<p>This work presents a robust nondimensionalization of the Hall effect thruster discharge and its facility effects. We define 23 nondimensional discharge parameters and five nondimensional facility effect parameters that are independent and are intended to capture all the relevant physics of the discharge and facility effects for atomic propellants. From the nondimensional discharge parameters, we find that exact similarity is only possible between thrusters if wall effects are negligible. We also define novel approximate similarity laws based on three nondimensional parameters in addition to geometric similarity that can cover a larger range of discharge powers and operation on different propellants. The similarity laws for different propellants suggest that HETs may not need to be redesigned to operate at similar operating conditions between different propellants. We then develop a nondimensional framework to analyze existing studies of vacuum facilities that combines the nondimensional facility parameters with the nondimensional discharge parameters and defines five new nondimensional facility effect parameters. We use this framework to reduce the dimensionality of existing data of thrust as a function of facility background pressure at different discharge conditions. Specifically, at low background pressures, the normalized thrust as a function of certain nondimensional facility effects is within measurement uncertainty between discharge conditions that differ by either anode mass flow rate, discharge voltage, or cathode flow fraction. This nondimensional framework could be used to design experiments to find a correction for thrust at a given background pressure that can be applied to a wide range of discharge conditions.</p>