ordreindeks

Ordreindeks, or order index, is a term used in mathematics, statistics, and computer science to denote the position of an element within a totally ordered sequence. Given a finite collection of values {x1, x2, ..., xn}, the values can be arranged in nondecreasing order: x(1) <= x(2) <= ... <= x(n). The order index of an element is the position it occupies in this sorted sequence. In many treatments, the i-th smallest value is called the i-th order statistic, denoted X_(i), while the concept of the order index focuses on the position rather than the value itself.

Handling of ties: when equal values occur, multiple elements share the same value. Different conventions exist for assigning an order index to tied elements. One common approach is to assign the index based on the first occurrence (the minimum rank), another is to assign the last occurrence (the maximum rank), or to use an average rank for all tied elements. The chosen convention affects downstream calculations that rely on ranks or positions.

Relation to rank and statistics: the order index is closely related to the notion of rank. In statistics, order statistics describe the values at specified positions in the sorted sample, while ranks describe positions possibly with tie-breaking rules. The order index is a foundational concept for computing percentiles, quantiles, and various nonparametric procedures.

Example: for the set {7, 2, 5, 5, 9}, sorted nondecreasingly as {2, 5, 5, 7, 9}, the 2nd and 3rd values are both 5, and their order indices can be 2 and 3 (with a chosen convention for ties). This illustrates how the order index combines position in a sorted order with handling of duplicates.

See also: order statistics, rank, percentile, sorted order.