Scatter DiagramsScatter Diagrams

A scatter plotscatter plot is a graph that may be used to represent the relationship between two variables. Also referred to as a scatter diagramscatter diagram.

Dependent and Dependent and Independent VariablesIndependent Variables

A dependent variabledependent variable is the variable to be predicted or explained in a regression model. This variable is assumed to be functionally related to the independent variable.

Dependent and Dependent and Independent VariablesIndependent Variables

An independent variableindependent variable is the variable related to the dependent variable in a regression equation. The independent variable is used in a regression model to estimate the value of the dependent variable.

Two Variable Two Variable RelationshipsRelationships

(Figure 11-1)(Figure 11-1)

X

Y

(a) Linear(a) Linear

Two Variable Two Variable RelationshipsRelationships

(Figure 11-1)(Figure 11-1)

X

Y

(b) Linear(b) Linear

Two Variable Two Variable RelationshipsRelationships

(Figure 11-1)(Figure 11-1)

X

Y

(c) Curvilinear(c) Curvilinear

Two Variable Two Variable RelationshipsRelationships

(Figure 11-1)(Figure 11-1)

X

Y

(d) Curvilinear(d) Curvilinear

Two Variable Two Variable RelationshipsRelationships

(Figure 11-1)(Figure 11-1)

X

Y

(e) No (e) No RelationshipRelationship

CorrelationCorrelation

The correlation coefficientcorrelation coefficient is a quantitative measure of the strength of the linear relationship between two variables. The correlation ranges from + 1.0 to - 1.0. A correlation of 1.0 indicates a perfect linear relationship, whereas a correlation of 0 indicates no linear relationship.

CorrelationCorrelation

SAMPLE CORRELATION COEFFICIENTSAMPLE CORRELATION COEFFICIENT

where:r = Sample correlation coefficientn = Sample sizex = Value of the independent variabley = Value of the dependent variable

])(][)([

))((22 yyxx

yyxxr

CorrelationCorrelation(Example 11-1)(Example 11-1)

Sales Years with MidwestSales 1Years with Midwest 0.832534056 1

Excel Correlation OutputExcel Correlation Output

(Figure 11-5)(Figure 11-5)

Correlation between Years and Sales

CorrelationCorrelation

TEST STATISTIC FOR CORRELATIONTEST STATISTIC FOR CORRELATION

where:t = Number of standard deviations r

is from 0r = Simple correlation coefficientn = Sample size

21 2

nr

rt

2ndf

228.2025. t0

Correlation Significance Correlation Significance TestTest

(Example 11-1)(Example 11-1)

Rejection Region /2 = 0.025

Since t=4.752 > 2.048, reject H0, there is a significant linear relationship

228.2025. t

Rejection Region /2 = 0.025

05.0

0.0:

)(0.0:0

AH

ncorrelationoH

752 . 4

106931 . 0 1

8325 . 0

21

2

nr

rt

CorrelationCorrelation

Spurious correlationSpurious correlation occurs when there is a correlation between two otherwise unrelated variables.