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The correlation coefficient cannot exceed an absolute value of 1

This is well-known. But why is that the case? How can we proof it? This post gives one explanation using the Cauchy-Schwarz inequality.

Here’s one version of the definition of correlation:

$$r=\frac{\sum \left(\Delta x\Delta y\right)}{\sqrt{\sum \Delta x^2}\sqrt{\sum \Delta y^2}}$$

where $\Delta x$ and $\Delta y$ are the differences of $x_i$ and $\overline{x}$, that is: $\Delta x_i=x_i-\overline{x}$, and similarly for $\Delta y_i$.

For the ease of notation, let’s proceed with the understanding that $x$ stands for the differences, ie $\Delta x$ (and similarly for $y$):

$$r=\frac{\sum \left(xy\right)}{\sqrt{\sum x^2}\sqrt{\sum y^2}}$$

Now, we conjecture that

$$r=\frac{\sum \left(xy\right)}{\sqrt{\sum x^2}\sqrt{\sum y^2}}\le 1$$

Let’s multiply the equation by the denominator of the LHS:

$$\sum \left(xy\right)\le \sqrt{\sum x^2}\cdot \sqrt{\sum y^2}$$

The Cauchy Schwarz inequality states that

$$|⟨x,y⟩|\le ||x||\cdot ||y||$$

In words, the inner product $⟨x,y⟩$ (in its positive variant, ie $>0$) is smaller or equal to the product of the vector norms.

Stated differently:

$$\sum xy\le \sqrt{\sum x^2}\cdot \sqrt{\sum y^2}$$

Which is what we wanted to proof in the first place.

Here’s a quite nice intuition on the Cauchy Schwarz inequality.