  Many experiments in physics involve searching for a localized excess over
background expectations in an observed spectrum. If the background is known and
there is Gaussian noise, the amount of excess of successive observations can be
quantified by the runs statistic taking care of the look-elsewhere effect. The
distribution of the runs statistic under the background model is known
analytically but the computation becomes too expensive for more than about a
hundred observations. This work demonstrates a principled high-precision
extrapolation from a few dozen up to millions of data points. It is most
precise in the interesting regime when an excess is present. The method is
verified for benchmark cases and successfully applied to real data from an
axion search. The code that implements our method is available at
https://github.com/fredRos/runs .
