The so-called 'energy test' is a frequentist technique used in experimental particle physics to decide whether two samples are drawn from the same distribution. Its usage requires a good understanding of the distribution of the test statistic, T , under the null hypothesis. We propose a technique which allows the extreme tails of the T -distribution to be determined more efficiently than possible with present methods. This allows quick evaluation of (for example) 5-sigma confidence intervals that otherwise would have required prohibitively costly computation times or approximations to have been made. Furthermore, we comment on other ways that T computations could be sped up using established results from the statistics community. Beyond two-sample testing, the proposed biased bootstrap method may provide benefit anywhere extreme values are currently obtained with bootstrap sampling. * GAM Systematic | Cantab † University of Cambridge 1 In brief, the Markov chain history is used to generate an estimator for the underlying Markov chain transition matrix, which is itself used to derive the covariance matrix for every pair of bin counts prior to the weighting and normalisation step. The final uncertainties (after the re-weighting and normalisation process, which introduces further correlations between bins) may then be written as a simple function of the former set of covariances provided that (as has already been assumed) the Markov chain has circulated through most of its domain 'a few' times.