Bivariate Linear Correlation:
Study behaviour of 2 variables simultaneously.
Correlation may be classified based on following criterias:
1) Number of variable
2 variables == simple/bivariate
More than 2 == multiple
(partial correlation )(crops == rainfall , temperature , money)
2) Degree of correlation:
If the 2 variables have exact relation (change is proportional), linear == > y = a +bx
3) Types (positive , negative or zero):
If both the variables increase together == positive
If one variable increase and the other decreases == negative.
Interpretation of correlation:
Correlation correlation (between -1 and +1)
Spurious(nonsense) correlation
Rainfall, birth rate
Methods:
1) Scatter Diagram
2) Karl Pearson’s product moment correlation
3) Spearman Rank Correlation.
Scatter Diagram:
The pairs of values of X and Y are represented by a point/dot on the
graph paper, such graphs are called scatter diagram.
Karl Pearson’s Product moment correlation:
1) Correlation coefficient gives numerical measure of nature and extent
of correlation
2) Does not depend on units used for measurements.
3) Lies b/w - 1 and +1
4) Independent of change of origin and scale.
Rank correlation(Spearman’s correlation)
Non Repeated rank:
1.
X Y R1 R2 d = R 1 - R2 d2
84 65 3 3 0 0
89 75 2 1 1 1
72 58 6 5 1 1
75 60 5 4 1 1
90 70 1 2 -1 1
62 54 8 7 1 1
64 51 7 8 -1 1
78 57 4 6 -2 4
10
R = 0.8809
2.
d = R1 -
X Y R1 R2 d2
R2
67 78 1 2 -1 1
42 80 8 1 7 49
53 72 6 5 1 1
65 75 2 4 -2 4 R=
62
60
68
63
3
4
6
7
-3
-3
9
9
0.07143
54 77 5 3 2 4
44 60 7 8 -1 1
78
Problems for practice:
1) Find the product moment correlation coefficient for following
data:
Σx =260
Σy =450
Σxy =7050
Σx2 =4720
Σy2 =12230
n=20
2) From the following data , find the coefficient of correlation:
No of pairs of observation=12
Sum of X values = 35
Sum of Y values = 60
Sum of product of X and Y values =105
Sum of square of X values =148
Sum of square of Y values =450
3)
X Y
60 50
30 25
37 33
30 27
42 40
37 33
55 50
45 42
4)Find the rank correlation coefficient
X Y
52 65
33 59
47 72
65 72
43 82
33 60
54 57
66 58
75 72
70 90
5)Find the rank correlation coefficient
X Y
20 15
22 17
18 16
17 10
10 5
25 19
7 4
15 8
