How to interpret a p-value for a correlation?

Question

I calculated the correlation between two variables with R. The result was $-0.465$ with a p-value of $\text{5.797e-10}$ I don't understand this strange p-value. I guess it the correlation is significant. But by how much. Is $\text{5.797e-10}$ just a very small number? How do I have to interpret this result?

Answer 1

Regarding that "e" in the number, that is a common way for software to report scientific notation. I know R does it this way.

$$AeN = A\times 10^N$$

So $\text{5.797e-10} = 5.797 \times 10^{-10} = 0.0000000005797$.

This is a useful notation for very large and very small numbers. Who wants to write the Planck length, $1.6\times 10^{−35}$, with all of those decimal places? Another advantage is that we often care about differences in orders of magnitude, and we will consider $1.6\times 10^{−35}$ and $2.3\times 10^{−35}$ to be "close enough", while $1.6\times 10^{−35}$ versus $1.6\times 10^{−33}$, a difference of a factor of $100$, is the kind of difference of interest.

Answer 2

It means that you'd be extremely unlikely to observe as extreme of a correlation as you observed if the variables were indeed uncorrelated.

This is the same interpretation that the p-value always has: the probability of observing a result as or more extreme as you observed, if the null hypothesis is true.

What the p-value does not tell you (and never does tell you) if if that amount of correlation, $-0.465$, is a "lot" of correlation. The statistical significance (p-value) is not to be confused with the practical significance (effect size), and you're not a bad statistician or bad scientist to see a tiny p-value but also a tiny effect size and say, "Okay...they're a little bit correlated (different, whatever), but that's not enough for me to care."

(I think that, in practice, a null hypothesis of $H_0:\theta=0$ is more like $H_0:\theta=0\text{-ish}$, but that's a separate discussion.)

Of possible interest: https://stats.stackexchange.com/a/483569/247274.