Calculating the central point with minimal average distance to other points I work at an office with colleagues coming from all over the country. Our office is quite centrally located, but some colleagues have to travel quite a lot further than others. I often wondered how I could calculate a central point which minimizes the average traveling distance for each employee (traveling as the crow flies). So if we're ever going to relocate, that would be the ideal spot; we could all save on time and fuel.
Look at this image:

Here we can see that the employees live in different places. With 2 employees, it's rather simple. Let's pick John and Pete. The point halfway between John and Pete would be the perfect spot for them. But how about when we include a 3rd person or an n'th? I'm kind of lost there.
Bonus points for explaining it in a way an average but not expert mathematician understands. :)

The average distance is the sum of distances divided by the number of colleagues. Since the latter is fixed, you can as well ask for the point which minimizes the sum. Which, by the way, indicates that for two employees the situation would be not as simple as you make it to be, since any point on the connecting line will satisfy the requirement.
Taking the terms “point minimize distance sum” to Wikipedia, you can find that such a point is called the geometric median of your set of employee locations. If you continue reading, you will find that computing it might be tricky, but there has been work on the subject.
