In the area of correlation analysis, two separate cases or approaches present themselves:
Let us consider these two situations in turn.
In this instance, we need to determine the direction as well as the strength (i.e., the degree of closeness) of the relationship between the random variables X and Y, where X and Y follow a “joint bivariate distribution.” This will be accomplished by first extracting a sample of points (Xi, Yi), i = 1,. . .,n, from the said distribution. Then once we compute the sample correlation coefficient, we can determine whether or not it serves as a “good” estimate of the underlying degree of covariation within the population.
To this end, let X and Y be random variables that follow a joint bivariate distribution. Let: E(X) and E(Y) depict the means of X and Y, respectively; S(X) and S(Y) represent the standard deviations of X and Y, respectively; and COV(X,Y) denotes the covariance between X and Y.2 Then the population ...