Summary: Neutrino oscillation experiments have demonstrated evidence for three distinct neutrino masses. However, whether there are two light neutrinos and one heavy neutrino (normal), or the other way around (inverted), known as the neutrino mass ordering, remains undetermined. This thesis presents a search for indications of the neutrino mass ordering in 6511 live-days (484 kiloton-years) of atmospheric neutrino data collected with the Super-Kamiokande (SK) detector between 1996 and 2020. The data set is a 30 % increase in exposure since the previous published analysis, and the analysis methodology includes improvements to the separation of neutrino and anti-neutrino data. This thesis also presents an analysis of the SK data with constraints on neutrino oscillation parameters from reactor neutrino experiments and the T2K long-baseline experiment. The constraints from the T2K experiment include, for the first time, an anti-neutrino-enhanced data sample. The atmospheric neutrino analysis favors the normal neutrino mass ordering, rejecting the inverted ordering at the 92.3 % (1.43 σ) level. The inclusion of external constraints from T2K data increases the rejection to the 97.9 % (2.03 σ) level.