Illustration - why median gives shortest sum of distance?

So while LeetCoding, I encountered problems like 296. Best Meeting Point where you use the fact that median gives you the shortest sum of distance from other nodes to it. And I am going to show your why, without boring math equations.

Graph time

Suppose we have a 1D axis, and a bunch of points on it: (Sorry about the M$ Paint quality)

We randomly pick a point x on this axis to see if this is the best point with the minimal sum of distance to other points. And lets say there are m points to the left of x, and n points to the right of x.

Let's try to minimize the sum

What would happen to the sum of distance if we move x to the right by 1 unit?

Well, since we are 1 unit further away from all of the m points on the left, the sum increase by m. And since we are 1 unit closer to all of the n points on the right, the sum also decrease by n. In other words: new_sum = old_sum + m - n.

Apparently, if there are more points on the right than these on the left, we should keep moving to the right to minimize the sum.

And we've reached median

But when do we stop moving? Thats where m == n, i.e, there are equal number of points on either left or right. Because m - n == 0 and we can't decrease the sum any further. And this point, is the median (because thats the definition of median: The median is the value separating the higher half from the lower half of a data sample)