09_MEASURE OF CORRELATION

 

 

 

ECONOMICS (CLASS-XI)

 

Chapter-9                                 Measure of Correlation

 

 

Introduction

          The term correlation indicates the relationship between two variables in which changes in the values of one variable, the value of the other variable, the value of the other variable also changes.

Kinds of Correlation

          Positive & Negative Correlation – When both the variable changes in one direction i.e. both increases or decreases, the relationship between the two is called positive or direct.

          When the variable changes in the opposite direction i.e. one increase and other decreases, the correlation is negative or inverse.

          Linear & Curvilinear– When the ratio of change between the two variable is uniform, than there exists linear correlation between them.

            If the ratio of change in two variable is non-uniform, then there exists non-linear or curvilinear correlation.

Simple, Multiple & Partial Correlation :When only two variables is studied, the analysis of relationship between them is called simple correlation.

          When three or more variables are studied, the relationship be multiple correlation.

Partial correlation is the study of relationship of two or more variables influencing each other, effect of other influencing factor being kept constant. e.g. amount of rainfall, production of wheat and constant temperature.

 

Degree of Correlation

• Perfect Correlation – Perfect correlation is that where changes in the two relation variables are exactly proportional, if these changes are in the same direction, there is perfect positive correlation (+1); if the changes are in the reverse direction, there is a perfect negative correlation (-1).
• Zero Correlation –Zero correlation means two values are uncorrelated and there is no linear correlation between them.
• Limited Degree of Correlation – Correlation is said to be limited positive, when there are unequal changes in the two variables in the same direction.

Correlation is said to be limited negative when there are unequal changes in the two variables in the opposite direction.

Scatter Diagram Method

• Scatter Diagram –Scatter diagram is the method for getting same idea about the presence of correlation. The values are plotted on the graph paper. The cluster of point on graph paper is called scatter diagram.

 

Merits & Demerits of Scatter Diagram

Merits

(1)      It is very easy to draw.

(2)      It is on attractive and non-mathematics method of measuring the correlation.

(3)      It is not affected by extreme values.

(4)      It gives a visual picture of the proportionate change in the value of X & Y.

 

Demerits

(1)      The degree of correlation between 2 variables cannot be known in magnitude.

(2)      It is not useful in case of more than 2 variables.

(3)      Definite conclusions cannot be drawn by examining the diagram.

 

Speakman’s Rank Correlation

         

N = No. of pairs.

D Þ R₁ – R₂

Three kinds of Data can be present

(1)      When the ranks are given.

(2)      When the ranks are not given.

(3)      When equal ranks are given

 

Merits & Demerits of Rank Correlation

Merits

(1)      It is easy to calculate and simple to understand.

(2)      It is used in case of qualitative dates.

(3)      It is a useful method, when the ranks of the different is given.

 

Demerits

(1)      This method is useful only in the individual series rather than in the frequency      test.

(2)      This method is not suitable in case the number of observations is to large.

(3)      All the information given is not used in this method.

 

Karl Pearson’s Cofficient Of Correlation

This method is based on arithmetic mean S.D.

 

Where

 

 

Covariance

There are 4 methods of measuring correlation =

1. Actual Mean

                  

2. Direct Method

                  

3. Assumed Method

                  

4. Step-Deviation Method

                  

Merits

(1)      This method gives definite quantitative figure which is easy to interpret.

(2)      It gives direction as well as degree of the relationship between 2 variables.

 

Demerits

(1)      The value of the coefficient is affected by extreme items.

(2)      The calculation is difficult and time consuming.

(3)      It is assumed that there is a linear relationship between 2 variables.

 

 

 

 

 

 

 

 

Coefficient of Correlation By Rank Differences

                  

D = R₁ – P₂

N = No. of observations.

 

When Ranks are given

Indicates	Judge-1	Judge-2	(R­1­ – R2) = D	D2
A B C D E F G H I J	1 2 3 4 5 6 7 8 9 10	10 9 8 7 6 5 4 3 2 1	-9 -7 -5 -3 -1 1 3 5 7 9	81 49 25 9 1 1 9 25 49 81
				330

 

 

 

When Ranks are not given

 

X	Y	R1	R2	R1 – R2	D2
3 13 10 24 36 75 46 34 21 17	12 25 36 4 10 27 14 32 72 44	1 3 2 6 8 10 9 7 5 4	3 5 8 1 2 6 4 7 10 9	-2 -2 -6 5 -6 4 5 0 -5 -5	4 4 36 25 36 16 25 0 25 25
					196

 

 

 

 

 

 

Equal Rank

Y	R1	R2	R1 – R2	D2
6 11 7 7 3 5	1 2 3 4.5 4.5 5	3 6 4.5 4.5 1 2	-2 -4 -1.5 0 -3.5 4	4 16 2.25 0 12.25 16
				50.50

Assign the rank of the given data q in case of equal ranks, assign average ranks.

Calculate D = R₁ – R₂

Apply the formula

 

 

 

 

approx.

m represents = no. of times rank repeated is to be added to 6 SD² as many times as the no. of groups of repeated ranks.

 

Kakl’s Pehrson’s Cofficient of Correlation

 

(Covariance or  )

There are 4 methods of calculating coefficient of correlation

(a)      Actual Mean Method

(i)       Calculate

(ii)      Calculate

(iii)     Calculate x²& y²

(iv)     Multiply x² y² calculate Sxy

(v)      Apply the formula

 

Ans.

X	Y			xy	y2	x2
4 7 10 12 2	7 3 4 4 2	-3 0 3 5 -5	-3 -1 0 0 -2	9 0 0 0 10	9 1 0 0 4	9 0 9 25 25
35	20			19	14	68

 

 

 

 

 

1. Calculate Correlation

No. of persons                           15      15

Arithmetic mean                       25      18

Square of the deviation from      136    138

Arithmetic mean

Sxy = 122

 

Ans. 6.891 approx.

 

X	Y	r – 1	Y – 4	xy	x2	y2
10 10 11 12 12	5 6 4 3 2	-1 -1 0 1 1	1 2 0 -1 -2	-1 -2 0 -1 -2	1 1 0 1 4	1 4 0 1 4
55					14	10

 

Ans. 0.949

 

4. Two Series × 24 with 50 items each have standard deviation 4.5 2365 respectively. If the sum of the product of X 24 series from arithmetic mean is 420. Find the r.

 

Sxy = 420

 

24. Find the coefficient of correlation by x 24. Variance between X 24 is 10 and te variance of X 241629 respectively.