Math 211
Introduction to Statistics
Chapter 6
Curve Fitting and the Method of Least Squares
•
Relationship between variables
Example 1.
Weights of adults depend on some degree on their heights.
The circumferences of circles depend on their radii.
The pressure of a given mass of gas depends on its temperature and volume.
This relation can be expressed in mathematical form by determining an equation that connects the
variables.
•
Curve Fitting
To determine an equation,
Step1
.
Collect data that shows corresponding values of the variables
(for example; x denotes the height and denotes the weight.)
Then choose a sample of N individuals
N
x
x
x
,...,
,
2
1
, and
.
,...,
,
2
1
N
y
y
y
Step2.
plot points
(
29
(
29
(
29
N
N
y
x
y
x
y
x
,
,
......
,
,
2
,
2
1
on a rectangular coordinate system.
The resulting figure is called ‘
SCATTER DIAGRAM’.
From the scatter diagram, it is often possible to see a curve approximating data. Such a curve is
called ‘
APPROXIMATE CURVE’
.
In general, the problem of finding equations of approximating curves that fit given set of data is
called ‘
CURVE FITTING’.
.
Sonuc Zorlu
Lecture Notes
1
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Math 211
Introduction to Statistics
•
Equations of Approximating Curves
Straight line
:
X
a
a
Y
1
0
+
=
For any two points
(
29
1
1
,
y
x
and
(
29
2
2
,
y
x
, the equation of the straight line is
(
29
.
1
1
2
1
2
1
x
X
x
x
y
y
y
Y



=

Parabola or quadratic curve
:
2
2
1
0
X
a
X
a
a
Y
+
+
=
ndegree curve
:
N
N
X
a
X
a
X
a
a
Y
+
+
+
+
=
......
2
2
1
0
•
The method of Least Squares
Definition.
{Best fitting curve}
Of all curves approximating a given set of data points, the curve
having the property that
2
2
2
2
1
.....
N
D
D
D
+
+
+
is minimum is called a ‘
BEST FITTING CURVE
’ .
•
The Least Squares Line
The least squares line approximating the set of points
(
29
1
1
,
y
x
,
(
29
2
2
,
y
x
,….,
(
29
N
N
y
x
,
has the
equation
X
a
a
Y
1
0
+
=
,
REGRESSION LINE of
Y
on
X
, where the constants
0
a
and
1
a
are
determined by solving the equation,
∑
∑
∑
∑
∑
+
=
+
=
2
1
0
1
0
X
a
X
a
XY
X
a
N
a
Y
(X is independent variable).
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 Spring '11
 YucelTANDOGDU
 Regression Analysis, xy, squares line, Sonuc Zorlu

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