An Alternate Proof of Samuelson's Inequality and its Extensions

Abstract

Paul Samuelson and other authors have shown that no element in a population of n items can lie farther than (n — 1)^½ standard deviations from the mean. Other researchers extended this by giving upper and lower bounds for the k th largest item in terms of the population mean and standard deviation. We provide an alternate proof of these results. The technique used in the proof gives insight into the conditions under which the bounds are attained and can be used to improve bounds when additional information is available.

Key Words:

• Bounds on order statistics
• Outliers