# correlation and independence: common mistake?

Almost everywhere I read

If the variables are independent, Pearson's correlation coefficient is 0

I understand that correlation gives information only for linear dependence/independence variables and, if the two values are linearly independent, then their correlation is zero. Instead, the term independent is often used as statistical independence (also in https://en.wikipedia.org/wiki/Correlation_and_dependence where in correspondence of independence there is a link to statistical independence) but linear independence is not the same of statistical independence (statistical ind. $\implies$ linear ind. but not but not vice versa) so, is there a common error (also in wikipedia) or something is not clear to me?

## 1 Answer

"If two variables are independent, their correlation is 0" is correct, at least except for random correlation. That is, the correlation won't be exactly 0 and, in 5% of the cases, it will be significantly different (at 5%) from 0, but it is 0 in the population.

However "if two variables have correlation 0, they are independent" is not necessarily correct, as they could depend on each other in a nonlinear way.

"Independence" includes "no linear relationship" as well as "no quadratic relationship", "no cubic relationship" etc.

• By not stating explicitly whether you are talking about data or random variables here, this post is potentially quite confusing and could be misread by many. You seem to have in mind the former, but the concept of "independent" applies only to the latter. – whuber Sep 13 '18 at 15:26