Beyond the GUM Normal Approximation: Hybrid Analytical–Numerical Uncertainty Propagation for OIML R111 Mass Calibration

The probability distribution of a measurand is rarely available in closed form for realistic measurement models, because the model often involves nonlinear algebraic combinations of random variables. Conventional mass calibration provides a representative case: the OIML R111 model combines reciprocal, subtraction, multiplication, and addition terms through the buoyancy correction and the conventional-mass equation. This paper proposes a hybrid analytical–numerical method to obtain distribution-based confidence intervals for conventional mass. The proposed approach derives the probability function and distribution function of the buoyancy correction analytically under independent uniform input assumptions. The resulting buoyancy-correction distribution is then propagated numerically through the remaining algebraic operations to obtain the conventional-mass distribution and its confidence interval. The computed intervals are benchmarked against Monte Carlo simulation and the normal-approximation approach recommended in the Guide to the Expression of Uncertainty in Measurement (GUM). In the illustrative example, the hybrid method agrees closely with Monte Carlo and departs from the GUM approximation when the output distribution is skewed.