Correlation and Covariance

1. Correlation and Covariance R. F. Riesenfeld (Based on web slides by James H. Steiger)

2. Goals • Introduce concepts of • Covariance • Correlation • Develop computational formulas CS5961 Comp Stat

3. Covariance • Variables may change in relation to each other • Covariancemeasures how much the movement in one variable predicts the movement in a corresponding variable CS5961 Comp Stat

4. Smoking and Lung Capacity Example: investigate relationship between cigarette smokingand lung capacity Data: sample group response data on smoking habits, andmeasured lung capacities, respectively CS5961 Comp Stat

5. Smoking v Lung Capacity Data CS5961 Comp Stat

6. Smoking and Lung Capacity

7. Smoking v Lung Capacity • Observe that as smoking exposure goes up, corresponding lung capacity goes down • Variables covaryinversely • Covarianceand Correlation quantify relationship CS5961 Comp Stat

8. Covariance • Variables that covaryinversely, like smoking and lung capacity, tend to appear on opposite sides of the group means • When smoking is above its group mean, lung capacity tends to be below its group mean. • Average product of deviationmeasures extent to which variables covary, the degree of linkage between them CS5961 Comp Stat

9. The Sample Covariance • Similarto variance, for theoretical reasons, average is typically computed using (N -1), notN. Thus, CS5961 Comp Stat

10. Calculating Covariance CS5961 Comp Stat

11. Calculating Covariance CS5961 Comp Stat

12. Evaluation yields, CS5961 Comp Stat

13. Covariance under Affine Transformation CS5961 Comp Stat

14. CS5961 Comp Stat

15. (Pearson) Correlation Coefficient rxy • Like covariance, but uses Z-values instead of deviations. Hence, invariant under linear transformation of the raw data. CS5961 Comp Stat

16. Alternative (common) Expression CS5961 Comp Stat

17. Computational Formula 1 CS5961 Comp Stat

18. Computational Formula 2 CS5961 Comp Stat

19. Table for Calculating rxy CS5961 Comp Stat

20. Computingrxyfrom Table CS5961 Comp Stat

21. Computing Correlation CS5961 Comp Stat

22. Conclusion rxy= -0.96 implies almost certainty smoker will have diminish lung capacity Greater smoking exposure implies greater likelihood of lung damage CS5961 Comp Stat

23. End Covariance & Correlation Notes CS5961 Comp Stat