Are two variables that are dependent also correlated? / Can two dependent variables not be correlated? [duplicate]

I cannot find the answer to my question anywhere, I hope someone can help.

I did a Chi-Square test with several variables to see if they were dependent or independent. The result showed that they were dependent.

After that I did a Spearman´s rho test to see if they were correlated or not (looking at the P-value). All of them were correlated except one. Can this be correct?

Can two dependent variables not be correlated?

• Can you say more about the variables in question that were dependent on a chi-squared test but uncorrelated w/ Spearman's rho? What were these variables? Can you post the data? Were they ordinal or just nominal? Apr 29 '15 at 23:19
• Dependent variables can be uncorrelated, but you're not necessarily in that situation; failure to reject the null in the Spearman doesn't necessarily mean the variables are uncorrelated. Surely your power isn't 1... Apr 30 '15 at 2:12
• As things currently stand I think this question is a duplicate of Does causation imply correlation? but if the post was edited to show us some data and some of the other information requested, then a more individual reply might be possible. I agree with @Glen_b that your test results don't necessarily imply this is the situation you're in. Apr 30 '15 at 7:27

Yes this can indeed be the case. A necessary (but insufficient) condition for independence is $cov(x,y) = 0$ alternatively $corr(x,y)=0$. In the sense that independence gives 0 correlation. Another way to think about it is that correlation measures only linear dependence, such that your variables are correlated in non-linear way (or that the test you used is somehow flawed).
Formally for X and Y to be independent we would require that the product, of their marginal distribution, would yield their joint distribution (which is really what the $\chi^2$-test, tests).