New bounds for the extreme values of a finite sample of real numbers (Q1916724)

This paper deals with the problem of estimating the smallest value \(x_{\min}\) and the largest value \(x_{\max}\) of a finite collection of unknown real numbers. The only information available on this collection is its average and its standard deviation. The authors derive new bounds by solving two optimization problems, one of them being convex and the other nonconvex. They show that the pair \((x_{\min}, x _{\max})\) lies in a region bounded by an ellipse and a hyperbola whose Cartesian equations are given in terms of the average and standard deviation of the collection.