Correlation has a dictionary meaning (two things are related to one another) and a math meaning (having to do with a mathematical relationship between two variable being well-approximated by a line). It is important to note that two variables may be perfectly correlated in the dictionary sense, but be mathematically completely uncorrelated.

Consider y=x^2. There aren’t even any random variables in this one. The variable y is completely determined by x. And yet, if you calculate the correlation coefficient between x and y on the interval [-1,1], you will find that it is exactly zero. This is a wonderful example of the fact that not only is correlation not causation, causation may not imply correlation! This insight has been brought to you by my former officemate, a statistician who occasionally would let slip wonderful fun facts like this one.

* Of course, x^2 and y are perfectly correlated even though x and y are not, but we’d need to use a different instrument variable s=x^2 to find that correlation.