Projects - Cengage Learning

C-28 Appendix C
Figure B and the accompanying calculations show two examples using this
definition. For the data point (x 1 , y 1 ) (4, 3), the result e 1 2 does represent
the vertical distance between the data point and the line. For the data point
(8, 2), the deviation is e 2 1, and the absolute value of this gives the vertical
distance between the data point and the line.
y
4
3
Data point (4, 3)
(8, 3)
1
ƒ=
2x-1
For the data point {⁄, › }=(4, 3):
e 1 =›- f (⁄)
1
=3-” (4)-1’ =2
2
Figure B
2
1
(4, 1)
Data point (8, 2)
2 4 6 8
x
For the data point {¤, fi }=(8, 2):
e 2 =fi-f (x 2 )
1
=2 _ ” (8) _ 1’ = _1
2
Now suppose we have a given data set and a line. One way to measure how
closely this line fits the data would be to simply add up all the deviations. The
problem with this, however, is that positive and negative deviations would tend
to cancel one another out. For this reason, we use the sum of the squares of the
deviations as the measure of how closely a line fits a data set. (The sum of the
absolute values of the deviations would be another alternative, but in calculus,
squares are easier to work with than absolute values.)
As an example, suppose we have the following data set:
x 1 2 3
y 2 4 5
or equivalently
(x 1 , y 1 ) (1, 2)
(x 2 , y 2 ) (2, 4)
(x 3 , y 3 ) (3, 5)
Definition
Line of Best Fit
Suppose we have a set of n data points (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ),..., (x n , y n ).
For any line, let E denote the sum of the squares of the deviations:
E e 2 1 e 2 2 e 2 3 p e 2 n
Then the line of best fit is that line for which E is as small as possible. (It can
be shown that there is only one such line.) The line of best fit is also known as
the least-squares line or the regression line.