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Chapter:4 Correlation Analysis
I
...
Statistics has originated as a
science of statehood and found applications slowly and steadily in Agriculture,
Economics, Commerce, Biology, Medicine, Industry, planning, education and so on
...

➢ Meaning of Statistics
Statistics is concerned with scientific methods for collecting, organizing,
summarizing, presenting and analysing data as well as deriving valid conclusions and
making reasonable decisions on the basis of this analysis
...

The word ‘statistic’ is used to refer to
1
...

2
...

➢ Definition
Statistics may be defined as the science of collection, presentation analysis and
interpretation of numerical data from the logical analysis
...


Correlation
If the changes in the values of two variables are simultaneous and when the changes
in one are due to the changes in other, the variables are said to be correlated
...

(i) Perfect Positive, Perfect Negative and no correlation:
• Perfect Positive correlation: Sometimes the changes in the values of two
variables are in the same direction i
...
when the values of one variable
increase, the values of the other variable also increase and when the values of
one variable decrease, the values of other variable also decrease, the
correlation between them is said to be positive
...

Age of
husband
Age of wife
•

25

40

44

60

65

22

32

40

55

64

Perfect Negative correlation: when the changes in the values of two
variables are in the opposite direction i
...
when the values of one variable
increase, the values of the other variable decreases and when the values of

one variable decrease, the values of other variable increase, the correlation
between them is said to be negative
...

Price
3
5
7
10
12
Demand
•

15

12

10

9

7

No correlation: No correlation exists when there is no relationship between
two variables
...
e
...

For example: There is no relationship between the amount of tea drunk and
level of intelligence
...
When only two variable are
studied correlation is said to be simple correlation
...

(ii) Linear or Non-linear correlation: If the amount of change in one variable tends
to bear a constant ratio to the amount of change in the other variable then the
correlation is said to be linear correlation
...

If the amount of change in one variable does not tend to bear a constant ratio to
the amount of change in the other variable then the correlation is said to be Nonlinear correlation or curly linear correlation
...

(i) Scatter Diagram
(ii) Karl Pearson’s method
(iii) Spearman’s method of rank correlation
➢ Scatter diagram: This is a very simple method studying the relationship between
two variables
...


X

Figure 1(Strong Positive)

Figure 2(Strong Negative)

Figure 3 (Strong positive correlation)

Figure 4 (Strong negative correlation)

•

•

•

From above scatter diagrams if all the points lies on the straight line in an
increasing order as shown in figure 1, are known as Perfect positive
correlation
...

From above scatter diagrams if the points are not in one straight line but
lie more or less around some straight line (see figure 3), are known as
strong positive correlation if the points show an increasing trend
...


No correlation
12
10

8
6
4
2
0

10

20

30

Axis Title

If the points of the scatter diagram are distributed randomly (see figure 5)
and there exists absence of the relationship between two variables
...
Prepare a
scatter diagram and discuss the relationship between two variables
...

Illustration 2: The following are the values of two variables x and y
...


X

1

2

3

4

5

6

7

8

y

8

7

6

5

4

3

2

1

y
9
8
7
6
5
4
3
2
1
0

1

2

3

4

5

6

7

8

9

Hence, the above scatter diagram shows that the perfect negative correlation
because all the points are lies on the straight line and they show a decreasing
trend
...

(3) If one of the pairs of values is extreme, it does not influence much in deriving the
conclusion
...

Demerits: This method gives an idea about the direction and to some extent the
degree of relationship between the variables, but does not give the exact measure of
the relationship between the variables
...
It is denoted by r
There are two methods of correlation coefficient:

1) Step Deviation Method
...

1) Step Deviation Method:
Step-wise explanation:
Let x and y be any two independent variables (Given)
−

−

Step-1: Find x and y
Step-2: Construct a table:

y

x

(x

yi − y

xi − x
2

i

− x

)

2

(y

i

− y

)

2

( x − x )( y − y )
i

i

2

− 
−
−
−



Step-3: Find   xi − x  ,   yi − y  and   xi − x  yi − y 








Step-4: Apply the below given formula for find the value of r

𝑟=

∑(𝑥 − 𝑥̅ ) (𝑦 − 𝑦̅)
√∑(𝑥 − 𝑥̅ )2 √∑(𝑦 − 𝑦̅)2

•

Properties of the coefficient of correlation:
(1) The coefficient of correlation always lies between -1 and 1 including -1
and 1
...
e
...

(3) The correlation coefficient is an absolute number and it is independent of units of
measurements
...

r = −1  Perfect negative correlation
...

r = near to + 1  Strong positive correlation
...


Example 1: Find the Pearson’s Correlation Coefficient of the following data:
x
y

100
98

101
99

102
99

102
97

100
95

99
92

97
95

98
94

96
90

95
91

Solution:
Step-1:
_

x=

 x = 990

= 99
10
_
 y = 950 = 95
y=
n
10
n

Step-2: Construction of table
x

y

100
101
102
102
100
99
97
98
96
95
x

98
99
99
97
95
92
95
94
90
91

=990

=950

x−x
1
2
3
3
1
0
-2
-1
-3
-4

( x − x ) ( y − y ) ( x − x )( y − y )

y− y

2

3
4
4
2
0
-3
0
-1
-5
-4

1
4
9
9
1
0
4
1
9
16

2

9
16
16
4
0
9
0
1
25
16

3
8
12
6
0
0
0
1
15
16

 y  ( x − x)  ( y − y)  ( x − x)2  ( y − y )  ( x − x )( y − y )
2

=0

=0

=54

=96

=61

Step-3: From table:

 ( x − x) = 54,  ( y − y )
2

2

=96 and

 ( x − x )( y − y ) =61

_
_



x
−
x
y
−
y
  

61

 =
= 0
...

Example 2: The data below gives the marks obtained by 10 students taking Math’s and
Physics tests of 30 marks
...
71
Step-4: Correlation Coefficient , r =
_ 2
_ 2
360
250

 

 x − x  y − y

 


Since, the value of r is near to +1, we can say that there is a strong positive correlation
between the variables
...

Age(X) 56
42
72
36
63
47
55
49
38
42
68
60
B
...
(Y) 147 125 160 118 149 128 150 145 115 140 152 155
Find the correlation coefficient between X and Y

Merits and Limitations of Pearson’s correlation coefficient:
Karl Pearson’s coefficient of correlation is the best measure for representing the relationship
between two variables
...

Limitations
(1) If is based on the assumption of linearity of relationship between the variables
...

(3) The correlation coefficient is highly influenced by extreme pairs of observations
...

•

Spearman’s Rank Correlation coefficient
Calculated by following formula:
6 ∑ 𝑑2
𝑟=1−
,
𝑛(𝑛 2 − 1)

𝑤ℎ𝑒𝑟𝑒 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑖𝑟𝑠

In case finding out rank correlation coefficient when the observations are paired the
above formula can be written as:

𝑟 =1−

(

𝑚
𝑚
6 {∑ 𝑑 2 + 12 (𝑚2 − 1) + 12 (𝑚2 − 1) + ⋯ … … …
...
The
12
value of correlation coefficient by Spearman’s method also lies between -1 and +1
...
Hence ∑ 𝑑 2=0
and the value of r = +1, which shows that perfect positive correlation between the two
variables
...

In

d

2

,

Example 1: Two judges have given ranks to 10 students for their honesty
...
30 = −0
...
Find the rank correlation coefficient of the
following data:
1st Judge

1

3

5

6

2

4

2nd judge

6

2

3

4

1

5

Solution:
Rank given
by 1st judge
1
3
5
6
2
4

Rank given
by 2nd judge
6
2
3
4
1
5

Difference

d2

-5
1
2
2
1
-1

25
1
4
4
1
1
d 2
=36

6 ∑ 𝑑2

6×36

216

𝑟 = 1 − 𝑛(𝑛2−1) = 1 − 216−6 = 1 − 210 = −0
...
5
6
5
8
...
5
5
...
5
5
...
5
7
...
25
1
2
...
25
10
...
25
42
...
25

Difference d
9
6
0
...
5
0
...
5
-5
-0
...
5

m
m
m
m


6  d 2 + ( m2 − 1) + ( m2 − 1) + ( m2 − 1) + ( m2 − 1) 
12
12
12
12

Rank Correlation , r = 1 − 
2
n ( n − 1)
6 200
...
5 + 0
...
5 + 0
...
227

= 1−

Example 4: Find the Coefficient of rank correlation of the following data:
x

25
y
55
Solution:

45
80

x

y

25
45
35
40
15
19
35
42

55
80
30
35
40
42
36
48

35
30
Ranks in x
6
1
4
...
5
2

40
35

15
40

Ranks in y
2
1
8
7
5
4
6
3

19
42

35
36

Difference d
4
0
3
...
5
-1

42
48

d2
16
0
12
...
25
1
 d 2 =200
...
5 + 0
...
786 = 0
...

(2) When the data are of qualitative nature like honesty, beauty, intelligence etc
...

(3) When the dispersion in a series is more this method is useful
...

•

Limitations:

(1) This method does not give accurate results as compared to Pearson’s method
...

(3) The method cannot be used for data given in a bivariate frequency distribution
...
Use
rank correlation coefficient to determine which of the two judges have similar approach

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