Correlation_and_Regression_Presentation.pptx

Correlation_and_Regression_Presentation.pptx

1. Correlation and Regression Understanding relationship between two variables and predicting one from another.

2. Meaning of Correlation Correlation measures the strength and direction of a linear relationship between two variables.

3. Types of Correlation Positive: both increase Negative: one increases, other decreases Zero: no relationship

4. Methods of Studying Correlation 1. Scatter Diagram 2. Karl Pearson’s Coefficient 3. Spearman’s Rank Correlation

5. Scatter Diagram A graphical representation showing the relationship between two variables.

6. Interpretation of Scatter Diagram Close dots: strong correlation Upward slope: positive Downward slope: negative Random: no correlation

7. Karl Pearson’s Coefficient r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)² Σ(y−ȳ)²] Range: -1 ≤ r ≤ +1

8. Interpretation of r r = +1 → Perfect positive r = -1 → Perfect negative r = 0 → No correlation

9. Properties of Correlation Coefficient 1. Unit-free 2. Symmetrical 3. Lies between -1 & +1 4. Independent of origin & scale

10. Spearman’s Rank Correlation rₛ = 1 - (6Σd²)/(n(n²−1))

11. Interpretation of Rank Correlation rₛ = +1 → Perfect positive rₛ = -1 → Perfect negative rₛ = 0 → No agreement

12. Example of Rank Correlation X:10,8,6; Y:20,15,25 Ranks and differences can be computed.

13. Probable Error of Correlation PE = 0.6745 × (1−r²)/√n Used to test significance.

14. Interpretation of Probable Error |r| > 6×PE → Significant |r| < PE → Not significant

15. Regression Analysis Regression studies dependence of one variable on another.

16. Types of Regression 1. Simple Regression 2. Multiple Regression

17. Simple Linear Regression Equation Y = a + bX Y: Dependent, X: Independent, a: Intercept, b: Slope

18. Regression Coefficients bᵧₓ = r(σᵧ/σₓ), bₓᵧ = r(σₓ/σᵧ)

19. Properties of Regression Coefficients Same sign, geometric mean = r, not symmetric, independent of origin

20. Regression Lines Y on X: Y = a + bX X on Y: X = a' + b'Y

21. Derivation of Regression Coefficients bᵧₓ = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² bₓᵧ = Σ(x−x̄)(y−ȳ)/Σ(y−ȳ)²

22. Example of Regression Calculation Given X:2,4,6,8,10 and Y:5,7,9,8,11, find Y = a + bX.

23. Steps to Solve Example 1. Find means 2. Compute deviations 3. Calculate b 4. Substitute in equation

24. Relation Between Correlation and Regression r² = bₓᵧ × bᵧₓ

25. Uses of Regression 1. Prediction 2. Cause-effect analysis 3. Forecasting

26. Limitations Correlation ≠ causation, linear only, sensitive to outliers

27. Comparison Table Correlation: relationship measure Regression: prediction model

28. Graphical Representation Scatter diagram showing regression lines intersecting at (x̄,ȳ).

29. Summary Correlation: relationship strength Regression: prediction tool

30. References Gupta & Kapoor (2020) Spiegel (2013) Croxton & Cowden