Probability Content of Regions Under Spherical Normal Distributions, II: The Distribution of the Range in Normal Samples

Abstract

Numerous investigations, both theoretical and numerical, have been made of the distribution of the range in normal samples. One of the first investigators was Student [1] who examined the distribution on an empirical basis. Somewhat earlier, Tippett [2] presented tables and charts for the mean, standard deviation, and of the measures of skewness and kurtosis $(\beta_1 \text{and} \beta_2)$ for the range; further studies of the moments were made by E. S. Pearson [3], Hartley and Pearson [6] and by Ruben [7]. Tables of the moment constants are available in [6] and in Pearson and Hartley's book of statistical tables [10] (Tables 20 and 27), while, more recently, tables of the moment constants were provided by Harter and Clemm [11]. Approximations to the probability integral and percentage points of the distribution were suggested by E. S. Pearson [4], Cox [12], Patnaik [14] and Tukey [15]. Pearson's approximations were based on Pearsonian distributions of Type I and VI, while Cox used the Gamma Function (i.e., a multiple of $\chi^2$ with fractional degrees of freedom) and Patnaik a multiple of $\chi$ with fractional degrees of freedom as the basis for their approximations. These last two approximations have been compared by Pearson [5]. An approximation of a different type was derived by Johnson [16] who gave a series expansion for the probability integral suitable for low sample sizes and low values of the range. The behavior of the distribution for large sample size has been studied by Gumbel [17], [18], [19], Elfving [20], Cox [13] and Harley and Pearson [21]. For theoretical studies of the exact distribution reference is made to McKay and Pearson [22], Pillai [23], [24] and Cadwell [25] (see also Hartley [26]). Finally, tables of the probability integral and percentage points of the distribution have been provided by Hartley and Pearson [27], [10] (Tables 23 and 22), as well as by Harter and Clemm [11]. In the present paper, the following new results relating to the range distribution will be obtained: (i) The latter function may be expressed as the product of the sample size and the probability content of a certain parallelotope relative to a hyperspherical normal distribution, and (ii) the function can be evaluated as an infinite series involving the even moments of a sum of independent truncated normal variables. The distribution function may also be directly related to the moment generating function of the square of the sum of these truncated variables.