Measurements of hailstone diameters and kinetic energy, collected by the Swiss network of automatic hail sensors, are available in three regions of Switzerland for the period between September 2018 and August 2023. In this study, we propose the use of double-moment normalization for modeling the hail size number distribution (HSND), which is defined as the number of hailstone impacts measured, for each diameter size, by one instrument during one hail event. This method uses two of the empirical moments of the HSND to compute a normalized distribution. While the HSND is dependent on the duration and intensity of the event and on the detection area of the sensor, we show that the normalized distribution has limited variability across the three geographical regions of deployment of the sensors. Thanks to its invariance in space and time, a generalized gamma function is used to model the normalized distribution, and its parameters have been determined through a fit over approximately 70 % of the events. The fitted model and the previously chosen pair of empirical moments can be used to reconstruct the HSND at any location in Switzerland. The accuracy of the reconstruction has been estimated over the remaining 30 % of the dataset. An additional evaluation has been performed on an independent HSND, made of estimates of hail diameters measured by drone photogrammetry during a single event. This HSND has a much larger number of hailstone impacts (18 000) than those of the hail sensor events (from 30 to 400). The double-moment normalization is able to reproduce well the HSND recorded by the hail sensors and the drone, albeit with an underestimation of the number of impacts at small diameters. These results highlight the invariance of the normalized distribution and the adaptability of the method to different data sources.