The Distribution of Quasi-Ranges in Samples From a Normal Population

Abstract

A method is developed for the evaluation of the probability density function of the statistic: $w_r = x_{n-r} - x_{r+1}$ where $x_1, x_2, \cdots, x_n$ are ordered values in a sample of $n$ from a normal population. It is shown that, up to $n = 17, w_0$ is the most efficient statistic of this type for the estimation of population standard deviation. Beyond this point $w_1$ is optimum up to $n = 31,$ where $w_2$ becomes better. Tables of moment constants and percentage points are given for $w_1$ over the range 10 to 30. Similar methods are used to determine the efficiencies of two estimates of the form $w_r + \lambda w_s.$ The approximation used is compared with three other published approximations in the case of range $(r = 0).$ Godwin [5] and Nair [11] have discussed problems of this kind for sample sizes up to 10, using exact values of the first two moments. Karl Pearson [12], Mosteller [10] and Jones [9] have considered the large sample case. The methods of the present paper go some way towards filling the gap between these approaches. Moreover, they are not restricted to consideration of mean and variance only.