0
$\begingroup$

I understand that one can calculate the correlation coefficient $r_{xy}$ between observations $x_i$ and $y_i$ with

$$ r_{xy} = \frac{\sum_{i=1}^n (x_i - x_m)(y_i - y_m)}{\sqrt{(\sum_{i=1}^n (x_i - x_m)^2)(\sum_{i=1}^n (y_i - y_m)^2)}}$$

where $x_m,y_m$ is the mean value. I have read that if the absolute value from $r_{xy}$ is close to 1 then this means that the observations are probably correlated, and if the value is close to 0 then its not correlated:

enter image description here

However, this is not obvious to me when looking at the formula. Can someone explain to me why it follows from the formula that $$|r_{xy}| \approx 1$$ means correlation and $$|r_{xy}| \approx 0$$ means no correlation?

$\endgroup$
  • $\begingroup$ Isn't that the definition of correlation? $\endgroup$ – Ivan Neretin Apr 2 '18 at 10:00
  • $\begingroup$ $r$ lies in the interval $[-1,1]$, the proof of which will tell you that $|r|=1$ implies perfect linear relationship between $x$ and $y$. By writing $r$ in terms of the usual definition of covariance and variance you would see that $r=0$ implies covariance is $0$, which is the definition of uncorrelated (i.e. no linear relation). $\endgroup$ – StubbornAtom Apr 2 '18 at 10:09
2
$\begingroup$

You're computing the cosine of the angle between two vectors; this is obvious if you know dot products well. A high positive correlation means the vectors are nearly parallel; a very negative correlation means they're nearly antiparallel.

$\endgroup$
  • $\begingroup$ Wow, this is mind blowing to me! Thank you!! $\endgroup$ – Adam Apr 2 '18 at 10:17

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.