0
$\begingroup$

I'm trying to understand what it means exactly for $p$ to be equal to $1$ or $-1$. I know that the following is true:

$$-1\le \frac{cov(X,Y)}{SD[X]·SD[Y]}\le 1$$

Where the central term is the Pearson correlation coefficient and $SD$ stands for standard deviation.

Now, let's say $corr(X,Y)=1$. First, what does that imply graphically? Does it mean that, if we were to plot $X$ and $Y$ together on a graph, all pairs $(x,y)$ would be sitting on a straight line with some (any) slope greater than zero? Is this interpretation correct? If so, is there any way to intuitively understand why $cov(X,Y)=SD[X]·SD[Y]$ implies this perfect alignment of all $(x,y)$? I know that a positive $cov(X,Y)$ means that, on average, when $X$ is above its expectation so is $Y$, if it's of any use. Any clarification will be appreciated. Thanks in advance.

$\endgroup$
  • $\begingroup$ Do you know about Regression Lines??? $\endgroup$ – Croma14 Mar 30 '17 at 12:40
  • $\begingroup$ I know they're designed to best fit a given set of data. I haven't quite studied the subject yet, but that's ok if your answer relates to it. I will study it this semester in college, so it will still be of great help. $\endgroup$ – daniels Mar 30 '17 at 13:02
1
$\begingroup$

It is true that $|corr(X, Y)| = 1$ if and only if all the points $(x_i, y_i)$ fall on a straight line. (I'm ignoring the case where either all the $x_i$ are equal or all the $y_i$ are equal, because in those cases the correlation coefficient isn't defined.) This can be proven from the definition of the correlation coefficient by doing a bunch of algebra, although I can't imagine this is particularly enlightening.

$\endgroup$

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.