Formulas for the n-th smallest term of a finite sequence of real numbers

Abstract

Considering the need for closed-form expressions to identify the n-th smallest element in a finite sequence of real numbers, this study presents three alternative methods: the MIN-MAX Method, the MAX-MIN Method, and the Recursive Method. The objective is to provide closed formulas to compute the desired term efficiently. To this end, the work develops and analyzes mathematical expressions for each approach, including a detailed assessment of their computational complexity in terms of binary MAX and MIN operations. As a result, it is observed that the Recursive Method requires significantly more operations than the other two, which show equal computational efficiency. These findings allow us to conclude that both the MIN-MAX and MAX-MIN methods are preferable in practical applications due to their reduced computational costs. This work also contributes by presenting a direct formula to compute the median, a special case of the n-th smallest term.