A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus".

In the discrete setting, a weight function ${\displaystyle w\colon A\to \mathbb {R} ^{+}}$ is a positive function defined on a discrete set ${\displaystyle A}$, which is typically finite or countable. The weight function ${\displaystyle w(a):=1}$ corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.

If the function ${\displaystyle f\colon A\to \mathbb {R} }$ is a real-valued function, then the unweighted sum of ${\displaystyle f}$ on ${\displaystyle A}$ is defined as

but given a weight function ${\displaystyle w\colon A\to \mathbb {R} ^{+}}$, the weighted sum or conical combination is defined as

One common application of weighted sums arises in numerical integration.

If B is a finite subset of A, one can replace the unweighted cardinality |B| of B by the weighted cardinality

If A is a finite non-empty set, one can replace the unweighted mean or average

by the weighted mean or weighted average