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Recently, I learnt that one can also calculate a p-value for the Pearson correlation coefficient. My question is not about computation of this p-value. (Based on what I read from other posts, we first calculate the t-statistic and then convert it to the p-value.) I would like to ask about the meaning of this value because from what I know, the value of the correlation coefficient already tells us the strength of the correlation. Suppose that based on sample data for variables X and Y, we find that they are strongly positively correlated with the value of r close to 1. Then, what new information does the p-value I calculate for this coefficient tell me? I'm not looking for an answer that just tells me "p-value tells us how significant the correlation is" because I'm not really able to distinguish between "strength of the correlation" and "significance of the correlation". To any lay person, probably they would think that they mean the same thing.

Specifically, I'm referring to the p-values calculated using this formula:

$$ t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$

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    $\begingroup$ It depends on what p-value you are referring to. The default is a hypothesis test of $H_0:\rho=0$ vs. $H_A:\rho\ne 0,$ but there are versions of the test of the form $H_0:\rho=\alpha$ where $\alpha$ is specified. This is no different than any other point hypothesis test. The generalities are explained at stats.stackexchange.com/questions/31. $\endgroup$
    – whuber
    Commented Aug 4, 2023 at 13:05
  • $\begingroup$ @whuber If $H_{0}$ is r = 0, then it is likely that a high $|r|$ value would also mean a low p-value? $\endgroup$ Commented Aug 4, 2023 at 13:18
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    $\begingroup$ Yes. And, generally, because the null distribution is unimodal (although generally not symmetric), higher values of $|r-\rho|$ correspond to smaller p-values. If you wish to explore and understand this better, use the Fisher Z transformation. I explain this at stats.stackexchange.com/a/368095/919 and provide some theoretical motivation at stats.stackexchange.com/a/180809/919. $\endgroup$
    – whuber
    Commented Aug 4, 2023 at 14:27

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A correlation coefficient is something you estimate using data.

If you have data on every observation in the population you are trying to generalize about them you don't need to talk about P values. The correlation coefficient you estimated is the "true" value. End of story

But if those data were collected from a random sample, rather than the full population, then your estimate of the correlation coefficient might be wrong. It might be too high or too low, just due to dumb luck, because random samples are often not representative of the population they are drawn from.

Put very simply, a p value is one way of quantifying how wrong that a particular estimate (in your case a correlation coefficient) is likely to be. More specifically, a p value is associated with a particular hypothesis that two things are different from one another. So you might hypothesize that the correlation between two different variables is "significantly different" from zero. The answer you get in your data is bigger than zero...but maybe it really is zero and you just got unlucky in your sample. To test that hypothesis you compute a t value, and then a p value.

Say you do this and the p value it comes out as .05. That means that there is only 5% chance that the true value is zero (due to sampling error) and your data are incorrectly telling you that the correlation is not zero. Of course, there are other reasons why the true value might be zero (biased samples, fake data, etc.) but the p value only tells you the chance that you are wrong due to sampling error.

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The p value does not tell you how strong the relationship is. Not for correlations or anything else. That's what parameter estimates do. The p value answers this question (general case, with your case in brackets, assuming your null is that r = 0):

If, in the population from which this sample was randomly taken, the null hypothesis is true [r = 0] what are the chances that I would get a test statistic (r) at least as extreme [far from 0] as I got, in a sample the size of the one I have?

For large samples, a small r would be significant, for small samples, a relatively large one would not be. R code to illustrate this (anything after a # is a comment):

set.seed(1234)
x1 <- rnorm(1000) #large sample from random normal
x2 <- x1 *2 + rnorm(1000,0, 50) #lots of noise
cor.test(x1,x2)  #r = 0.097, p = 0.002

set.seed(1234)
x1 <- rnorm(20) # small  sample from random normal
x2 <- x1 *2 + rnorm(20,0, 5) #much less noise
cor.test(x1,x2)  #r = 0.23, p = 0.31
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    $\begingroup$ "The p value does not tell you how strong the relationship is." I mostly agree. I certainly cannot deduce one from the other. However l've noticed in some simulations that they appear to have mutual information. Not enough to make precise inferences, but a reproducible relationship is there. $\endgroup$
    – Galen
    Commented Aug 6, 2023 at 0:15
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It seems that p-value means more for the intermediate correlation coefficient values.

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