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1. Welcome To A Session On Correlation Analysis www.AssignmentPoint.com

2. What is meant by correlation? It is viewed as a statistical tool with the help of which the relationship between two or more than two variables is studied. Correlation analysis refers to a technique used in measuring the closeness of the relationship between the variables. If two quantities vary in such a way that movements in one are accompanied by movements in the other, these quantities are said to be correlated. Continued….. www.AssignmentPoint.com

3. What is meant by Correlation? • Examples: • Relationship between family income and expenditure • on luxury items • Price of a commodity and amount demanded • Increase in rainfall up to a point and production of rice • Increase in the number of a television licenses and • number of cinema admissions. Continued….. www.AssignmentPoint.com

4. What are the major issues of analyzing the relation between different series ? • Determining whether a relation exists and, if it does, one has to measure it ; • Testing whether it is significant; • Establishing the cause –and- effect relations www.AssignmentPoint.com

5. What is the significance of the study of correlation? • The study of correlation is of immense use in practical life because of the following reasons: • Most of the variables show some kind of relationship. With the help of correlation analysis one can measure in one figure the degree of relationship existing between the variables. • Once two variables are closely related, one can estimate the value of one variable given the value of another. Continued…….. www.AssignmentPoint.com

6. What is the significance of the study of correlation? • Correlation analysis contributes to the economic behavior, aids in locating the critically important variables on which others depend. This may reveal the connection by which disturbances spread and suggest the paths through which stabilizing forces become effective. • Progressive development in the methods of science and philosophy has been characterized by increase in the knowledge of relationship or correlations. • It should be noted that coefficient of correlation is one of the most widely used tool and also one of the most widely abused statistical measures. www.AssignmentPoint.com

7. Example: Advertisement expenditure (Tk lakhs) 25 35 45 55 65 Sales (Tk. Crores) 120 140 160 180 200 The above data show a perfect positive relationship between advertisement expenditure and sales. But such a situation is rare in practice. Continued……. www.AssignmentPoint.com

8. Does correlation always signify a cause, and effect relationship between variables? If not, Why? • Both the correlated variables may be influenced by one or more other variables: A high degree of correlation between the yield per acre of the rice and tea may be due to the amount of rainfall. But none of the two variables is the cause of the other. • Both the variables may be mutually influencing each other so that neither can be designated as cause and the other the effect. • Variables like demand and supply, price and production, etc. mutually interact. Continued……. www.AssignmentPoint.com

9. Example: As the price of a commodity increases, its demand goes down and so price is the cause and demand the effect. But it is also possible that increased demand of a commodity due to the growth of population or other reasons may force its price up. Now the cause is the increased demand, the effect the price. Thus at times it may become difficult to explain from the two correlated variables which is the cause and which is the effect because both may be reacting on each other. The above points clearly show that correlation does not manifest causation or functional relationship. By itself, it establishes only covariation. Continued……. www.AssignmentPoint.com

10. What are various types of correlation? Correlation is classified in several different ways. The most important types of correlation are: • Positive and negative correlation • Simple, partial and multiple correlation • Linear and non–linear correlation Continued……… www.AssignmentPoint.com

11. What are various types of correlation? • Positive correlation: If both the variables vary in the same direction i.e. if one variable increases, the other on average also increases, or if one variable decreases, the other on average also decreases, correlation is said to be positive. Continued……… www.AssignmentPoint.com

12. Example: Positive correlation y 15 20 22 25 37 X 10 12 11 18 20 X 80 70 60 40 30 y 50 45 30 20 10 www.AssignmentPoint.com

13. What are the various types of correlation? • Negative correlation: If the variables vary in opposite directions i.e. if one variable increases the other decreases or vice versa, correlation is said to be negative. Continued……… www.AssignmentPoint.com

14. Example: Example: Negative correlation X 20 30 40 60 80 y 40 30 22 15 16 X 100 90 60 40 30 y 10 20 30 40 50 Continued……… www.AssignmentPoint.com

15. What are the various types of correlation? • Simple correlation: When only two variables are studied, it is a problem of simple correlation. Continued……… www.AssignmentPoint.com

16. What are the various types of correlation? • Multiple correlation: When three or more variables are studied simultaneously, it is a problem of multiple correlation. • Example 1: When we study the relationship between the yield of rice per acre and both the amount of rainfall and the amount of fertilizers used, it is problem of multiple correlation. • Example 2: The relationship of plastic hardness, temperature and pressure . Continued……… www.AssignmentPoint.com

17. What are the various types of correlation. • Partial correlation: In partial correlation, we recognise more than two variables. But, when only two variables are considered to be influencing each other and the effect of other influencing variable is kept constant, it is a problem of partial correlation. • Example: If we limit our correlation analysis of yield of rice per acre and rainfall to periods when a certain average daily temperature existed, it becomes a problem of partial correlation. Continued……… www.AssignmentPoint.com

18. What are various types of correlation? • Linear (curvilinear) correlation: If the amount of change in one variable tends to bear constant ratio to the amount of change in other variable, then the correlation is said to be linear. Continued……… www.AssignmentPoint.com

19. Example: It is clear that the ratio of change between two variables is the same. If these variables are plotted on a graph paper, all the plotted points would fall on a straight line. Continued……… www.AssignmentPoint.com

20. Example: Positive Liner Correlation Continued……… www.AssignmentPoint.com

21. What are the various types of correlation. • Non–linear (curvilinear) correlation: If the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable, then the correlation is said to be non–linear (curvilinear). • Example: If the amount of rainfall is doubled, the production of rice or wheat, etc. would not necessarily be doubled. In most practical cases, we find a non-linear relationship between the variables. • But, since techniques of analysis for measuring non-linear correlation are very complicated, the relationship between the variables is assumed to be of the linear type. Continued……… www.AssignmentPoint.com

22. Example: Curvilinear Correlation Y X Continued……… www.AssignmentPoint.com

23. What are the methods of studying correlations? • Scatter Diagram Method • Karl Pearson’s coefficient of correlation • Spearman’s Rank correlation coefficient Continued……… www.AssignmentPoint.com

24. What is Scatter Diagram ? A scatter diagram refers to a diagram in which the values of the variables are plotted on a graph paper in the form of dots i.e. for each pair of X and Y values. If we put dots and thus obtain as many points as the number of observations, the diagram of dots, so obtained is known as scatter diagram. Continued……… www.AssignmentPoint.com

25. How can scatter diagram method (Dot chart method) be used to study correlation? In this method, the given data are plotted on a graph paper in the form of dots. From scatter diagram i.e. by looking to the scatter of the various points, we can form a fairly good, though vague, idea whether the variables are correlated or not, e.g., if the points are dense, i.e. very close to each other, we should expect a fairly good amount of correlation between the variables and if the points are widely scattered, a poor correlation is expected. Continued……… www.AssignmentPoint.com

26. When is correlation said to be perfectly positive or and perfectly negative? If all the points lie on a straight line rising from the lower left hand corner to the upper right hand corner, correlation is said to be perfectly positive ( i.e. r = +1) If all the points lie on a straight line falling from the upper left hand corner to the lower right hand corner of the diagram, correlation is said to be perfectly negative (i.e. r = - 1). Continued……… www.AssignmentPoint.com

27. Example: Perfect Positive Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X Continued……… www.AssignmentPoint.com

28. Perfect Negative Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X www.AssignmentPoint.com

29. If the plotted points fall in a narrow band, there would be a high degree of correlation between the variables. Correlation shall be positive if the points show a rising tendency from upper left-hand corner to the right hand corner of the diagram, and negative if the points show a declining tendency from upper left hand corner to the lower right hand corner of the diagram. Continued……… www.AssignmentPoint.com

30. Strong Positive Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X www.AssignmentPoint.com

31. High degree of Negative Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X www.AssignmentPoint.com

32. When will there be low degree of correlation between two variables ? If the points are widely scattered over the diagrams, it indicates very low degree of relationship between the variables. This correlation shall be positive if the points rise from the lower left-hand corner to the upper right-hand corner, and negative if the points run from the upper left–hand side to the lower right hand side to the diagram. www.AssignmentPoint.com

33. Low degree of positive correlation × × × × × × × × × × × × × × × www.AssignmentPoint.com

34. Low degree of negative correlation × × × × × × × × × × × × www.AssignmentPoint.com

35. When will there be no correlation between two variables? If the plotted points lie on a straight line parallel to the X- axis, or in a haphazard manner, it shows the absence of any relationship between the variables (i.e. r = 0) www.AssignmentPoint.com

36. Zero Correlation 10 9 8 7 6 5 4 3 2 1 0 Y 0 1 2 3 4 5 6 7 8 9 10 X www.AssignmentPoint.com

37. Example: The following pairs of values are given: • Make a scatter diagram • Do you think that there is any correlation between profits and capital employed? www.AssignmentPoint.com

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39. Do you think that there is any correlation between profits and capital employed? • It appears from the above diagram that the variables – profits and capital employed are correlated. • Correlation is positive because the trend to the points is upward rising from the lower left hand corner to the upper right–hand corner. • The degree of relationship is high because the plotted points are in a narrow band which shows that it is a case of high degree of positive correlation. www.AssignmentPoint.com

40. What are the merits of scatter diagram method studying of correlation? • Merits: • It is a simple and non–mathematical method of studying correlation between the variables. Hence, it can be easily understood and rough idea can quickly be formed as to whether or not the variables are related. • It is not influenced by the size of extreme values whereas, most of the mathematical methods of finding correlation are influenced by extreme values. • Making a scatter diagram usually is the first step in investigating the relationship between the variable. www.AssignmentPoint.com

41. What are the limitations of Scatter diagram method of studying correlation? • Limitations: • It is not possible to establish the exact degree of correlation between the variables as is possible by applying the mathematical method. www.AssignmentPoint.com

42. What is meant by coefficient of correlation? • The coefficient of correlation (r) is a measure of the strength of the linear relationship between two or more variables.This summarizes in one figure the direction and degree of correlation. • Designated r, it is often referred to as Pearson’s ‘r’ • It can assume any value from –1.00 to +1.00 inclusive. A correlation co-efficient of –1.00 or +1.00 indicates perfect correlation. • If there is absolutely no relationship between the two sets of variables, Pearson’s r is zero. • It requires interval or ratio-scaled data (variables). Continued……. www.AssignmentPoint.com

43. What is meant by Coefficient of Correlation? • Negative values indicate an inverse relationship and positive values indicate a direct relationship. • If there is absolutely no relationship between the two sets of variables, Pearson’s r is zero. A coefficient of correlation r close to o (say, 0.08). shows that the linear relationship is very weak. The same conclusion is drawn if r = - 0.08 . Continued……. www.AssignmentPoint.com

44. What is meant by Coefficient of Correlation? • Coefficients of –0.91 and + 0.91 have equal strength, both indicate very strong correlation between the two variables. Thus, the strength of correlation does not depend on the direction (either – or +). • If the correlation is weak, there is considerable scatter about a line drawn through the center of the data. • For the scatter diagram representing a strong relationship, there is very little scatter about the line. • The following drawing shows the strength and direction • of the coefficient of correlation: Continued……. www.AssignmentPoint.com

45. Perfect negative correlation Perfect positive correlation No correlation Moderate positive correlation Moderate negative correlation Strong positive correlation Weak negative correlation Strong negative correlation Weak positive correlation - 0.50 0 -1.00 + 0.50 + 1.00 Negative correlation Positive correlation www.AssignmentPoint.com Continued……

46. The coefficient of correlation describes not only the magnitude of correlation but also its direction. Thus, + 0.8 would mean that correlation is positive and the magnitude of correlation is 0.8. www.AssignmentPoint.com

47. What are the properties of the co– efficient of correlation? • The following are the important properties of the co – efficient of correlation: • The co– efficient of correlation lies between - 1 and + 1. Symbolically, - 1 ≤ r< +1 or │r ≤ 1 • The co–efficientof correlation is independent of change of origin and scale. • The co–efficientof correlation is the geometric mean of two regression co-efficient • If X and Y are independent variables then co – efficient of correlation is zero. However, the converse is not true. Continued……. www.AssignmentPoint.com

48. Prove that the co–efficient of correlation lies between - 1 and +1. Symbolically, Or r ≤ Solution: Continued….. www.AssignmentPoint.com

49. Continued……. www.AssignmentPoint.com

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