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The weighted arithmetic mean is similar to an ordinary
arithmetic mean In mathematics and statistics, the arithmetic mean (, stress on first and third syllables of "arithmetic"), or simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the ...
(the most common type of average), 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 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 and statistics, the arithmetic mean (, stress on first and third syllables of "arithmetic"), or simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the ...
. 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.


Examples


Basic example

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were: :Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98 :Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99 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 combinationImage:Convex combination illustration.svg, Given three points x_1, x_2, x_3 in a plane as shown in the figure, the point P ''is'' a convex combination of the three points, while Q is ''not.'' (Q is however an affine combination of the three points, a ...
. Using the previous example, we would get the following weights: : \frac = 0.4 : \frac = 0.6 Then, apply the weights like this: : \bar = (0.4\times80) + (0.6\times90) = 86.


Mathematical definition

Formally, the weighted mean of a non-empty finite
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a Set (mathematics), set that, unlike a set, allows for multiple instances for each of its Element (mathematics), elements. The number of instances given for each elem ...
of data \, with corresponding non-negative weights \ 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' = \frac. Using the normalized weight yields the same results as when using the original weights: :\begin \bar &= \sum_^n w'_i x_i= \sum_^n \frac x_i = \frac \\ & = \frac. \end The ordinary mean \frac \sum_^n is a special case of the weighted mean where all data have equal weights. The ''standard error of the weighted mean (unit input variances)'', \sigma_ can be shown via uncertainty propagation to be: : \sigma_ = \left(\sqrt \right)^


Statistical properties

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. 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, 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' = \frac.


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 and 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 ...
with known
variance In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...
\sigma_i^2, 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 of the mean of the probability distributions under the assumption that they are independent and normally distributed 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: :\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 (squared), \sigma_^2 = \sigma^2/n, formulated in terms of the sample standard deviation (squared), :\sigma^2 = \frac .


Bootstrapping validation

It has been shown by bootstrapping methods that the following is an accurate estimation for the square of the standard error of the mean (general case): : \sigma_^2 = \frac \left sum (w_i x_i - w_s \bar)^2 - 2 \bar \sum (w_i - w_s)(w_i x_i - w_s\bar) + \bar^2 \sum (w_i - w_s)^2 \right where w_s = \sum w_i. Further simplification leads to : \sigma_^2 = \frac \sum w_i^2(x_i - \bar)^2


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 concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...
and standard deviation 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 sample variance \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 Independent and identically distributed random variables, iid Gaussian observations. For small samples, it is customary to use an unbiased estimator 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 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: : \begin s^2\ &= \frac \end 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 : \begin s^2\ &= \frac \sum_^N w_i \left(x_i - \mu^*\right)^2 \end 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 Standard score, standardized nor Normalization (statistics), normalized, these processes changing the data's mean and variance and thus leading to a Base rate fallacy, 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, : \begin \operatorname [\hat \sigma^2] &= \frac N \\ &= \operatorname [(X - \operatorname[X])^2] - \frac \operatorname [(X - \operatorname[X])^2] \\ &= \left( \frac N \right) \sigma_^2 \\ \operatorname [\hat \sigma^2_\mathrm] &= \frac \\ &= \operatorname[(X - \operatorname[X])^2] - \frac \operatorname[(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(1 - \frac\right) , analogous to the \left( \frac \right) bias in the unweighted estimator (also notice that \ V_1^2 / V_2 = N_ is the effective sample size#weighted samples, effective sample size). This means that to unbias our estimator we need to pre-divide by 1 - \left(V_2 / V_1^2\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 \\ &= \frac \end , where \operatorname[s^2_] = \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 \textstyle \textbf_ (each set of single observations on each of the ''K'' random variables) is assigned a weight \textstyle w_i \geq0. Then the weighted mean vector \textstyle \mathbf is given by : \mathbf=\frac. And the weighted covariance matrix is given by: : \begin \mathbf &= \frac . \end 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: : \begin \mathbf &= \frac . \end Note that this estimator can be unbiased only if the weights are not Standard score, standardized nor Normalization (statistics), normalized, these processes changing the data's mean and variance and thus leading to a Base rate fallacy, 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 Normalizing constant, normalized: : 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' = \frac Then the weighted mean vector \textstyle \mathbf can be simplified to : \mathbf=\sum_^N w_i \mathbf_i. and the ''unbiased'' weighted estimate of the covariance matrix \textstyle \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(\mathbf_i - \mu^*\right)^T \left(\mathbf_i - \mu^*\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. \textstyle 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 estimate. We simply replace the variance \sigma^2 by the covariance matrix \mathbf and the arithmetic inverse by the matrix inverse (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 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(\mathbf_1^ + \mathbf_2^\right)^ \left(\mathbf_1^ \mathbf_1 + \mathbf_2^ \mathbf_2\right) \\[5pt] & =\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=[x_1,\dots,x_n]^T, \mathbf is the covariance matrix 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 [1, ..., 1]^T (of length n). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by: :\sigma^2_\bar=(\mathbf^T \mathbf \mathbf)^, and :\bar = \sigma^2_\bar (\mathbf^T \mathbf \mathbf), 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/(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 \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.


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


References


Further reading

* *


External links

*
Tool to calculate Weighted Average
{{DEFAULTSORT:Weighted Mean Means Mathematical analysis Summary statistics