# The Collaborative Dictionary

Weight \Weight\, v. t. [imp. & p. p. Weighted; p. pr. & vb. n. Weighting.] [1913 Webster]
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

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

• /ˈweɪˌtɛd/

## Homophones

1. Having weights on it
She wore a weighted dress so it woldn't blow in the wind.
2. biased, so as to favour one party.
The competition was weighted so he'd be the clear favourite to win.

## Verb

weighted
1. 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).
If A is a finite non-empty set, one can replace the unweighted mean or average
\frac \sum_ f(a)
\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.

## General definition

If f: \Omega \to is a real-valued function, then the unweighted integral
\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.