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Chemistry 223 – Appendix 1
1
APPENDIX 1: LEAST SQUARES AND STATISTICS
I.A. Least Squares.
In this course we will use a very simple version of a widely used algorithm for finding the “best
fit” parameters to describe experimental data by a theoretical model. The approach is called
linear least squares and consists of minimizing an error sum between the experimental data and
the theoretical model. Furthermore, this approach is applicable not just to data which can be
modeled as a straight line, but more generally to any data which can be described as
linear in the
parameters
. For example, the generic equation for a straight line is,
y=mx + b
where the slope,
m
, and intercept,
b
, can be calculated from the following equations:
m
=
N
x
i
y
i
"
x
i
y
i
#
#
#
( )
N
x
i
2
"
#
x
i
#
( )
2
$
%
’
(
)
(1)
b
=
y
i
x
i
2
"
x
i
y
i
x
i
#
#
#
#
( )
N
x
i
2
"
x
i
#
( )
2
#
(2)
r
=
N
x
i
y
i
"
x
i
y
i
#
#
#
N
x
i
2
"
x
i
#
( )
2
#
$
%
’
(
)
1/2
N
y
i
2
"
y
i
#
( )
2
#
$
%
’
(
)
1/2
(3)
where
N
= number of data points,
x
i
= individual x values,
y
i
= individual y values, and
r
2
=
squared correlation coefficient. The
r
2
value gives an indication of how well the data fit the
calculated line, with a perfect fit given by r
2
= l.00000.
.. The results of these calculations can
be compared with values of
m
and
b
obtained graphically.
The generic linear least squares approach is also applicable to data models like the third order
polynomial,
y
=
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
where each of the parameters to be fit, the
a
i
, appears as a linear term in the equation, even
though the independent variable,
x
, clearly appears nonlinearly.
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2
I.B. Sample calculation.
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This note was uploaded on 01/06/2011 for the course CHEM 223 taught by Professor Scheeline during the Fall '08 term at University of Illinois at Urbana–Champaign.
 Fall '08
 SCHEELINE

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