Let be $X$ and $Y$ two random variables which are respectively continuous and binary. Assume that we have a sample $(X\_i, Y\_i)\_{1\leq i\leq n}$. We define the point-biserial correlation coefficient as below :
$$ r\_{X,Y} = \frac{m\_1 - m\_0}{\hat{s}\_X} \sqrt{\frac{n\_0n\_1}{n(n-1)}} $$
$m\_k$ is the mean of the $n\_k$ elements $x\_i$ whose corresponding $y$ values are $y\_i=k$ , $k \in \{0,1\}$. $\hat{s}\_X = \sqrt{\frac{1}{n-1} \sum\_{i=1}^n (x\_i - \bar{x})^2}$ and $\bar{x} = \frac{1}{n} \sum\_{i=1}^n x\_i$.   

I have some questions about this coefficient. I got it on the web, but it looks strange for me. Why it is $m\_1-m\_0$ and not $m\_0-m\_1$ ? I think that they forget the absolute value, but I do not know. The fact is that there more than one website that write this coefficient without an absolute value.   
 Also, can someone explain how to test if this coefficient is not zero ?   

Thanks in advance :)



Hypothesis test and point-biserial correlation coefficient
First here, when the null hypothesis is that the population correlation is zero, the choice of the sign before $m\_1-m\_0$ does not really matter.  

Next, as pointed out in [this Wikipedia article](https://en.wikipedia.org/wiki/Point-biserial_correlation_coefficient) (where the point-biserial correlation coefficient is denoted by $r\_{pb}$), the simple increasing transform, 
$$r\_{pb}\sqrt{\frac{n\_0+n\_1-2}{1-r\_{pb}^2}},
$$
of $r\_{pb}$ has the Student $t$-distribution with $n\_0+n\_1-2$ degress of freedom (when the distribution of $X$ is normal or approximately $t$ when $n$ is large). This of course allows a simple test of the null hypothesis that the correlation is zero in the population. Such tests are included into standard statistical packages such as R, SPSS, SAS. 


