Geoscience Reference
In-Depth Information
and
n
∂
∂
c
e
i
0
=
i
=
1
t
i
)
2
n
∂
∂
=
(
mx
i
+
c
−
c
i
=
1
n
=
2(
mx
i
+
c
−
t
i
)
(A4.5)
i
=
1
Rearranging Eqs. (A4.4) and (A4.5)gives
n
n
n
x
i
(A4.6)
x
i
t
i
=
m
+
c
x
i
i
=
1
i
=
1
i
=
1
and
n
n
t
i
=
m
x
i
+
nc
(A4.7)
i
=
1
i
=
1
Equations (A4.6) and (A4.7) are simultaneous equations, which are solved to give
m
and
c
:
n
i
=
1
x
i
t
i
−
i
=
1
x
i
i
=
1
t
i
n
i
=
1
x
i
−
i
=
1
x
i
2
m
=
(A4.8)
i
=
1
t
i
i
=
1
x
i
−
i
=
1
x
i
i
=
1
x
i
t
i
n
i
=
1
x
i
−
i
=
1
x
i
2
c
=
(A4.9)
The standard errors in these values of
m
and
c
are
m
and
c
(these are
one-standard-deviation, 1
σ
,errors), given by
n
i
=
1
e
i
(
n
−
2)
n
i
=
1
x
i
−
i
=
1
x
i
2
m
)
2
(
=
(A4.10)
and
i
=
1
x
i
i
=
1
e
i
(
n
−
2)
n
i
=
1
x
i
−
i
=
1
x
i
2
c
)
2
(
=
(A4.11)
Equations (A4.8)-(A4.11) can easily be programmed. Two-standard-deviation, 2
σ
,errors are
generally quoted in geochronology.
The least-squares method can be applied also to curve fitting in exactly the same way as is
shown here for straight lines. However, it becomes more difficult to solve the simultaneous
equations when more than two coefficients need to be determined.
