The influence of parent population distribution on D-squared values

The two measures of dispersion which are used in statistical quality control are rho, the expected value of the range of samples of a uniform size n, and sigma, the standard deviation of the population from which the samples are taken. Conversion of one into the other is facilitated by use of the quantity called D-squared which is taken to be the ratio of rho to sigma. Tables of D-squared versus n are reproduced in all texts on statistical quality control. These tables are predicated upon the parent population being normally distributed, an assumption which may be unjustified in many instances. To understand the consequences of this assumption of normality, a short table of D-squared values based upon five distributions is presented. The five distributions are: normal, uniform, triangular, Erlang upsilon = 1 (negative exponential), and Erlang upsilon = 2. The sample size varies from two through 12, plus 15, 20, and 50. It was observed that for many values of n the normal distribution produced the largest value of D-squared, while the negative exponential distribution produced the smallest value. Depending upon the intended use, and whether Type I or Type II error is of more concern, one may wish to use D-squared vlaues based upon other than the normal distribution. Some recommendations are made. A derivation of the formula for computing rho as a function of n which is dependent upon the parent population distribution is given in an appendix.