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# Linear correlation

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### Linear correlation

1. 1. CORRELATION (LINEAR)3 September 2012 1
2. 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.3 September 2012 2
3. 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.Conner3 September 2012 3
4. 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.3 September 2012 4
5. 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 uncertainty3 September 2012 5
6. 6. CORRELATION AND CAUSATION1. The correlation may be due to pure chance especially in a small sample. Income(rs) 500 600 700 800 900 Weight(lbs) 120 140 160 180 200 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.3 September 2012 6
7. 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 correlation3 September 2012 7
8. 8. TYPES OF CORRELATION Positive or negative Linear and non linear correlation Simple , partial and multiple correlation3 September 2012 8
9. 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 negative3 September 2012 9
10. 10. Positive correlation X 10 12 15 18 20 Y 15 20 22 25 37 Y-Values 40 35 30 25 20 15 10 5 0 0 5 10 15 20 253 September 2012 10
11. 11. Negative correlation X 20 30 40 60 80 Y 40 30 22 15 10 Y-Values 45 40 35 30 25 20 15 10 5 0 0 20 40 60 80 1003 September 2012 11
12. 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.3 September 2012 12
13. 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. X 10 20 30 40 50 Y 70 140 210 280 350 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.3 September 2012 13
14. 14. 400 350 300 250 200 150 100 50 0 0 10 20 30 40 50 603 September 2012 14
15. 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.3 September 2012 15
16. 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 squares3 September 2012 16
17. 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.3 September 2012 17
18. 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) 8 7 6 5 4 3 2 1 0 0 2 4 6 83 September 2012 18
19. 19. If all the points are lying on a straight line rising from the upper left handcorner to the lower right hand corner of the diagram correlation is said tobe perfectly negative. 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 83 September 2012 19
20. 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. 10 10 8 8 6 6 4 4 2 2 0 0 0 5 10 0 5 10 HIGH DEGREE OF POSITIVE CORRELATION LOW DEGREE OF POSITIVE CORRELATION3 September 2012 20
21. 21. If the plotted points lie in a haphazard manner it shows the absence of any relationship between the variables 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 143 September 2012 21
22. 22. EXAMPLE: X 2 3 5 6 8 Y 6 5 7 8 12 14 12 10 8 6 4 2 0 0 2 4 6 8 103 September 2012 22
23. 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.3 September 2012 23
24. 24. MERITS AND LIMITATIONSMERITS• 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.3 September 2012 24
25. 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.3 September 2012 25
26. 26. Year Average income Average expenditure 1979 100 90 1980 102 91 1981 105 93 1982 105 95 1983 101 92 1984 112 943 September 2012 26
27. 27. 120 INCOME 100 EXPENDITURE 80 60 Series 1 Series 2 40 20 0 1979 1980 1981 1982 1983 19843 September 2012 27
28. 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 r3 September 2012 28
29. 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) = σXY ∕ σX σY• This is also called as PEARSON CORRELATION COEFFICIENT ρ= ∑xy ∕ √ (∑x2 * ∑y2) = ∑xy ∕ N σX σY , where x=(X-X’) ; y=(Y-Y’) σX = Standard Deviation of X and σY = Standard Deviation of Y N = no of pairs of observation3 September 2012 ρ = correlation coefficient 29
30. 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., Σx2• Take the deviations of Y from the mean of Y and denote by y• Square these deviations and obtain the total i.e., Σy2• Multiply the deviations of X and Y and obtain the total i.e., Σxy• Substitute the values in the formula3 September 2012 30
31. 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 45SOLUTION: Let marks in Accountancy be denoted by X and Statistics by Y3 September 2012 31
32. 32. Roll no X (X-34) x2 Y (Y-35) y2 xy x y 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 ∑x2=776 ∑Y=175 ∑y=0 ∑y2=550 ∑xy=2803 September 2012 32
33. 33. • The Pearson’s coefficient of correlation is ρ= ∑xy ∕ √(∑x2 *∑y2) where x=(X-X’); y=(Y-Y’) , X= ∑X ∕ N; Y’=∑Y ∕ N ∑xy=280 ∑x2=776 ∑y2=550 ρ = 280 ∕ √ (776 * 550) = 0.4963 September 2012 33
34. 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 correlated3 September 2012 34
35. 35. 3 September 2012 35