# How to interpret a p-value for a correlation? [closed]

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?

• I've posted two answers, one for each of the questions I think you might be asking.
– Dave
Sep 3, 2020 at 14:32
• Does this answer your question? Understanding the significance of very small p-values Sep 3, 2020 at 17:03
• The need for two different answers implies a fortiori the need to clarify this question.
– whuber
Sep 3, 2020 at 18:44

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.

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.

• Sorry I don't really understand your answer.So is the correlation significant or not?
– user283542
Sep 3, 2020 at 14:07
• Is your question about what the "e" in the number means?
– Dave
Sep 3, 2020 at 14:09