Why is correlation only defined between two variables? I am an MBA student is taking statistics courses.
Our statistics prof was teaching us about correlations in statistics. We learned about Pearson's Correlation of Coefficient which is defined as the correlation between only two variables.
I am asked the prof if the correlation can be calculated between more than two variables and he said "no". But I am struggling to understand why this is the case?
For example: Hurricanes are correlated with Wind Speed, Hurricanes are also correlated with Temperature - Although I am not a meteorologist don't actually know if this is true (I just assumed this to show a point), we can see that in theory more than two variables can be correlated.
Therefore, why do we only evaluate the correlation between two variables and not more than two variables?

One of the interpretations of Pearson's correlation coefficient is the proportionate reduction of error (PRE): Assume a variable vector $y$, about which nothing further is known. Then, if you have to predict a missing $y_i$, your best estimator would be the arithmetic mean $\bar{y}$, and the error would be $E_1 = SS_{total} = \sum_{i=1}^n(y_i - \bar{y})^2$.
Assume now that you have a second data set $x$, which can be used to predict $y$, in the simplest case (but without limitation of generality) by linear regression $\hat{y}_i = a + b x_i$. Then the error becomes $E_2 = SS_{residual} = \sum_{i=1}^n(y_i - \hat{y}_i)^2$, and the coefficient of determination (fraction of variability of $y$ that is explained by $x$) is $r^2 = \frac{E_1 - E_2}{E_1} = 1 - \frac{E_2}{E_1}$. The $R^2$ mentioned by others is a generalisation of this approach for several explaining variables. There are other correlation coefficients with PRE-interpretation for non-cardinal data (Guttman's λ, Freeman's θ, Wilson's e, the point-biserial correlation, $\eta^2$). See Freeman, L.C.: Elementary applied statistics, New York, London, Sidney (John Wiley and Sons) 1965 for a very readable introduction.
Thus, you should think about what PRE is in your particular case to construct a suitable correlation coefficient.
