1 Formula for the Sample Linear Correlation Coefficient
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1 Formula for the Sample Linear Correlation Coefficient
sample linear correlation coefficient. • rXY measures the strength and direction of
linear association between two quantitative variables X and Y. Formula for the ...
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
