Which mean to calculate for a series of points (x,y)?

Question

I have a series of points in space with $(x_i,y_i)$ coordinates. And I would like to represent the average/mean point of this series:

--> Should I go for an arithmetic or a geometric mean of $x$ and $y$ coordinates ?

Thanks a lot to answer to my naive question :)

Answer 1

Depends on the situation. e.g the center of mass in case of uniform mass density $\rho$ would be $$ (X,Y) = \left( \frac{ \sum_{i=1}^n \rho \cdot X_i }{\sum_{i=1}^n \rho}, \frac{ \sum_{i=1}^n \rho \cdot Y_i }{\sum_{i=1}^n \rho} \right) = \left( \frac{ \sum_{i=1}^n \cdot X_i }{n}, \frac{ \sum_{i=1}^n Y_i }{n} \right) = (\bar{X}, \bar{Y}) $$ where the bar designates the arithmetic mean.

For other problems the geometric or harmonic mean (exists, if all values are positve) might be more useful.

All three are related: $$ \bar{x}_H \le \bar{x}_G \le \bar{x}_A $$