The Use of Sample Quasi-Ranges in Estimating Population Standard Deviation

Abstract

The use of sample quasi-ranges in estimating the standard deviation of normal, rectangular, and exponential populations is discussed. For the normal population, the expected value and the variance of the $r$th quasi-range for samples of size $n$ are tabulated for $r = 0(1) 8$ and $n = (2r + 2) (1) 100$. The efficiency of the unbiased estimate of population standard deviation based on one sample quasi-range is tabulated for the same values of $r$, with $n = (2r + 2) (2) 50 (5) 100$. Estimates based on a linear combination of two quasi-ranges are considered and a method is given for determining the weighting factor which maximizes the efficiency. The most efficient unbiased estimates based on one quasi-range for $n = 2 (1) 100$ and on linear combinations of two adjacent quasi-ranges and of two quasi-ranges among those with $r < r' \leqq 8$ for $n = 4 (1) 100$ are tabulated, along with their efficiencies. An example illustrates the use of these estimates. For the rectangular population, the efficient estimate of population standard deviation, which is based on the sample range, is tabulated for $n = 2(1) 100$. The bias, when estimates which assume normality are used, is tabulated for $n = 2(1) 100$ for rectangular and exponential populations.