# How can I simply prove that the pearson correlation coefficient is between -1 and 1?

For building a recommendation system, I also use the Pearson correlation coefficient. This is the definition:

$r(x, y)=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2 \cdot \sum_{i=1}^n (y_i-\bar{y})^2}}$

$x$ and $y$ are part of $\mathbb{R}$.

Now for coding, it is important to take care of all potential outcomes. For example, if the denominator is zero, you will have to filter that or throw an exception.

I came up with some arguments, one of them being that if all values of $x_i$ and/or $y_i$ were equal to the average of $x$ and/or $y$, then the denominator would be zero.

But how can I prove that the coefficient is either undefined (zero denominator) or in between -1 and 1? What is the best approach?

• It is the Schwartz Inequality (but I keep misspelling his name). Depending where you are from, you might call it Cauchy Schwartz, or throw in Bunyakovsky, maybe others. – André Nicolas Nov 13 '13 at 0:50

First of all Pearson's correlation coefficient is bounded between -1 and 1, not 0 and one. It's absolute value is bounded between 0 and 1, and that useful later.

Pearson's correlation coefficient is simply this ratio:

$\rho = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$

Both of the variances are non-negative by definition, so the denominator is $\ge 0$. The only way a singularity can occur is if one of the variables has 0 variance.

If two random variables are perfectly uncorrelated, (i.e. independent) then their covariance is 0. So 0 is a valid lower bound.

This can be shown like so:

$Cov(X,Y) = E[(X-\bar{X})(Y-\bar{Y})] = E[XY] - E[X]E[Y]$

if two random variables are independent, then $E[XY]=E[X]E[Y]$, and

$Cov(X,Y) = E[XY] - E[X]E[Y] = E[X]E[Y] - E[X]E[Y] = 0$.

Now for the upper bound. Here we apply the Cauchy-Schwarz inequality.

$|Cov(X,Y)|^2 \le Var(X)Var(Y)$

$\therefore |Cov(X,Y)| \le \sqrt{Var(X)Var(Y)}$

plug this result from the Cauchy-Schwarz inequality into the formula for $\rho$, and we get:

$|\rho| = |\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}| \le \frac{\sqrt{Var(X)Var(Y)}}{\sqrt{Var(X)Var(Y)}} = 1$

Thus we have the absolute value of the correlation is bounded below by 0 and above by 1.