Correlation Analysis
Chapter 10
Correlation Analysis
J.C. Wang
JC Wang (WMU)
Stat2160
S2160, Chapter 10
1 / 12
Correlation Analysis
Goal and Objective
To learn about the correlation coefficient as measure of strength of the
linear association between two variables.
JC Wang (WMU)
Stat2160
S2160, Chapter 10
2 / 12
Correlation Analysis
Outline
1
Correlation Analysis
Purpose of Correlation Analysis
Correlation Coefficient
Sample Correlation Coefficient
JC Wang (WMU)
Stat2160
S2160, Chapter 10
3 / 12
Correlation Analysis
Outline
1
Correlation Analysis
Purpose of Correlation Analysis
Correlation Coefficient
Sample Correlation Coefficient
JC Wang (WMU)
Stat2160
S2160, Chapter 10
4 / 12
Correlation Analysis
frame title here
The correlation measures the strength of the linear relationship
between numerical variables, for example, the height of men and their
shoe size or height and weight. In these situations the goal is not to
use one variable to predict another but to show the strength of the
linear relationship that exists between the two numerical variables.
JC Wang (WMU)
Stat2160
S2160, Chapter 10
5 / 12
Correlation Analysis
Correlation Coefficient
definition
The strength of linear association between two numerical variables in a
population is determined by the correlation coefficient, ρ, whose range
is −1 to +1.
Linear Relationship as Measured by ρ
strong ←− weak no weak −→
−
−
−
0
+
+
−1
−.65 −.35 0
.35
.65
strong
+
+1
The sign (positive or negative) of the correlation
= the sign of the slope of straight line.
JC Wang (WMU)
Stat2160
S2160, Chapter 10
6 / 12
Correlation Analysis
Graphical Examples
r = sample correlation
r=1
●
12
y
16
●
r = −1
●
●
●
8
●
●
r = 0.87
●
12
y
r = −0.87
●
16
4
●
●
●
●
8
●
4
●
●
2
3
4
5
6
7
82
3
x
JC Wang (WMU)
4
5
6
7
8
x
Stat2160
S2160, Chapter 10
7 / 12
Correlation Analysis
Graphical Examples
continued
r=0
16
r=0
●
●
12
y
●
●
●
●
8
●
16
4
●
r=0
●
●
● ● ● ● ●
●
12
3
4
5
6
7
8
●
x
●
8
y
2
●
●
●
●
●
●
●
● ● ● ● ●
●
●
4
●
2
3
4
5
6
7
8
x
JC Wang (WMU)
Stat2160
S2160, Chapter 10
8 / 12
Correlation Analysis
Sample Correlation Coefficient
Since our interest is the regression analysis, the sample correlation
coefficient (r ) is derived from the coefficient of determination (R 2 , to be
discussed in Chapter 11).
SSR
regressionSumOfSquares
=
totalSumOfSquares
SST
√
The correlation coefficient r = ± R 2 where r takes the sign of the
slope.
R2 =
JC Wang (WMU)
Stat2160
S2160, Chapter 10
9 / 12
Correlation Analysis
Sample Correlation Coefficient
continued
Correlation coefficient can also be calculated by
P
(xi − x)(yj − y )
q
r=p
(xi − x)2 (yj − y )2
JC Wang (WMU)
Stat2160
S2160, Chapter 10
10 / 12
Correlation Analysis
Example
r = 0.9526
x
y
−2
0
2
3
Data
5 −1
10 1
6
15
STAT → EDIT L1 and L2
STAT → TESTS ↓ LinRegTTest
JC Wang (WMU)
Stat2160
S2160, Chapter 10
11 / 12
Correlation Analysis
iClicker Question 10.1
iClicker Question 10.1
JC Wang (WMU)
Stat2160
S2160, Chapter 10
12 / 12