Geoscience Reference
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Fig. 3.9 Standard deviation plotted against mean for copper in different rock types of the Muskox
intrusion. Two curves (linear and quadratic best ts) are also shown. The pattern in this diagram
indicates that, to a rst approximation, standard deviation is proportional to mean. Rock types with
olivine are:
DN
dunite,
PD
peridotite,
FPI
feldspatic peridotite,
TRPD
troctolitic peridotite,
PC
picrite,
CPX
olivine clinopyroxenite,
PCWB
picritic websterite,
OGB
olivine gabbro,
BGB
bronzite gabbro,
ANGB
anorthositic gabbro; and rock types without olivine are:
OPX
orthopyroxenite,
WB
websterite,
GB
gabbro,
GR-GB
granophyre-bearing gabbro,
GRGB
grano-
phyric gabbro,
MGR
mac granophyre,
GR
granophyre (Source: Agterberg
1974
, Fig. 29)
The Muskox intrusion has been sampled in detail both at the surface and by
drilling. Average and standard deviation for copper in 17 different rock types have
been plotted in Fig.
3.9
. There is positive correlation between these two statistical
parameters (
r
0.9002). A straight line and a parabola were tted by least squares,
taking the mean as the independent variable. To a rst approximation, the standard
deviation is proportional to the mean. This implies a constant coefcient of
variation. The plots of Fig.
3.9
suggest
¼
0.8.
From the basic properties of variance it follows that
ʳ
2
(
X
) where
a
and
b
are two constants. Further, if
f
(
x
) is a function of
x
that is approximately
linear in a range of variation of
x
, then, according to Taylor's formula:
f
(
x
)
2
(
a
+
bX
)
b
2
˃
¼
˃
f
(
c
)
c
)·
f
0
(
c
) where
c
is another constant. Consequently:
+(
x
n
2
2
f
0
2
fX
2
X
˃
ðfg
ðÞ
c
: ˃
ðÞ
Application to
f
(
X
)
¼
log
e
X
gives:
2
d
log
e
X
ð
ðÞ
Þ
2
2
2
X
˃
f
log
e
X
ðÞ
g
x
0
: ˃
ðÞ
dx
x
¼
where
x
0
is a constant value for
x
used in truncated Taylor series approximation.
Setting
x
0
¼
ʼ
(
X
) it is derived that
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