Statistics of Range of a Set of Normally Distributed Numbers

We consider a set of numbers independently drawn from a normal distribution. We investigate the statistical properties of the maximum, minimum, and range of this set. We find that the range is closely related to the standard deviation of the original population. In particular, we investigate the use of the online positioning user service (OPUS) global positioning system (GPS) precise position utility, which produces three estimates of each coordinate and reports the range of these three estimates. We find that the range divided by 1.6926 is an unbiased estimate of the standard deviation of a single coordinate estimate, and that the variance of this estimate is 0.2755 ${\sigma }^2$ . We compare this to the more conventional method of estimating the standard deviation of a single observation from the sum of squares of residuals, which is shown to have a variance 0.2275 ${\sigma }^2$ .