Summation of errors for a small number of constituents

Abstract

One special aspect of the summation of errors involving a small number of constituents (possibly nonuniform in magnitude) is the indeterminacy of the resultant probability-dens ity distribution law. The assumption of a normal law for the final leads to serious errors in regards the confidence interval. In this case the latter may best be determined by setting up a composition of the probability-dens ity distributions of the constituents. The composition of one dominant error (distributed in accordance with a normal or Student-type law) with the sum of the others (for which a normal distribution law was assumed) was considered earlier [1]. Tables were also given in [2] together with the corresponding curves for determining the confidence interval (with a 0.997 probability) of the sum of a normally-distrib uted error having an arbitrary number of constituents of uniform magnitude distributed in accordance with an arcsin law. In this paper we shall consider a means of determining the integrated probability distribution for the sum of an arbitrary number n --> 2 of independent components, based on a composition of the distribution laws. We shall consider symmetrical distributions- normal, uniform, triangular (Simpson), trapezoidal, and arcsin-type, since these are most frequently encountered in measuring practice. The results may be used for the summation of errors and also for estimating how close the resultant distribution is to a normal law. TABLE 1 Probability-dens ity distribution law Mean square Coefficients of the expansion t(x) deviation o Normal 1 x | 2o s