  We provide supporting details for the search for a 3+1 sterile neutrino using
data collected over eleven years at the IceCube Neutrino Observatory. The
analysis uses atmospheric muon-flavored neutrinos from 0.5 to 100\, TeV that
traverse the Earth to reach the IceCube detector, and finds a best-fit point at
$\sin^2(2\theta_{24}) = 0.16$ and $\Delta m^{2}_{41} = 3.5$ eV$^2$ with a
goodness-of-fit p-value of 12\% and consistency with the null hypothesis of no
oscillations to sterile neutrinos with a p-value of 3.1\%. Several improvements
were made over past analyses, which are reviewed in this article, including
upgrades to the reconstruction and the study of sources of systematic
uncertainty. We provide details of the fit quality and discuss stability tests
that split the data for separate samples, comparing results. We find that the
fits are consistent between split data sets.
