The Chemistry Maths Book, Second Edition

622 Chapter 21Probability and statistics

This is the example shown in Figure 21.10. The error bars correspond toσ 1 = 1 0.25so

that the linear least-squares fit in this case is

y 1 = 1 mx 1 + 1 c, where m 1 = 1 −0.1691 1 ± 1 0.0158, c 1 = 1 3.2214 1 ± 1 0.1704

0 Exercises 26, 27

Chi-square fitting

When the (known) errors in the values of yare not all the same, withσ

i

fory

i

, the

simple least-squares method is replaced by the more general weighted least-squares

method:

(21.57)

Each contribution to the sum is weighted by the factor 12 σ

i

2

, and this has the effect

of giving greater weight to the more precise values of y(for the smaller values ofσ

i

).

This situation arises, for example, when the precision with which the measurements

can be made varies over the range of yvalues, or when the data points come from

measurements on several different instruments. This fitting method reduces to

simple least squares when all theσ

i

are equal.

The minimization ofχ

2

for a straight-line fit gives the same equations as before,

(21.54) for mand cand (21.55) forσ

m

andσ

c

, except that the averages are replaced

by weighted averages, for example

(21.58)

and, in equations (21.55),

(21.59)

Fits to other types of function are obtained in the same way, but lead to more

complicated formulas. The method can also be generalized for more than two variables.

The justification of the use of equation (21.57) comes from a consideration of the

normal distribution (21.39) for random errors. If the errors iny

i

are random and if

f(x

i

)is the (unknown) truevalue of ywhenx 1 = 1 x

i

then

(21.60)

ρε

σ

εσ

()

i

i

e

ii

=

−

1

2

22

2

π

σ

σ

2

1

2

1

1

N

i

N

i

is replaced by

=

∑

















y

N

yy

y

i

N

i

i

N

i

i

==

















==

∑∑

1

11

2

is replaced by

σ

ii

N

i

=

∑

















1

2

1

σ

χ

σ

2

1

2

=

−;

















=

=

∑

i

N

ii

i

yfx()a

minimum