On Hotelling's Weighing Problem

Abstract

The paper contains some solutions of the weighing problems proposed by Hotelling [1]. The experimental designs are applicable to a broad class of problems of measurement of similar objects. The chemical balance problem (in which objects may be placed in either of the two pans of the balance) is almost completely solved by means of designs constructed from Hadamard matrices. Designs are provided both for a balance which has a bias and for one which has no bias. The spring balance problem (in which objects may be placed in only one pan) is completely solved when the balance is biased. For an unbiased spring balance, designs are given for small numbers of objects and weighing operations. Also the most efficient designs are found for the unbiased spring balance, but it is shown that in some cases these cannot be used unless the number of weighings is as large as the binomial coefficient $\binom {p}{\frac{1}{2}p}$ or $\binom {p}{\frac{1}{2}(p + 1)}$ where $p$ is the number of objects. It is found that when $p$ objects are weighed in $N \geq p$ weighings, the variances of the estimates of the weights are of the order of $\sigma^2/N$ in the chemical balance case $(\sigma^2$ is the variance of a single weighing), and of the order of $4\sigma^2/N$ in the spring balance case.