The Sample Linear Correlation Coefficient
rXY (or just r for short) is the sample linear correlation coefficient.
rXY measures the strength and direction of linear association between two quantitative variables X and Y .
rXY =
Pn
i=1 (Xi
− X̄)(Yi − Ȳ )/(n − 1)
,
sX sY
where n is number of pairs of observations, X̄ is the sample average of the X data, Ȳ is the sample average of the Y
data, sX is the sample standard deviation of the X data, and sY is the sample standard deviation of the Y data.
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 ith family. Let Yi denote the height (in inches) of
the sister in the ith family.
i
1
2
3
4
5
6
7
8
9
10
11
Xi
71
68
66
67
70
71
70
73
72
65
66
759
Yi
69
64
65
63
65
62
65
64
66
59
62
704
Xi − X̄
2
-1
-3
-2
1
2
1
4
3
-4
-3
0
X̄ = 759/11 = 69
rXY =
Pn
Yi − Ȳ
5
0
1
-1
1
-2
1
0
2
-5
-2
0
(Xi − X̄)(Yi − Ȳ )
10
0
-3
2
1
-4
1
0
6
20
6
39
Ȳ = 704/11 = 64
i=1 (Xi
SX =
s
(Xi − X̄)2
4
1
9
4
1
4
1
16
9
16
9
74
74
11 − 1
SY =
s
(Yi − Ȳ )2
25
0
1
1
1
4
1
0
4
25
4
66
66
11 − 1
− X̄)(Yi − Ȳ )/(n − 1)
3.9
≈ 0.558
=p
sX sY
(7.4)(6.6)
rXY estimates the population linear correlation coefficient ρXY .
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 means there is no linear association between X and Y .
Write the letter for each pair of variables on the number line to indicate the value of rXY that you would expect to see.
Pair
A
X
Stalk Diameter of Corn Plant
Y
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
−1
0
1