weighted average

TheInfoList

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), 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): $\bar = \frac = 86.$ 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": :$\bar = \frac = 86.$ 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 a
convex 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: :$\frac = 0.4$ :$\frac = 0.6$ Then, apply the weights like this: :$\bar = \left(0.4\times80\right) + \left(0.6\times90\right) = 86.$

# Mathematical definition

Formally, the weighted mean of a non-empty finite
multiset 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 $\,$ 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 ...
$\$ is :$\bar = \frac,$ which expands to: :$\bar = \frac.$ 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.: :$\sum_^n = 1$. For such normalized weights the weighted mean is then: :$\bar = \sum_^n$. Note that one can always normalize the weights by making the following transformation on the original weights: :$w_i\text{'} = \frac$. Using the normalized weight yields the same results as when using the original weights: :$\begin \bar &= \sum_^n w\text{'}_i x_i= \sum_^n \frac x_i = \frac \\ & = \frac. \end$ 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 ...
$\frac \sum_^n$ 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 $\sigma^2$, the ''standard error of the weighted mean'', $\sigma_$, 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: :$\sigma_ = \sigma \sqrt$

# Statistical properties

## Expectancy

The weighted sample mean, $\bar$, 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 $E(\bar) = \sum_^n .$ In particular, if the means are equal, $\mu_i=\mu$, then the expectation of the weighted sample mean will be that value, $E(\bar)= \mu.$

## Variance

### Simple i.i.d case

When treating the weights as constants, and having a sample of ''n'' observations from
uncorrelated 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 ...
): : $\operatorname\left(\bar y_w\right) = \frac \frac$ With $\hat \sigma_y^2 = \frac$, $\bar = \frac$, and $\overline = \frac$ 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 arbitrary
sampling 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 ($\hat N$). 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 $P\left(I_i=1 \mid \text n \right) = \pi_i$, and the one-draw probability of selection is $P\left(I_i=1 , \text\right) = p_i \approx \frac$ (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: = y_i \pi_i; and variance: = 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 $\pi$-expanded ''y'' values, i.e.: $\check y_i = \frac$. A related quantity is $p$-expanded ''y'' values: $\frac = n \check y_i$. As above, we can add a tick mark if multiplying by the indicator function. I.e.: $\check y\text{'}_i = I_i \check y_i = \frac$ 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.: $w_i = \frac \approx \frac$.

### Variance of the weighted sum (''pwr''-estimator for totals)

If the population size ''N'' is known we can estimate the population mean using $\hat_ = \frac \approx \frac$. If the
sampling 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: : $\operatorname \left\left( \hat_ \right\right) = \frac \frac \sum_^n \left\left( w_i y_i - \overline \right\right)^2$ 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: $\operatorname(\hat \bar Y_) = \frac \sum_^n \sum_^n \left( \check_ \check_i \check_j \right)$ With $\check_i = \frac$. Also, $C\left(I_i, I_j\right) = \pi_ - \pi_\pi_ = \Delta_$ where $\pi_$ is the probability of selecting both i and j. And $\check_ = 1 - \frac$, and for i=j: $\check_ = 1 - \frac = 1- \pi_$. If the selection probability are uncorrelated (i.e.: $\forall i \neq j: C\left(I_i, I_j\right) = 0$), and when assuming the probability of each element is very small, then: : $\operatorname\left(\hat \bar Y_\right) = \frac \sum_^n \left\left( w_i y_i \right\right)^2$

### 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 ($\hat Y$) 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.: $\hat N$). The estimation of $N$ can be described as the sum of weights. So when $w_i = \frac$ we get $\hat N = \sum_^n w_i I_i = \sum_^n \frac = \sum_^n \check 1\text{'}_i$. When using notation from previous sections, the ratio we care about is the sum of $y_i$s, and 1s. I.e.: $R = \bar Y = \frac = \frac = \frac$. We can estimate it using our sample with: $\hat R = \hat = \frac = \frac = \frac = \frac = \bar y_w$. As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: $\bar y_w = \frac$. This will be the
estimand 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 $\bar y_w = \frac$. 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: : $\widehat = \frac \sum_^n w_i^2 \left(y_i - \bar y_w\right)^2$. We note that if $\pi_i \approx p_i n$, then either using $w_i = \frac$ or $w_i = \frac$ 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
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.

### Bootstrapping validation

It has been shown, by Gatz et. al. (1995), that in comparison to
bootstrapping 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): : where $\bar = \frac$. Further simplification leads to :$\widehat = \frac \sum w_i^2\left(x_i - \bar_w\right)^2$ 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 and
Bootstrapping 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 $\sigma^2_i$, the variance of the weighted sample mean is : $\sigma^2_ = \sum_^n$ whose square root $\sigma_$ can be called the ''standard error of the weighted mean (general case)''. Consequently, if all the observations have equal variance, $\sigma^2_i= \sigma^2_0$, the weighted sample mean will have variance : $\sigma^2_ = \sigma^2_0 \sum_^n ,$ where $1/n \le \sum_^n \le 1$. The variance attains its maximum value, $\sigma_0^2$, 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 $\sigma_ = \sigma_0 / \sqrt$, i.e., it degenerates into 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. 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 $w_i\text{'} = \frac$.

# 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 different
probability 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 ...

$\sigma_i^2$, all having the same mean, one possible choice for the weights is given by the reciprocal of variance: :$w_i = \frac.$ The weighted mean in this case is: :$\bar = \frac,$ and the ''standard error of the weighted mean (with variance weights)'' is: :$\sigma_ = \sqrt,$ Note this reduces to $\sigma_^2 = \sigma_0^2/n$ when all $\sigma_i = \sigma_0$. It is a special case of the general formula in previous section, :$\sigma^2_ = \sum_^n = \frac.$ The equations above can be combined to obtain: :$\bar = \sigma_^2 \sum_^n \frac.$ 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 $\chi^2$ is too large. The correction that must be made is :$\hat_^2 = \sigma_^2 \chi^2_\nu$ where $\chi^2_\nu$ is the
reduced 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 ...
: :$\chi^2_\nu = \frac \sum_^n \frac;$ The square root $\hat_$ can be called the ''standard error of the weighted mean (variance weights, scale corrected)''. When all data variances are equal, $\sigma_i = \sigma_0$, they cancel out in the weighted mean variance, $\sigma_^2$, 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), $\sigma_^2 = \sigma^2/n$, 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), :$\sigma^2 = \frac .$

# Related concepts

## Weighted sample variance

Typically when a mean is calculated it is important to know the
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
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 $\mu^*$ is used, the variance of the weighted sample is different from the variance of the unweighted sample. The ''biased'' weighted
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$\hat \sigma^2_\mathrm$ is defined similarly to the normal ''biased'' sample variance $\hat \sigma^2$: :$\begin \hat \sigma^2\ &= \frac N \\ \hat \sigma^2_\mathrm &= \frac \end$ where $\sum_^N w_i = 1$ for normalized weights. If the weights are ''frequency weights'' (and thus are random variables), it can be shown that $\hat \sigma^2_\mathrm$ is the maximum likelihood estimator of $\sigma^2$ 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: :$s^2\ = \frac$ This effectively applies Bessel's correction for frequency weights. For example, if values $\$ 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 $\$ with corresponding weights $\$, and we get the same result either way. If the frequency weights $\$ are normalized to 1, then the correct expression after Bessel's correction becomes :$s^2\ = \frac \sum_^N w_i \left\left(x_i - \mu^*\right\right)^2$ where the total number of samples is $\sum_^N w_i$ (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 not
standardized 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 $\mu$ and actual variance $\sigma_^2$, taking expectations we have, :
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_$V_1_=_\sum_^N_w_i$_and_$V_2_=_\sum_^N_w_i^2$._Therefore,_the_bias_in_our_estimator_is_$\left\left(1_-_\frac\right\right)_$,_analogous_to_the_$_\left\left(_\frac__\right\right)$_bias_in_the_unweighted_estimator_(also_notice_that_$\_V_1^2_/_V_2_=_N__$_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 $V_1 = \sum_^N w_i$ and $V_2 = \sum_^N w_i^2$. Therefore, the bias in our estimator is $\left\left(1 - \frac\right\right)$, analogous to the $\left\left( \frac \right\right)$ bias in the unweighted estimator (also notice that $\ V_1^2 / V_2 = N_$ 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 $1 - \left\left(V_2 / V_1^2\right\right)$, 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: :$\begin s^2_\ &= \frac \\\left[4pt\right] &= \frac , \end$ where $\operatorname\left[s^2_\right] = \sigma_^2$. 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 $\mathbf_$ (each set of single observations on each of the ''K'' random variables) is assigned a weight $w_i \geq0$. 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 $\mathbf$ is given by :$\mathbf=\frac.$ And the weighted covariance matrix is given by: :$\mathbf = \frac .$ 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 $\textstyle \mathbf$, with Bessel's correction, is given by: :$\mathbf = \frac .$ Note that this estimator can be unbiased only if the weights are not
standardized 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 are
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 ...
: : $V_1 = \sum_^N w_i = 1.$ (If they are not, divide the weights by their sum to normalize prior to calculating $V_1$: : $w_i\text{'} = \frac$ 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 $\mathbf$ can be simplified to :$\mathbf=\sum_^N w_i \mathbf_i.$ and the ''unbiased'' weighted estimate of the covariance matrix $\mathbf$ is:Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi
GNU Scientific Library - Reference manual, Version 1.15
2011.

/ref> :$\begin \mathbf &= \frac \sum_^N w_i \left\left(\mathbf_i - \mu^*\right\right)^T \left\left(\mathbf_i - \mu^*\right\right) \\ &= \frac . \end$ 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: : $\mathbf=\frac.$ 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 a
maximum 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 $\sigma^2$ 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 ...
$\mathbf$ 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: $\mathbf_i = \mathbf_i^.$ The weighted mean in this case is: $\bar = \mathbf_ \left(\sum_^n \mathbf_i \mathbf_i\right),$ (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: $\mathbf_ = \left(\sum_^n \mathbf_i\right)^,$ 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 : $\mathbf_1 := \begin1 & 0\end^\top, \qquad \mathbf_1 := \begin1 & 0\\ 0 & 100\end$ : $\mathbf_2 := \begin0 & 1\end^\top, \qquad \mathbf_2 := \begin100 & 0\\ 0 & 1\end$ then the weighted mean is: : $\begin \bar & = \left\left(\mathbf_1^ + \mathbf_2^\right\right)^ \left\left(\mathbf_1^ \mathbf_1 + \mathbf_2^ \mathbf_2\right\right) \\\left[5pt\right] & =\begin 0.9901 &0\\ 0& 0.9901\end\begin1\\1\end = \begin0.9901 \\ 0.9901\end \end$ 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 $\mathbf=\left[x_1,\dots,x_n\right]^T$, $\mathbf$ is 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 ...
relating the quantities $x_i$, $\bar$ is the common mean to be estimated, and $\mathbf$ is a design matrix equal to a vector of ones $\left[1, \dots, 1\right]^T$ (of length $n$). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by: :$\sigma^2_\bar=\left(\mathbf^T \mathbf \mathbf\right)^,$ and :$\bar = \sigma^2_\bar \left(\mathbf^T \mathbf \mathbf\right),$ where: :$\mathbf = \mathbf^.$

## 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 $m$. :$z_k=\sum_^m w_i x_.$

## 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 $0<\Delta<1$ at each time step. Setting $w=1-\Delta$ we can define $m$ normalized weights by : $w_i=\frac ,$ where $V_1$ is the sum of the unnormalized weights. In this case $V_1$ is simply : $V_1=\sum_^m = \frac ,$ approaching $V_1=1/\left(1-w\right)$ 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 $\left(1-w\right)^$, the weight approximately equals $\left(1-w\right)=0.39\left(1-w\right)$, the tail area the value $e^$, the head area $=0.61$. The tail area at step $n$ is $\le$. 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 Katz
First Systems of Weighted Differential and Integral Calculus''
, 1980.

* 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 ...