# Why does the correlation coefficient work?

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:

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?

• Isn't that the definition of correlation? – Ivan Neretin Apr 2 '18 at 10:00
• $r$ 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). – StubbornAtom Apr 2 '18 at 10:09