Correlation
 A measure of the strength of the linear association between two numerical variables.
1

Sample Covariance
 Measure of the co-variability between two numerical variables.
  x
 x
 y
 y
 n

1
2

Sample Correlation
 The sample covariance scaled to account for variation in the x ’s and y ’s.
r



 x n

 x
1
 s
 x y s y
 y

3

Properties
 The value of the correlation coefficient, r , is always between

–1: a perfect negative linear relationship
 +1: a perfect positive linear relationship
4

Properties
 r = 0: There is no linear relationship between the two numerical variables.
 Random scatter
 There could be a relationship, but not one that is linear.
5

Properties
 The correlation coefficient, r , does not have any units.
 Changing the scales of the numerical variables will not change the value of the correlation coefficient.
6

CO
2 r
 and Temperature


 x n

 x
1
 s
 y s
 y
 x y r r


19

16 .
63 .
68808
32108

0 .
22878


0 .
8977
7

CO
2 and Temperature
 There is a strong, positive linear association between the carbon dioxide concentration and the temperature.
8

CO
2 and Temperature
 Is the linear association between the carbon dioxide concentration and temperature statistically significant?
9

Step 1: Hypotheses
H
H
0
A
:

:



0
0
(no linear
(linear associatio associatio n) n)
10

Step 2: Test statistic t t
 r

0
1
 n
 r
2
2

8 .
64
P
 value

1
0


0 .
0001
.
8977
0 .
8059
18
11

Step 3: Decision
 Reject the null hypothesis because the P-value is so small.
12

Step 4: Conclusion
 There is a statistically significant linear association between carbon dioxide concentration and temperature.
13

Connection
 The test for the statistical significance of correlation is exactly the same as the test for the statistical significance of the estimated slope.
14

Connection
R
2
R
2


 
2

0 .
8977

2 
0 .
8059
15

Difference
 R 2 can be interpreted as a % of total variation.
 r has a sign (+/ –) that matches the direction of the association and cannot be interpreted as a %.
16

JMP
 Analyze – Multivariate
Methods – Multivariate
 Y, Columns: CO2, Temp
 Multivariate
 Pairwise correlations
17

Multivariate
Correlations
CO2
Temp
CO2
1.0000
0.8977
Temp
0.8977
1.0000
Scatterplot Matrix
370
360
350
340
330
320
CO2
310
14.6
14.5
14.4
14.3
14.2
14.1
14.0
13.9
13.8
310 320 330 340 350 360 370 13.8
14.0
Pairwise Correlations
Variable
Temp by Variable
CO2
Correlation
0.8977
Count
20
Signif Prob
<.0001*
Temp
14.2
14.4
14.6
-.8 -.6 -.4 -.2 0 .2 .4 .6 .8
18