Why is the Pearson correlation coefficient between -1 and 1?

Question

I know how to interpret it and how to calculate it

But why does the sum of the products of the standardized values divided by n will always be between -1 and 1?

I would like to understand this mathematical property in an intuitive way, please.

I searched a lot for this answer but couldn't find an answer suitable for someone who is not a mathematician like me.

Thanks in advance.

Answer 1

It's not clear to me if you are asking a purely technical question (to which the answer "it drops out of Cauchy Schwarz" is acceptable) or a motivational question (why would we expect it to be between $\pm1$, or how can we understand, in the context of applications, what the $\pm1$ bound "means").

So here is a stab at the 2nd kind of answer. In the context of least squares fitting of straight lines to data scatter plots, where you try to fit an affine function of $X$ to observed $Y$ values, one can ask for the fraction of variance in $Y$ "explained" by the affine function of $X$. The wikipedia article explains that this is the square of the correlation coefficient.