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.

  • $\begingroup$ 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

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.


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