Linear correlation

1. CORRELATION (LINEAR) 1 September 2012 1

2. CORRELATION If two quantities vary in such a way that movements of one are accompanied by movements of others then these quantities are said to be correlated. Ex: relationship between price of commodity and amount demanded, Increased in amount of the rainfall and the production of rice The degree of relationship between variables under consideration is measured through the correlation analysis. The measure of correlation is called the correlation coefficient or correlation index ( usually denoted by r or ρ ) The correlation analysis refers to the techniques used in measuring the closeness of the relationship between the variables. 1 September 2012 2

3. DEFINITIONS Correlation analysis deals with the association between two or more variables. Simpson and Kafka Correlation is an analysis of co variation between two or more variables. A.M.Tuttle If two or more quantities were in sympathy so that the movement of one tend to be accompanied by the corresponding movements in the other then they are said to be correlated L.R.Conner 1 September 2012 3

4. ANALYSIS The problem of analyzing the relation between different series should be broken down in to three steps 1. Determining whether a relation exists and if it does, measuring it. 2. Testing whether it is significant. 3. Establishing the cause and effect relation if any. 1 September 2012 4

5. SIGNIFICANCE OF THE STUDY OF CORRELATION Most of the variables show some kind of relationship Once we know that two variables are closely related we can estimate the value of one variable given the value of another. Correlation analysis contributes to the understanding of the economic behavior The effect of correlation is to reduce the range of uncertainty 1 September 2012 5

6. CORRELATION AND CAUSATION 1. The correlation may be due to pure chance especially in a small sample. The above data show a perfect positive relationship between income and weight i.e., as the income is increasing the weight is increasing and the rate of change between two variables is the same. Income(rs) 500 600 700 800 900 Weight(lbs) 120 140 160 180 200 1 September 2012 6

7. 2. Both the correlated variables may be influenced by one or more other variables. 3. Both the variables may be mututally influencing each other so that neither can be designated as the cause and the other the effect. Correlation observed between variables that cannot conceivably be casually related is called spurious or nonsense correlation 1 September 2012 7

8. TYPES OF CORRELATION Positive or negative Linear and non linear correlation Simple , partial and multiple correlation 1 September 2012 8

9. POSITIVE OR NEGATIVE CORRELATION Whether the correlation is positive or negative would depend up on the direction of the change of the variable. If both the variables are varying in the same direction , then the correlation is said to be positive . If the variables are varying in opposite direction the correlation is said to be negative 1 September 2012 9

10. Positive correlation X 10 12 15 18 20 Y 15 20 22 25 37 1 September 2012 10

11. Negative correlation X 20 30 40 60 80 Y 40 30 22 15 10 1 September 2012 11

12. SIMPLE PARTIAL AND MULTIPLE CORRELATION The distinction between simple partial and multiple correlation is based up on the number of variables studied. When only two variables are studied it is a problem of simple correlation When three or more variable are studied it is problem of either multiple or partial correlation. In multiple correlation three or more variables are studied simultaneously. On the other hand in partial correlation we recognize more than two variables but consider only two variables to be influencing each other the effect of other influencing variable kept constant . 1 September 2012 12

13. LINEAR AND NONLINEAR(CURVILINEAR) CORRELATION Distinction between linear and non linear correlation is based up on the constancy of the ratio of change between the variables. If the amount of change in one variable tends to bear constant ratio to the amount of change in the other variable then the correlation is said to be linear. It is clear that the ratio of change between the two variables is the same. If such variables are plotted on the graph paper all the plotted points would fall on a straight line. X 10 20 30 40 50 Y 70 140 210 280 350 1 September 2012 13

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15. Correlation would be called non linear or curvilinear if the amount of change in one variable does not bear a constant ratio with the amount of change in the other variable. 1 September 2012 15

16. METHODS OF STUDYING CORRELATION 1. Scatter diagram 2. Graphic method 3. Karl Pearson’s coefficient of correlation. 4. Concurrent Deviation Method 5. Method of least squares 1 September 2012 16

17. SCATTER DIAGRAM METHOD The simplest device for ascertaining if the two variables are related is to prepare a dot chart called scatter diagram . When this method is used the given data are plotted on a graph paper in the form of dots. I.e., for each pair of X and Y values we put a dot and thus obtain as many points as the number of observations. By looking to the scatter of the various points we can form an idea as to whether the variables are related or not. The greater the scatter of the plotted points on the chart the lesser is the relationship between the two variables The more closely the points come to the straight line higher the degree of relationship. 1 September 2012 17

18. If all the points lie on a straight line falling from the lower left hand corner to the upper right hand corner the correlation is said to be perfectly positive(r=1) 1 September 2012 18

19. If all the points are lying on a straight line rising from the upper left hand corner to the lower right hand corner of the diagram correlation is said to be perfectly negative . 1 September 2012 19

20. If the plotted points fall in a narrow band there would be a high degree of correlation between the variables. If the points are widely scattered over the diagram it indicates very little relation ship between the variables. HIGH DEGREE OF POSITIVE CORRELATION LOW DEGREE OF POSITIVE CORRELATION 1 September 2012 20

21. If the plotted points lie in a haphazard manner it shows the absence of any relationship between the variables 1 September 2012 21

22. EXAMPLE: X 2 3 5 6 8 Y 6 5 7 8 12 1 September 2012 22

23. By looking at the scattered diagram we can say that the variables x and y are correlated. Further the correlation is positive because the trend of the points is upward rising from the lower left hand corner to the upper right hand corner of the diagram. It also indicates that the degree of relationship is higher because the plotted points are near to the line which shows perfect relationship between the variables. 1 September 2012 23

24. MERITS AND LIMITATIONS It is a simple and non mathematical method of studying correlation between variables. As such it can be easily understood and a rough idea can very quickly be formed as to whether or the variables are related. It is the first step in investigating relationship between 2 variables. LIMITATIONS: By applying this method we can get an idea about the direction of correlation and also whether it is high or low But we cannot establish the exact degree of correlation between the variables as is possible by applying the mathematical methods. MERITS 1 September 2012 24

25. GRAPHIC METHOD When this method is used the individual values of the two variables are plotted on the graph paper. We thus obtain 2 curves. One for x variable and another for y variable. By examining the direction and closeness of the two curves so drawn we can infer if the variables are related or not. If both the curves drawn on the graph are moving in the same direction (either upward or downward)then the correlation is said to be positive. On the other hand if the curves are moving in the opposite direction correlation is said to be negative. 1 September 2012 25

26. Year Average income Average expenditure 1979 100 90 1980 102 91 1981 105 93 1982 105 95 1983 101 92 1984 112 94 1 September 2012 26

27. INCOME 1 September 2012 27

28. KARL PEARSON’S COEFFICIENT OF CORRELATION Among several mathematical methods of measuring correlation, the Karl Pearson’s method, popularly known as Pearson’s coefficient of correlation, is most widely used in practice It is denoted by the symbol ρ or r 1 September 2012 28

29. CORRELATION COEFFICIENT If [X,Y] is a two dimensional random variable, the correlation coefficient, denoted r, is ρ = Cov (X,Y) ∕ Var (X) . Var (Y) = σ XY ∕ σ X σ Y This is also called as PEARSON CORRELATION COEFFICIENT ρ = ∑ xy ∕ √ (∑x 2 * ∑y 2 ) = ∑ xy ∕ N σ X σ Y , where x=(X-X’) ; y=(Y-Y’) σ X = Standard Deviation of X and σ Y = Standard Deviation of Y N = no of pairs of observation ρ = correlation coefficient 1 September 2012 29

30. STEPS TO CALCULATE CORRELATION COEFFICIENT Take the deviations of X from the mean of X and denote by x Square these deviations and obtain the total i.e., Σ x 2 Take the deviations of Y from the mean of Y and denote by y Square these deviations and obtain the total i.e., Σ y 2 Multiply the deviations of X and Y and obtain the total i.e., Σ xy Substitute the values in the formula 1 September 2012 30

31. EXAMPLE Calculate the Karl Pearson’s Correlation Coefficient from the following data and interpret it’s value Roll no of students: 1 2 3 4 5 Marks in Accountancy : 48 35 17 23 47 Marks in Statistics: 45 20 40 25 45 SOLUTION: Let marks in Accountancy be denoted by X and Statistics by Y 1 September 2012 31

32. Roll no X (X-34) x x 2 Y (Y-35) y y 2 xy 1 48 14 196 45 10 100 140 2 35 1 1 20 -15 225 -15 3 17 -17 289 40 5 25 -85 4 23 -11 121 25 -10 100 110 5 47 13 169 45 10 100 130 ∑X=170 ∑x=0 ∑x 2 =776 ∑Y=175 ∑y=0 ∑y 2 =550 ∑ xy =280 1 September 2012 32

33. The Pearson’s coefficient of correlation is ρ = ∑ xy ∕ √(∑x 2 *∑y 2 ) where x=(X-X’); y=(Y-Y’) , X'= ∑X ∕ N; Y’=∑Y ∕ N ∑ xy =280 ∑x 2 =776 ∑y 2 =550 ρ = 280 ∕ √ (776 * 550) = 0.496 1 September 2012 33

34. DEGREE OF CORRELATION The value of ρ always lies between -1 and 1. If ρ lies between 0 and 1, it is positive . Else, if it lies between -1 and 0 , it is negative If ρ =1, then the two variables are said to be perfect positively correlated If ρ =-1 , then the two variables are said to be perfect negatively correlated If ρ =0 , then the two variables are not correlated 1 September 2012 34

35. THANK YOU!!! 1 September 2012 35