Smile

Jerry 10

Elan 6

George 8

Newman 9

Kramer 7

Smile

Jerry 10

Elan 6

George 8

Newman 9

Kramer 7

You can calculate:

Central tendency

Variability

You could graph the data

Talk

Jerry 5

Elan 1

George 3

Newman 4

Kramer 2

You can calculate:

Central tendency

Variability

You could graph the data

Bivariate Distribution

Smile Talk

Jerry 10 5

Elan 6 1

George 8 3

Newman 9 4

Kramer 7 2

Positive Correlation

Smile Talk

Jerry 10 5

Elan 6 1

George 8 3

Newman 9 4

Kramer 7 2

Positive Correlation

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

Correlation

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

r = 1.00

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

. . .. .

r = .64

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

. .. .

r = .64

.

Practice

Smile Talk

Jerry 9 5

Elan 2 1

George 5 3

Newman 4 4

Kramer 3 2

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

.

.. ..

Regression Line

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Smil

e

.

.. ..

Frown Talk

Jerry 10 2

Elan 6 6

George 8 4

Newman 9 3

Kramer 7 5

Frown Talk

Jerry 10 2

Elan 6 6

George 8 4

Newman 9 3

Kramer 7 5

Negative Correlation

Negative Correlation

0

2

4

6

8

10

12

2 3 4 5 6

Talk

Fro

wn

r = - 1.00

Negative Correlation

0

2

4

6

8

10

12

1 2 3 4 5

Talk

Fro

wn

.

.

. .. r = - .85

Gas in car Talk

Jerry 10 8

Elan 6 9

George 8 3

Newman 9 4

Kramer 7 3

Gas in car Talk

Jerry 10 8

Elan 6 9

George 8 3

Newman 9 4

Kramer 7 3

Zero Correlation

Zero Correlation

0

2

4

6

8

10

12

3 4 5 6 7 8 9

Talk

Gas

in c

ar

.

... .r = .00

Correlation Coefficient

• The sign of a correlation (+ or -) only tells you the direction of the relationship

• The value of the correlation only tells you about the size of the relationship (i.e., how close the scores are to the regression line)

Excel Example

• Which is a bigger effect?

r = .40 or r = -.40

How are they different?

Interpreting an r value

• What is a “big r”

• Rule of thumb:

Small r = .10

Medium r = .30

Large r = .50

Practice

• Do you think the following variables are positively, negatively or uncorrelated to each other?

• Alcohol consumption & Driving skills• Miles of running a day & speed in a foot race• Height & GPA• Forearm length & foot length• Test #1 score and Test#2 score

Statistics Needed

• Need to find the best place to draw the regression line on a scatter plot

• Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)

Covariance• Correlations are based on the statistic called

covariance

• Reflects the degree to which two variables vary together– Expressed in deviations measured in the original

units in which X and Y are measured

1

))((

N

YYXXCOVXY

• Note how it is similar to a variance– If Ys were changed to Xs it would be s2

• How it works (positive vs. negative vs. zero)

1

))((

N

YYXXCOVXY

Computational formula

1

NN

YXXY

COVXY

Smile Talk

Jerry 9 5

Elan 2 1

George 5 3

Newman 4 4

Kramer 3 2

1

NN

YXXY

COVXY

Ingredients:

∑XY

∑X

∑Y

N

Smile (Y)

Talk (X)

XY

Jerry 9 5

Elan 2 1

George 5 3

Newman 4 4

Kramer 3 2

Smile (Y)

Talk (X)

XY

Jerry 9 5 45

Elan 2 1 2

George 5 3 15

Newman 4 4 16

Kramer 3 2 6

Smile (Y)

Talk (X)

XY

Jerry 9 5 45

Elan 2 1 2

George 5 3 15

Newman 4 4 16

Kramer 3 2 6

∑ = 23

∑ = 15

∑ = 84

N = 5

∑XY = 84

∑Y = 23

∑X = 15

N = 5

1

NN

YXXY

COVXY

∑XY = 84

∑Y = 23

∑X = 15

N = 5

1

84

NN

YX

COVXY

∑XY = 84

∑Y = 23

∑X = 15

N = 5

1

)23(1584

NNCOVXY

∑XY = 84

∑Y = 23

∑X = 15

N = 5

155

)23(1584

XYCOV

∑XY = 84

∑Y = 23

∑X = 15

N = 5

155

)23(1584

75.3

Problem!

• The size of the covariance depends on the standard deviation of the variables

• COVXY = 3.75 might occur because– There is a strong correlation between X and

Y, but small standard deviations

– There is a weak correlation between X and Y, but large standard deviations

Solution

• Need to “standardize” the covariance

• Remember how we standardized single scores

Correlation

YX

XY

SS

COVr

Smile (Y)

Talk (X)

XY

Jerry 9 5

Elan 2 1

George 5 3

Newman 4 4

Kramer 3 2

SY =2.70 SX =1.58

Correlation

YX

XY

SS

COVr

Correlation

)70.2(58.1

75.388.

Practice

• You are interested in if candy intake is related to childhood depression. You collect data from 5 children.

Practice

Candy Depression

Charlie 5 55

Augustus 7 43

Veruca 4 59

Mike 3 108

Violet 4 65

Scandy = 1.52 Sdepression = 24.82

Practice

Candy

(X)

Depression

(Y)

XY

Charlie 5 55 275

Augustus 7 43 301

Veruca 4 59 236

Mike 3 108 324

Violet 4 65 260

∑

Practice

Candy

(X)

Depression

(Y)

XY

Charlie 5 55 275

Augustus 7 43 301

Veruca 4 59 236

Mike 3 108 324

Violet 4 65 260

∑ 23 330 1396

∑XY = 1396

∑Y = 330

∑X = 23

N = 5

1

NN

YXXY

COVXY

∑XY = 1396

∑Y = 330

∑X = 23

N = 5

155

)330(231396

5.30

Correlation

YX

XY

SS

COVr

COV = -30.5

Sx = 1.52

Sy = 24.82

Correlation

)82.24(52.1

5.3081.

COV = -30.5

Sx = 1.52

Sy = 24.82

Hypothesis testing of r

• Is there a significant relationship between X and Y (or are they independent)– Like the X2

Steps for testing r value

• 1) State the hypothesis

• 2) Find t-critical

• 3) Calculate r value

• 4) Calculate t-observed

• 5) Decision

• 6) Put answer into words

Practice

• Determine if candy consumption is significantly related to depression.– Test at alpha = .05

Practice

Candy Depression

Charlie 5 55

Augustus 7 43

Veruca 4 59

Mike 3 108

Violet 4 65

Scandy = 1.52 Sdepression = 24.82

Step 1

• H1: r is not equal to 0

– The two variables are related to each other

• H0: r is equal to zero

– The two variables are not related to each other

Step 2

• Calculate df = N - 2

• Page 747– First Column are df– Look at an alpha of .05 with two-tails

t distributiondf = 3

0

t distribution

tcrit = 3.182tcrit = -3.182

0

t distribution

tcrit = 3.182tcrit = -3.182

0

Step 3

)82.24(52.1

5.3081.

COV = -30.5

Sx = 1.52

Sy = 24.82

Step 4

• Calculate t-observed

21

2

r

Nrt

Step 4

• Calculate t-observed

2)81.(1

2581.

t

Step 4

• Calculate t-observed

2)81.(1

2581.39.2

Step 5

• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:

– Fail to reject H0

t distribution

tcrit = 3.182tcrit = -3.182

0

t distribution

tcrit = 3.182tcrit = -3.182

0

-2.39

Step 5

• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tIf tobsobs does not fall in the critical region: does not fall in the critical region:

– Fail to reject HFail to reject H00

Step 6

• Determine if candy consumption is significantly related to depression.– Test at alpha = .05

• Candy consumption is not significantly related to depression– Note: this finding is due to the small sample

size

Practice

• Is there a significant (.05) relationship between aggression and happiness?

Aggression Happiness

Mr. Blond 10 9

Mr. Blue 20 4

Mr. Brown 12 5

Mr. Pink 16 6

Mean aggression = 14.50; S2aggression = 19.63

Mean happiness = 6.00; S2happiness = 4.67

Answer

• Cov = -7.33• r = -.76

• t crit = 4.303

• Thus, fail to reject Ho

• Aggression was not significantly related to happiness

)76.(1

2476.65.1

2

Practice

• Situation 1• Based on a sample of 100 subjects you find the

correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.

• Situation 2• Based on a sample of 600 subjects you find the

correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.

Step 1• Situation 1

• H1: r is not equal to 0– The two variables are related to each other

• H0: r is equal to zero– The two variables are not related to each other

• Situation 2

• H1: r is not equal to 0– The two variables are related to each other

• H0: r is equal to zero– The two variables are not related to each other

Step 2

• Situation 1

• df = 98

• t crit = +1.985 and -1.984

• Situation 2

• df = 598

• t crit = +1.96 and -1.96

Step 3

• Situation 1

• r = .15

• Situation 2

• r = .15

Step 4

• Situation 1

• Situation 2

2)15(.100

210015.5.1

215.1

260015.71.3

Step 5

• Situation 1• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:– Fail to reject H0

• Situation 2• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:– Fail to reject H0

Step 6

• Situation 1• Based on a sample of 100 subjects you find the

correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.

• There is not a significant relationship between extraversion and happiness

• Situation 2• Based on a sample of 600 subjects you find the

correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.

• There is a significant relationship between extraversion and happiness.