A statistical analysis of measurement results obtained from nonlinear physical laws

Abstract

Statistical properties of quantities obtained from measurements based on some fundamental physical laws are analyzed in this paper, using methods for expressing measurement uncertainty of indirectly measured quantities. Nonlinear laws are considered, with repeated measurements of input quantities providing identical readings on respective digital instruments. Under such conditions, input quantities are assigned uniform distributions. It is shown that in addition to the asymmetry arising in the probability density function (PDF) of the output quantity, its mean and nominal value also differ. Resistance obtained from Ohm’s law and power measured using three alternative forms of Joule’s law are investigated in detail. Some characteristic shapes of PDFs are obtained by a Monte Carlo method (MCM). It is demonstrated that the mean value of the measured resistance is greater than its nominal value. It is also proved that for two forms of Joule’s law the mean value of the measured power is larger than its nominal value, while the third variant of the law renders the mean and the nominal power equal. Analytical expressions for the deviations of mean from nominal values are derived. It is suggested that the presented analysis can readily be adapted to many other nonlinear physical laws.