An Upper Bound on the Value of the Standard Deviation

Abstract

Summary This article derives a simple upper bound for the sample standard deviation that could be useful in guarding against gross errors of calculation.

On the Attainability of Bounds on the Standard Deviation

“…Croucher (2004), Petocz (2005) and Eisenhauer (1993) all discuss a prototype data sample containing n − 1 identical values and a single datum with a unique value. This sample provides an upper bound for ‘the most deviant Z ‐score’ calculable from the data.…”

“…Although the upper bound in (1) still applies when n is odd, this bound is no longer attainable. This can be easily demonstrated in the case when n = 3 by adapting an example used by Croucher (2004), who constructed a prototype sample containing a triplet of observations ( x 1 , x 2 , x 3 ) equal to (0, a , R ), where 0 ≤ a ≤ R . For these data…”

The Monty Hall Three Doors Problem

“…In a recent contribution to Teaching Statistics, John Croucher (2004) provides a clever proof that, for a data set with n = 3 observations, the ratio of the sample standard deviation (s) to the range (r)what I have previously called the relative dispersion -cannot exceed 1/√3. Readers may wish to note that an alternative proof can be derived from an earlier article by Macleod and Henderson (1984).…”

Standard Deviation for Small Samples

“…Shiffler and Harsha (1980) have formulated an upper bound for the sample standard deviation ( s ) in terms of the sample range d , while Macleod and Henderson (1984) have determined a lower bound for s in terms of d that also follows from Thomson (1955). Recently, a better upper bound for standard deviation has been derived by Croucher (2004) but for a sample of size 3. This is further discussed by Eisenhauer (2005) and Petocz (2005).…”