L12.8 The Correlation Coefficient
The covariance between two random variables tells us something about the strength of the dependence between them. But it is not so easy to interpret qualitatively. For example, if I tell you that the covariance of X and Y is equal to 5, this does not tell you very much about whether X and Y are closely related or not. Another difficulty is that if X and Y are in units, let's say, of meters, then the covariance will have units of meters squared. And this is hard to interpret. A much more informative quantity is the so-called correlation coefficient, which is a dimensionless version of the covariance. It is defined by this formula here. We just take the covariance and divide it by the product of the standard deviations of the two random variables. Now, if X has units of meters, then the standard deviation also has units of meters. And so this ratio will be dimensionless. And it is not affected by the units that we're using. The same is true for this ratio here, and ...