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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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    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*.2012-06-28