# 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?

• If you have some kind of absolute sense of "low" and "high" values of $r$, then what's the point of computing a p-value in the first place? Perhaps you might enjoy reading the thread at stats.stackexchange.com/questions/31/….
– whuber
Commented Apr 14, 2014 at 18:20
• You have a weak correlation, but it's definitely not zero. Where's the puzzle? If the correlation were higher, the P-value would be even smaller. Commented Apr 14, 2014 at 18:25
• "My statistic isn't very big but the p-value is tiny! What gives?" is an incredibly common question here (what gives is usually large sample size). The question usually indicates (i) some lack of understanding of what a p-value is; and (ii) a problem you probably shouldn't have been calculating p-values for in the first place (if small p-values on small effects seem wrong, it's because you're probably actually interested in effect size). Significance tests are vastly overused. Commented Apr 15, 2014 at 6:27

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:

The reason you had a low $p$-value anyway is presumably due to high $N$.