Covariance and correlation, and how are they related?

I get that corellation is the covariance divided by the multiplie variance of the two, uh, things.

What i don't get is why they are divided by the multiplied variance, and why that limits the value to the range -1 : 1.

I suppose i'm really looking for a logical explanation, although a mathematical one wold welcome nontheless.

• Do you know about the Cauchy-Schwarz-inequality? – Michael Hoppe Oct 1 '15 at 11:55
• @MichaelHoppe no, i just had my first stats lecture today. Lots of algorithms, but not much rationale behind them. Is it relevant, this inequality? – bharal Oct 1 '15 at 12:00
• Sort of; from there the it's clear why the range is $[-1,1]$. BTW, the covariance is divided by the product of the standard deviations, not by the variances. – Michael Hoppe Oct 1 '15 at 12:03
• You should read the Wikipedia page entitled "Covariance" – Alec Teal Oct 1 '15 at 12:21

As Michael said, the fact that the correlation coefficient is between $-1$ and $1$ can be shown as a special case of the Cauchy-Schwarz inequality (see https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Rn ).