Uncertainty of the conversion function caused by systematic effects in measurements of input and output quantities

The paper presents an evaluation with the Type A and B methods for standard uncertainties of coefficients of a polynomial function of order 𝑘 determined by 𝑛 points obtained by measurement of input and output quantities. A method for deriving a posteriori distributions of function coefficients based on the transformation of estimator distributions without assuming any a priori distributions is presented. It was emphasized that since the correct values of the standard uncertainty of type A depend on the √ 𝑛 − 𝑘 − 3 and not on the √ 𝑛 − 𝑘 − 1, therefore, with a small number of measurement points, the use of the classical approach leads to a significant underestimation of uncertainty. The relationships for direct evaluation with the type B method of uncertainties caused by uncorrected systematic additive (offset error) and multiplicative (gain error) effects in the measurements of both input and output quantities are derived. These standard uncertainties are determined on the basis of the manufacturers’ declared values of the maximum permissible errors of the measuring instruments used. A Monte Carlo experiment was carried out to verify the uncertainties of the coefficients and quadratic function, the results of which fully confirmed the results obtained analytically.