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

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.

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.

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.

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 CorrelationLow Degree of Correlation:- When Correlation exists in very small magnitude, then it is called as low degree of Correlation.

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:-

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

SCATTER DIAGRAM

Y

O X XO XO

Y Y

(A) r = +1 (B) r = -1 (C) High Positive

(C) High Negative (E) No CorrelationX XO 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

SCATTER DIAGRAM

O

O

O

O

O

O1O 2O 3O 4O 5O 6O 7O

5O

10O

15O

20O

X

Y

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

ALGEBRIC METHOD

Cov (X,Y)=∑(X - ˉ)(Y -ˉ ) =∑XY - ˉ ˉ Y Y XYN 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.

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=35 N=7

∑X=0 ∑x²= 28

∑Y=70

∑y=0 ∑y²=130

∑xy=58

Xˉ=∑X/N=35/7 ,Yˉ=∑Y∕N=70/7=10R=∑XY/√∑X ²X ∑Y² = 58/√3640=58/60.33 = +.96ASSUMED 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

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

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.

|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.

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

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.

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.

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

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