Interpreting p-value significance from Pearson correlation

I used matlab corr() function to identify correlation of 236 samples. Pearson correlation is selected, and the output return r and p-value. Two sets of samples returned different r & p-value.

May I know how to interpret the significance of correlation with the results below?

(a) The data has strong negative correlation, and it's significant as p-value is a lot lesser than 0.05 ( p << 0.05 )

r = -0.9383
p = 6.7415e-110

(b) the data has weak positive correlation, and it's insignificant as p-value > 0.05.

r = 0.06800
p = 0.2981

Am I right?

• Is this a homework problem? Is so, please add the HW tag. Note that HW questions get special treatment here (see our FAQ). – gung Feb 2 '13 at 15:27
• No, this is not a homework problem (I'm probably too old even for graduate coursework). Anyway, I'm self-studying machine learning, and try to check out Kaggle Energy Load Forecasting data to find out correlation between Energy Load vs. Weather Stations (in order to imply which weather station data should be used to forecast energy load from specific Energy Station). Since I'm alone, I'd need some peer to verify my thought from my readings about pearson correlation and significance test. – twfx Feb 3 '13 at 14:04
• Especially for a challenge only considering 0.05 as "THE value" may be too simple. – Dennis Jaheruddin Feb 7 '13 at 13:12
• ^ That much is true but I don't think I've ever seen research with a significance level more lenient than 0.1 – readyready15728 Mar 25 '18 at 11:28

Personally, I am often not a fan of the standard way of thinking about p-values. (Mounting soapbox...) Firstly, it's worth considering that there are several valid ways to look at p-values. Fisher thought of them as a continuous measure of evidence against the null hypothesis, and Neyman & Pearson used them as the hub around which the decision making process turned. The most common way p-values seem to be used is not valid under either approach. The Neyman-Pearson framework has much to speak for it, in my opinion, but is primarily applicable in situations where there are theories that clearly posit two possible values, a null value (which could be $r_{null}=0$, but could be another number), and an alternative value ($r_{alt}$). In such a case, you could design your whole investigation around differentiating between those two values. This would entail specifying, among other things, $\alpha$ (the long-run type I error rate you're willing to live with), $\beta$ (the long-run type II error rate you're willing to live with), $N$ (the sample size), etc. In that context, it makes sense to me to say that something is 'significant' or 'not-significant'. However, I believe those situations are the minority of the cases. For example, for your second sample, I would say that you cannot conclude with more than 70% confidence that the correlation is positive. You will also want to examine your data and think about possible non-linearities and range restriction. (Stepping down from soapbox...)
Consistent with @gung, I believe that hypothesis testing is very problematic, especially in this particular correlation assessment setting. It is far more useful, and will get us into less trouble, to think of this as an estimation problem: we are estimating the unitness strength of association of two variables. We can compute an uncertainty interval for the estimate, e.g. a 0.95 confidence interval. If this interval is $[a,b]$ we can roughly say that our data are consistent with an underlying true correlation between $a$ and $b$ with 0.95 "confidence", if model and random sampling assumptions are true. You can also compute a worst-case (which happens with the true correlation is zero) margin of error for the estimating the correlation. This is described in BBR Section 8.5.2.