# Pearson Product Moment Correlation Coefficient for linear relationship only?

Pearson Product Moment Correlation Coefficient method is used only if variables are linearly correlated.
But if they are linearly correlated, then correlation coefficient $$r=\pm 1$$ only.
Then why we find out r by this method and get something like $r=0.6$?

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If two random variables are linearly dependent then you will find $r=\pm 1$. To be linearly dependent means that one is a linear function of the other:

$$Y = a + bX$$

However, you can have a linear correlation without linear dependence, for example

$$Y = a + bX + \epsilon$$

where $\epsilon$ is some other random variable which is independent from $X$. In this case you will find $r^2<1$, and how much less than $1$ depends on the variance of $\epsilon$.

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But If we have only two variables, then also we use pearson's method to find $r$, which can evaluate to $<1$. So how is it possible ? –  Happy Mittal Jun 28 '12 at 8:41
What do you mean? If you have two random variables, then the only way to get $r=\pm 1$ is if one of them is an exact linear multiple of the other one. In any other situation, you will have $-1<r<1$. –  Chris Taylor Jun 28 '12 at 8:43
Yes, but in other situation i.e. if they are not linearly dependent, then also we use pearson method, but pearson method is used only if two variables are linearly dependent. Isn't it contradictory ? –  Happy Mittal Jun 28 '12 at 8:48
No. You can always use the method. However, you need to be aware that it only reveals the strength of linear relationships. In particular, there can be a very strong nonlinear relationship, but you still might get a low value for $r^2$. I think you need to look up the difference between linear dependence and linear correlation. –  Chris Taylor Jun 28 '12 at 8:55