New Bounds for the Extreme Values of a Finite Sample of Real Numbers

We derive new bounds for the smallest value xmin and the largest value xmax of a finite sample x1, . . . , xn of real numbers. Our bounds are obtained by solving two optimization problems, one of them being convex and the other nonconvex. We show that the pair xmin , xmax.lies in a region bounded by an ellipse and an hyperbola. The corresponding cartesian equations are given in terms of the average and the standard deviation of the sample. Q 1996 Academic Press, Inc.