  The energy test is a powerful binning-free, multi-dimensional and
distribution-free tool that can be applied to compare a measurement to a given
prediction (goodness-of-fit) or to check whether two data samples originate
from the same population (two-sample test). In both cases the distribution of
the test statistic under the null hypothesis H_0, (correct prediction, same
population) has to be obtained by simulation. This poses computational problems
if the data samples are large, but the difficulty can be overcome with the help
of a scaling property which relates the distribution of small samples to the
distribution of large samples. Scaling has been made plausible in Ref. W.
Barter et al. JINST 13 P04011 by extensive simulations. In this article an
analytic proof is presented which makes the calculation of p-values obtained by
scaling more reliable.
