Find the minimum sum of distances between sets of points to a straight line in a plane

Question

Given $n$ dots on a plane, such as: n couples ($x_i$,$y_i$)
I would like to find a line parallel to y-axis ( $x=b$ ), such that the sum of all of the point's distances from that line will be minimal

In order to do that, I need to write an alogrithem with a linear run-time ( $O(n)$ )

MY METHOD
I relate only for the $x$ values of each point as an element in an Array called $A$
So I used the Select(A,left, right, p) & Partition Alogorithm in order to find the median of medians of the array

pseudo-code:

Select(A,left,right,p)
 n<-right-left+1
 if n=1 then
  return A[left]
 m<-⌈n/5⌉
 let B array with length m
 for i <-1 to m
  B[i]=medianOf5(A,left,right,x)
 x<-Select(B,1,m, ⌈m/2⌉) 
 q <- partition(A,left,right,x)
 k <-q-left+1
 if p<k then
  return Select(A,left,q,p)
 if p>k 
  return Select(A,q,right,p)

But, using median of medians seems unnecessary
If so, is there an easier way to that? if not ( or if yes for that matter ) was my way correct?

Answer 1

Given a line $L = b$, the distance from any point $(x, y)$ to $L$ is $\left| y - b\right|$.

The sum of a distances for a set of points $S = \{(x_1, y_1), \dots, (x_n, y_n)\}$ is then $$\left| y_1 - b \right| + \cdots + \left| y_n - b \right|$$ which is minimized for which $b$?