The Weighted Mean Center is the central point calculated by assigning importance or influence (weights) to each point. While the regular mean center treats all points equally, the weighted mean center allows greater influence to be reflected for specific points, making it more suitable for realistic spatial analysis. In practice, various real-world factors such as population, sales volume, or the number of visitors can be used as weights.
$$ \bar{X}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i},\quad \bar{Y}_w = \frac{\sum_{i=1}^{n} w_i y_i}{\sum_{i=1}^{n} w_i} $$
$ \text{Here, } x_i \text{ and } y_i \text{ are the coordinates of point } i,\ \text{and } w_i \text{ is its weight.} $
Characteristics of the Weighted Mean Center
- Points with larger weights have a greater influence on the calculation of the mean center.
- The overall center tends to shift toward the points with higher weights.
- In real analyses, various real-world factors such as population, sales, or visitor numbers can be applied as weights.
Application Examples
- Identifying the actual customer center by setting the number of customers at each location as weights
- Selecting the optimal service location (e.g., schools, hospitals, fire stations) by considering positions and population
- Determining the optimal placement for logistics hubs using shipment volumes as weights
For example, one can identify the actual customer center by weighting each point by the number of customers, or select a service center location by considering both facility locations and population size. Likewise, shipment volumes can be used as weights to determine the optimal logistics hub location.