# Do two variables need to be independent in order to obtain a correlation? [closed]

I would like to find the correlation between two variables. It was suggested to me that two variables should be independent; otherwise it is not meaningful statistically to calculate a correlation. For example, variable is x and another is $y$ which won't be calculated from $x$, e.g., $y=ab/c+x$. $a$, $b$, and $c$ are some constant value that are the same for all $y$.

### Questions

1. Is what I understand correct?
2. The data set is $y = P+Q+x$. $P$ and $Q$ depends on each $x$, and the same $x$ value can have different value of $P$ and $Q$. Is it meaningful to find a correlation given this data?
• I tried to transform a question to the form that it is answerable. Please confirm that nothing is lost from the original form. Mar 7 '11 at 12:53
• @mpiktas Thank you for your valiant effort to rescue this question. However, the changes were so substantial that significant and potentially important parts of the original question disappeared or were altered beyond recognition. Therefore I have rolled this back to the original question. Let's try to make sense of it as it stands.
– whuber
Mar 7 '11 at 15:35
• We can try to fix the English--that shouldn't be a problem--but there are problems with the mathematical expressions in the questions. When $y = ab/c + x$, obviously $y$ is "calculated from" $x$. In this case the correlation of a set of $(x,y)$ would equal $1$. In the second question it is mysteriously redundant to write $y=P+Q+x$; you might just as well write $y=P(x)$. These solecisms suggest you might not have properly expressed the questions that you really want answered. Please clarify these points or we'll have to close the question as unanswerable.
– whuber
Mar 7 '11 at 15:39
• @whuber the OP says he had a suggestion that it would only be meaningful statistically if he attempted to calculate correlation on variables that are independent. Is this suggestion right? Doesnt 'Independence of X and Y mean they are uncorrelated'? Oct 31 '13 at 11:42

What was suggested to you is absolutely wrong. Correlation is calculated to investigate if there is some relation between two variables. If the two variables (lets call them $x$ and $y$) are independent, then the correlation is zero, for sure, so no need to compute it. That conclusion cannot be inverted, though, even if the correlation is zero, they might be dependent, in some non-linear way.
• I don't think this is correct. OP's original phrasing implied that y was deterministically dependent on x. In which case, he/she is right to say that this correlation is not statistically meaningful (r=1). Of course you are right that x and y should be partially dependent, but I don't think that's what OP was asking.
However, perhaps the relation between $y$ and $x$ is not deterministic after all. It could be that the functions $P(x)$ and $Q(x)$ are themselves stochastic, in the sense that they contain some error component (additive, multiplicative or otherwise). In that case, it absolutely makes sense to study the correlation between $y$ and $x$. The way to proceed largely depends on whether the models $P(x)$ and $Q(x)$ are known or not. If so, it may be possible to determine the correlation analytically. If not, (polynomial) linear regression is likely the best approach to reconstruct a good model from the data.