Dependency between quantitative and categorial variable I have a data set with a categorical variable $C$ (approx. 8 levels) and a quantitative variable $X$. I suspect that $X$ and $C$ are strongly dependent. How can I verify this hypothesis? 
My approach would be to 'bin' $X$ in order to obtain about the same number of categories as for $C$ and then use a test for contingency tables. 
Once I have established depedency I would like to find 'the natural order' of the categories of $C$ implied by $X$, or even to assign distances between the levels of $C$, based on the numeric scale of $X$. Can this idea be made rigorous? Does it make sense?
 A: ALL  EDITED:
The poster has made clear in the comments below that they are looking for a test of association that doesn't assume one variable is independent and one is dependent.
This is probably a duplicate of this Cross Validated question.  There is some helpful discussion there.  There are some good option at this other CV question.
To meet the Poster's criteria, maybe one approach would be to use an extended version of the Cochran-Armitage test that allows for more than two categories.  For this test, one variable is ordinal and the other is nominal, arranged in a contingency table, so it would still require reducing the quantitative variable to an ordinal one, perhaps with some binning.  But I think this approach is closer than reducing the continuous variable to a categorical one.  From what I read, it's a test of association that doesn't assume that one variable is dependent.
An appropriate measure of association for the ordinal/nominal case may be Freeman's theta.
An appropriate measure of association for the interval/nominal case may be eta-squared.
