Tool to calculate weighted means. The weighted mean of a statistical value related to a list of numbers that are associated with a coefficient: their weight.

Weighted Mean of Numbers - dCode

Tag(s) : Mathematics, Data Processing

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The weighted mean of a statistical value related to a list of numbers that are associated with a coefficient: their weight, a number which therefore takes up more or less value in the calculated mean.

Take a list of $ n $ values $ X = \{x_1, x_2, \dots, x_n\} $ associated with weights $ W = \{ w_1, w_2, \dots, w_n\} $. The **weighted arithmetic mean** is defined by the sum of values multiplied by their weight, divided by the sum of weights. Formula: $$ \bar{x} = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} $$

__Example:__ The list of 3 numbers $ 12 $ (coefficient $ 7 $), $ 14 $ (coefficient $ 2 $) and $ 16 $ (coefficient $ 1 $) has for weighted mean $ (12 \times 7 + 14 \times 2 + 16 \times 1) / (7 + 2 + 1) = 12.8 $

Take a list of n values $ X = \{x_1, x_2, \dots, x_n\} $ associated with weights $ W = \{ w_1, w_2, \dots, w_n\} $. The **weighted geometric mean** is defined by the pth root of the product of values, where p is the weight's sum. Formula: $$ \bar{x}^G = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} = \quad \exp \left( \frac{1}{\sum_{i=1}^n w_i} \; \sum_{i=1}^n w_i \ln x_i \right) $$

Take a list of n values $ X = \{x_1, x_2, \dots, x_n\} $ associated with weights $ W = \{ w_1, w_2, \dots, w_n\} $. The **weighted harmonic mean** is defined by the ratio of p (the weight sum) to the sum of the ratio of each weigth over the values. Formula: $$ \bar{x}^H = \sum_{i=1}^n w_i \bigg/ \sum_{i=1}^n \frac{w_i}{x_i} $$

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*Weighted Mean of Numbers* on dCode.fr [online website], retrieved on 2024-06-14,

moyenne,weighted,pondered,nombre,chiffre,arithmetic,harmonic,geometric,proportional,list

https://www.dcode.fr/weighted-mean

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