Measures of correlation and the methods in it

1. CORRELATION
 Correlation analysis deals with the association between
two or more variables.
Simphson &
Kafka
 If two or more quantities vary in sympathy, so that
movement in one tend to be accompanied by
corresponding movements in the other , then they are
said to be correlated.
Conner
 Correlation analysis attempts to determine the degree
of relationship between variables .
Ya- Lun Chou

2. UTILITY OF CORRELATION
The study of correlation is of immense significance in
statistical analysis and practical life , which is clear from
the following points:-
 With the help of correlation analysis we can measure
the degree of relationship in one figure between different
variables like supply and price , income and expenditure ,
etc.
 The concept of regression is based on correlation.
 An economist specifies the relationship between
different variables like demand and supply , money
supply and price level by way of correlation.
 A trader makes the estimation of costs, prices etc, with
the help of correlation and makes appropriate plans.

3. TYPES OF CORRELATION
Main types of correlation are given below:-
Positive Correlation :- If two variables X and Y move in
the same direction ,ie , if one rises, other rises too and
vice versa, then it is called as positive correlation.
Negative Correlation:- If two variables X and Y move in
the opposite direction ,ie , if one rises, other falls and if
one falls, other rises , then it is called as negative
correlation.
Linear correlation:-If the ratio of change of tqwo variables
remains constant through out ,then they are said to be
lineally correlated.
Curvi- Linear correlation:- If the ratio of change between
the two variables is not constant but changing ,
Correlation is said to be Curvi-linear.

4. TYPES OF CORRELATION
Simple Correlation :- When we study the relation between
two variables only, then it is called simple Correlation.
Partial Correlation:- When three or more variables are
taken but relationship between any two of variables is
studied ,assuming other variables as constant ,then it is
called partial Correlation.
Multiple Correlation:- When we study the relationship
among three or more variables then it is called multiple
Correlation.

5. DEGREE OF CORRELATION
Degree of correlation can be known by coefficient of
correlation ( r ). Following are the types of the degree of
the Correlation:-
Perfect Correlation :- When two variables vary at
constant ratio in the same direction, then it is called
perfect positive Correlation.
High Degree of Correlation:- When correlation exists in
very large magnitude, then it is called high degree of
correlation.
Moderate Degree of Correlation:- Correlation coefficient
,on being within the limits +0.25 and +0.75 is termed as
moderate degree of Correlation
Low Degree of Correlation:- When Correlation exists in
very small magnitude, then it is called as low degree of
Correlation.

6. DEGREE OF CORRELATION
Absence of Correlation :- When there is no relationship between the
variables ,then correlation is found to be absent . In case of absence of
Correlation ,the value of correlation coefficient is zero.
Ser
No
Degree of Correlation Positive Negative
01 Perfect correlation +1 -1
02 High degree of Correlation Between +0.75
to +1
Between -0.75 to
-1
03 Moderate Degree of
Correlation
Between +0.25
to 0.75
Between -0,25 to
-0.75
04 Low Degree of Correlation Between 0 to
+0.25
Between 0 to -
0.25
05 Absence of Correlation 0 0
The degree of correlation on the basis of the value of
correlation coefficient can be summarized with the following
table:-

7. METHODS OF STUDYING CORRELATION
Correlation can be determined by the following
methods:-
(1) Graphic Method (2) Algebraic Method
 Scatter Diagram Karl persons Coefficient
 Correlation Graph Spearman's Rank Correlation
Concurrent Deviation Method
Graphic Method Algebraic Method
Scatter Diagram Correlation Graph
Karl Person
Coefficient
Rank
Correlation
Concurrent
Deviation
Methods of studying Correlation

8. SCATTER DIAGRAM
Y
O X X
O X
O
Y Y
(A) r = +1 (B) r = -1 (C) High Positive
(C) High Negative (E) No Correlation
X X
O O
Y Y
X: 10 20 30 40 50 60
Y: 25 50 75 100 125 150
Given the following pairs of values of the variables X and Y

9. SCATTER DIAGRAM
O
O
O
O
O
O
1O 2O 3O 4O 5O 6O 7O
5O
10O
15O
20O
X
Y

10. ALGEBRIC METHOD
Karl Pearson’s coefficient :- This is the best method of
working out correlation coefficient. This method has the
following main characteristics:-
Knowledge of Direction of Correlation:- By this method
direction of correlation is determined whether it is
positive or negative.
Knowledge of Degree of Relationship:- By this method ,it
becomes possible to measure correlation quantitatively.
The coefficient of correlation ranges between -1 and
+1.The value of the coefficient of correlation gives
knowledge about the size of relation.
Ideal Measure:- It is considered to be an ideal measure of
correlation as it is based on mean and standard deviation.
Covariance:- Karl Pearson’s method is based on co-
variance.The formula for co-variance is as follows

11. ALGEBRIC METHOD
Cov (X,Y)=∑(X - ˉ)(Y -ˉ ) =∑XY - ˉ ˉ
Y Y
XY
N N
The magnitude of co- variance can be used to express correlation between two
variables. As the magnitude of co-variance becomes greater , higher will be the
degree of correlation , otherwise lower. With positive sign of covariance, correlation
will be positive . On the contrary ,correlation will be negative if the sign of
covariance is negative .
Calculation of Karl Pearson’s coefficient of correlation:-
(a) Calculation of Coefficient of correlation in the case of individual series or un
grouped data.
(b) Calculation of coefficient of correlation in the case of grouped data.
(1) Actual Mean Method:-
This method is useful when arithmetic mean happens to be in whole numbers or
integers . This method involves the following steps:-
(1) First , we compute the arithmetic mean of X and Yseries , ie Xˉ and Yˉ are
worked out.

12. ALGEBRIC METHOD
(2) Then from the arithmetic mean of the two series , deviation of the individual
values are taken.the deviations of X-series are denoted by x and of the Y-series by
y ,i.e x=X-Xˉ and y=Y-Yˉ.
3. Deviations of the two series are squared and added upto get ∑X² and ∑Y².
4. The coressponding deviations of the two series are multiplied and summed upto
get ∑xy.
5 correlation coefficient=
R=∑xy/√∑x²X∑y²
Following example will illustrate this:
X X-Xˉ x² Y Y-Yˉ y² xy
2 -3 9 4 -6 36 +18
3 -2 4 7 -3 9 +6
4 -1 1 8 -2 4 +2
5 0 0 9 -1 1 0
6 +1 1 10 0 0 0
7 +2 4 14 4 16 +8
∑X=3
5
N=7
∑X=0 ∑x²=
28
∑Y=70 ∑y=0 ∑y²=
130
∑xy=
58

13. Xˉ=∑X/N=35/7 ,Yˉ=∑Y∕N=70/7=10
R=∑XY/√∑X ²X ∑Y² = 58/√3640
=58/60.33 = +.96
ASSUMED MEAN METHOD:-
(1) ANY VALUES ARE TAKEN AS THEIR ASSUMED MEAN AX AND AY.
(2) DEVIATIONS OF THE INDIVIDUAL SERIES OF BOTH THE SERIES ARE
WORKED OUT FROM THEIR ASSUMED MEANS.DEVIATIONS OF X
SERIES (X-AX)ARE DENOTED BY DX AND OF Y SERIES(Y-AY) BY DY.
(3) DEVIATIONS ARE SUMMED UP TO GET ∑DX AND ∑DY.
(4)THEN,SQUARES OF THE DEVIATIONS DX² AND DY² ARE WORKED OUT
AND SUMMED UP TO GET ∑DX² AND ∑DY².
(5)EACH DX IS MULTIPLIED BY THE CORRESPONDING DY AND PRODUCTS
ARE ADDED UPTO GET ∑DXDY.
R=∑DXDY-∑DX.∑DY/N/√∑DX²-(∑DX)²/N √∑DY²-(∑DY)²/N

14. PROPERTIES OF COEFFICIENT OF
CORRELATION
(1)limits of coefficient of correlation:-1≤r≤+1
(2) change of origin and scale:shifting the origin or scale
does not affect in any way the value of correlation
(3)geometric mean of regression coefficients: r=√bxy.byx
(4)if x and y are independent variables,then coefficient of
correlation =0

15. INTERPRETING THE COEFFICIENT OF
CORRELATION
 If r=+1,then there is perfect correlation
 If r=0,then there is absence of linear correlation
 If r =+0.25,then there will be low degree of positive
correlation
 If r=+0.50,then there is moderate degree of
correlation
 If r=+0.75,then there is high degeree of positive
correlation.

16. |r| > 6 P.E. Significance of r
|r| < 6 P.E. Insignificance of r
TO TEST THE RELIABILITY OF KARL PEARSONS
CORRELATION COEFFICIENT(r), PROBABLE ERROR
(P. E.) IS USED.

17.  SPEARMANS RANK CORRELATION METHOD
 R=1-6∑D²/N(N²-1)
X 0 -4 4 -2 2
Y 1 3 -3 0 -1
X X^2 Y Y^2 XY
0 0 1 1 0
-4 16 3 9 -12
4 16 -3 9 -12
-2 4 0 0 0
2 4 -1 1 -2
∑X = 0 ∑X^2 =
40
∑Y=0 ∑Y^2=20 ∑XY=-26

18. APPLYING FORMULA:
R= ∑XY/ √{(∑X^2)ₓ(∑Y^2)}
= −0.9192
NOW P.E. IS
P.E.= −0.6745 ₓ {(1 − R^2)/ √N}
= 0.04677
FOR SIGNIFICANCE OF ‘R’
|R|/ P.E.= 0.9192/ 0.04677
=19.65
|R|= 19.65 P. E.
AS THE VALUE OF |R| IS MORE THAN 6 TIMES THE
P.E., SO ‘R’ IS HIGHLY SIGNIFICANT.

19. CONCURRENT DEVIATION METHOD
 UNDER this method,whatever the series x
and y are to be studied for correlation,each
item of the series are compared with its
preceeding item.if the value is more than its
preeceding value then it is assigned with a
positive sign and if less then a negative
sign.
 The deviations of X and Y series(dx) and
(dy) are multiplied to get dxdy.product of
similar signs will be positive n opposite
signs will be negative.

20. (+) (+)=+
(-) (-)= +
(0) (0)=+
(-) (+)= -
(+) (-)= -
(0) (-)= -
(-) (0)= -
(0) (+)= -
:-ALL POSITIVE SIGNS ARE ADDED AND NEGATIVE SIGNS
ALSO
:-R=±√±{2C – N/N}

21. X 85 91 56 72 95 76 89 51 59 90
Y 18.3 20.8 16.9 15.7 19.2 18.1 17.5 14.9 18.9 15.4
X DEVIATION SIGNS
(DX)
Y DEVIATION SIGNS
(DY)
DXDY
85 18.3
91 + 20.8 + +
56 - 16.9 - +
72 + 15.7 - -
95 + 19.2 + +
76 - 18.1 - +
89 + 17.5 - -
51 - 14.9 - +
59 + 18.9 + +
90 + 15.4 - -
N=9 n=9 C=6
√(2x6-9)/9=+√3/9=0.577