Relation

N

YXn

iYiXi

XY

1

))((

1

))((1

n

YYXXs

n

iii

XY 1*

n

ns

XYXY

Correlation coefficient

Unit-free and always between -1 (perfectly negative linear relationship) and +1 (perfectly positive linear relationship)

The greater the absolute value of the correlation coefficient, the stronger the linear relationship.

Population Correlation

Sample Correlation

YX

XYXY

YX

XY

ss

sr

Least squares line

A Unique line that describes the relationship between two variables, when one causes the other. It has the smallest sum of squared error!

0 1 2 3 4 5 6 7 8 90

20

40

60

80

100

120

140

160

180

f(x) = 18.6246719160105 x + 16.3328083989501

Boiling time and Ounces of water

Boiling time (sec.)Linear (Boiling time (sec.))

Ounces of water

Boilin

g t

ime (

sec.)

Sum of Squared Error

2

1

)'( i

n

ii YYSSE

iY is the observed value and

'iY is the predicted value.