# What can be inferred from a 95% confidence interval on a correlation coefficient?

If the only information you have for a pearson's correlation is the 95% confidence interval, what can you infer from that data?

For example, if you had a correlation coefficient of (0.24;0.78) what would be the best inference to make?

I don't have a strong background in stats so if someone could explain it without lots of equations that would be preferable, thanks!

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possible duplicate of What, precisely, is a confidence interval? –  Nick Stauner Mar 25 '14 at 0:10
Thanks! Unfortunately that post doesn't really answer my question :/ –  user42458 Mar 25 '14 at 0:13
If that post doesn't really answer your question, @Mark, it would be helpful to use it to clarify how your Q is distinct from it. I think the answer below is somewhat ambiguous, & I worry that you may take away the wrong lesson. It would be best if you could read that thread thoroughly, & then edit your Q to state what you now understand & what you still need to know. Then you can get the best information. –  gung Mar 25 '14 at 0:43
I have a basic understanding of what a confidence interval is, but am wondering more about the interpretation of confidence intervals, as apposed to the definition. –  user42458 Mar 25 '14 at 1:01
We have a few hundred posts discussing the interpretation of confidence intervals. –  whuber Mar 25 '14 at 16:22

## 1 Answer

All you can say is the sample Pearson's correlation coefficient (r) in contained in the interval from 0.24 to 0.78. You are 95% confident that you will detect a significantly different correlation when testing values outside this interval. What this means is that variable X has some degree of positive linear relationship to variable Y in your sample. (I hesitate to use qualitative descriptors of this "strength" of the relationship because: 1) this is somewhat an outdated way to think of it, 2) what may be a strong correlation in one discipline may be weak in another, and 3) I have no idea of the sample size used to calculate the correlation coefficient.) If this experiment were conducted several independent times, with random sampling over the same population, then 95% (in the long run) will contain the population parameter, rho.

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Thanks! Exactly what I was looking for :) I thought it was as much, but I've been struck with a question that made it look like you could infer a lot more! –  user42458 Mar 25 '14 at 0:16
Confidence has a specific statistical meaning that I would not equate with "certainty". Also, what Pearson's correlation coefficient are you referring to? If you mean the population parameter $\rho$, you're wrong. If you mean the sample statistic $r$ for future samples drawn randomly from the same population, that should be specified. –  Nick Stauner Mar 25 '14 at 0:22
The question I'm looking at doesn't give any other information. It's asking what can you infer from that small amount of data –  user42458 Mar 25 '14 at 0:42
@Nick Stauner - edited for clarity. –  leonardo Mar 25 '14 at 0:42
Confidence intervals like this are consistent with about $n = 27, r = 0.571$, but don't take this more precisely than it deserves. I assume Fisher z transformation. –  Nick Cox Mar 25 '14 at 9:42