Pearson correlation coefficient test: low $r$ and low p-value I am running a correlation coefficient test, and the results were: $r = 0.382, p = 2.76 \times 10^{-13}$.  So the $r$ value is not that impressive (usually we see $r>.5$), but the $p$-value is still significant.  Usually I would think a low $r$ value would mean high $p$-value (no significant correlation), or vice versa (low $p$-value would mean a high $r$ value). 
Could anyone please explain what this means?
 A: This is a frequently asked question.  What constitutes a "low" (or 'weak', etc.) correlation is subject specific.  I'll take your word for it that in your field, $r = .382$ is "low".  The reasons why this might have turned out to be lower than you expected can be any number of possibilities including:  


*

*bad luck

*range restriction

*higher than normal error of some type (e.g., measurement error)

*the correlation varies according to some factor you aren't aware of that happened to obtain in this case but doesn't typically


The reason you had a low $p$-value anyway is presumably due to high $N$.  
A: You can look at a critical value table for Pearson's correlation to determine significance. You will need to look at your df, which is the number of participants minus 2 for Pearson's correlation.  For example, if you have 27 participants, your df is 25 (27-2). When looking at the critical value table, you need to find your df (25) at your set alpha level. If your alpha level is .05, then your r value will need to be higher than .381.  If it is, then you have significance, and you can say p <.05, or that your results have less than a 5% chance of error.  If your r value is lower than .381, then you do not have significance. Therefore, it is possible to have a low p value with a low r value because you are looking at the critical value table to tell if you have significance, not the p value. The p value is saying that you are 95% correct that your r value carries significance based on the critical value table. I obtained my understanding from the following website: http://www.gifted.uconn.edu/siegle/research/correlation/alphaleve.htm.
A: Basically you're explaining a situation where the relationship between your variables is very weak, but you have sufficient sample size or power to draw significant conclusions about it anyways. Depending on the question you want to explore, I would say what you have is a weak (albeit statistically significant) relationship, and if that relationship is between something like migration timing for salmonids and condition indexes, your relationship will not help you draw any biologically relevant information. You are likely missing a part of the picture when you have these results and testing correlation and covariates then running a multiple regression with un-related independent variables may really help. Be very careful how you interpret these results. Statements like "highly significant relationships" are a little misleading if not applied with the appropriate caveats.
