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Weighted Mean of Numbers

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.

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Weighted Mean of Numbers -

Tag(s) : Mathematics, Data Processing

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# Weighted Mean of Numbers

## Mean Calculator

 Calculate Weighted Arithmetic Mean (Classic) Weighted Geometric Mean Weighted Harmonic Mean

### What is a weighted mean? (Definition)

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.

### How to compute a weighted arithmetic 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 $(12 \times 7 + 14 \times 2 + 16 \times 1) / (7 + 2 + 1) = 12.8$

### How to compute a weighted geometric 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 geometric mean is defined by the pth root of the product of values, where p is the weight's sum. Formula: $$\bar{x}^G = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} = \quad \exp \left( \frac{1}{\sum_{i=1}^n w_i} \; \sum_{i=1}^n w_i \ln x_i \right)$$

### How to compute a weighted harmonic 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 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 2024-09-14, https://www.dcode.fr/weighted-mean

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