Experiment planning: Optimal measurement by the minimal sample dispersion criterion

The criteria of optimum measurements based on a minimum dispersion and the variational problem of optimum distribution of measurements are formulated. An analytical solution for the density of measurement distribution in the case of optimum measurement is obtained and the discrete equivalent of the optimal measurement is constructed. Measurements with optimal distribution have the most stable characteristics. They provide the minimum loss of accuracy when the numbers of measurements are both changed and redistributed. An optimal measurement of any portion of measurements (reading of any detector) has the same contribution to the determination of the accuracy of measurements, i.e., optimal measurements provide equivalence (homogeneity) of the measurements at different points. Optimal measurements provide the best measure of not only the values of the function, but also of the values of an arbitrary parameter of this function. In this case, the best of the quadratic forms to determine an arbitrary parameter of measured function is the expression for the distribution function χ 2 . Continuous representation of discretely defined distribution function χ 2 , corresponding exactly to optimal distribution of the measurements is proposed. An example of solving the problem of planning the experiment is given: measuring the spatial parameters of the laser beam with the lowest standard deviation of a limited number of photodetectors.