1. CORRELATION

2. Correlation  If two variables vary in such a way that movement in one is accompanied by the movement in other, the variables are said to be correlated.

3. Correlation The manager of the business environment of today is very often interested in finding out whether there is any association between two or more variables and if it is true, he would like to know the strength of relationship between the variables. The strength of relationship is also known as the degree of relationship.. The degree of relationship between two variables can be elegantly worked out by correlation coefficient.

4. Examples  What is the correlation between demand and price of a product?  A marketing manager would like to know the degree of relationship between advertising expenditure and the sales volume.  Relationship between I.Q. and Performance in Entrance Examination to MBA  Relationship between Speed of Conveyor Belt, in a factory and Percentage of Defectives in the output.

5. Scatter Diagram  It is a way to plot the bi-variate data on a two dimensional graph so that pattern can be revealed.

6. Visual Displays and Correlation Analysis Begin the analysis of bivariate data (i.e., two variables) with a scatter plot. Begin the analysis of bivariate data (i.e., two variables) with a scatter plot. A scatter plot - displays each observed data pair (x i, y i ) as a dot on an X/Y grid - indicates visually the strength of the relationship between the two variables A scatter plot - displays each observed data pair (x i, y i ) as a dot on an X/Y grid - indicates visually the strength of the relationship between the two variables McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

7. Correlation Analysis The sample correlation coefficient (r) measures the degree of linearity in the relationship between X and Y. The sample correlation coefficient (r) measures the degree of linearity in the relationship between X and Y. -1 < r < +1 r = 0 indicates no linear relationship r = 0 indicates no linear relationship In Excel, use =CORREL(array1,array2), where array1 is the range for X and array2 is the range for Y. In Excel, use =CORREL(array1,array2), where array1 is the range for X and array2 is the range for Y. McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Strong negative relationship Strong positive relationship

8. McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Correlation Analysis

9. Scatter Plot McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Strong Positive Correlation

10. Scatter Plot McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Weak Negative Correlation Strong Negative Correlation

11. Scatter Plot McGraw-Hill/Irwin© 2007 The McGraw-Hill Companies, Inc. All rights reserved. No Correlation Nonlinear Relation

12. Linear Correlation  Correlation is said to be linear if the plotted graph resembles more or less a straight line.

13. Non- Linear Correlation  Correlation is said to be Non-linear if the plotted graph doesn't resembles a straight line.

14. No Correlation  If the plotted graph doesn't resembles any uniform figure it can be considered as if there is no correlation.

15. Positive Correlation  If both the variables either increase or decrease simultaneously, they are called positively correlated.

16. Negative Correlation  If the increase in one variable causes decrease in another variable and vice-versa, variables are said to be negatively correlated.

17. Pearson’s correlation coefficient

18. Co-Variance  Cov(x,y) = (X - µ x )(Y - µ y )/N  Where µ x & µ y are means of X and Y series respectively.

19. Pearson’s correlation coefficient

20. Correlation Coefficient Interpretation  The value of correlation coefficient r lies between –1 and +1.  The value +1 implies that there is a perfect positive correlation,.  The value –1 implies that there is a perfect negative correlation.  In real life situations, the correlation coefficient does not equals the value 1( +1 or –1). Its value nearing 1, indicates strong linear relationship, and its value nearing 0 indicates no correlation.

21. Question 1 XY 69 211 510 48 87

22. Question 2  Ad ExpenseSales 3947 6553 6258 9086 8262 7568 2560 9891 3651 7884

23. Question 3  Age of Husband Age of wife 2318 2722 2823 2924 3025 3126 3328 3529 3630 3932

24. Question 4 Sales Incentives 5011 5013 5514 6016 6516 6515 6014 6013 5013

25. Coefficient of Determination  The square of the correlation coefficient r, expressed as r 2, is known as the coefficient of determination.  It indicates the extent to which variation in one variable is explained by the variation in the other.

26. Coefficient of Determination  For example, let two variables, say x and y, be inter- dependent and variation in x cause variation in y.  Further, let the correlation coefficient work out to be, say, 0.9. The coefficient of determination, in this case, is square of 0.9 i.e. 0.81.  It implies that 0.81 or 81% of the variation in y is due to variation in x or explained by the variation in x.- the remaining 19 % ( = 100 % – 81%) is due to or explained by some other factors.  ( 1 – r 2 ) is referred to as coefficient of alienation, and gives the percentage of variation in the dependent variable, not explained by the independent variable.