Stat 104 – Lecture 6
Correlation
• Linear Association
–How closely do the points on the
scatter plot represent a straight line?
–The correlation coefficient gives the
direction of and quantifies the
strength of the linear association
between two quantitative variables.
1
Correlation
• Standardize y
• Standardize x
zy =
y− y
sy
zx =
x−x
sx
2
Standardized Bite Force
Bite Force vs Body Mass of Canidae
3
2
1
0
-1
-1
0
1
2
Standardized Body Mass
3
3
1
Stat 104 – Lecture 6
Correlation Coefficient
zx z y
∑
r=
n −1
( x − x )( y − y )
r=∑
(n − 1)s x s y
4
Correlation Coefficient
• Body mass and Bite force
zx z y
∑
r=
n −1
=
26 .4796
27
• r = 0.9807
5
Correlation Coefficient
• There is a very strong positive
correlation, linear association,
between the body mass and
bite force for the various
species of Canidae.
6
2
Stat 104 – Lecture 6
JMP
• Analyze – Multivariate
methods – Multivariate
• Y, Columns
–
–
Body mass
BF ca (Bite force at the
canine)
7
Multivariate
Correlations
Body Mass (kg)
BFca (N)
Body Mass (kg)
1.0000
0.9807
BFca (N)
0.9807
1.0000
Scatterplot Matrix
40
35
30
25
Body
20
Mass (kg)
15
10
5
500
400
300
BFca (N)
200
100
8
5
10 15
20
25
30 35
40
100
200
300
400
50
Correlation Properties
• The sign of r indicates the direction of
the association.
• The value of r is always between
–1 and +1.
• Correlation has no units.
• Correlation is not affected by changes
of center or scale.
9
3
Stat 104 – Lecture 6
Algebra Review
• The equation of a straight line
• y = mx + b
– m is the slope – the change in y
over the change in x – or rise
over run.
– b is the y-intercept – the value
where the line cuts the y axis.
10
y = 3x + 2
15
10
y
5
0
-5
-10
-15
-5
-4
-3
-2
-1
0
1
2
x
3
4
5
11
Review
• y = 3x + 2
–x = 0
y = 2 (y-intercept)
–x = 3
y = 11
–Change in y (+9) divided by the
change in x (+3) gives the slope, 3.
12
4