18.2  Correlation Coefficient of X and Y
18.2  Correlation Coefficient of X and YThe covariance of \(X\) and \(Y\) necessarily reflects the units of both random variables. It is helpful instead to have a dimensionless measure of dependency, such as the correlation coefficient does.
 Correlation Coefficient

Let \(X\) and \(Y\) be any two random variables (discrete or continuous!) with standard deviations \(\sigma_X\) and \(\sigma_Y\), respectively. The correlation coefficient of \(X\) and \(Y\), denoted \(\text{Corr}(X,Y)\) or \(\rho_{XY}\) (the greek letter "rho") is defined as:
\(\rho_{XY}=Corr(X,Y)=\dfrac{Cov(X,Y)}{\sigma_X \sigma_Y}=\dfrac{\sigma_{XY}}{\sigma_X \sigma_Y}\)
Example 181 (continued)
Suppose that \(X\) and \(Y\) have the following joint probability mass function:
What is the correlation coefficient of \(X\) and \(Y\)?
On the last page, we determined that the covariance between \(X\) and \(Y\) is \(\frac{1}{4}\). And, we are given that the standard deviation of \(X\) is \(\frac{1}{2}\), and the standard deviation of \(Y\) is the square root of \(\frac{1}{2}\). Therefore, it is a straightforward exercise to calculate the correlation between \(X\) and \(Y\) using the formula:
\(\rho_{XY}=\dfrac{\frac{1}{4}}{\left(\frac{1}{2}\right)\left(\sqrt{\frac{1}{2}}\right)}=0.71\)
So now the natural question is "what does that tell us?". Well, we'll be exploring the answer to that question in depth on the page titled More on Understanding Rho, but for now let the following interpretation suffice.
Interpretation of Correlation
On the page titled More on Understanding Rho, we will show that \(1 \leq \rho_{XY} \leq 1\). Then, the correlation coefficient is interpreted as:
 If \(\rho_{XY}=1\), then \(X\) and \(Y\) are perfectly, positively, linearly correlated.
 If \(\rho_{XY}=1\), then \(X\) and \(Y\) are perfectly, negatively, linearly correlated.
 If \(\rho_{XY}=0\), then \(X\) and \(Y\) are completely, unlinearly correlated. That is, \(X\) and \(Y\) may be perfectly correlated in some other manner, in a parabolic manner, perhaps, but not in a linear manner.
 If \(\rho_{XY}>0\), then \(X\) and \(Y\) are positively, linearly correlated, but not perfectly so.
 If \(\rho_{XY}<0\), then \(X\) and \(Y\) are negatively, linearly correlated, but not perfectly so.
So, for our example above, we can conclude that \(X\) and \(Y\) are positively, linearly correlated, but not perfectly so.