Question

The least absolute deviation line equation for the data in the table is m = 0.1 x + 2.9

X 1,6,0.5,4,7.5,12,10,11 y 3,2.5,8,1,1.5,9,13,4

What is the sum of the absolute deviations?

Answer 1

First find the absolute deviation for each coordinate.
(1,3), (6,2.5), (0.5,8), (4,1), (7.5,1.5), (12,9), (10,13), (11,4).
I will do the 1st coordinate: (1,3)

Least absolute deviation line equation: m(x)= 0.1 x + 2.9
If you plot all 8 coordinates onto a xy graph, m(x) is the line of best fit.
The distance from one coordinate's y-value from the line m(x) is the deviation.

Plug in x from coordinate (1,3) into the equation m(1):
At x = 1, m(1)= 0.1(1) + 2.9 = 3.0
This is (1,3) on the m(x) line.

Find the distance of coordinate (1,3) from (1,3):
Absolute Deviation = |m(1) - y| = |3 - 3| = 0

Repeat for 2nd coordinate: (6, 2.5)
m(6) = 0.1(6) + 2.9 = 3.5
(6,3.5) on the m(x) line.
Absolute Deviation: |m(6) - y| = |3.5 - 2.5| = 1.0

Repeat this again for the other six coordinates.
Add up the eight absolute deviations to find the sum.