To load with a weight or weights; to load down;
to make heavy; to attach weights to; as, to weight a horse or a
jockey at a race; to weight a whip handle. [1913 Webster] The
arrows of satire, . . . weighted with sense. --Coleridge. [1913
Webster]

(Astron. & Physics) To assign a weight to; to
express by a number the probable accuracy of, as an observation.
See Weight
of observations, under Weight. [1913 Webster]

(Dyeing) To load (fabrics) as with barite, to
increase the weight, etc. [Webster 1913 Suppl.]

(Math.) to assign a numerical value expressing
relative importance to (a measurement), to be multiplied by the
value of the measurement in determining averages or other aggregate
quantities; as, they weighted part one of the test twice as heavily
as part

[PJC] [1913 Webster]

# Word Net

weighted adj1 made heavy or weighted down with weariness;
"his leaden arms"; "weighted eyelids" [syn: leaden]

2 adjusted to reflect value or proportion; "votes
weighted according to the size of constituencies"; "a law weighted
in favor of landlords"; "a weighted average"

# English

## Pronunciation

- /ˈweɪˌtɛd/

## Homophones

## Adjective

- past of weight

A weight function is a mathematical device used
when performing a sum, integral, or average in order to give some
elements more of a "weight" than others. They occur frequently in
statistics and
analysis,
and are closely related to the concept of a measure.
Weight functions can be constructed in both discrete and continuous
settings.

# Discrete weights

## General definition

In the discrete setting, a weight function \scriptstyle w: A \to ^+ is a positive function defined on a discrete set A, which is typically finite or countable. The weight function 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 \scriptstyle f: A \to is a
real-valued
function,
then the unweighted sum of f
on A is defined as

- \sum_ f(a);

but given a weight function \scriptstyle w: A \to
^+, the weighted sum is defined as

- \sum_ f(a) w(a).

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

- \sum_ w(a).

- \frac \sum_ f(a)

by the weighted
mean or weighted
average

- \frac.

In this case only the relative weights are
relevant.

## Statistics

Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f measured multiple independent times f_i with variance \scriptstyle\sigma^2_i, the best estimate of the signal is obtained by averaging all the measurements with weight \scriptstyle w_i=\frac 1 , and the resulting variance is smaller than each of the independent measurements \scriptstyle\sigma^2=1/\sum w_i. The Maximum likelihood method weights the difference between fit and data using the same weights w_i .## Mechanics

The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights \scriptstyle w_1, \ldots, w_n (where weight is now interpreted in the physical sense) and locations :\scriptstyle\boldsymbol_1,\ldots,\boldsymbol_n, then the lever will be in balance if the fulcrum of the lever is at the center of mass- \frac,

which is also the weighted average of the
positions \scriptstyle\boldsymbol_i.

# Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain \Omega,which is typically a subset of an Euclidean space \scriptstyle^n, for instance \Omega could be an interval [a,b]. Here dx is Lebesgue measure and \scriptstyle w: \Omega \to \R^+ is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.- \int_\Omega f(x)\ dx

can be generalized to the weighted integral

- \int_\Omega f(x) w(x)\, dx

Note that one may need to require f to be
absolutely integrable with respect to the weight w(x) dx in
order for this integral to be finite.

## Weighted volume

If E is a subset of \Omega, then the volume vol(E) of E can be generalized to the weighted volume- \int_E w(x)\ dx.

## Weighted average

If \Omega has finite non-zero weighted volume, then we can replace the unweighted average- \frac \int_\Omega f(x)\ dx

by the weighted average

- \frac

## Inner product

If \scriptstyle f: \Omega \to and \scriptstyle g: \Omega \to are two functions, one can generalize the unweighted inner product- \langle f, g \rangle := \int_\Omega f(x) g(x)\ dx

to a weighted inner product

- \langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x) dx

See the entry on Orthogonality
for more details.

weighted in German: Gewichtung