Correlation Coefficient of X and Y

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The covariance of X and Y neccessarily reflects the units of both random variables.  It is helpful instead to have a dimensionless measure of dependency, such as the correlation coefficient does.

Definition.  Let X and Y be any two random variables (discrete or continuous!) with standard deviations σX and σY, respectively. The correlation coefficient of X and Y, denoted Corr(X,Y) or rho of x y (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 (continued)

Suppose that and Y have the following joint probability mass function:

table

What is the correlation coefficient of X and Y?

Solution. On the last page, we determined that the covariance between X and Y is 1/4. And, we are given that the standard deviation of X is 1/2, and the standard deviation of Y is the square root of 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:

  1. If \(\rho_{XY}=1\), then X and Y are perfectly, positively, linearly correlated.
  2. If \(\rho_{XY}=-1\), then X and Y are perfectly, negatively, linearly correlated.
  3. If \(\rho_{XY}=0\), then X and Y are completely, un-linearly 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.
  4. If \(\rho_{XY}>0\), then X and Y are positively, linearly correlated, but not perfectly so.
  5. 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 and are positively, linearly correlated, but not perfectly so.