Most of the variables show some kind of relationship. For example, there is relationship between profits and dividends paid, income and expenditure, etc. with the help of correlation analysis we can measure in one figure the degree of relationship existing between the variables. Correlation analysis contributes to the understanding of economic behaviour, aids in locating the critically important variables on which others depend, may reveal to the economist the connection by which disturbances spread and suggest to him the paths through which stabilizing forces may become effective.

1. Bivariate Data Analysis
Correlation and regression -Definition,
Explanation of concepts, Karl Pearson
and Spearman’s rank correlation;
Curve Fitting - Linear and quadratic.

2. •Correlation coefficient: statistical index of the degree to
which two variables are associated, or related.
•When the relationship is of a quantitative nature, the
appropriate statistical tools for discovering and
measuring the relationship and expressing it in brief
formula is known as correlation
CORRELATION

3. EXAMPLE OF CORRELATION
Is there an association between:
Children’s IQ and Parents’ IQ
Degree of social trust and number of membership in voluntary
association ?
Urban growth and air quality violations?
GRA funding and number of publication by Ph.D. students
Number of police patrol and number of crime
Grade on exam and time on exam

4. Significance of the study of correlation
• Most of the variables show some kind of relationship. For example,
there is relationship between profits and dividends paid, income and
expenditure, etc. with the help of correlation analysis we can
measure in one figure the degree of relationship existing between
the variables.
• Correlation analysis contributes to the understanding of economic
behaviour, aids in locating the critically important variables on which
others depend, may reveal to the economist the connection by
which disturbances spread and suggest to him the paths through
which stabilizing forces may become effective.

5. NOTE
• Correlation lies between +1 to -1
• A zero correlation indicates that there is no relationship
between the variables
• A correlation of –1 indicates a perfect negative correlation
•A correlation of +1 indicates a perfect positive correlation

6. TYPES OF CORRELATION

7. Positive and negative correlation
•If two related variables are such that when
one increases (decreases), the other also
increases (decreases) – Positive Correlation
•If two variables are such that when one
increases (decreases), the other decreases
(increases) – Negative Correlation

9. Simple, Partial and Multiple Correlation
•When only two variables are studied it is a
problem of simple correlation.
•When three or more variables are studied it is a
problem of either multiple or partial correlation.
In multiple correlation three or more variables are
studied simultaneously.

10. Linear and Non-linear Correlation
•Linear Correlation: When plotted on a
graph it tends to be a perfect line
•Non-linear Correlation: When plotted on
a graph it is not a straight line

14. Karl Pearson’s
Coefficient of Correlation

15. Methods of studying correlation
1. Karl Pearson’s Coefficient of Correlation
2. Spearman’s rank correlation

16. Karl Pearson’s Coefficient Correlation
The formula for computing Pearson Coefficient Correlation (r) is:
  
   
2 2
2 2
.
n xy x y
r
n x x n y y
   

     
Karl Pearson
27 March 1857 – 27 April 1936

17. The value of the coefficient of correlation as obtained
by the formula shall always lie between ±1. When r =
+1, it means there is perfect positive correlation
between the variables. When r = -1, it means there is
perfect negative correlation between the variables.
When r = 0, it means there is no relationship between
the two variables.

18. Calculating a Correlation Coefficient
In Words In Symbols
x

y

xy

2
x

2
y

  
   
2 2
2 2
.
n xy x y
r
n x x n y y
   

     
1. Find the sum of the x-values.
2. Find the sum of the y-values.
3. Multiply each x-value by its
corresponding y-value and find the
sum.
4. Square each x-value and find the sum.
5. Square each y-value and find the sum.
6. Use these five sums to calculate
the correlation coefficient.

19. Example 1:
Calculate the correlation coefficient r for the following data.
  
   
2 2
2 2
n xy x y
r
n x x n y y
   

     
x y xy x2 y2
1 – 3 – 3 1 9
2 – 1 – 2 4 1
3 0 0 9 0
4 1 4 16 1
5 2 10 25 4
15
x
  1
y
   9
xy
  2
55
x
  2
15
y
 
  
 2
2
5(9) 15 1
5(55) 15 5(15) 1
 

  
60
50 74
 0.986

There is a strong positive linear correlation
between x and y.

20. Example 2: A sample of 6 children was selected, data about their age in years and weight in kilograms was
recorded as shown in the following table . It is required to find the correlation between age and weight.
Weight
(Kg)
Age
(years)
serial
No
12
7
1
8
6
2
12
8
3
10
5
4
11
6
5
13
9
6

21. y2
x2
xy
Weight
(Kg)
(y)
Age
(years)
(x)
Sl
no.
144
49
84
12
7
1
64
36
48
8
6
2
144
64
96
12
8
3
100
25
50
10
5
4
121
36
66
11
6
5
169
81
117
13
9
6
∑y2=
742
∑x2=
291
∑xy=
461
∑y=
66
∑x=
41
Total
  
   
2 2
2 2
n xy x y
r
n x x n y y
   

     
r = 0.759
strong direct correlation

22. Example 3:
The following data represents the number of hours 12 different students watched
television during the weekend and the scores of each student who took a test the
following Monday. Calculate the correlation coefficient r.
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50

23. Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score, y 96 85 82 74 95 68 76 84 58 65 75 50
xy 0 85 164 222 285 340 380 420 348 455 525 500
x2 0 1 4 9 9 25 25 25 36 49 49 100
y2 9216 7225 6724 5476 9025 4624 5776 7056 3364 4225 5625 2500
54
x
  908
y
  3724
xy
  2
332
x
  2
70836
y
 
  
   
2 2
2 2
n xy x y
r
n x x n y y
   

     
  
 2
2
12(3724) 54 908
12(332) 54 12(70836) 908


 
0.831
 
There is a strong negative linear correlation.
As the number of hours spent watching TV increases, the test
scores tend to decrease.

24. Example 4: Relationship between Anxiety and Test Scores
Anxiety
(X)
Test
score (Y)
X2 Y2 XY
10 2 100 4 20
8 3 64 9 24
2 9 4 81 18
1 7 1 49 7
5 6 25 36 30
6 5 36 25 30
∑X = 32 ∑Y = 32 ∑X2 = 230 ∑Y2 = 204 ∑XY=129

25.   
94
.
)
200
)(
356
(
1024
774
32
)
204
(
6
32
)
230
(
6
)
32
)(
32
(
)
129
)(
6
(
2
2








r
r = - 0.94
Indirect strong correlation

26. Spearman’s
rank correlation

27. 10.09.1863-17.09.1945
 Charles Edward Spearman, FRS was an
English psychologist known for work in
statistics, as a pioneer of factor analysis,
and for Spearman's rank correlation
coefficient.
 Known for: g factor, Spearman's rank
correlation coefficient, Factor analysis

29. It is a non-parametric measure of correlation.
This procedure makes use of the two sets of ranks that may be
assigned to the sample values of x and Y.
Spearman Rank correlation coefficient could be computed in
the following cases:
Both variables are quantitative.
Both variables are qualitative ordinal.
One variable is quantitative and the other is qualitative
ordinal.

30. PROCEDURE
1. Rank the values of X from 1 to n where n is the
numbers of pairs of values of X and Y in the sample.
2. Rank the values of Y from 1 to n.
3. Compute the value of di for each pair of observation
by subtracting the rank of Yi from the rank of Xi (Xi-
Yi)
4. Square each di and compute ∑𝑑𝑖2
which is the sum
of the squared values.
5. Apply the following formula
1)
n(n
(di)
6
1
r 2
2
s




The value of rs denotes the magnitude and nature of
association giving the same interpretation as simple r.

31. Example 1: In a study of the relationship between level education and income the
following data was obtained. Find the relationship between them and comment.
Income
(Y)
level education
(X)
sample
numbers
25
Preparatory.
A
10
Primary.
B
8
University.
C
10
secondary
D
15
secondary
E
50
illiterate
F
60
University.
G

32. di2
di
Rank
Y
Rank
X
(Y)
(X)
4
2
3
5
25
Preparatory
A
0.25
0.5
5.5
6
10
Primary.
B
30.2
5
-5.5
7
1.5
8
University.
C
4
-2
5.5
3.5
10
secondary
D
0.25
-0.5
4
3.5
15
secondary
E
25
5
2
7
50
illiterate
F
0.25
0.5
1
1.5
60
university.
G
∑ di2=64
1
.
0
)
48
(
7
64
6
1 




s
r
There is an indirect weak
correlation between level
of education and income.

33. Problems
on
Spearman’s
rank correlation

34. Marks
English 56 75 45 71 62 64 58 80 76 61
Math 66 70 40 60 65 56 59 77 67 63
Example 2:

35. English
(mark)
Math
(mark)
Rank
(English)
Rank
(math)
d d2
56 66 9 4 5 25
75 70 3 2 1 1
45 40 10 10 0 0
71 60 4 7 3 9
62 65 6 5 1 1
64 56 5 9 4 16
58 59 8 8 0 0
80 77 1 1 0 0
76 67 2 3 1 1
61 63 7 6 1 1
This indicates a strong positive relationship between the ranks
individuals obtained in the math and English exam. That is, the higher
you ranked in math, the higher you ranked in English also, and vice
versa.

36. Example 3: The following table provides data about the percentage of students who have free university
meals and their CGPA scores. Calculate the Spearman’s Rank Correlation between the two and
interpret the result.
State University
% of students having
free meals
% of students scoring
above 8.5 CGPA
Pune 14.4 54
Chennai 7.2 64
Delhi 27.5 44
Kanpur 33.8 32
Ahmedabad 38.0 37
Indore 15.9 68
Guwahati 4.9 62

37. State
University
dX = RanksX dY = RanksY d = (dX – dY) d2
Pune 3 4 -1 1
Chennai 2 6 -4 16
Delhi 5 3 2 4
Kanpur 6 1 5 25
Ahmedabad 7 2 5 25
Indore 4 7 -3 9
Guwahati 1 5 -4 16
Σd2
= 96
1)
n(n
(di)
6
1
r 2
2
s




= 𝟏–
𝟔. 𝟗𝟔
𝟕. (𝟒𝟗– 𝟏)
= 𝟏–
𝟓𝟕𝟔
𝟑𝟑𝟔
= −𝟎. 𝟕𝟏𝟒