The weighted arithmetic mean is similar to an ordinary

Poisson sampling
In survey methodology
Survey methodology is "the study of survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on ...

), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that.

GNU Scientific Library - Reference manual, Version 1.15

2011.

/ref> :$\backslash begin\; \backslash mathbf\; \&=\; \backslash frac\; \backslash sum\_^N\; w\_i\; \backslash left(\backslash mathbf\_i\; -\; \backslash mu^*\backslash right)^T\; \backslash left(\backslash mathbf\_i\; -\; \backslash mu^*\backslash right)\; \backslash \backslash \; \&=\; \backslash frac\; .\; \backslash end$ The reasoning here is the same as in the previous section. Since we are assuming the weights are normalized, then $V\_1\; =\; 1$ and this reduces to: : $\backslash mathbf=\backslash frac.$ If all weights are the same, i.e. $w\_\; /\; V\_1=1/N$, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.

First Systems of Weighted Differential and Integral Calculus''

, 1980.

Tool to calculate Weighted Average

{{DEFAULTSORT:Weighted Mean Means Mathematical analysis Summary statistics

arithmetic mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

(the most common type of average
In colloquial language, an average is a single number taken as representative of a non-empty list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divide ...

), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics
A descriptive statistic (in the count noun
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The trad ...

and also occurs in a more general form in several other areas of mathematics.
If all the weights are equal, then the weighted mean is the same as the arithmetic mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox
Simpson's paradox, which also goes by several other names, is a phenomenon in probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theo ...

.
Examples

Basic example

Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students): $$\backslash bar\; =\; \backslash frac\; =\; 86.$$ Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight": :$\backslash bar\; =\; \backslash frac\; =\; 86.$ Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed.Convex combination example

Since only the ''relative'' weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called aconvex combination
In convex geometryIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...

.
Using the previous example, we would get the following weights:
:$\backslash frac\; =\; 0.4$
:$\backslash frac\; =\; 0.6$
Then, apply the weights like this:
:$\backslash bar\; =\; (0.4\backslash times80)\; +\; (0.6\backslash times90)\; =\; 86.$
Mathematical definition

Formally, the weighted mean of a non-empty finitemultiset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of data $\backslash ,$
with corresponding non-negative weights
Weight is a measurement of the gravitational force acting on an object. In non-scientific contexts it may refer to an object's mass (quantity of matter). Figuratively, it refers to the seriousness or depth of an idea or thought, or the danger and u ...

$\backslash $ is
:$\backslash bar\; =\; \backslash frac,$
which expands to:
:$\backslash bar\; =\; \backslash frac.$
Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).
The formulas are simplified when the weights are normalized such that they sum up to $1$, i.e.:
:$\backslash sum\_^n\; =\; 1$.
For such normalized weights the weighted mean is then:
:$\backslash bar\; =\; \backslash sum\_^n$.
Note that one can always normalize the weights by making the following transformation on the original weights:
:$w\_i\text{'}\; =\; \backslash frac$.
Using the normalized weight yields the same results as when using the original weights:
:$\backslash begin\; \backslash bar\; \&=\; \backslash sum\_^n\; w\text{'}\_i\; x\_i=\; \backslash sum\_^n\; \backslash frac\; x\_i\; =\; \backslash frac\; \backslash \backslash \; \&\; =\; \backslash frac.\; \backslash end$
The ordinary mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$\backslash frac\; \backslash sum\_^n$ is a special case of the weighted mean where all data have equal weights.
If the data elements are independent and identically distributed random variables
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces i ...

with variance $\backslash sigma^2$, the ''standard error of the weighted mean'', $\backslash sigma\_$, can be shown via uncertainty propagation
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

to be:
:$\backslash sigma\_\; =\; \backslash sigma\; \backslash sqrt$
Statistical properties

Expectancy

The weighted sample mean, $\backslash bar$, is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one). If the observations have expected values $$E(x\_i\; )=,$$ then the weighted sample mean has expectation $$E(\backslash bar)\; =\; \backslash sum\_^n\; .$$ In particular, if the means are equal, $\backslash mu\_i=\backslash mu$, then the expectation of the weighted sample mean will be that value, $$E(\backslash bar)=\; \backslash mu.$$Variance

Simple i.i.d case

When treating the weights as constants, and having a sample of ''n'' observations fromuncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname- \operatorname \operatorname is zero. If two variables are uncorrelated, t ...

random variables
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. ...

, all with the same variance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

and expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* Expectation (alb ...

(as is the case for i.i.d random variables), then the variance of the weighted mean can be estimated as the multiplication of the variance by Kish's design effect (see proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

):
: $\backslash operatorname(\backslash bar\; y\_w)\; =\; \backslash frac\; \backslash frac$
With $\backslash hat\; \backslash sigma\_y^2\; =\; \backslash frac$, $\backslash bar\; =\; \backslash frac$, and $\backslash overline\; =\; \backslash frac$
However, this estimation is rather limited due to the strong assumption about the ''y'' observations. This has led to the development of alternative, more general, estimators.
Survey sampling perspective

From a ''model based'' perspective, we are interested in estimating the variance of the weighted mean when the different $y\_i$ are not i.i.d random variables. An alternative perspective for this problem is that of some arbitrarysampling design In the theory of finite population
Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their j ...

of the data in which units are selected with unequal probabilities (with replacement).
In Survey methodology
Survey methodology is "the study of survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on Survey (human research) ...

, the population mean, of some quantity of interest ''y'', is calculated by taking an estimation of the total of ''y'' over all elements in the population (''Y'' or sometimes ''T'') and dividing it by the population size – either known ($N$) or estimated ($\backslash hat\; N$). In this context, each value of ''y'' is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The survey sampling In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...

procedure yields a series of BernoulliBernoulli can refer to:
People
*Bernoulli family of 17th and 18th century Swiss mathematicians:
** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle
** Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...

indicator values ($I\_i$) that get 1 if some observation ''i'' is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: Poisson sampling
In survey methodology
Survey methodology is "the study of survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on ...

). The probability of some element to be chosen, given a sample, is denoted as $P(I\_i=1\; \backslash mid\; \backslash text\; n\; )\; =\; \backslash pi\_i$, and the one-draw probability of selection is $P(I\_i=1\; ,\; \backslash text)\; =\; p\_i\; \backslash approx\; \backslash frac$ (If N is very large and each $p\_i$ is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as cluster sampling
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...

design).
Since each element ($y\_i$) is fixed, and the randomness comes from it being included in the sample or not ($I\_i$), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: $y\text{'}\_i\; =\; y\_i\; I\_i$. With the following expectancy: $E;\; href="/html/ALL/s/\text{\'}\_i.html"\; ;"title="\text{\'}\_i">\text{\'}\_i$= y_i \pi_i; and variance: $V;\; href="/html/ALL/s/\text{\'}\_i.html"\; ;"title="\text{\'}\_i">\text{\'}\_i$= y_i^2 \pi_i(1-\pi_i).
When each element of the sample is inflated by the inverse of its selection probability, it is termed the $\backslash pi$-expanded ''y'' values, i.e.: $\backslash check\; y\_i\; =\; \backslash frac$. A related quantity is $p$-expanded ''y'' values: $\backslash frac\; =\; n\; \backslash check\; y\_i$. As above, we can add a tick mark if multiplying by the indicator function. I.e.: $\backslash check\; y\text{'}\_i\; =\; I\_i\; \backslash check\; y\_i\; =\; \backslash frac$
In this ''design based'' perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.: $w\_i\; =\; \backslash frac\; \backslash approx\; \backslash frac$.
Variance of the weighted sum (''pwr''-estimator for totals)

If the population size ''N'' is known we can estimate the population mean using $\backslash hat\_\; =\; \backslash frac\; \backslash approx\; \backslash frac$. If thesampling design In the theory of finite population
Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their j ...

is one that results in a fixed sample size ''n'' (such as in pps sampling), then the variance of this estimator is:
: $\backslash operatorname\; \backslash left(\; \backslash hat\_\; \backslash right)\; =\; \backslash frac\; \backslash frac\; \backslash sum\_^n\; \backslash left(\; w\_i\; y\_i\; -\; \backslash overline\; \backslash right)^2$
An alternative term, for when the sampling has a random sample size (as in Poisson sampling
In survey methodology
Survey methodology is "the study of survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on ...

), is presented in Sarndal et. al. (1992) as:
$$\backslash operatorname(\backslash hat\; \backslash bar\; Y\_)\; =\; \backslash frac\; \backslash sum\_^n\; \backslash sum\_^n\; \backslash left(\; \backslash check\_\; \backslash check\_i\; \backslash check\_j\; \backslash right)$$
With $\backslash check\_i\; =\; \backslash frac$. Also, $C(I\_i,\; I\_j)\; =\; \backslash pi\_\; -\; \backslash pi\_\backslash pi\_\; =\; \backslash Delta\_$ where $\backslash pi\_$ is the probability of selecting both i and j. And $\backslash check\_\; =\; 1\; -\; \backslash frac$, and for i=j: $\backslash check\_\; =\; 1\; -\; \backslash frac\; =\; 1-\; \backslash pi\_$.
If the selection probability are uncorrelated (i.e.: $\backslash forall\; i\; \backslash neq\; j:\; C(I\_i,\; I\_j)\; =\; 0$), and when assuming the probability of each element is very small, then:
: $\backslash operatorname(\backslash hat\; \backslash bar\; Y\_)\; =\; \backslash frac\; \backslash sum\_^n\; \backslash left(\; w\_i\; y\_i\; \backslash right)^2$
Variance of the weighted mean (-estimator for ratio-mean)

The previous section dealt with estimating the population mean as a ratio of an estimated population total ($\backslash hat\; Y$) with a known population size ($N$), and the variance was estimated in that context. Another common case is that the population size itself ($N$) is unknown and is estimated using the sample (i.e.: $\backslash hat\; N$). The estimation of $N$ can be described as the sum of weights. So when $w\_i\; =\; \backslash frac$ we get $\backslash hat\; N\; =\; \backslash sum\_^n\; w\_i\; I\_i\; =\; \backslash sum\_^n\; \backslash frac\; =\; \backslash sum\_^n\; \backslash check\; 1\text{'}\_i$. When using notation from previous sections, the ratio we care about is the sum of $y\_i$s, and 1s. I.e.: $R\; =\; \backslash bar\; Y\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac$. We can estimate it using our sample with: $\backslash hat\; R\; =\; \backslash hat\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash bar\; y\_w$. As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: $\backslash bar\; y\_w\; =\; \backslash frac$. This will be theestimand An estimand is a quantity that is to be estimated in a statistical analysis. The term is used to more clearly distinguish the target of inference
Inferences are steps in reasoning
Reason is the capacity of consciously applying logic
Logic ...

for specific values of y and w, but the statistical properties comes when including the indicator variable $\backslash bar\; y\_w\; =\; \backslash frac$.
This is called Ratio estimator The ratio estimator is a statistical parameterIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or soc ...

and it is approximately unbiased for ''R''.
In this case, the variability of the ratio
In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily Taylor series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

first-order linearization, asymptotics, and bootstrap/jackknife. The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be relatively accurate even for medium sample sizes. For when the sampling has a random sample size (as in Poisson sampling
In survey methodology
Survey methodology is "the study of survey
Survey may refer to:
Statistics and human research
* Statistical survey
Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on ...

), it is as follows:
: $\backslash widehat\; =\; \backslash frac\; \backslash sum\_^n\; w\_i^2\; (y\_i\; -\; \backslash bar\; y\_w)^2$.
We note that if $\backslash pi\_i\; \backslash approx\; p\_i\; n$, then either using $w\_i\; =\; \backslash frac$ or $w\_i\; =\; \backslash frac$ would give the same estimator, since multiplying $w\_i$ by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size ''N'', the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator.
We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is known). For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in Bootstrapping validation

It has been shown, by Gatz et. al. (1995), that in comparison tobootstrapping
In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input.
Etymology
Tall boot
A boot, plural boots, is a type of specific footwear
Footwear refers to garments wor ...

methods, the following (variance estimation of ratio-mean using Taylor series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):
:$\backslash widehat\; =\; \backslash frac\; \backslash left;\; href="/html/ALL/s/sum\_(w\_i\_x\_i\_-\_\backslash bar\_\backslash bar\_w)^2\_-\_\; 2\_\backslash bar\_w\_\backslash sum\_(w\_i\_-\_\backslash bar)(w\_i\_x\_i\_-\_\backslash bar\_\backslash bar\_w)\; +\_\backslash bar\_w^2\_\backslash sum\_(w\_i\_-\_\backslash bar)^2\_\backslash right.html"\; ;"title="sum\; (w\_i\; x\_i\; -\; \backslash bar\; \backslash bar\_w)^2\; -\; 2\; \backslash bar\_w\; \backslash sum\; (w\_i\; -\; \backslash bar)(w\_i\; x\_i\; -\; \backslash bar\; \backslash bar\_w)\; +\; \backslash bar\_w^2\; \backslash sum\; (w\_i\; -\; \backslash bar)^2\; \backslash right">sum\; (w\_i\; x\_i\; -\; \backslash bar\; \backslash bar\_w)^2\; -\; 2\; \backslash bar\_w\; \backslash sum\; (w\_i\; -\; \backslash bar)(w\_i\; x\_i\; -\; \backslash bar\; \backslash bar\_w)\; +\; \backslash bar\_w^2\; \backslash sum\; (w\_i\; -\; \backslash bar)^2\; \backslash right$
where $\backslash bar\; =\; \backslash frac$. Further simplification leads to
:$\backslash widehat\; =\; \backslash frac\; \backslash sum\; w\_i^2(x\_i\; -\; \backslash bar\_w)^2$
Gatz et. al. mention that the above formulation was published by Endlich et. al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size.., based on the formulation published by Cochran (1977), as an approximation to the ratio mean. However, Endlich et. al. didn't seem to publish this derivation in their paper (even though they mention they used it), and Cochran's book includes a slightly different formulation.Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Nashville, TN: John Wiley & Sons. Still, it's almost identical to the formulations described in previous sections.
Replication based estimators

Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the Jackknife andBootstrapping
In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input.
Etymology
Tall boot
A boot, plural boots, is a type of specific footwear
Footwear refers to garments wor ...

.
Other notes

For uncorrelated observations with variances $\backslash sigma^2\_i$, the variance of the weighted sample mean is : $\backslash sigma^2\_\; =\; \backslash sum\_^n$ whose square root $\backslash sigma\_$ can be called the ''standard error of the weighted mean (general case)''. Consequently, if all the observations have equal variance, $\backslash sigma^2\_i=\; \backslash sigma^2\_0$, the weighted sample mean will have variance : $\backslash sigma^2\_\; =\; \backslash sigma^2\_0\; \backslash sum\_^n\; ,$ where $1/n\; \backslash le\; \backslash sum\_^n\; \backslash le\; 1$. The variance attains its maximum value, $\backslash sigma\_0^2$, when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have $\backslash sigma\_\; =\; \backslash sigma\_0\; /\; \backslash sqrt$, i.e., it degenerates into thestandard error of the mean
The standard error (SE) of a statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population
Pop ...

, squared.
Note that because one can always transform non-normalized weights to normalized weights all formula in this section can be adapted to non-normalized weights by replacing all $w\_i\text{'}\; =\; \backslash frac$.
Occurrences of using weighted mean

Variance weights

For the weighted mean of a list of data for which each element $x\_i$ potentially comes from a differentprobability distribution
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

with known variance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

$\backslash sigma\_i^2$, all having the same mean, one possible choice for the weights is given by the reciprocal of variance:
:$w\_i\; =\; \backslash frac.$
The weighted mean in this case is:
:$\backslash bar\; =\; \backslash frac,$
and the ''standard error of the weighted mean (with variance weights)'' is:
:$\backslash sigma\_\; =\; \backslash sqrt,$
Note this reduces to $\backslash sigma\_^2\; =\; \backslash sigma\_0^2/n$ when all $\backslash sigma\_i\; =\; \backslash sigma\_0$.
It is a special case of the general formula in previous section,
:$\backslash sigma^2\_\; =\; \backslash sum\_^n\; =\; \backslash frac.$
The equations above can be combined to obtain:
:$\backslash bar\; =\; \backslash sigma\_^2\; \backslash sum\_^n\; \backslash frac.$
The significance of this choice is that this weighted mean is the maximum likelihood estimator
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...

of the mean of the probability distributions under the assumption that they are independent and normally distributed
In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
:
...

with the same mean.
Correcting for over- or under-dispersion

Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that $\backslash chi^2$ is too large. The correction that must be made is :$\backslash hat\_^2\; =\; \backslash sigma\_^2\; \backslash chi^2\_\backslash nu$ where $\backslash chi^2\_\backslash nu$ is thereduced chi-squared
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares.
It ...

:
:$\backslash chi^2\_\backslash nu\; =\; \backslash frac\; \backslash sum\_^n\; \backslash frac;$
The square root $\backslash hat\_$ can be called the ''standard error of the weighted mean (variance weights, scale corrected)''.
When all data variances are equal, $\backslash sigma\_i\; =\; \backslash sigma\_0$, they cancel out in the weighted mean variance, $\backslash sigma\_^2$, which again reduces to the standard error of the mean
The standard error (SE) of a statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population
Pop ...

(squared), $\backslash sigma\_^2\; =\; \backslash sigma^2/n$, formulated in terms of the sample standard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

(squared),
:$\backslash sigma^2\; =\; \backslash frac\; .$
Related concepts

Weighted sample variance

Typically when a mean is calculated it is important to know thevariance
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

and standard deviation
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

about that mean. When a weighted mean $\backslash mu^*$ is used, the variance of the weighted sample is different from the variance of the unweighted sample.
The ''biased'' weighted sample variance
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...

$\backslash hat\; \backslash sigma^2\_\backslash mathrm$ is defined similarly to the normal ''biased'' sample variance $\backslash hat\; \backslash sigma^2$:
:$\backslash begin\; \backslash hat\; \backslash sigma^2\backslash \; \&=\; \backslash frac\; N\; \backslash \backslash \; \backslash hat\; \backslash sigma^2\_\backslash mathrm\; \&=\; \backslash frac\; \backslash end$
where $\backslash sum\_^N\; w\_i\; =\; 1$ for normalized weights. If the weights are ''frequency weights'' (and thus are random variables), it can be shown that $\backslash hat\; \backslash sigma^2\_\backslash mathrm$ is the maximum likelihood estimator of $\backslash sigma^2$ for iid
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces i ...

Gaussian observations.
For small samples, it is customary to use an unbiased estimator
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, Prejudice, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual ...

for the population variance. In normal unweighted samples, the ''N'' in the denominator (corresponding to the sample size) is changed to ''N'' − 1 (see Bessel's correction
In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a Sample (statistics), sample. This method co ...

). In the weighted setting, there are actually two different unbiased estimators, one for the case of ''frequency weights'' and another for the case of ''reliability weights''.
Frequency weights

If the weights are ''frequency weights'' (where a weight equals the number of occurrences), then the unbiased estimator is: :$s^2\backslash \; =\; \backslash frac$ This effectively applies Bessel's correction for frequency weights. For example, if values $\backslash $ are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample $\backslash $ with corresponding weights $\backslash $, and we get the same result either way. If the frequency weights $\backslash $ are normalized to 1, then the correct expression after Bessel's correction becomes :$s^2\backslash \; =\; \backslash frac\; \backslash sum\_^N\; w\_i\; \backslash left(x\_i\; -\; \backslash mu^*\backslash right)^2$ where the total number of samples is $\backslash sum\_^N\; w\_i$ (not $N$). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if $w\_i$ has a different meaning other than frequency weight. Note that the estimator can be unbiased only if the weights are notstandardized
Standardization or standardisation is the process of implementing and developing technical standard
A technical standard is an established norm (social), norm or requirement for a repeatable technical task which is applied to a common and repeat ...

nor normalized
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...

, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
Reliability weights

If the weights are instead non-random (''reliability weights''), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean $\backslash mu$ and actual variance $\backslash sigma\_^2$, taking expectations we have, :$\backslash begin\; \backslash operatorname;\; href="/html/ALL/s/hat\_\backslash sigma^2.html"\; ;"title="hat\; \backslash sigma^2">hat\; \backslash sigma^2$X_-_\operatorname[X^2">.html"_;"title="X_-_\operatorname[X">X_-_\operatorname[X^2-_\frac_\operatorname_X_-_\operatorname[X^2.html"_;"title=".html"_;"title="X_-_\operatorname[X">X_-_\operatorname[X^2">.html"_;"title="X_-_\operatorname[X">X_-_\operatorname[X^2\\
&=_\left(1_-_\frac\right)_\sigma_^2
\end
where_$V\_1\_=\_\backslash sum\_^N\_w\_i$_and_$V\_2\_=\_\backslash sum\_^N\_w\_i^2$._Therefore,_the_bias_in_our_estimator_is_$\backslash left(1\_-\_\backslash frac\backslash right)\_$,_analogous_to_the_$\_\backslash left(\_\backslash frac\_\_\backslash right)$_bias_in_the_unweighted_estimator_(also_notice_that_$\backslash \_V\_1^2\_/\_V\_2\_=\_N\_\_$_is_the_effective_sample_size#weighted_samples.html" "title="">X - \operatorname[X^2">.html" ;"title="X - \operatorname[X">X - \operatorname[X^2- \frac \operatorname X_-_\operatorname[X^2.html" ;"title=".html" ;"title="X - \operatorname[X">X - \operatorname[X^2">.html" ;"title="X - \operatorname[X">X - \operatorname[X^2\\
&= \left(1 - \frac\right) \sigma_^2
\end
where $V\_1\; =\; \backslash sum\_^N\; w\_i$ and $V\_2\; =\; \backslash sum\_^N\; w\_i^2$. Therefore, the bias in our estimator is $\backslash left(1\; -\; \backslash frac\backslash right)$, analogous to the $\backslash left(\; \backslash frac\; \backslash right)$ bias in the unweighted estimator (also notice that $\backslash \; V\_1^2\; /\; V\_2\; =\; N\_$ is the effective sample size#weighted samples">effective sample sizeIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

). This means that to unbias our estimator we need to pre-divide by $1\; -\; \backslash left(V\_2\; /\; V\_1^2\backslash right)$, ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution.
The final unbiased estimate of sample variance is:
:$\backslash begin\; s^2\_\backslash \; \&=\; \backslash frac\; \backslash \backslash [4pt]\; \&=\; \backslash frac\; ,\; \backslash end$
where $\backslash operatorname[s^2\_]\; =\; \backslash sigma\_^2$.
The degrees of freedom of the weighted, unbiased sample variance vary accordingly from ''N'' − 1 down to 0.
The standard deviation is simply the square root of the variance above.
As a side note, other approaches have been described to compute the weighted sample variance.
Weighted sample covariance

In a weighted sample, each row vector $\backslash mathbf\_$ (each set of single observations on each of the ''K'' random variables) is assigned a weight $w\_i\; \backslash geq0$. Then theweighted mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shap ...

vector $\backslash mathbf$ is given by
:$\backslash mathbf=\backslash frac.$
And the weighted covariance matrix is given by:
:$\backslash mathbf\; =\; \backslash frac\; .$
Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights.
Frequency weights

If the weights are ''frequency weights'', the ''unbiased'' weighted estimate of the covariance matrix $\backslash textstyle\; \backslash mathbf$, with Bessel's correction, is given by: :$\backslash mathbf\; =\; \backslash frac\; .$ Note that this estimator can be unbiased only if the weights are notstandardized
Standardization or standardisation is the process of implementing and developing technical standard
A technical standard is an established norm (social), norm or requirement for a repeatable technical task which is applied to a common and repeat ...

nor normalized
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...

, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
Reliability weights

In the case of ''reliability weights'', the weights arenormalized
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...

:
: $V\_1\; =\; \backslash sum\_^N\; w\_i\; =\; 1.$
(If they are not, divide the weights by their sum to normalize prior to calculating $V\_1$:
: $w\_i\text{'}\; =\; \backslash frac$
Then the weighted mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shap ...

vector $\backslash mathbf$ can be simplified to
:$\backslash mathbf=\backslash sum\_^N\; w\_i\; \backslash mathbf\_i.$
and the ''unbiased'' weighted estimate of the covariance matrix $\backslash mathbf$ is:Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice RossiGNU Scientific Library - Reference manual, Version 1.15

2011.

/ref> :$\backslash begin\; \backslash mathbf\; \&=\; \backslash frac\; \backslash sum\_^N\; w\_i\; \backslash left(\backslash mathbf\_i\; -\; \backslash mu^*\backslash right)^T\; \backslash left(\backslash mathbf\_i\; -\; \backslash mu^*\backslash right)\; \backslash \backslash \; \&=\; \backslash frac\; .\; \backslash end$ The reasoning here is the same as in the previous section. Since we are assuming the weights are normalized, then $V\_1\; =\; 1$ and this reduces to: : $\backslash mathbf=\backslash frac.$ If all weights are the same, i.e. $w\_\; /\; V\_1=1/N$, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.

Vector-valued estimates

The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide amaximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating
Estimation (or estimating) is the process of finding an estimate, or approximation
An approximation is anything that is intentionally similar but not exactly equa ...

estimate. We simply replace the variance $\backslash sigma^2$ by the covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between eac ...

$\backslash mathbf$ and the arithmetic inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

by the matrix inverse
In linear algebra, an ''n''-by-''n'' square matrix is called invertible (also nonsingular or nondegenerate), if there exists an ''n''-by-''n'' square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the ''n''-by-''n'' identit ...

(both denoted in the same way, via superscripts); the weight matrix then reads:
$$\backslash mathbf\_i\; =\; \backslash mathbf\_i^.$$
The weighted mean in this case is:
$$\backslash bar\; =\; \backslash mathbf\_\; \backslash left(\backslash sum\_^n\; \backslash mathbf\_i\; \backslash mathbf\_i\backslash right),$$
(where the order of the matrix-vector product is not commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

), in terms of the covariance of the weighted mean:
$$\backslash mathbf\_\; =\; \backslash left(\backslash sum\_^n\; \backslash mathbf\_i\backslash right)^,$$
For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then
: $\backslash mathbf\_1\; :=\; \backslash begin1\; \&\; 0\backslash end^\backslash top,\; \backslash qquad\; \backslash mathbf\_1\; :=\; \backslash begin1\; \&\; 0\backslash \backslash \; 0\; \&\; 100\backslash end$
: $\backslash mathbf\_2\; :=\; \backslash begin0\; \&\; 1\backslash end^\backslash top,\; \backslash qquad\; \backslash mathbf\_2\; :=\; \backslash begin100\; \&\; 0\backslash \backslash \; 0\; \&\; 1\backslash end$
then the weighted mean is:
: $\backslash begin\; \backslash bar\; \&\; =\; \backslash left(\backslash mathbf\_1^\; +\; \backslash mathbf\_2^\backslash right)^\; \backslash left(\backslash mathbf\_1^\; \backslash mathbf\_1\; +\; \backslash mathbf\_2^\; \backslash mathbf\_2\backslash right)\; \backslash \backslash [5pt]\; \&\; =\backslash begin\; 0.9901\; \&0\backslash \backslash \; 0\&\; 0.9901\backslash end\backslash begin1\backslash \backslash 1\backslash end\; =\; \backslash begin0.9901\; \backslash \backslash \; 0.9901\backslash end\; \backslash end$
which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1].
Accounting for correlations

In the general case, suppose that $\backslash mathbf=[x\_1,\backslash dots,x\_n]^T$, $\backslash mathbf$ is thecovariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square Matrix (mathematics), matrix giving the covariance between eac ...

relating the quantities $x\_i$, $\backslash bar$ is the common mean to be estimated, and $\backslash mathbf$ is a design matrix equal to a vector of ones $[1,\; \backslash dots,\; 1]^T$ (of length $n$). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:
:$\backslash sigma^2\_\backslash bar=(\backslash mathbf^T\; \backslash mathbf\; \backslash mathbf)^,$
and
:$\backslash bar\; =\; \backslash sigma^2\_\backslash bar\; (\backslash mathbf^T\; \backslash mathbf\; \backslash mathbf),$
where:
:$\backslash mathbf\; =\; \backslash mathbf^.$
Decreasing strength of interactions

Consider the time series of an independent variable $x$ and a dependent variable $y$, with $n$ observations sampled at discrete times $t\_i$. In many common situations, the value of $y$ at time $t\_i$ depends not only on $x\_i$ but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean $z$ for a window size $m$. :$z\_k=\backslash sum\_^m\; w\_i\; x\_.$Exponentially decreasing weights

In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction $0<\backslash Delta<1$ at each time step. Setting $w=1-\backslash Delta$ we can define $m$ normalized weights by : $w\_i=\backslash frac\; ,$ where $V\_1$ is the sum of the unnormalized weights. In this case $V\_1$ is simply : $V\_1=\backslash sum\_^m\; =\; \backslash frac\; ,$ approaching $V\_1=1/(1-w)$ for large values of $m$. The damping constant $w$ must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step $(1-w)^$, the weight approximately equals $(1-w)=0.39(1-w)$, the tail area the value $e^$, the head area $=0.61$. The tail area at step $n$ is $\backslash le$. Where primarily the closest $n$ observations matter and the effect of the remaining observations can be ignored safely, then choose $w$ such that the tail area is sufficiently small.Weighted averages of functions

The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus.Jane Grossman, Michael Grossman, Robert KatzFirst Systems of Weighted Differential and Integral Calculus''

, 1980.

See also

* Average * Central tendency * Mean * Standard deviation * Summary statistics * Weight function * Weighted average cost of capital * Weighted geometric mean * Weighted harmonic mean * Weighted least squares * Weighted median * Weighting * Binomial proportion confidence interval#Standard error of a proportion estimation when using weighted data, Standard error of a proportion estimation when using weighted data *Ratio estimator The ratio estimator is a statistical parameterIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or soc ...

References

Further reading

* *External links

*Tool to calculate Weighted Average

{{DEFAULTSORT:Weighted Mean Means Mathematical analysis Summary statistics