Pairing of Observations in Ordered Samples from: A Normal Population

Abstract

GIVEN an ordered sample of n observations Xl <: X 2 <: <: X n drawn at random from a normal population with known standard deviation 0", let m be the (maximum) number of pairs (Xi, Xt+l) that can be formed so that for each pair the difference (Xt+l -Xi) is not greater than a given quantity d, with the restriction that no observation is used more than once. In the present note we consider the mean value of m, for d = O'}O" or d = 0'050" and values of n up to 200, andgive an approximate solution obtained by sampling from a table of random normal deviates.We consider the following problem. Given an ordered sample of n observations drawn at random from a normal population with known standard deviation 0", let m be the (maximum) number of pairs (Xi, Xt+l) that can be formed so that for each pair the difference (Xi+l -Xi) is not greater than a given quantity d, with the restriction that no observation is used more than once. It is required to determine the distribution of m, and in particular to obtain the mean value of m. Numerical solution for the moments of m with any particular values of d, nand 0" would appear to be rather difficult and, so far as we know, the problem has not yet been treated in the literature: but a definite practical requirement for its solution has arisen in our own department. The practical problem relates to manufacture of a quartz dumb-bell which is the test piece in a magnetic oxygen meter. The dumb-bell, consisting of two hollow spheres of diameter about 2· 7 mm., connected by a rod of length about 4· 6 mm., has to be made by hand, and it is not practicable to make a single sphere of any predetermined weight. Within certain comfortably wide limits the weight of either sphere alone (about 2 mg.) is immaterial, but the weights of the two spheres for anyone dumb-bell must not differ by more than 0·04 mg. (and preferably by not more than 0'02 mg.). The problem is to determine the number of spheres which must be made in order that a given number of such pairs will be obtained with reasonable certainty. It is necessary, of course, to make some assumption concerning theform of the weight distribution that will be obtained in practice, and it seems reasonable to assume, at least as a first approximation, that this distribution will not be substantially different from normal. From a sample of 28 spheres the mean weight was 2'04 mg. and the standard deviation was estimated to be 0·41 mg. (Since the coefficient of variation is relatively large, it might be better to assume a log-normal distribution of weight: but from the 28 sample values there was no definite evidence of non-normality.)In the present note we consider.the mean value of m in a sample of size n, and give an approximate solution obtained by repeated sampling from a table of random normal deviates (Wold, 1948). 'It is not suggested that these results have any substantial theoretical value: but it is felt that they may be of some general interest. We ourselves were somewhat surprised at the relatively small variance of m. We hope tha...