Range is a term used in different branches of mathematics including algebra, set theory and probability theory. It is usually first encountered in algebra in the study of functions, and is typically defined as the set of all values attained by a given function throughout its domain. One can think of a function, denoted by the letter f, as a rule that takes a given value and yields another, unique value. Within algebra this rule takes the form of an algebraic expression. For instance, the expression "x2" transforms the number values of "x" into another set of number values. This rule can be denoted by the equation y = x2. In general, a function f can be thought of as consisting of all the values that x possesses, as well as all the values that y possesses.
A function has a domain and a range. The domain is the set of all values of "x", and the range is the set of all values of "y". Because the values of y are calculated from the values of x, x is called the independent variable and y the dependent variable. The following example uses the equation y = x2 in order to illustrate the relationship between a function and its domain and range: for x = -2, then x2 = 4; for x = 0, then x2 = 0; and for x = 2, then x2 = 4. In this illustration, the domain of x consists of the set {-2, 0, 2}, while the range of y is the set {0, 4}. Obviously the domain could have many more elements than just three; indeed, in some cases the domain and range of a function form infinite sets. In this example, if the domain of f is allowed to be the set of integers, then by definition the domain is an infinite set, and the range of f (calculated by f(x) = y = x2) is also an infinite set, the values of which are all positive (except for zero).
The description of function, range, and domain given above is essentially the same as the one provided by German mathematician Peter Gustav Lejeune Dirichlet (1805-1859) in the mid-nineteenth century. However, a more general meaning for range was developed with the advent of set theory in the latter part of the nineteenth century. In order to motivate this 'set-theoretic' concept of range, a closer look at the concepts of 'relation' and 'function' as found in set theory is presented.
Consider the set A consisting of the first three lowercase letters of the alphabet; i.e., A = {a, b, c}. Generally speaking, the order in which the elements of a set are written is inconsequential, so that A = {a, b, c} = {b, c, a} = {c, a, b}, etc. There are, however, sets whose elements are arranged with respect to one another; such sets are called "ordered". Ordered sets having two elements are called "ordered pairs". An ordered pair is denoted by parentheses, as in (a, b). Here "a" is the first element of the ordered pair and "b" the second element. A relation is a type of set whose elements are all ordered pairs. A relation R on a set S is a set of ordered pairs of S. For instance, for set A = {a, b, c}, the set {(a, b), (a, c)} is a set of ordered pairs of A and therefore constitutes a relation on A. The set {(c, b)} is a different relation on set A. Indeed, any collection of ordered pairs of set A constitutes a relation on A. Using these ideas and definitions from set theory, range is then defined as: the range of a relation R is the set of all the "second elements" of the ordered pairs in R. The domain of R is the set of all the "first elements" of the ordered pairs in R.
From this concept of relation a function is defined as a "right unique" relation. A right unique relation is one that assigns to each element of the domain one, and only one, element in the range. Referring to the function f defined by the example above (f(x) = x2), f is the set of ordered pairs given by f = {(-2, 4), (0, 0), (2, 4)}. Note that each element of the function's domain corresponds to only one element of the range. For example, the domain element "-2" is paired only with "4", and with no other element of the range. The same is true for the domain elements "0" and "2". As an everyday example of a relation and its range, consider a set P of people, each of whom has a spouse within P. The relationship of marriage is then a relation on set P and consists of married couples. One could define this relation R as: R = {(x, y) | where x is a husband with wife y}. The range of this relation R is the set of all wives in P while the domain of R is the set of all husbands in P. For a set P of monogamous people, R is a function because each husband in the domain has only one wife. However, in some societies polygamy is practiced. In those cases R is not necessarily a function because to each husband x in the domain there could be multiple wives.
This treatment of range within the context of set theory has shown a broader, more general view of function, namely that a function is simply a particular type of relation (specifically, a right unique relation), meaning that a function is a set of ordered pairs whose range and domain consist of the sets of the "second" and "first" elements of these pairs, respectively.
Although the term range is most often used in mathematics with respect to relations and functions, it is also used in statistical distributions as the name given to limits between which the probability takes on non-zero values. Thus the range of a sample is the difference between the highest and lowest observations. Range is an elementary measure of dispersion and, in terms of the mean range in repeated sampling, provides a reasonable estimate of the population standard deviation.
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