The use of mean range as an estimator of variance in statistical tests.

The distribution of the range in random samples from a normal population has been studied numerically by Tippett (1925) and Pearson (1926, 1932). McKay & Pearson (1933) have obtained the general expression for the distribution. Hartley (1942) determined a simpler expression for the probability integral and with Pearson (1942) tabulated this for samples of sizen= 2to 20. Only the first two moments of the range* have so far been fully calculated and are available numerically (Table A, Pearson, 1932), although values of fl1, /2 are known exactly for n up to 6 and, approximately, after 6 (Pearson, 1926, p. 192). It is possible, however, to compute tables of the exact moments from those of the probability integral by quadrature.t The use of the range in place of the root-mean-square estimate of the population standard deviation has become important in several fields, particularly in quality control. Although the range furnishes a less efficient estimate, its simplicity and ease of application in routine work, such as production-line inspection, makes it an important statistic in practice. It is known that the distribution of the range in normal samples depends on the sample size n, is independent of the population mean, but dependent on its standard deviation, a. Moreover, for large n (say n > 20), even small departures from normality in the tails of the parental distribution have a considerable effect on the distribution of range. Furthermore, the relative efficiency of the range as an estimator of a. decreases as n increases. Hence, in practice, when n exceeds say 12, it is preferable to divide the sample into a number of groups and take a weighted mean of the several group ranges. The question of the subdivision which will give the smallest variance of the estimate has been considered by Pearson (1932) and by Grubbs & Weaver (1947). It is found that groups of seven or eight are the best. It will generally be advantageous to have equal groups of about this size and use the average of the group ranges, viz. the mean range. The exact distribution of mean range cannot be obtained in a simple form, except in the trivial case of two samples of two observations in each. We shall, however, obtain an approximation to the distribution and, from this, derive the distribution of the ratio of a normal variate to mean range and compare it with the results of Lord (1947), who has studied the position numerically. We will also find an approximate distribution of the ratio of an individual range to mean range. These distributions will be used to obtain certain statistical tests. 2. APPROXIMATION TO THE DISTRIBUTION OF MEAN RANGE