The Distribution of the Range

Abstract

The asymptotic distribution of the range $w$ for a large sample taken from an initial unlimited distribution possessing all moments is obtained by the convolution of the asymptotic distribution of the two extremes. Let $\alpha$ and $u$ be the parameters of the distribution of the extremes for a symmetrical variate, and let $R = \alpha(w - 2u)$ be the reduced range. Then its asymptotic probability $\Psi(R)$ and its asymptotic distribution $\psi(R)$ may be expressed by the Hankel function of order one and zero. A table is given in the text. The asymptotic distribution $g(w)$ of the range proper is obtained from $\psi(R)$ by the usual linear transformation. The initial distribution and the sample size influence the position and the shape of the distribution of the range in the same way as they influence the distribution of the largest value. If we take the parameters from the calculated means and standard deviations, the asymptotic distribution of the range gives a good fit to the calculated distributions for normal samples from size 6 onward. Consequently the distribution of the range for normal samples of any size larger than 6 may be obtained from the asymptotic distribution of the reduced range. The asymptotic probabilities and the asymptotic distributions of the mth range and of the range for asymmetrical distributions are obtained by the same method and lead to integrals which may be evaluated by numerical methods.