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Weighted arithmetic mean

The weighted arithmetic mean, also known as the weighted average, is a generalization of the standard arithmetic mean that accounts for the relative importance of each data point by assigning a weight to it. It is computed using the formula \bar{x} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}, where x_i represents the data points and w_i > 0 are the corresponding positive weights, ensuring that more significant observations have greater influence on the result.^[1]^[2] This measure of central tendency is particularly valuable in scenarios where equal treatment of data would be inappropriate, such as when combining measurements with differing levels of precision or reliability. For instance, in statistics, weights are often chosen as the inverse of the variance to emphasize more accurate observations, yielding an optimal estimator under certain assumptions.^[3] In finance, it facilitates calculations like the weighted average cost of capital by incorporating the proportions of different funding sources.^[4] Similarly, in engineering and physics, it determines centers of mass or centroids by weighting component masses or areas.^[5] When all weights are equal, the weighted arithmetic mean simplifies to the ordinary arithmetic mean, highlighting its role as a flexible extension of basic averaging.^[6] Key properties include its linearity—allowing decomposition into weighted sums—and its status as a convex combination when weights are normalized to sum to 1, which preserves bounds between the minimum and maximum values. These attributes make it robust for applications in optimization, index construction (e.g., consumer price indices), and decision-making under uncertainty, though care must be taken to select appropriate weights to avoid bias.^[7]^[8]

Fundamentals

Definition

The weighted arithmetic mean is a generalization of the arithmetic mean that accounts for the relative importance of each data point by assigning positive weights to the observations. For a finite set of values x_1, x_2, \dots, x_n with corresponding weights w_1, w_2, \dots, w_n, the weighted arithmetic mean \bar{x} is given by the formula

\bar{x} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i},

where the denominator provides normalization to ensure the result is scale-invariant with respect to the weights.^[9]^[10] The weights w_i are typically non-negative (w_i \geq 0) to maintain the interpretive consistency of the mean as a convex combination, ensuring it lies within the convex hull of the x_i values. Negative weights can lead to sign-related issues that place the mean outside the observed range, while zero weights effectively exclude those observations without distortion. Non-negative weights are standard in statistical contexts.^[9]^[10] Normalization variants arise depending on whether the weights are pre-scaled: if \sum_{i=1}^n w_i = 1, the formula simplifies to \bar{x} = \sum_{i=1}^n w_i x_i; otherwise, the division by the sum of weights is essential for proper averaging.^[11] The concept originated in 7th-century India with Brahmagupta's Brāhma Sphuṭa Siddhānta (628 CE), where it served as a statistical tool for estimating central values in applications like irregular excavations, extending basic averaging to handle unequal importance.^[12]

Relation to Unweighted Mean

The weighted arithmetic mean serves as a generalization of the unweighted arithmetic mean, allowing for the incorporation of varying levels of importance or reliability among data points. The unweighted arithmetic mean, defined as \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i, treats each of the n observations equally by assigning an implicit weight of 1 to every x_i.^[10] This formula arises directly as a special case of the weighted arithmetic mean when all weights w_i are set equal to 1, reducing the general form \bar{x}_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} to the unweighted expression.^[10]^[7] To derive this connection explicitly, substitute w_i = 1 for all i into the weighted formula: the numerator becomes \sum_{i=1}^n 1 \cdot x_i = \sum_{i=1}^n x_i, and the denominator simplifies to \sum_{i=1}^n 1 = n, yielding \bar{x}_w = \frac{\sum_{i=1}^n x_i}{n} = \bar{x}.^[10] This substitution highlights the equal weighting inherent in the unweighted mean, where no differentiation is made based on external factors such as precision or frequency.^[7] In interpretive terms, the unweighted mean represents an egalitarian or "democratic" averaging process, assigning identical influence to each data point regardless of context.^[13] By contrast, the weighted mean adjusts the contribution of each x_i according to specified weights, enabling importance-adjusted summaries in applications like survey analysis or experimental data aggregation.^[13] Notably, the two means coincide precisely when all weights are equal, underscoring the unweighted form's role as the baseline case.^[7] This relationship assumes familiarity with the basic arithmetic mean but clarifies its position within the broader framework of weighted averages.^[10]

Examples and Illustrations

Basic Example

To illustrate the computation of the weighted arithmetic mean, consider the values x = [2, 4, 6] with corresponding weights w = [1, 2, 3]. The numerator is the sum of the weighted values: $1 \cdot 2 + 2 \cdot 4 + 3 \cdot 6 = 2 + 8 + 18 = 28. The denominator is the sum of the weights: $1 + 2 + 3 = 6. Thus, the weighted mean is \bar{x} = 28 / 6 \approx 4.67. In contrast, the unweighted arithmetic mean of these values is (2 + 4 + 6)/3 = 4. The weighted mean exceeds this because the highest weight (3) is assigned to the largest value (6), which pulls the result upward toward that value.^[7] A practical application arises in education, such as averaging grades weighted by course credit hours; for example, grades of 80 in a 2-credit course and 90 in a 3-credit course yield a weighted mean of (2 \cdot 80 + 3 \cdot 90)/(2 + 3) = 430/5 = 86, reflecting the greater influence of the higher-credit course.^[7]

Convex Combination Example

The weighted arithmetic mean can be expressed as a convex combination when the weights are normalized to sum to 1 and are nonnegative, providing a geometric and probabilistic perspective on the concept. In this form, the weighted mean \bar{x} of values x_1, x_2, \dots, x_n is given by

\bar{x} = \sum_{i=1}^n p_i x_i,

where p_i \geq 0 for all i and \sum_{i=1}^n p_i = 1. This representation aligns with the definition of a convex combination in convex analysis, where the result is a point within the convex hull of the original points.^[14] Furthermore, in the context of affine geometry, these weights p_i correspond to barycentric coordinates, which parameterize positions relative to the vertices of a simplex, such as a line segment or triangle.^[15] Consider a simple numerical example with two values, x_1 = [1](/page/1) and x_2 = [3](/page/3), and corresponding weights p_1 = 0.4 and p_2 = 0.6. The weighted mean is then \bar{x} = 0.4 \cdot [1](/page/1) + 0.6 \cdot [3](/page/3) = 2.2. Geometrically, this places \bar{x} on the line segment joining 1 and 3, specifically at a position closer to 3 due to the larger weight p_2, illustrating how the convex combination divides the segment in the ratio of the weights.^[7] Probabilistically, the weights p_i can be interpreted as probabilities of a discrete random variable X taking values x_i, making the weighted mean equivalent to the expected value E[X] = \sum_{i=1}^n p_i x_i. This connection underscores the mean's role in summarizing the central tendency of a probability distribution.^[16] A key property is that the weighted mean, as a convex combination, uniquely lies within the convex hull of the points \{x_1, \dots, x_n\}, ensuring it remains bounded by the extremal values and preserving the geometric structure of the set.^[17]

Types of Weights

Variance-Defined Weights

Variance-defined weights assign greater importance to observations with lower variability, using the inverse of the variance as the weight for each data point. Specifically, for a set of measurements x_i with associated variances \sigma_i^2, the weights are given by w_i = 1 / \sigma_i^2. These weights are then normalized such that they sum to 1, yielding the weighted arithmetic mean as

\bar{x} = \frac{\sum_i (x_i / \sigma_i^2)}{\sum_i (1 / \sigma_i^2)}.

This approach stems from the principle of optimal estimation in statistics, where the goal is to combine independent estimates to minimize the overall variance of the resulting mean.^[18] The rationale for inverse-variance weighting lies in its ability to produce the minimum-variance unbiased estimator when the observations are uncorrelated and normally distributed; a full derivation of this variance minimization appears in the statistical properties section.^[18] Consider two measurements of a quantity: x_1 = 5 with standard error \sigma_1 = 1 (variance 1) and x_2 = 6 with \sigma_2 = 2 (variance 4). The weights are w_1 = 1/1 = 1 and w_2 = 1/4 = 0.25, with total weight 1.25. The weighted mean is (5 \cdot 1 + 6 \cdot 0.25)/1.25 = 6.5/1.25 = 5.2. This result leans toward the more precise measurement (x_1), reflecting its lower uncertainty.^[3] In applications, inverse-variance weighting is foundational in meta-analysis for pooling effect sizes from multiple studies, where each study's contribution is weighted by the precision of its estimate. This method was formalized by William G. Cochran in the 1950s for combining estimates from different experiments, emphasizing weights inversely proportional to their variances to achieve efficient aggregation.^[19]^[20]

Frequency Weights

In the context of the weighted arithmetic mean, frequency weights arise when data points represent multiplicities or counts of occurrences, such as in grouped or summarized datasets where individual observations are not listed separately. The weight w_i for each distinct value x_i is simply the frequency f_i, the number of times x_i appears in the full dataset. The resulting mean is given by the formula

\bar{x} = \frac{\sum f_i x_i}{\sum f_i},

where the denominator normalizes by the total number of observations.^[21]^[22] A practical example occurs in survey data aggregation, such as election polling where candidates A and B receive 10 and 20 votes, respectively, from a sample of 30 respondents. Assigning frequency weights of 10 to A and 20 to B yields a weighted mean vote share of \frac{10A + 20B}{30}, reflecting the proportional support without needing to replicate entries in the dataset. This approach is computationally equivalent to expanding the dataset by replicating each x_i exactly f_i times and then computing the unweighted arithmetic mean, but it avoids redundant data storage and processing, making it efficient for large-scale frequency tables.^[22] Frequency weights have been employed historically in demography to compute population-level averages from census data, such as age distributions or vital rates, where counts from tabulated returns serve as weights; for instance, analyses of 19th-century U.S. frontier populations used such weighted averages to derive demographic indicators like migration rates from grouped census figures.^[23]

Statistical Properties

Expectation

The expected value of the weighted arithmetic mean can be derived using the linearity of expectation. Consider independent random variables X_1, X_2, \dots, X_n with respective expected values \mu_1, \mu_2, \dots, \mu_n, and fixed positive weights w_1, w_2, \dots, w_n such that W = \sum_{i=1}^n w_i. The weighted mean is defined as \bar{X} = \frac{1}{W} \sum_{i=1}^n w_i X_i. By the linearity of expectation, which applies regardless of dependence among the X_i,

E[\bar{X}] = E\left[ \frac{1}{W} \sum_{i=1}^n w_i X_i \right] = \frac{1}{W} \sum_{i=1}^n w_i E[X_i] = \frac{\sum_{i=1}^n w_i \mu_i}{W}.

This result follows from the property that for constants a_i and random variables Y_i, E[\sum a_i Y_i] = \sum a_i E[Y_i], extended here with a_i = w_i / W and Y_i = X_i.^[24] In the special case where all \mu_i = \mu for some common mean \mu, the weighted mean simplifies to E[\bar{X}] = \mu, indicating that \bar{X} is an unbiased estimator of \mu. This unbiasedness holds provided the weights are non-random constants; if the weights were themselves random variables, the expectation would not generally equal the weighted average of the individual expectations, though such scenarios are not addressed here.^[18]

Variance in Independent Cases

When the random variables X_i are independent with known variances \sigma_i^2 > 0, the variance of the weighted arithmetic mean \bar{X} = \frac{\sum_{i=1}^n w_i X_i}{\sum_{i=1}^n w_i} is

\Var(\bar{X}) = \frac{\sum_{i=1}^n w_i^2 \sigma_i^2}{\left( \sum_{i=1}^n w_i \right)^2},

where the w_i > 0 are fixed weights. This expression arises from the general property that the variance of a linear combination of independent random variables is the weighted sum of the individual variances, with the coefficients for \bar{X} being w_i / \sum w_i. To derive it, note that \Var\left( \sum a_i X_i \right) = \sum a_i^2 \Var(X_i) for independent X_i, so substituting a_i = w_i / W with W = \sum w_i yields the formula after simplification. In the special case where all variances are equal (\sigma_i^2 = \sigma^2 for all i) and the weights are equal (w_i = [1](/page/1)), the formula simplifies to \Var([\bar{X}](/page/Var)) = \sigma^2 / n, which is the familiar variance of the unweighted arithmetic mean of n observations. This reduction highlights how the weighted mean generalizes the unweighted case, preserving the same form when precision is uniform across observations. The variance \Var([\bar{X}](/page/Var)) can be minimized by choosing weights w_i \propto 1 / \sigma_i^2, known as inverse-variance weighting; without loss of generality, set w_i = 1 / \sigma_i^2. Under these optimal weights, the minimum variance simplifies to \Var([\bar{X}](/page/Var)) = \left( \sum_{i=1}^n 1 / \sigma_i^2 \right)^{-1}. This choice is derived by minimizing the variance expression subject to \sum w_i = 1 using Lagrange multipliers: let L = \sum (w_i^2 \sigma_i^2) + \lambda (1 - \sum w_i); setting partial derivatives to zero gives w_i = 1 / (\lambda \sigma_i^2), implying the proportionality, and substituting back yields the minimized value. Such weighting is particularly useful when combining independent estimates of a common parameter, as it allocates greater influence to more precise observations while ensuring the estimator remains unbiased for the expectation.

Variance in Sampling Contexts

In survey sampling, the weighted arithmetic mean is frequently employed to estimate the population mean under designs with unequal inclusion probabilities \pi_i, where \pi_i denotes the probability that unit i is selected into the sample. A key estimator in this context is the \pi-estimator (also called the Hajek estimator), defined as

\bar{x}_\pi = \frac{\sum_{i \in s} x_i / \pi_i}{\sum_{i \in s} 1 / \pi_i},

where s is the realized sample. This form arises as the ratio of Horvitz-Thompson estimators for the population total and size, providing approximate unbiasedness for the population mean when inclusion probabilities are bounded away from 1.^[25] The variance of \bar{x}_\pi is derived using linearization or delta method approximations, incorporating the first- and second-order inclusion probabilities \pi_i and \pi_{ij} (for i \neq j). Exact computation requires joint probabilities, but practical estimation often relies on design effects, which measure the ratio of the design-based variance to that under simple random sampling, adjusting for clustering, stratification, or unequal probabilities inherent in the sampling scheme.^[26] The Horvitz-Thompson estimator itself targets the population total unbiasedly as \hat{Y} = \sum_{i \in s} x_i / \pi_i, with adaptation to the mean via \bar{x}_{HT} = \hat{Y} / N when the population size N is known; its variance formula,

V(\hat{Y}) = \sum_i \sum_j (\pi_i \pi_j - \pi_{ij}) \frac{Y_i}{\pi_i} \frac{Y_j}{\pi_j},

similarly involves design effects for estimation and highlights efficiency gains or losses relative to equal-probability sampling. When N is unknown, the \pi-estimator replaces it with \hat{N} = \sum_{i \in s} 1 / \pi_i, yielding a consistent mean estimator under standard conditions.^[25] A practical illustration occurs in stratified sampling, where the population is partitioned into H mutually exclusive strata, with independent simple random samples drawn from each. Here, weights are set as w_i = N_h / n_h for unit i in stratum h (with N_h and n_h the stratum population and sample sizes), so the weighted mean is \bar{x} = \sum_i w_i x_i / N. An approximate variance formula, useful for initial assessments or when finite population corrections are negligible, is V(\bar{x}) \approx \sum_h \left( \frac{N_h}{N} \right)^2 \frac{S_h^2}{n_h}, where S_h^2 is the population variance within stratum h. More precise estimation incorporates stratum-specific sample variances s_h^2 = \frac{1}{n_h - 1} \sum_{i \in h} (x_i - \bar{x}_h)^2 and finite population corrections, yielding \hat{V}(\bar{x}) = \sum_h \left( \frac{N_h}{N} \right)^2 \frac{s_h^2}{n_h} \left(1 - \frac{n_h}{N_h}\right), reflecting the design's impact on overall variability.^[27] For complex designs where analytical variance formulas are intractable, bootstrapping offers a resampling-based validation: replicates are generated by mimicking the original sampling process, and the empirical variance across replicates approximates the design variance without deriving joint probabilities. This approach, as in the Rao-Wu method, is particularly effective for stratified or multistage schemes, ensuring robust inference in practice.^[28]

Weighted Sample Variance and Covariance

In statistics, the weighted sample variance extends the unweighted sample variance to account for varying importance or precision of observations through weights. For reliability weights, which reflect the relative precision or inverse variance of each data point (often used in contexts like meta-analysis or regression diagnostics), the population weighted variance is defined as

\sigma_w^2 = \frac{\sum_{i=1}^n w_i (x_i - \bar{x}_w)^2}{\sum_{i=1}^n w_i},

where \bar{x}_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} is the weighted mean, and w_i > 0 are the reliability weights.^[29] This formula provides a biased estimator of the population variance when used as a sample estimator, analogous to dividing by n in the unweighted case. To obtain an unbiased estimator analogous to Bessel's correction (dividing by n-1), the denominator is adjusted to \sum w_i \left(1 - \frac{\sum w_i^2}{(\sum w_i)^2}\right), yielding

s_w^2 = \frac{\sum_{i=1}^n w_i (x_i - \bar{x}_w)^2}{\sum_{i=1}^n w_i - \frac{(\sum_{i=1}^n w_i^2)}{\sum_{i=1}^n w_i}}.

This correction accounts for the effective degrees of freedom in the weighted setting, reducing bias particularly when weights are unequal.^[29] In contrast, for frequency weights f_i (non-negative integers representing the number of times each x_i is replicated in the sample, common in survey data or grouped observations), the weighted sample variance treats the data as an expanded dataset of size \sum f_i. The unbiased estimator is then

s_f^2 = \frac{\sum_{i=1}^n f_i (x_i - \bar{x}_f)^2}{\sum_{i=1}^n f_i - 1},

where \bar{x}_f = \frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}. This directly parallels the unweighted sample variance formula applied to the full replicated sample.^[29] The distinction between reliability and frequency weights is crucial to avoid biased variance estimates: reliability weights model varying precisions without implying replication, leading to the adjusted denominator for unbiasedness, whereas frequency weights assume actual multiplicity, preserving the simple N-1 correction where N = \sum f_i. Misapplying one type for the other can inflate or deflate the variance, affecting downstream inferences like confidence intervals.^[30] The weighted sample covariance similarly generalizes the unweighted version to paired observations (x_i, y_i) with weights w_i. For reliability weights, it is given by

\text{cov}_w(X, Y) = \frac{\sum_{i=1}^n w_i (x_i - \bar{x}_w)(y_i - \bar{y}_w)}{\sum_{i=1}^n w_i},

where \bar{y}_w is the weighted mean of the y_i. This measures the weighted linear association between X and Y, with an unbiased version obtainable via a similar degrees-of-freedom correction in the denominator. For frequency weights, the formula becomes

\text{cov}_f(X, Y) = \frac{\sum_{i=1}^n f_i (x_i - \bar{x}_f)(y_i - \bar{y}_f)}{\sum_{i=1}^n f_i - 1}.

These estimators are essential for analyzing heteroscedastic data or weighted least squares, ensuring covariance reflects the intended weighting scheme.^[31]

Vector and Function Weighted Averages

The weighted arithmetic mean extends naturally to vectors in a multivariate setting by applying the scalar formula component-wise to each dimension of the input vectors. For a set of vectors \mathbf{x}_i = (x_{i1}, x_{i2}, \dots, x_{id}) \in \mathbb{R}^d with corresponding positive weights w_i > 0 for i = 1, \dots, n, the weighted vector mean is given by

\bar{\mathbf{x}} = \frac{\sum_{i=1}^n w_i \mathbf{x}_i}{\sum_{i=1}^n w_i} = \left( \frac{\sum_{i=1}^n w_i x_{i1}}{\sum_{i=1}^n w_i}, \frac{\sum_{i=1}^n w_i x_{i2}}{\sum_{i=1}^n w_i}, \dots, \frac{\sum_{i=1}^n w_i x_{id}}{\sum_{i=1}^n w_i} \right).

This operation produces a vector in the same space \mathbb{R}^d and represents a convex combination when the weights are normalized to sum to 1.^[9] The resulting \bar{\mathbf{x}} is the balance point or barycenter of the weighted points, preserving the affine structure of the space. In vector spaces, this mean is a linear operator, meaning that if \mathbf{x}_i = a \mathbf{u}_i + b \mathbf{v}_i for scalars a, b and vectors \mathbf{u}_i, \mathbf{v}_i, then the weighted mean of the \mathbf{x}_i equals a times the weighted mean of the \mathbf{u}_i plus b times the weighted mean of the \mathbf{v}_i.^[32] A prominent application arises in geometry and physics, where the weighted vector mean computes the centroid of a system of points with masses proportional to the weights w_i. For position vectors \mathbf{r}_i, the center of mass is \bar{\mathbf{r}} = \frac{\sum w_i \mathbf{r}_i}{\sum w_i}, which balances the system under gravitational forces.^[9] In machine learning, this formulation underpins weighted k-means clustering, where cluster centroids are updated as the weighted means of assigned data points, with weights reflecting sample importance, frequency, or density to handle imbalanced or noisy datasets. For instance, in gene expression analysis, genetic weighted k-means uses such centroids to group large-scale vector data while accounting for varying feature reliabilities.^[33] The weighted arithmetic mean also generalizes to averaging functions, treating them as elements in a function space. For a collection of functions f_i: \mathcal{X} \to \mathbb{R} defined on a domain \mathcal{X} with weights w_i > 0, the weighted functional average is the function

\bar{f}(x) = \frac{\sum_{i=1}^n w_i f_i(x)}{\sum_{i=1}^n w_i}, \quad \forall x \in \mathcal{X}.

This pointwise operation yields a new function \bar{f} that interpolates the inputs according to their weights, analogous to the continuous case \bar{f}(x) = \frac{\int f(x) w(x) \, dx}{\int w(x) \, dx} for density-based weighting.^[34] Such averages are used in approximation theory and numerical analysis to blend basis functions, ensuring the result lies in the convex hull of the f_i. Unlike the vector case, where linearity holds unconditionally in the vector space, functional averages may exhibit non-commutativity when composed with nonlinear operators, such as when averaging compositions f_i \circ g differs from composing the average with g, highlighting the need for careful handling in nonlinear applications.^[34]

Advanced Weighting Techniques

In scenarios where observations exhibit pairwise correlations, such as in experimental measurements sharing common systematic errors, the standard weighted arithmetic mean assuming independence can be biased or inefficient. The optimal weights are derived from the inverse of the covariance matrix V, given by w_i \propto \sum_k (V^{-1})_{ik}, with the variance of the estimator being $1 / (1^T V^{-1} 1). For the special case of equal variances and constant pairwise correlation \rho, the variance of the mean approximates \frac{\sigma^2}{n} [1 + (n-1)\rho], effectively reducing the total weight by the factor $1 + (n-1)\rho.^[35] For time series data where recent observations are more relevant, exponentially decreasing weights provide a mechanism to emphasize recency without abrupt cutoffs. The weights are defined as w_i = \alpha^{t-i} for observation at time i, with decay parameter $0 < \alpha < 1 and t the current time, normalized such that \sum w_i = 1. This scheme underlies the exponentially weighted moving average (EWMA), originally proposed for forecasting and control charts, where the resulting mean converges to a recursive form \bar{x}_t = \alpha x_t + (1 - \alpha) \bar{x}_{t-1}. Seminal work established its properties for detecting shifts in means with minimal lag.^[36] When the observed dispersion in data exceeds expectations under assumed variances \sigma_i^2, a correction scales the weights to address over- or under-dispersion. The dispersion factor is computed as \phi = \frac{\sum w_i (x_i - \bar{x})^2}{n-1}, where \bar{x} is the preliminary weighted mean, n is the number of observations, and weights w_i are typically $1/\sigma_i^2; if \phi > 1, weights are scaled by $1/\phi to inflate variances appropriately. This parameter, common in weighted regression diagnostics, quantifies deviation from the assumed error model and adjusts for heteroscedasticity or model misspecification in generalized linear models.^[37] In network analysis, where node importance diminishes with structural distance, weights proportional to the inverse of distance w_i \propto 1 / d_i (with d_i the graph distance from a reference node) enable localized averages that prioritize nearby interactions. This inverse distance weighting (IDW) approach, applied to weighted networks for centrality or aggregation, assumes similarity decays with separation, yielding a mean that interpolates values across connected components. The method, foundational in spatial and graph interpolation, uses a power parameter to tune decay sharpness.

References

[1]
Formula | How to Calculate Weighted Mean? - Cuemath
The weighted mean is defined as the summation of the product of weights and quantities, divided by the summation of weights. The concept of the weighted mean is ...Missing: authoritative source
[2]
Weighted Arithmetic Mean - an overview | ScienceDirect Topics
The weighted arithmetic mean is defined as a measure of central tendency that takes into account the relative importance or weight of each value in a ...Missing: authoritative | Show results with:authoritative
[3]
[PDF] Data Analysis Toolkit #12: Weighted averages and their uncertainties
Weighted averages use measurements (xi) and their weights (wi) to calculate the mean. Weights can be based on importance or uncertainty, with inverse variance ...
[4]
Weighted Mean - Definition, Uses, and Practical Example
The weighted mean is a type of mean that is calculated by multiplying the weight (or probability) associated with a particular event or outcome with its.
[5]
7.1 Weighted Averages - Engineering Statics
Weighted averaging is used to find centroids, centers of gravity and centers of mass, the subject of this chapter.
[6]
[PDF] Weighted Means and Means as Weighted Sums
A weighted mean is a sum of coefficients times numbers, where the coefficients, called weights, sum to 1. The ordinary mean is a special case.
[7]
Weighted averages - Department of Mathematics at UTSA
Oct 24, 2021 · The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points ...Examples · Basic example · Convex combination example · Mathematical definition
[8]
[PDF] ELEMENTARY INDEXES FOR A CONSUMER PRICE INDEX
With arithmetic mean (Laspeyres) indexes, the quantity-weighted arithmetic mean (1a) gives the same result as the expenditure-weighted arithmetic mean (1b) ...<|control11|><|separator|>
[9]
Weighted Mean -- from Wolfram MathWorld
Weighted means are also commonly used in statistics, for instance, in population studies. See also. Arithmetic Mean, Mean. This entry contributed by David ...
[10]
[PDF] weighted mean
Mar 18, 1997 · WEIGHTED MEAN. DATAPLOT Reference Manual. March 18, 1997. 2-65. WEIGHTED MEAN. PURPOSE. Compute the weighted mean of a variable. DESCRIPTION.
[11]
Formula: How to Find Weighted Mean - Statistics How To
The image above is the technical formula for the weighted mean. In simple terms, the formula can be written as: Weighted mean = Σwx/Σw.
[12]
Weighted Arithmetic Mean in Ancient India - Bhāvanā
From the early seventh century, the weighted arithmetic mean was conceived in its statistical avatar by Brahmagupta and subsequent Indian mathematicians, and ...
[13]
[PDF] NUREG-1475, Revision 1, "Applying Statistics".
Notice that the weighted mean is identical to the unweighted mean of all nine values. In this case, the weighted mean is simply another way to calculate the.
[14]
cc_convex
A convex combination is a weighted average in which the weights are nonnegative and add to $ 1. The term convex combination comes from the connection with ...Missing: arithmetic | Show results with:arithmetic
[15]
[PDF] Barycentric Coordinates; Parameterizations - UT Computer Science
Weighted average. Still in convex hull. Scaling the masses doesn't affect CoM ... Calculate barycentric coordinates. 3. Check coords valid. 4. Linearly ...
[16]
SticiGui The Long Run and the Expected Value - Stat.berkeley.edu
The Expected Value of a Discrete Random Variable. The expected value of a discrete random variable X, denoted E(X), is a weighted average of the values that ...
[17]
[PDF] Convex sets - CMU School of Computer Science
Convexity is very closely related to the notion of means. For example, the point (1−α)x+αy is just the (weighted) arithmetic mean of x and y. Over the ...
[18]
Minimizing the Variance of a Weighted Average
To minimize the variance of a weighted average, use weights that are the inverse of the variance of each estimator, specifically wi=1Var(Xi)∑nj=11Var(Xj).Missing: formula | Show results with:formula
[19]
The combination of estimates from different experiments.
A sampling investigation of the efficiency of weighting inversely as the estimated variance. W. G. CochranSarah Porter Carroll. Mathematics. 1953. Data of this ...
[20]
Chapter 10: Analysing data and undertaking meta-analyses
The inverse-variance method is so named because the weight given to each study is chosen to be the inverse of the variance of the effect estimate (i.e. 1 over ...Missing: 1950s | Show results with:1950s
[21]
What types of weights do SAS, Stata and SPSS support? - OARC Stats
frequency weights – Frequency weights are whole numbers (i.e., integers) ... analytic weights – Analytic weights are used when the cases are actually an average.Missing: definition | Show results with:definition<|separator|>
[22]
Weights in Statistics: What Do People Often Get Wrong?
Apr 30, 2024 · Lumley and Gelman identify four primary types of weights in statistics: sampling weights, precision weights, frequency weights, and importance ...Sampling Weights · Precision Weights · Frequency Weights
[23]
An analysis of nineteenth-century frontier populations
*Range of percentages, 1840-60. NOTE.-The average of these points is not the average given in Table 1. Table 1 shows weighted average percent- ages and ...
[24]
[PDF] 3.2.1 Linearity of Expectation
Right now, the only way you've learned to compute expectation is by first computing the PMF of a random variable pX(k) and using the formula E [X] = Pk∈ΩX.
[25]
[PDF] Horvitz-Thompson-1952-jasa.pdf
Author(s): D. G. Horvitz and D. J. Thompson. Source: Journal of the American Statistical Association, Vol. 47, No. 260 (Dec., 1952), pp. 663-. 685.
[26]
A Generalization of Sampling Without Replacement from a Finite ...
This paper presents a general technique for the treatment of samples drawn without replacement from finite universes when unequal selection probabilities are ...
[27]
[PDF] Chapter 4 Stratified Sampling - IIT Kanpur
Moreover, the variance of the sample mean not only depends on the sample size and sampling fraction ... mean is defined as the weighted arithmetic mean of stratum ...
[28]
[PDF] With-replacement bootstrap variance estimation for household ...
Jan 6, 2022 · As explained by Rao and Wu (1988), the rescaled bootstrap may be applied to a variety of sampling designs including two-stage sampling and with/ ...
[29]
Weighted Variance, Standard Deviation, and Covariance
Describes how to calculate the weighted variance, standard deviation, and covariance in Excel for both reliability and frequency weights.
[30]
Weights in statistics - Biased and Inefficient
Aug 4, 2020 · There are three main types of weights: precision weights (observation precision), frequency weights (cell sizes), and sampling weights (sample ...
[31]
WEIGHTED CORRELATION, WEIGHTED COVARIANCE ...
Nov 8, 2018 · Description: Given paired response variables x and y of length n and a weights variable w, the weighted covariance is computed with the formula.
[32]
[PDF] 3 Random vectors and multivariate normal distribution
Formally, the average may be thought of as a “weighted” average, where each possible value is represented in accordance to the probability with which it ...
[33]
Genetic weighted k-means algorithm for clustering large-scale gene ...
In this paper, we propose a genetic weighted K-means algorithm (denoted by GWKMA), which solves the first two problems and partially remedies the third one.
[34]
[PDF] Weighted Averages - MIT OpenCourseWare
Weighted Averages. A weighted average is calculated by dividing the weighted total value of a fraction by the total of the weighting function: / b f(x)w(x) ...Missing: arithmetic | Show results with:arithmetic
[35]
[PDF] Averaging Correlated Data
Nov 21, 1994 · This article proposes a scheme for averaging such data and illustrates its properties by applying it to these examples. The suggested procedure ...
[36]
Charles Holt's report on exponentially weighted moving averages
Charles Holt's classic paper on exponentially weighted moving averages appeared as Report ONR 52 from the Office of Naval Research in 1957.
[37]
13.1 - Weighted Least Squares | STAT 501
The method of weighted least squares can be used when the ordinary least squares assumption of constant variance in the errors is violated (which is called ...