Weighing designs look at how to optimally combine small objects whose weights are to be determined into larger collections placed simultaneously onto a beam- or spring-balance. With an analytical treatment of the summed measurements, the individual weights can be estimated with smaller relative errors. The techniques are equally applicable to measuring a collection of summed low intensities, whose magnitudes can be estimated with improved signal-to-noise ratio. This treatment is taken directly from Harwit and Sloane (1979).

Suppose we need to weigh four objects, with true but unknown weights ? ₁, ? ₂, ? ₃, ? ₄, with a beam-balance that shows a significant error e. The balance has been calibrated as well as possible, such that the average error is zero, and a large number of repeated measurements will actually deliver the object s true weight. If we weigh each object separately, we obtain four readings ? ₁, ? ₂, ? ₃, ? ₄, and four errors e ₁, e ₂, e ₃, e ₄. The readings must be taken as the best estimate of the true weights. The expectation value of the errors themselves is zero ( E{ e} = 0), while the expectation value of the squares of the errors ( E{ e ²}), is just ? ², the variance of the measurements, or the mean square error in the weight estimates.

Now suppose that instead we weigh groupings of objects...