Agenda• Correlation

CORRELATION

• Co-relation

• 2 variables tend to “go together”

• Does knowing a person’s score on one variable give you an idea of their score on other variable?

• Predict the degree of co-occurrence or association between 2 variables

Correlation• A measure of association between

– Two ordinal variables– An ordinal and an interval/ratio variables– Two variables

• Correlation analysis examines if one variable changes by a certain amount, how much the other variable would change in which direction

Example: Test Score Ten students

took a Chemistry class and a Biology class together

Compared final exam scores in two classes

J

I

H

G

F

E

D

C

B

A

9894

9390

9089

8388

8580

8378

8276

6972

6870

6562

BiologyChemistry

Example: Scatter Plot

60

65

70

75

80

85

90

95

100

60 65 70 75 80 85 90 95 100

Chemistry

Bio

logy

Students who get higher score in the Chemistry class also get higher score in the Biology class

Positive Correlation• When scores of two variables move

together in the same direction, we say that these variables are positively (or directly) correlated

• There is a positive correlation between between the chemistry final score and the biology final score

• When two variables are positively correlated, the scatter plot shows a trend line that runs from lower-left to upper-right

Example: Test Score The same

students also took the Art class

Compared final exam scores in Chemistry and Art

J

I

H

G

F

E

D

C

B

A

7194

7390

7489

7588

7780

7778

7876

7472

8070

9062

ArtChemistry

Example: Scatter Plot

60

65

70

75

80

85

90

95

60 65 70 75 80 85 90 95 100

Chemistry

Art

Students who get higher score in the Chemistry class get lower score in the Art Classc

Negative Correlation• When scores of two variables move in

opposite directions, we say that these variables are negatively (or inversely) correlated

• There is a negative correlation between between the chemistry final score and the art final score

• When two variables are negatively correlated, the scatter plot shows a trend line that runs from upper-left to lower-right

Example: Test Score The same ten

students also took an English class together

Compare the English final score with the Chemistry final score

J

I

H

G

F

E

D

C

B

A

7694

8990

6089

7588

9080

8078

7576

9072

6070

8562

EnglishChemistry

Example: Test Score

60

65

70

75

80

85

90

95

60 65 70 75 80 85 90 95 100

Chemistry

Engl

ish

Score in Chemistry and Score in English are not related

No Correlation• When the change in one variable does not

affect the change in another variable, we say these variables have no correlation

• There is no correlation between the chemistry final score and the English final score

• When two variables have no correlation, the scatter plot shows the dots scattered throughout the grids

SIGN

• 0: No relationship

• Positive: + • As one variable gets bigger, so does

the 2nd

• Negative: -• As one variable gets

bigger, the 2nd gets smaller

Should your height in inches have anything to do with….• How much you weigh?• What size shoe your wear?• How old you are?• What portion of your tuition you pay?• Your college GPA?• Your skill in basketball?• Your attractiveness to the opposite sex?• Your eye color?

TASK 1• Get in groups of 8-10

• Line up in order of height

• Name Height

• Line up in order of shoe size (men, add 2 sizes)

• Name Shoe size

• Line up in order of how much $ you spent the last time you bought something

• Name $ spent

• Where any of the 3 orders similar?

Do they go together?

• Draw a scatterplot• Each person has 2 scores, 1 on each variable

• Plot each person’s score

• Is there a pattern? E.g. as one variable gets bigger, does the other get bigger too? (Or smaller?)

• A strong relationship makes a diagonal line

Shoe size by height

Height in inches

80706050

Sh

oe

size

(w

om

en

's s

ize

s)16

14

12

10

8

6

4

Weight by height

Height in inches

80706050

We

igh

t in

po

un

ds

220

200

180

160

140

120

100

80

GPA by height

Height in inches

80706050

Gra

de

po

int

ave

rag

e4.5

4.0

3.5

3.0

2.5

2.0

Height in CM by Height in Inches

Height in inches

80706050

He

igh

t in

ce

ntim

ete

rs200

190

180

170

160

150

140

Draw a line through the dots

Height in Cm by Height in Inches

60.00 65.00 70.00 75.00

Height in inches

150.00

160.00

170.00

180.00

190.00

Hei

gh

t in

cen

tim

eter

s

Shoe size by Height

60.00 65.00 70.00 75.00

Height in inches

6.0

8.0

10.0

12.0

14.0S

ho

esiz

e (w

om

en's

siz

es)

GPA by Height

60.00 65.00 70.00 75.00

Height in inches

2.50

3.00

3.50

4.00

Gra

de

po

int

aver

age

Scatter of midterm scores by time to complete test

Fall 2002

Midterm Score

706050403020

# of Minutes to complete test

70

60

50

40

30

20

r = - .4p = .01

Fall 2002

Midterm score

Minutes

Correlation between midterm score and time to finish test

CorrelationsMidterm Score Min. to finish test

Midterm Score Pearson Correlation 1 -0.4Sig. (2-tailed) . 0.01N 40 40

Min. to complete testPearson Correlation -0.4 1Sig. (2-tailed) 0.01 .N 40 40

* Correlation is significant at the 0.05 level (2-tailed).

Fall 2002

Correlation Co-efficient

• A number that indicates how strongly and in which direction 2 variables are correlated with each other

• A correlation co-efficient varies –1 to +1

• Indicated as r

• r = +1: Perfect positive correlation– If one variable increases by x%, another

variable also increases by x%

• r = - 1: Perfect negative correlation

• r = 0: No correlation

Correlation Co-efficient

+1-1 0Negative Positive

Stronger StrongerWeaker

Perfect PerfectNone

Significance Test• Correlation co-efficient also comes with

significance test (p-value)

• p=.05: .05 probability of no correlation in the population = 5% risk of TYPE I Error = 95% confidence level

• If p<.05, reject H0 and support Ha at 95% confidence level

Chemistry & BiologyChemistry Biology

A 62 65

B 70 68

C 72 69

D 76 82

E 78 83

F 80 85

G 88 83

H 89 90

I 90 93

J 94 98

r = .939 p = .000

Significant, Positive and

Strong Correlation

Chemistry & ArtChemistry Art

A 62 90

B 70 80

C 72 74

D 76 78

E 78 77

F 80 77

G 88 75

H 89 74

I 90 73

J 94 71

r = - .839 p = .002

Significant, Negative

and Strong Correlation

Chemistry & EnglishChemistry English

A 62 85

B 70 60

C 72 90

D 76 75

E 78 80

F 80 90

G 88 75

H 89 60

I 90 89

J 94 76

r = -.133 p = .714

Non-Significant

Correlation

SIZE / STRENGTH• Ranges from –1 to + 1• 0 or close to 0 indicates NO relationship• +/- .2 - . 39 weak • +/- .4 - .6 moderate• +/- .>6 - .8 strong• +/- .>8 - .9 very strong• +/- 1.00 perfectNegative relationships are NOT weaker!

USE THIS FOR YOUR ANALYSIS!

Types of Correlation r

• Use Spearman rho’s correlation if one or both of your variables are ordinal

• Use Pearson’s r correlation if both of your variables are interval or ratio

• You can interpret both kinds of correlation in the same way

STATISTICAL SIGNIFICANCE

• How different from 0 must r be for there to be some kind of relationship?

• Depends on size of sample, other factors

• IF statistically significant, safe to conclude there is a relationship

• If p < .05– May also be indicated by * next to the correlation

Limitations of r (correlation)

• Correlation does NOT equal cause

C

A B

• Linear relations only

A B

B A “Third variable”

?

?

“Problem of direction”

TASK 2 with your neighbor:

• Write down 3 examples.

1. 2 variables expect to have no association.

2. 2 variables expect to have positive association.

3. 2 variables expect to have negative association.

• Write down 2 examples.

1. 2 variables expect a weak to medium association.

2. 2 variables expect a strong or very strong association.

TASK: Graph job aptitude vs. performance Graph self-esteem vs. depression

100520John

91420Jake

84621Milly

83822Homer

67725Leslie

55827June

09829Bob

181028Jane

210930Joe

DepressionSelf-EsteemPerformance

Job AptitudeName

Job Aptitude by Performance

Supervisor's performance rating

11109876543

Job

ap

titu

de

te

st s

core

32

30

28

26

24

22

20

18

r = .84, p < .005

Self-Esteem by Depression

Depressive symptoms

121086420-2

Se

lf-e

ste

em

12

10

8

6

4

2

0

-2

r = -.92, p < .001

Example from class data

Height in inches, weight, age, GPA, # semesters completed, height in cm,

who pays tuition (high score = self), where sit in class (high score = towards front)

• For which pairs do you expect no relation? Positive? Negative? Strongest? Weakest?

1.00 .032-.021 .725 * .544 *1.00 *-.012Hgt cm

1.00-.018 .058 .144 .032 .234*Sems

1.00-.050-.115-.021 .075GPA

1.00 .724 * .725 * .016Shoe

1.00 .544 * .108Wt.

1.00-.012Ht.

1.00Age

Hgt cm

Sems

compGPAShoeWt.Ht. Age