numeriset

Numeriset is a concept used in some mathematical and educational contexts to denote a finite multiset of numbers, where elements may occur with multiplicity. Formally, a numeriset N can be described by a nonnegative multiplicity function f defined on a subset of numbers, such that N consists of each number x appearing f(x) times, and the total quantity |N| = sum f(x) is finite.

A numeriset can be represented as a collection with repetition, for example N = {1, 2, 2, 5}. The size of N is |N|, the total number of elements including repetitions. The sum of the numeriset is the weighted sum sum_x f(x)·x, and the average (mean) is the sum divided by |N|. Other descriptive statistics, such as the median and mode, are defined in the usual way for multisets. Variance can be defined as the average of squared deviations with multiplicity.

Examples illustrate these ideas. N = {1, 2, 2, 5} has |N| = 4 and sum 10, yielding a mean of 2.5. N = {−3, −3, 0, 4} has |N| = 4 and sum −2, yielding a mean of −0.5.

Relation to other concepts: a numeriset generalizes an ordinary set by allowing duplicates; it is equivalent to a finite multiset or a finite list of numbers. It is closely related to frequency distributions and to the multiset concept in combinatorics. In statistics and data analysis, a numeriset provides a precise model for a sample where repeated values matter. In number theory, numeriset-like constructions can encode multisets of residues or values in modular contexts.

Variants and limitations: If order is required, the object becomes a list or sequence rather than a numeriset. If duplicates are forbidden, the numeriset reduces to a standard set. Weighted or measure-valued variants extend the idea beyond finite multiplicities. See also: Multiset, Frequency distribution, Statistical moments, List.