What level of correlation coefficient suggests correlation? In Python (and Numpy), I recently did:-

a = np.random.randint(0,256,1000)

b = np.random.randint(0,256,1000)


and when I calculate the correlation coefficient $r$ using:-

np.corrcoef( a,b )


I get a value of 0.07944437.  Rounded, that's 0.08, 8% or about $1 \over 13$ th. 

$r$ is said to typically range -1.0 thru 0.0 to +1.0.  I've never had a 0.0 and don't think that it's possible as random correlations can always be found between any two series. It's like asking "How small is small?" 0.08 doesn't seem that small. Yet clearly $a$ is not correlated to $b$ as both originate from the good quality Mersenne Twister deep inside Python. 

So if $r = 0$ is empirically impossible, when does it actually become significant and suggest a real correlation?



Note 1.  There is of course the 68–95–99.7 rule. This covers a $3 \delta$ range of certainty. The atomic guys use $5 \delta$ to confirm a discovery. Do  signal guys do anything similar, and if so, how would they apply it?

Note 2. With the greatest of respect, How do I know quantitatively if the correlation of two time series is significant? seems ambiguous.
One way to look at it is that $r^2$ (yes, I do mean $r^2$, not $r$ or $|r|$) is the fraction of the variance of the target (the series A) that is explained by the estimator (a linear function $\alpha + \beta B$ of the series B).  Note that if Series A had variance $\sigma^2$, then the variance of $(A - (\alpha + \beta B))^2$ (what B cannot explain about A) is $(1-r^2)\sigma^2$.  So, if $|r| &lt; \frac{1}{\sqrt{2}} \approx 0.707\ldots$, less than half the variance has been explained, and so one viewpoint is that a correlation of less than $70\%$ or so suggests that the linear estimator being considered here is not all that great.