The Sample Linear Correlation Coefficient
Formula for the Sample Linear
Correlation Coefficient
• rXY (or just r for short) is the
sample linear correlation coefficient
rXY =
i
- X )(Yi - Y ) /(n - 1)
i =1
sX sY
66
64
Sister's Height (inches)
68
Scatter Plot of the Heights of a
Brother and Sister in 11 Families
62
For example, consider 11 families randomly
selected from the population of families with
one brother and one sister, both full grown.
Let Xi denote the height (in inches) of the
brother in the i-th family. Let Yi denote the
height (in inches) of the sister in the i-th
family.
n
60
• rXY measures the strength and direction
of linear association between two
quantitative variables X and Y.
? (X
66
68
70
72
Brother's Height (inches)
1
25
0
0
1
0
1
-3
9
1
1
2
68
64
-1
3
66
65
-3
4
67
63
-2
-1
2
4
5
70
65
1
1
1
1
1
6
71
62
2
-2
-4
4
4
7
70
65
1
1
1
1
1
8
73
64
4
0
0
16
0
9
72
66
3
2
6
9
4
10
65
59
-4
-5
20
16
25
66
62
-3
-2
6
9
4
759
704
0
0
39
74
66
11
X = 759 / 11 = 69
∑i =1
n
rXY =
Y = 704 / 11 = 64
( X i − X ) (Yi −Y ) /(n − 1)
s X sY
=
SX =
74
11 −1
SY =
Scatter Plot of the Heights of a
Brother and Sister in 11 Families
r=0.558
68
(Yi − Y )2
4
5
66
(Xi − X)2
10
2
64
( X i − X ) (Yi −Y )
Xi − X Yi − Y
69
Sister's Height (inches)
Yi
71
62
Xi
60
i
1
66
11 − 1
3. 9
≈ 0 .558
(7 .4 ) (6 .6 )
66
68
70
72
Brother's Height (inches)
Is a Bear's Neck Circumference Correlated with a Bear's Weight?
400
300
Weight in Pounds
200
100
• rXY estimates the population linear
correlation coefficient ?XY .
• rXY is dimensionless and is always
between 1 and -1.
• rXY =1 if and only if all data points fall
perfectly on a line with positive slope.
• rXY =-1 if and only if all data points fall
perfectly on a line with negative slope.
• rXY =0 suggests that there is no linear
association between X and Y.
500
Some Properties of rXY
10
15
20
25
30
Neck Circumference in Inches
2
Is Outside Temperature Associated with Crawling Time?
Nov
Oct
32
Dec
Apr
31
Jun
Feb
Jan
30
Average Crawling Age in Weeks
400
300
200
Aug
Mar
29
100
Weight in Pounds
Sep
Jul
r=0.934
33
500
34
Is a Bear's Neck Circumference Correlated with a Bear's Weight?
May
10
15
20
25
30
30
40
Neck Circumference in Inches
Sep
70
Apr
31
Jun
Feb
Mar
China
Russia Iraq
Morocco
Vietnam
S.Africa
Egypt
Pakistan
Haiti
Uganda
Madagascar
50
30
Jan
Life Expectancy in Years
32
Dec
60
33
Aug
Yemen
Cambodia
Angola
May
30
40
50
70
Japan
France
Canada,Australia
UK
US
Mexico Sri Lanka
29
Average Crawling Age in Weeks
r=-0.699
Nov
Oct
60
Does Watching TV Increase Life Expectancy?
80
34
Is Outside Temperature Associated with Crawling Time?
Jul
50
Average Monthly Temperature (F) in First 6 Months
60
Average Monthly Temperature (F) in First 6 Months
70
0
50
100
150
200
People Per Television
3
Guess the correlation for each pair of variables.
Japan
France
Canada,Australia
UK
US
Mexico Sri Lanka
r=-0.804
China
Russia Iraq
60
Morocco
Vietnam
S.Africa
Egypt
A) Stalk Diameter of Corn Plant
Weight of Corn Plant
B) Person’s Age
Person’s Year of Birth
C) Daily Dow Jones Industrial Average
Daily Rainfall in Seattle
D) # of Ultrasounds During Pregnancy
Birth Weight of Baby
E) U.S. Monthly Ice Cream Cone Sales
Drowning per Month in U.S.
F) Age of Wife
Age of Husband
Pakistan
Haiti
Uganda
Madagascar
50
Life Expectancy in Years
70
80
Does Watching TV Increase Life Expectancy?
Yemen
Cambodia
Angola
-1
0
50
100
150
0
1
200
People Per Television
4