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A complicated theory (and
advanced understanding) behind
simple steps
Use tabulated information!
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View Full Document Table 2. Critical values of Student’s
t
. The number of degrees of
freedom
ν
= n 1
Degrees
of
freedom
t
at a confidence level of
90%
95%
99%
1
6.314
12.706
63.657
2
2.920
4.303
9.925
3
2.353
3.182
5.841
4
2.132
2.776
4.604
5
2.015
2.571
4.032
6
1.943
2.447
3.707
7
1.895
2.365
3.500
8
1.860
2.306
3.355
9
1.833
2.262
3.250
10
1.812
2.228
3.169
15
1.753
2.131
2.947
20
1.725
2.086
2.845
25
1.708
2.068
2.787
30
1.697
2.042
2.750
40
1.684
2.021
2.704
60
1.671
2.000
2.660
∞
1.645
1.960
2.576
The number of degrees of
freedom
ν
= n 1
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View Full Document LINEAR REGRESSION (variables: y and x)
A line will be
the best straight line
drawn through the data (
x
i
,y
i
)
when the sum of squares of the vertical deviations between the
data points and the line (squares of
“residuals”) is minimized.
y depends on x!
Data scatter around the “best straight
line”?
A set of (
x
i
,
y
i
) data (data pairs) yields a straight line fit:
y = mx + c
where m is the slope, and
c
is the intercept.
We will first look at the distance vs. time and next electrolysis
mass vs. time plots
.
Harris pages: 83 – 95; use of a scientific calculator, or Excel
spreadsheet function “linest”.
“The Method of Least Squares” and “Calibration Curves”.
One minimizes the expression:
y
i
y
i
i
i n

=
=
∑
2
1
assuming the linear fit through individual points i is:
c
i
mx
+
=
i
y
ˆ
The form for differentiation is (by substitution):
y
i
mx
i
c
i
i n


=
=
∑
2
1
the differentiation is performed with respect to m and c.
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View Full Document
the leastsquares method
of the data
treatment.
The product is “the best straight line”.
How good is the best straight line is expressed by the
coefficient
of determination,
r
2
(the square of the correlation coefficient)
.
The closer to unity is
r
2
the better the fit.
In practice,
r
2
smaller than 0.9 makes the fitting procedure to a
straight line unreliable.
A higher order fit should then be
considered (binomial or polynomial).
A set of equations resulting from bringing the derivatives to
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This note was uploaded on 12/28/2010 for the course CHEMISTRY 222 taught by Professor Wieckowski during the Spring '10 term at University of Illinois at Urbana–Champaign.
 Spring '10
 WIECKOWSKI

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