  Statistical methods for the extraction of a small shift in broad data
distributions are examined by means of Monte Carlo simulations. This work was
originally motivated by the CERN neutrino beam to Gran Sasso (CNGS) experiment
for which the OPERA detector collaboration reported a time shift in a broad
distribution with an accuracy of $\pm 7.8\,$ns, while the fluctuation of the
average time turns with $\pm 23.8\,$ns out to be much larger. Although the
physical result of a big shift has been withdrawn, statistical methods that
make an identification in a broad distribution with such a small error possible
remain of interest.
