Is there only one formula for the statistical difference between a pair of distribution ensembles? Statistical closeness implies computational indistinguishability was recently posed.  It revolves around a numeric value $\Delta(n)$ of the statistical difference between a pair of distribution ensembles, as:-

$$ \Delta(n) = 1/2 \sum_{\alpha}|\mathbb{P}[X_n = \alpha] - \mathbb{P}[Y_n = \alpha]| $$

But statistical distances can also be measured via other techniques such as Pearson's chi-squared test, Kolmogorov-Smirnov &amp; Anderson–Darling tests and Kullback–Leibler. There are many other exotic tests as well.  My four examples are non parametric tests too that produce a numeric test statistic. And they are also used in many aspects of cryptography.

Is the $\Delta(n)$ formula specific to cryptography, and if so, why?
What we call "statistical distance" in cryptography is called total variation distance by statisticians. So it certainly exists outside of cryptography. 

I can't speak to its applications within statistics. But it certainly is the most natural metric for cryptography because it has an equivalent formulation in terms of distinguishing two source distributions given a sample, which is how we state many security properties. Restricting the distinguisher to be polynomial-time leads to natural and useful generalizations.