  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.
