Correlation is a Cosine

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You might have heard a statement of the kind “correlation is a cosine” but you have not bothered enough to investigate what it means precisely. It certainly sounds interesting. How can the simplest bivariate summary statistic be related to a trigonometric function from sixth grade? What is the relation between correlation and cosines?

Let me explain.

As with anything else, a Google search is your friend here, with many links to helpful Stack Overflow posts explaining this connection from all sorts of angles. However, I do find John D. Cook’s blog post most helpful, and I am following his exposition closely.

Piece #1: The Law of Cosines

The law of cosines states that in any triangle with sides x, y, and z and an angle (between x and y) \theta, we have:

(1)   \begin{equation*} z^2 = x^2 + y^2 - 2 x y cos(\theta), \end{equation*}

In the special case when \theta=\frac{\pi}{2}, the term on the right-hand side equals 0 and the equation reduces to the well-known Pythagorean Theorem.

Piece #2: The Variance of the Sum of Two Random Variables

Let’s imagine two random variables A, B. The variance of their sum is given by:

    \begin{equation*}var(A+B) = var(A)+var(B)+2 cov(A,B),\end{equation*}

where cov(\cdot), denotes covariance. We can substitute the last term with its definition as follows:

    \begin{equation*}var(A+B) = var(A)+var(B)+2 corr(A,B) sd(A) sd(B). \end{equation*}

Next, we know that var(\cdot)=sd^2(\cdot). Substituting, we get:

(2)   \begin{equation*}sd^2(A+B) = sd^2 (A)+ sd^2 (B)+2 corr(A,B) sd(A) sd(B).\end{equation*}

Piece #3: Putting the Two Equations Together

Setting x=sd(A), y=sd(B), and z=sd(A+B) in equation (1) gives the desired result. With one small caveat – the negative sign on the cosine term. To get around this we can simply look at the complementary angle \delta = \pi - \theta.

That is, we imagine a triangle with sides equal to sd(A), sd(B) and sd(A+B), where \theta is the angle between sd(A), sd(B). When this angle is small (\theta < \frac{pi}/{2}), the two sides point in the same direction and A and B are positively correlated. The opposite is true for theta > \frac{pi}/{2}. As mentioned above, theta = \frac{pi}/{2} kills the correlation term, consistent with A and B being independent.


In short, the formula for the variance of two random variables follows the law of cosines. Substituting and arranging terms shows the desired result.

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