Weighted Mean

Global Algorithm - One-Dimensional Algorithm

Weighted Mean of a data set is similar to the mean, but the values in the data set are weighted differently. The basic formula is stated as follows:

$$\mu_{\text{weighted}} = \frac{\sum_{i=1}^{N} w_i \cdot Y_i}{\sum_{i=1}^{N} w_i} \, \text{,}$$

where $Y_i$ is the $i$th value in the data set, $N$ is the length of the data vector and $\mu_{\text{weighted}}$ is the weighted mean value.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$Y$	$Y \in \mathbb R^N$	$N \in \mathbb{N}$	Input data vector of length $N$
$W$	$W \in \mathbb R^N$	$N \in \mathbb{N}$	Input weighting vector of length $N$	If $\sum w_i = 1$ , the algorithm simplifies to $\mu_{\text{weighted}} = \sum w_i \cdot Y_i$ .

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$\mu_{\text{weighted}}$	$\mu_{\text{weighted}} \in \mathbb R$		Weighted mean value of data $Y$	The result is sensitive to improper values such as ‘nan’, ‘inf’, ‘null’, etc.

Single Steps using the Algorithm

• Computing Weighted-Mean Value with Weighted Mean Algorithm
• Reconstructing Improper Values with Local Centered Moving Weighted Mean