Chi-squared & Pearson correlation coefficient I have just ended my math's assignment and my chi-squared test approved the null hypothesis that my data are independent; however, pearson's correlation coefficient is -0.23. 
Can they therefore be independent and negatively correlated at the same time?
 A: *

*No test "approves" the null - you either reject it or fail to reject it. Failing to reject the null (as with your chi-square result) does NOT mean the two variables are independent, it means there is insufficient evidence of their dependence to reject the null

*You don't give the CI or p for the correlation, but you state that it is negative and imply that it is significant. That does not contradict 1) because you are asking questions about different pairs of variables: One is categorical the other is continuous.  

*You state in a comment that you categorized the continuous variables in order to use chi-square. Binning a continuous variable is rarely correct. See here.

*The first reason given at that link is a loss of power, thus, it would not be surprising to find the more powerful test having more significant results - that's more or less what power means. 

*You ask which you should keep. Clearly, I  vote against the chi-square. But it is not necessarily the case that correlation is what you want, either. It depends on your question. 
