What Are Other Specific Points to Analyze in a 2D Coordinate Set?

[onako]

Given a set of 2D coordinates (real numbers, involves positives and negatives), I could calculate the (weighted) barycenter by simply using the logic with plain numbers. For the barycenter calculations, I sum all the values with respect to x-axis and y-axis separately, and then divide with the number of coordinates (in weighted case, the weight coordinate product is incorporated).
However, I wonder what other interesting points I might analyse. For median value calculation, I would have to select the middle one. But, in 2D world, do I select the median coordinate (if such a thing exists) based on median x-axis value and median y-axis value (usually not the same coordinate)?

Also, there are other specific points I might use based on "mean variations", but I wonder which of those could be translated to 2D world. For example, harmonic and geometric mean are related to the set of positive numbers. What would be the way to incorporate the ideas into the 2D world?

The question in the Probability and Statistics subforum, meaning that calculating other specific points, based on different measures, is an option I might want to consider. Any suggestions on which measure to consider is welcome. These might incorporate the value (coordinate) spread, multiple occurrences of (nearly) same coordinates, emphasizing coordinates "on the border"...
Thanks.

 

[onako]

To simplify; the barycenter of the coordinates is the point that minimizes the sum of squared distances; also, geometric median is the point that minimizes the sum of distances.
I'm interested about other specific points in 2D space with certain characteristic (in above examples, these would be "minimizing the sum of distances").