How accurate can combined measurements be -- experiment, simulation, and theory

Abstract

In this paper we investigate the question of how much combined measurements can increase the accuracy of additive quantities. Therefore, we consider a set of measurements from a selection of all possible combinations of the n labeled masses and then estimate the individual weights of the n masses by a linear regression approach.We present experimental results which motivate comprehensive simulation campaigns. These simulations provide valid statistical statements and reliable forecasts of the experimental results. A profound analytical treatment in turn supports these simulation outcomes with excellent consistency. One important achievement therein is a general analytical expression for the estimate's error, not only limited to the two particular weighing schemes presented.It turns out that combined measurements allow to estimate the weight of mass elements with an accuracy that under-runs by orders of magnitude the resolution of the scale used. As the error depends on the amount of measurements, one gains higher accuracy with increasing effort.In a broader sense, our work wants to promote the method and give inspirations to applications in various metrological fields beyond high-precision mass determination. Moreover, the novel simulations and analytic formulas enable the design of optimal experiments.

Improving the accuracy of a weighing scale by rigorously combining measurements

“…In our experiment (see section 3), the square root of the measurement variance, computed from the residuals with formula (19), has a value of 5.95 g ŝ = , which coincides quite well with the theoretical value σ = 5.77 g for the rectangular distribution over the interval [−10 g, + 10 g] (see (19)).…”

“…Our theory and simulation tool presented herein is generic enough to be applied to other weighing schemes. In [19] the case of combining a fixed number k out of n elements has been treated both theoretically and in simulation.…”

“…In [19], we demonstrated in a simulation that the assumption of a rectangular distribution of the rounding error of a scale is justified, if the standard deviation σ m of the ground-truth weights is larger than half of the scale resolution a, i.e. The residuals can furthermore be used to check the validity of the second Gauß-Markov condition of equal variance, the so-called homoscedasticity.…”

“…In [19], the solution of this linear equation with the Gauß-Jordan algorithm is derived together with a condition under which the solution becomes singular. We present here only the solution for the case that all N = 2 n possible combined measurements are performed, which is…”