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?

Correlations between multiple variables can be defined as a joint cumulant. In physics we call this a "connected correlation function". In statistics, one would call these quantities covariances instead of correlations as it's customary to normalize correlations so that they are between -1 and 1.
The connected correlation between $n$ variables is an effect that's due to all $n$ variables together that cannot be attributed to correlations between variables in proper subsets of the $n$ variables. Connected correlation functions are multilinear functions of their arguments.
The expectation value of any arbitrary product of random variables can be written as a sum of products of connected correlations involving all the ways one can partition the set of variables into disjoint subsets. This property yields a recursion for the joint correlation function, so it can be used to define it.
If we denote connected correlations as $C(X_1,X_2,\ldots,X_n)$, then we have:
$$C(X) = E(X)\tag{1}$$
which follows trivially from the recursion.
For two variables $X$ and $Y$ we have that the connected correlation $C(X,Y)$ is the covariance between $X$ and $Y$. This also follows easily from the recursion:
$$
E(XY) = C(X,Y) + C(X) C(Y)
$$
Using (1) we then get:
$$
C(X,Y)=E(XY) - E(X) E(Y)\tag{2}
$$
For 3 variables $X$, $Y$ and $Z$, the recursion is:
$$
E(XYZ) = C(X,Y,Z) + C(X,Y) C(Z) + C(X,Z) C(Y)+C(Y,Z) C(X) + C(X)C(Y)C(Z)
$$
Substituting the expression (1) and (2) for the connected correlations of one and two variables then yields:
$$C(X,Y,Z)=E(XYZ) - E(XY) E(Z) - E(XZ)E(Y) - E(YZ)E(X) + 2E(X)E(Y)E(Z)$$
