1 Dr. Jerrell T. Stracener EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Department of Engineering Management, Information and Systems Covariance & Correlation

2 Let X and Y be random variables with joint mass function p(x,y) if X & Y are discrete random variables or with joint probability density function f(x, y) if X & Y are continuous random variables. The covariance of X and Y is if X and Y are discrete, and if X and Y are continuous. Covariance of X and Y

3 The covariance of two random variables X and Y with means  X and  Y, respectively is given by Covariance of X and Y

4 Let X and Y be random variables with covariance  XY And standard deviation  X and  Y, respectively. The correlation coefficient X and Y is Correlation Coefficient

5 If X and Y are random variables with joint probability distribution f(x, y), then Theorem

6 If X and Y are independent random variables, then Theorem

7 Correlation Analysis - statistical analysis used to obtain a quantitative measure of the strength of the relationship between a dependent variable and one or more independent variables Correlation Analysis

8 Sample correlation coefficient Notes: -1  r  1 R=r 2  100% = coefficient of determination Correlation

9 Possible Relationship Between X and Y as Indicated by Scatter Diagrams

10 To test for no linear association between x & y, calculate Where r is the sample correlation coefficient and n is the sample size. Conclude no linear association if then treat y 1, y 2, …, y n as a random sample Correlation

11 otherwise conclude that there is linear association between x and y and proceed with regression analysis, where is the value of the t-distribution for which the probability of exceeding it is /2. Correlation

12 The data used for illustration are from a study of two methods of estimating tread wear of commercial tires. The data are shown here and plotted. The variable which is taken as the independent variable X is the estimated tread life in hundreds of miles by the weight-loss method. The associated variable Y is the estimated tread life by the groove-depth method. Correlation - Example

13 Correlation - Example

14 By plugging the data into the formula, Correlation - Example

15 and t= Correlation - Example

16 For a 95% confidence interval, Since is not between –2.145 and 2.145, we can conclude that there is linear association between x and y. Therefore proceed to regression analysis, otherwise treat the y values, y 1, y 2,…,y n as a random sample of size n and analyze the data using methods previously discussed. Correlation - Example