Height (m)
1·4
1·5
1·6
1·7
1·8
Weight (kg)
50
55
60
65
70
75
80
85
Time (hours)
1
2
3
4
5
6
Concentration
2
4
6
8
10
12
Regression:
Finding the equation of the line of best fit
Objectives:
To find the equation of the least squares regression line of y on x.
Background and general principle
The aim of regression is to calculate the
equation of the line of best fit on a scatter
graph.
Consider the scatter graph on the right.
One
possible line of best fit has been drawn on
the diagram.
Some of the points lie above
the line and some lie below it.
The vertical distance each point is above or
below the line has been added to the
diagram.
These distances are called
deviations
or
errors
– they are symbolised as
n
d
d
d
,...,
,
2
1
.
When drawing in a regression line, the aim is to make the line fit the points as closely as possible.
We
do this by making the
total of the squares of the deviations as small as possible
,
i.e. we minimise
2
i
d
∑
.
If a line of best fit is found using this principle, it is called the
leastsquares
regression
line
.
Example 1:
A patient is given a drip feed containing a particular chemical and its concentration in his blood is
measured, in suitable units, at one hour intervals.
The doctors believe that a linear relationship will
exist between the variables.
Time,
x
(hours)
0
1
2
3
4
5
6
Concentration,
y
2.4
4.3
5.0
6.9
9.1
11.4
13.5
We can plot these data on a scatter graph –
time would be plotted on the horizontal axis
(as it is the independent variable).
Time is
here referred to as a
controlled
variable
,
since the experimenter fixed the value of this
variable in advance (measurements were taken
every hour).
Concentration is the dependent variable as the
concentration in the blood is likely to vary
according to time.
The doctor may wish to estimate the
concentration of the chemical in the blood after 3.5 hours.
She could do this by finding the equation of the line of best fit.
There is a formula which gives the equation of the line of best fit.
ε
1
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The equation of the line is
bx
a
y
+
=
where
xx
xy
S
S
b
=
and
x
b
y
a

=
.
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 Spring '11
 brown
 Least Squares, Linear Regression, Regression Analysis, regression line

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