Describing Relationships: Correlation

RelationshipsRelationshipsPositive relationshipPositive relationship Pairs of scores vary in the same direction.Pairs of scores vary in the same direction. When one goes up, the other goes up; when one When one goes up, the other goes up; when one

goes down the other goes down.goes down the other goes down.

As temperature goes up, water consumption As temperature goes up, water consumption goes up.goes up.

As perceived status of a car goes up, the price As perceived status of a car goes up, the price goes up.goes up.

Chapter 6 2

RelationshipsRelationshipsNegative relationshipNegative relationship Pairs of scores vary in the opposite direction.Pairs of scores vary in the opposite direction. When one goes up, the other goes down; when When one goes up, the other goes down; when

one goes down the other goes up.one goes down the other goes up.

When the temperature goes down, gas usage for When the temperature goes down, gas usage for heating goes up.heating goes up.

As the altitude goes up, the percentage of As the altitude goes up, the percentage of oxygen in the air goes down.oxygen in the air goes down.

Chapter 6 3

RelationshipsRelationshipsNo RelationshipNo Relationship Pairs of scores vary independently of each other.Pairs of scores vary independently of each other.

The number of students in the Commons has no The number of students in the Commons has no relationship to the size of cargo barges on the relationship to the size of cargo barges on the Mississippi.Mississippi.

The amount of time studying Statistics has no The amount of time studying Statistics has no relationship to the on-time arrival percentage for relationship to the on-time arrival percentage for Southwest Airlines. Southwest Airlines.

Chapter 6 4

Scatter plotsScatter plots

Chapter 6 5

Scatter plotsScatter plots

Chapter 6 6

RelationshipsRelationships Strong or Weak relationship?Strong or Weak relationship?

The more closely the dot cluster The more closely the dot cluster approximates a straight line, the approximates a straight line, the stronger (the more regular) the stronger (the more regular) the relationship will be.relationship will be.

Chapter 6 7

Correlation coefficient (r)Correlation coefficient (r) A correlation coefficient is a number between -1 A correlation coefficient is a number between -1

and 1 that describes the relationship between and 1 that describes the relationship between variables.variables.

Named for Karl PearsonNamed for Karl Pearson

The sign of r indicates the type of linear The sign of r indicates the type of linear relationship, whether positive or negative.relationship, whether positive or negative.

The numerical value of r, without regard to sign, The numerical value of r, without regard to sign, indicates the strength of the linear relationship.indicates the strength of the linear relationship.

Chapter 6 8

Estimating correlationsEstimating correlations Correlation Example GeneratorCorrelation Example Generator

Chapter 6 9

Caution!!Caution!! The strength of a correlation The strength of a correlation

does NOT signify does NOT signify causalitycausality..

Page 137 !!!Page 137 !!!

Chapter 6 10

Now….the fun!!!Now….the fun!!! Calculating a correlation.Calculating a correlation.

Use the computational formula on Use the computational formula on page 143.page 143.

Chapter 6 11

Correlation calculation stepsCorrelation calculation steps1.1. Assign a value to n.Assign a value to n.

2.2. Sum all scores for x and y.Sum all scores for x and y.

3.3. Multiply each pair (x*y) and sum the products.Multiply each pair (x*y) and sum the products.

4.4. Square each x and find the sum of the squares.Square each x and find the sum of the squares.

5.5. Square each y and find the sum of the squares.Square each y and find the sum of the squares.

6.6. Calculate the intermediate values for:Calculate the intermediate values for:1.1. Sum of productsSum of products

2.2. Sum of squares for xSum of squares for x

3.3. Sum of squares for ySum of squares for y

7.7. Plug in intermediate values to solve for rPlug in intermediate values to solve for r

Chapter 6 12

Correlation CalculationCorrelation Calculation

r = SPr = SPxyxy

√ √ SSSSxxSSSSyy

Chapter 6 13

Correlation calculation Correlation calculation formulaformula

Sum of productsSum of products

SPSPxyxy= = ΣΣ XY - XY - ((ΣΣ X)( X)(ΣΣ Y) Y)

nn Sum of squares for X Sum of squares for X

SSSSx x = = Σ(X) Σ(X)22 – – (ΣX)(ΣX)2 2

nn

Sum of squares for Y Sum of squares for Y

SSSSyy= = Σ(Y)Σ(Y)22 – – (ΣY)(ΣY)2 2

nn

Chapter 6 14

PracticePractice Calculate the value of r, using the Calculate the value of r, using the

computational formula for the data computational formula for the data on page 143.on page 143.

PairPair Sent, XSent, X Received, YReceived, Y

DorisDoris 1313 1414

SteveSteve 99 1818

MikeMike 77 1212

AndreaAndrea 55 1010

JohnJohn 11 66

Answer on page 487

Chapter 6 15

PracticePractice Calculate the value of r, using the Calculate the value of r, using the

computational formula for the computational formula for the following data.following data.

PairPair XX YY

AA 11 22

BB 33 44

CC 22 33

DD 33 22

EE 11 00

FF 22 33 Answer on page 487

Chapter 6 16

PracticePractice CalculationsCalculations

PairPair XX YY XYXY XX22 YY22

AA 11 22 22 11 44

BB 33 44 1212 99 1616

CC 22 33 66 44 99

DD 33 22 66 99 44

EE 11 00 00 11 00

FF 22 33 66 44 99

ΣΣ 1212 1414 3232 2828 4242

Answer on page 487

Chapter 6 17

Correlation calculationCorrelation calculationSPSPxyxy= Σ XY - = Σ XY - (Σ X)((Σ X)(ΣΣ Y) Y) = 32 – = 32 – (12)(14)(12)(14) = 32- = 32-168168 = 32-28 = 4 = 32-28 = 4

nn 6 6 6 6

SSSSx x = Σ(X)= Σ(X)22 – – (ΣX)(ΣX)22 = 28 – = 28 – (144)(144) = 28-24 = 4 = 28-24 = 4 nn 6 6

SSSSyy= Σ(Y)= Σ(Y)22 – – (ΣY)(ΣY)22 == 42 – 42 – (196)(196) = 42-32.67 = 9.33 = 42-32.67 = 9.33 nn 6 6

r = r = SP SPxy xy = = 4 4 = 4 = 4 = 4 = 4

√√ SSSSxxSSSSyy √ √ (4)(9.33)(4)(9.33) √ 37.32 6.11 √ 37.32 6.11

= .65= .65

Chapter 6 18

Gather the following data from at Gather the following data from at least 10 people in the class and least 10 people in the class and calculate the correlation coefficient.calculate the correlation coefficient.

The number of miles you live from The number of miles you live from the campus and the average number the campus and the average number of hours you work each week.of hours you work each week.

Chapter 6 19

OutliersOutliers The best strategy is to report the The best strategy is to report the

correlation that includes and exclude correlation that includes and exclude the outliers.the outliers.

Chapter 6 20

Correlation MatrixCorrelation MatrixCorrelationsCorrelations

AgeAge College GPACollege GPA High School High School GPAGPA

GenderGender

AgeAge Pearson Pearson CorrelationCorrelation

1.001.0000

.2228.2228 -.0376-.0376 .0813.0813

Sig. (2-tailed)Sig. (2-tailed) ______ .000.000 .511.511 .138.138

NN 335335 333333 307307 335335

College College GPAGPA

Pearson Pearson CorrelationCorrelation

.222.22288

1.0001.000 .2521.2521 .2069.2069

Sig. (2-tailed)Sig. (2-tailed) .000.000 ______ .000.000 .000.000

NN 333333 335335 306306 334334

HS GPAHS GPA Pearson Pearson CorrelationCorrelation

-.037-.03766

.2521.2521 1.0001.000 .2981.2981

Sig. (2-tailed)Sig. (2-tailed) .511.511 .000.000 ______ .000.000

NN 307307 306306 307307 307307

GenderGender Pearson Pearson CorrelationCorrelation

.081.08133

.2069.2069 .2981.2981 1.0001.000

Sig. (2-tailed)Sig. (2-tailed) .138.138 .000.000 .000.000 ______

NN 335335 334334 307307 336336

Chapter 6 21