Weighing designs

The general weighing design (using either a spring balance or a chemical balance) is still described by an equation of the form... 

Enough is known about the answer to this question to solve any practical problem in weighing designs or optics. On the other hand a number of important mathematical questions still remain open. 

Hadatnard matrices make the best chemical balance weighing designs, i.e., the best matrices with entries +1, 0, and -1. If there are n unknowns and a Hadamard matrix of order n is used, the mean squared error in each unknown is reduced by a factor of n or in other words the signal-to-noise ratio is increased by a factor of VIT... 

Furthermore it can be shown that Hadamard matrices are also D-optimal (cf. ref. 11, Section 3.2.5 of ref. 2). Equations 11 and 12 describe a weighing design based on a Hadamard matrix of order 4. 

S-matrices make the best spring balance weighing designs and the best (0,l)-masks for multiplexing spectrometers. 

In this section we describe some of the more important matrices used in weighing designs and spectrometers. 

The construction gives symmetric, cyclic, conference matrices of orders 6, 14, 18, 30, 38, 42, 54, 62,", and symmetric conference matrices of orders 10, 26, 50,. Choosing the mask W = Cn we obtain a weighing design that has mean squared error... 

Banerjee, K. S. 1975, Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics, Marcel Dekker, H. f. 

A complete balance test consists of a systematic set of weighings designed to evaluate all aspects of balance pterformance. including repeatability, eccentric loading errors, linearity, and span calibration. For a balance to be declared in proper working condition, all test results must fall within the tolerance limits specified by the manufacturer. These tolerances normally apply only to new or newly serviced equipment. In other cases a so-called in-service tolerance—equal to twice the original tolerance—is customarily allowed. 

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