Population Correlation
Coefficient:
Sample Correlation Coefficient:
(b) If r = 1, all the points on the scatter diagram lie on the regression line of positive slope. It is called a ‘perfect positive correlation’.
(c) If r = 0, all the points on the scatter diagram are spread throughout the diagram indicating no correlation between x and y.
“Correlation coefficient is a measure of the closeness of linear relationship between the two variables.”
Correlation Coefficient and
Regression Coefficient:
or
or
Where
Properties of Coefficient of
Correlation:
r_{xy}
= r_{uv}
–1
≤ r ≤ +1
Example:
x 
3 
1 
1 
2 
4 
2 
3 
5 
2 
3 
y 
2 
4 
3 
2 
1 
2 
1 
3 
2 
1 
Required:
(a) Covariance of x and y,
(b) Standard deviation of x and y,
(c) Coefficient of correlation, and
(d) Scatter diagram.
Solution:
(a) Covariance of x and y:
x 
y 
x – μ_{x } 
x – μ_{y } 
(x
– μ_{x})( x – μ_{y}) 
(x – μ_{x})^{2 } 
(x – μ_{y})^{2 } 
3 
2 
0.4 
–0.1 
–0.04 
9 
4 
1 
4 
–1.6 
1.9 
–3.04 
1 
16 
1 
3 
–1.6 
0.9 
–1.44 
1 
9 
2 
2 
–0.6 
–0.1 
0.06 
4 
4 
4 
1 
1.4 
–1.1 
–1.54 
16 
1 
2 
2 
–0.6 
–0.1 
0.06 
4 
4 
3 
1 
0.4 
–1.1 
–0.44 
9 
1 
5 
3 
2.4 
0.9 
2.16 
25 
9 
2 
2 
–0.6 
–0.1 
0.06 
4 
4 
3 
1 
0.4 
–1.1 
–0.44 
9 
1 
26 
21 


–4.6 
82 
53 
(b) Standard deviation of x and y:
(c) Coefficient of correlation:
(d) Scatter diagram:
Example:
Calculate:
(a) Covariance of x and y,
(b) Variances of x and y,
(c) Coefficient of correlation, and
(d) Coefficient of determination.
For the following sample data:
x 
1 
2 
4 
6 
8 
10 
14 
15 
18 
20 
y 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
Solution:
(a) Covariance of x and y:
x 
y 


(
)(
) 
( )^{2 } 
( )^{2 } 
1 
10 
–8.8 
–45 
396 
77.44 
2025 
2 
20 
–7.8 
–35 
273 
60.84 
1225 
4 
30 
–5.8 
–25 
145 
33.64 
625 
6 
40 
–3.8 
–15 
57 
14.44 
225 
8 
50 
–1.8 
–5 
9 
3.24 
25 
10 
60 
0.2 
5 
1 
0.04 
25 
14 
70 
4.2 
15 
63 
17.64 
225 
15 
80 
5.2 
25 
130 
27.04 
625 
18 
90 
8.2 
35 
287 
67.24 
1225 
20 
100 
10.2 
45 
459 
104.04 
2025 
98 
550 


1820 
405.6 
8250 
(b) Variances of x and y:
(c) Coefficient of correlation:
(d) Coefficient of determination:
r^{2} = b × d
r^{2} = 4.48720 × 0.22059 = 0.9898 = 98.98%
P(–0.6745 ≤ z ≤ 0.6745) = 0.5
P.E.
= 0.6745 × σ_{r}
or
P.E.
= 0.6745 ×
P(–P.E.
≤ r ≤ P.E.) = 0.5
P(–3P.E.
≤ r ≤ 3P.E.) = 0.9544
Where d_{i} = x_{i} – y_{i} (the difference between the rankings).
Number
of ranks (n) 
Critical
value (r_{s}) 
5 
1.0 
6 
0.89 
7 
0.79 
8 
0.74 
9 
0.74 
10 
0.65 
20 
0.45 
25 
0.40 
50 
0.28 
Example:
Ranks of 9 students in a class in History (x) and Geography (y) are as follows:
Students 
I 
II 
III 
IV 
V 
VI 
VII 
VIII 
IX 
x 
1 
9 
7 
4 
5 
3 
8 
2 
6 
y 
4 
5 
6 
3 
7 
2 
8 
1 
9 
Calculate Spearman’s Rank Correlation Coefficient and test its significance.
Solution:
Students 
x 
y 
d
= x – y 
d^{2 } 
I 
1 
4 
–3 
9 
II 
9 
5 
4 
16 
III 
7 
6 
1 
1 
IV 
4 
3 
1 
1 
V 
5 
7 
–2 
4 
VI 
3 
2 
1 
1 
VII 
8 
8 
0 
0 
VIII 
2 
1 
1 
1 
IX 
6 
9 
–3 
9 
Total 
45 
45 
0 
42 
Where d_{i} = x_{i}
– y_{i}
Critical value of r_{s} for n = 9 and α = 0.05 is 0.74
Since 0.65 is less than the critical value of 0.74, r_{s} is insignificant.