In a recent paper in Teaching Statistics, Croucher (2004) discussed an upper bound on the standard deviation of a set of sample data in terms of the range of the data. He derived the result that for any three numbers the standard deviation can never exceed 1/sqrt(3) or 58% of the range. He also conjectured that the maximum value of standard deviation divided by range was a decreasing function of sample size. This conjecture can easily be shown to be incorrect since, for example, the maximum value of standard deviation divided by range in equal for samples of size 3 and 4. We prove a modified version of Croucher's conjecture, that the maximum value of standard deviation divided by range is a non-increasing function of sample size. Our derivation uses a simplified approach and analogous ideas of the physical meaning of standard deviation.<br><br>This article disproves a conjecture that the ratio of the maximum standard deviation to the range of a set of data decreases as the number of data points increases. It also provides an alternative and more general approach for examining the standard deviation as a function of the range.
The CAUSE Research Group is supported in part by a member initiative grant from the American Statistical Association’s Section on Statistics and Data Science Education