1. BUSINESS MATHEMATICS & STATISTICS

2. Module 6 Correlation ( Lecture 28-29) Line Fitting ( Lectures 30-31) Time Series and Exponential Smoothing ( Lectures 32-33)

3. LECTURE 28 Review Lecture 27 Measures of Dispersion Correlation Part 1

4. Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Variance For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

5. Most Important Measure of Variation Shows Variation About the Mean Same unit of measurement as the observations For the Population : For the Sample: Standard Deviation For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

6. Sample Standard Deviation For the Sample : use n - 1 in the denominator. Data: 10 12 14 15 17 18 18 24 s = n = 8 Mean =16 = 4.2426 s

7. Comparing Standard Deviations Value for the Standard Deviation is larger for data considered as a Sample. Data : 10 12 14 15 17 18 18 24 N= 8 Mean =16

8. Comparing Standard Deviations Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s =.9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C

9. Coefficient of Variation Measure of relative variation Always a % Shows variation relative to mean Used to compare 2 or more groups Formula ( for sample):

10. Comparing Coefficient of Variation Stock A: average price last year = Rs. 50 Standard deviation = Rs. 5 Stock B: average price last year = Rs. 100 Standard deviation = Rs. 5 Coefficient of Variation : Stock A: CV = 10% Stock B: CV = 5%

11. Shape Describes how data are distributed Measures of shape: Symmetric or skewed Right-Skewed Left-SkewedSymmetric Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean

12. DISPERSION OF DATA Mean Deviation About Mean MD(mean) = Sum(xi- mean)/n For Grouped data MD(mean) = Sum fi (xi - mean)/Sum fi Mean Deviation About Median MD(median) = Sum(xi-median)/n For Grouped data MD(median) = Sum fi (xi-median)/Sum fi

13. CORRELATION Types of regression models Determining the simple linear regression equation Measures of variation in regression and correlation Assumptions of regression and correlation

14. CORRELATION Residual analysis Inferences about the slope Estimation of predicted values Pitfalls in regression and ethical issues Correlation - measuring the strength of the association

15. Purpose of Regression Analysis Regression Analysis Is Used Primarily to Model Causality And Provide Prediction Predict the values of a dependent (response) variable based on values of at least one independent (explanatory variable) Explain the effect of the independent variables on the dependent variable

16. Purpose of Regression Analysis Correlation Analysis Is Used to Measure Strength of Association Between Numerical Variables Only Strength of the Relationship is Concerned No Causal Effect is Implied

17. The Scatter Diagram Plot of all (X i, Y i ) pairs

18. Types of Regression Models Positive Linear RelationshipNegative Linear Relationship

19. BUSINESS MATHEMATICS & STATISTICS

20. Types of Regression Models Relationship NOT Linear No Relationship

21. Simple Linear Regression Model Relationship Between Variables Is Described by A Linear Function The Change of One Variable Causes The Other Variable to Change A Dependency of One Variable on the Other

22. Average Value Population Regression Line Is A Straight Line that Describes The Dependence of The Average Value of One Variable on The Other Population Y intercept Population Slope Coefficient Random Error Dependent (Response ) Variable Independent (Explanatory) Variable Population Linear Regression Population Regression Line

23. = Random Error Y X Population Linear Regression Observed Value (continued)

24. Sample Linear Regression Sample Y Intercept Sample Slope Coefficient Sample Regression Line Provides an Estimate of The Population Regression Line as well as a Predicted Value of Y provides an estimate of Sample Regression Line Residual

25. Sample Linear Regression Y X Observed Value (continued)

26. Simple Linear Regression Equation: Example You wish to examine the relationship between the square footage of produce stores and their annual sales. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best Annual Store Square Sales Feet(1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

27. Scatter Diagram Example Excel Output

28. Equation for Sample Regression Line  From Excel Printout:

29. Graph of the Sample Regression Line Y i = 1636.415 +1.487X i 

30. Interpreting the Results Y i = 1636.415 +1.487X i The slope of 1.487 means that each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units. The model estimates that for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487. 

31. BUSINESS MATHEMATICS & STATISTICS