2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Do you know about the Cauchy-Schwarz-inequality? $\endgroup$ – Michael Hoppe Oct 1 '15 at 11:55
  • $\begingroup$ @MichaelHoppe no, i just had my first stats lecture today. Lots of algorithms, but not much rationale behind them. Is it relevant, this inequality? $\endgroup$ – bharal Oct 1 '15 at 12:00
  • $\begingroup$ 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. $\endgroup$ – Michael Hoppe Oct 1 '15 at 12:03
  • $\begingroup$ You should read the Wikipedia page entitled "Covariance" $\endgroup$ – Alec Teal Oct 1 '15 at 12:21
1
$\begingroup$

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 ).

There is a geometrical interpretation where you see random variables as vectors. Then the covariance is a scalar product, the standard deviations are the norms and the correlation coefficient is the cosine of the angle between the random variables.

But you can also see this as a scaling. Dividing by the standard deviations is a way to put everything on the same scale, so that you can compare variables that have different units for example.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.