In mathematics, the term "symmetric function" can mean two different concepts.
A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. While this notion can apply to any type of function whose n arguments live in the same set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. There is very little systematic theory of symmetric non-polynomial functions of n variables, so this sense is little-used, except as a general definition.
In algebra and in particular in algebraic combinatorics, the term "symmetric function" is often used instead to refer to elements of the ring of symmetric functions, where that ring is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric groups.
For these specific uses, see the articles symmetric polynomials and ring of symmetric functions; the remainder of this article addresses general properties of symmetric functions in n variables.
Symmetrization
Given any function f in n variables with values in an abelian group, it can be made into a symmetric function by averaging over permutations. Similarly, it can be made into an anti-symmetric function by averaging over even permutations and subtracting the average over odd permutations.
Applications
U-statistics
In statistics, an n-sample statistic (a function in n variables) that is obtained by bootstrapping symmetrization of a k-sample statistic, yielding a symmetric function in n variables, is called a U-statistic. Examples include the sample mean and sample variance.