of 10
Chapter
12
Statistics
and
other
damned
lies
The
remark
attributed
to
Disraeli
would
often
apply
with
justice
and
force:
'There
are
three
kinds
of
lies:
lies,
damned
lies,
and
statistics'.
Mark
Twain
['lies
, damn
lies
- and
statistics'
(sic)
-
usuall
y
attributed
to
Twain
(a
lie)
-or
to
Disraeli
(a
damn
lie),
as
Twain
took
trouble
to
do;
Lord
Blake,
Disraeli's
biographer,
thinks
that
this
is
most
unlikely
(statistics)].
Overview
Semantics,
rhetoric
,
logic
and
assumptions
play
a
large
role
in
science
but
are
usually
relegated
to
specialized
books
and
courses
on
philosophy
and
paradigms
.
Conventional
wisdom
is
often
the
controlling
factor
in
picking
and
solving
prob-
lem
.s.
This
chapter
is
a
detour
into
issues
that
may
be
holding
up
efforts
to
develop
a
theory
of
the E
arth,
one
that
is
as
paradox
free
as
possible.
A
large
part
of
petrology,
geochemistry
and
geophysics
is
about
sampling
the
Earth.
Sa
m-
pl
i
ng
t
heo
r y
is
a
branch
of
statistics
.
If
the
mantle
is
blobby
on
a
kilometer
scale
,
then
individual
rocks
will
exhibit
large
scatter
-
or
variance
-
but
the
mean
of
a
large
number
of
samples
will
eventually
settle
down
to
the
appro-
priate
mean
for
the
mantle;
the
same
mean
will
be
achieved
in
one
fell
swoop
by
a
large
vol-
cano
sampling
a
large
volume
of
the
same
man
-
tle
.
If
the
outputs
of
many
large
volcanoes
are
plotted
on
a
histogram
,
the
spread
will
be
much
smaller
than
from
the
rock
samples,
even
though
the
same
mantle
is
being
sampled
.
Midocean
ridges
sample
vast
volumes
of
the
mantle
and
mix
together
a
variety
of
components.
TI1ey
are
the
world's
largest
blenders.
Seamount
and
ocean
island
volcanoes
sample
much
smaller
volumes
.
This
is
a
situation
where
the
ce
nt
ra
l
l
imi
t
th
eo
r
em
(CLT)
applies.
The
central
limit
theorem
The
central
li
mi t
th
eorem
and
the
law
of
l
arge
numbers
state
that
variably
sized
samples
from
a
heterogenous
population
will
yield
the
same
mean
but
will
have
variances
that
decrease
as
n
(the
sample
size)
or
V
(the
volume
of
the
sampled
region)
incr
eases.
Small
sample
sizes
are
more
likely
to
have
extreme
values
than
samples
that
blend
components
from
a
large
volume
[C
LT
i
sotopes
].
One
expects
that
melt
inclusions
in
a lava
sample
will
have
greater
isotopic
diversity
than
that
sampled
in
whole
rocks.
An
amazing
thing
about
the
central
limit
the-
orem
is
that
no
matter
what
the
shape
of
the
original
distribution
-
bimodal,
uniform
, expo-
nential,
multiple
peaks
-
the
sampling
distribu-
tion
of
the
mean
approaches
a
normal
distri-
bution.
Furthermore,
a
normal
distribution
is
usually
approached
very
quicldy
as
n
increases;
144
STATISTICS
AND
OTHER
DAMNED
LIES
n
is
the
sample
size
for
each
mean
and
is
not
the
number
of
samples.
The
sample
size
is
the
number
of
measurements
that
goes
in
to
the
computation
of
each
mean.
In
the
case
of
sampling
by
a vol-
cano
of
a
heterogenous
mantle
,
the
volume
of
the
sample
space
is
equivalent
to
the
number
of
discrete
samples
that
are
averaged;
the
volcano
does
the
averaging
in
this
case.
Suppose
the
mantle
has
a
variety
of
discrete
components,
or
blobs
,
with
distinctive
isotope
ratios.
The
probability
density
function
has,
say,
five
peaks;
call
them
DM,
EM1,
EM2,
HIMU
and
Q
The
CLT
states
that
a
sufficiently
large
sam-
ple,
or
average,
from
this
population,
will
have
a
narrow
Gaussian
distribution
.
MORE
has
this
property;
it
has
nothing
to
do
with
the
homo-
geneity
of
the
mantle
or
convective
stirring.
MORE
is
best
viewed
as
an
average,
not
a
reservoir
or
component.
If
one
has
several
large
datasets
there
are
sta-
tistical
ways
to
see
if
they
are
drawn
from
the
same
population.
If
two
datasets
have
different
means
and
different
standard
deviations
they
may
still
be
drawn
from
the
same
population.
But
if
the
datasets
have
been
filtered,
trimmed
or
corrected,
then
statistics
cannot
be
applied.
Often
these
corrections
involve
some
hypothesis
about
what
the
data
should
look
like.
This
is a very
com-
mon
error
in
seismology
and
geoc
hemistry;
data
can
be
selected,
filtered
and
discarded
for
vari-
ous
reasons
but
there
are
formal
and
statistically
valid
ways
of
doing
this.
The
problems
of
scale
Volcanoes
sample
and
average
large
volumes
of
the
mantle
.
It
is
not
necessary
that
the
source
region
correspond
to
a
rock
type
familiar
in
hand-specimen
size.
The
'grains'
of
the
so
urc
e
may
be
kilometres
in
extent,
and
the
melt
from
one
source
region
may
rise
and
interact
with
melts
from
a
different
lithology,
or
smaller
degree
melts
from
the
same
lithology
.
The
prop-
erties
of
individual
crystals
are
greatly
washed
out
in
the
average.
The
hypothetical
rock
types-
pyrolite
and
piclogite
-
and
hypothetical
reser-
voirs
-
MORE
,
FOZO
-
may
only
exist
as
averages,
ra
t
her
than
as
distinct
10
em
by
10
em
'rocks'
that
might
be
found
in
a
rock
collection.
Magmas
average
over
a
large
volume
of
the
mantle,
just
as
river
sediments
can
average
over
a
large
area
of
a
continent.
Similarly,
seismic
waves
average
over
tens
to
hundreds
of
km
and
see
many
different
ldnds
of
lithologies;
they
do
not
see
the
extremes
in
properties
of
the
material
that
they
average.
Anisotropy
of
the
mantle
may
be
due
to
anisotropy
of
individual
mantle
minerals
or
due
to
large-scale
organized
heterogeneity.
A
1-second
P-wave
has
a
wavelength
of
about
10
km
and
therefore
does
not
usually
care
whether
it
is
microphysics
or
macrophysics
that
gives
the
fabric.
Similar
considerations
apply
to
seismic
anelasticity.
Geophysics
is
as
much
a
matter
of
composites,
and
averaging,
as
it
is
a
science
of
crystallography
and
mineral
physics.
The
prob-
lem
of
averaging
occurs
throughout
the
sciences
of
petrology,
geochemistry
and
geophysics.
Much
of
deep
Earth
science
is
about
unravelling
aver-
ages.
In
seismology
this
is
called
tomography.
The
scale
becomes
important
if
it
becomes
bi
g
enough
so
that
gravity
takes
over
from
diffu-
sion.
A
large
blob
behaves
differently,
over
time,
than
a
small
impurity,
be
ca
use
of
buoyancy
.
If
the
heterogeneities
are
large
, a
fertile
blob
can
be
confused
with
a
hot
plume.
Scrabble
statistics
The
central
limit
theorem
can
be
illustrated
with
the
ga
me
of
Scrabb
le,
which
has
a
well-
defined
distribution
function
of
letters
or
scores
(Figure
12.1).
Put
all
the
letter
tiles
into
a
can
and
draw
them
out
repeatedly.
Plot
the
number
of
times
each
letter
is
pulled
out
on
a
cumu-
lative
histogram.
The
histo
g
ram
is
very
ragged
;
this
is
what
the
population
-
the
world
of
let-
ters
-
looks
like.
Now
play
a
game
with
only
two-letter
com
bin
ations
-
words
-
allowed,
and
plot
the
average
scores.
Contin
u e
this
process
with
three-
and
four-
l
etter
words
(but
do
not
fil-
ter
out
the
offensive
words
or
words
that
do
not
make
sense).
The
histograms
get
smoother
and
smoother
and
narrower
and
narrower.
By
the
time
one
gets
to
three-
l
etter
words
one
a l
ready
has
nearly
a
Gaussian
distribution,
with
very
few
average
scores
of
4
or
5,
or
20
and
higher
.
If
this
SUMA-
SAMPLING
UPON
MELTING
AND
AVERAGING
145
1
Scrabble
game
18
-
average
by
ones
16
·
- -
average
by
two
s
14
-
average
by
threes
(f)
1-
12
z
::J
0
u
8
6
4
2
4
6
8
1
0
1 2
14
16
1 8
20
22 24
26
ABCDEFGHIJKLMNOPQRSTUVWXYZ
The
distribution
of
letters
,
and
their
assigned
numerical
values,
in
a
Scrabble
game.
Also
shown
are
average
values
for
two
and
three-letter
combinations,
illustrating
the
smoothing
effect
of
the
CLT
.
One
might
conclude
that
the
various
combinations
represent
different
bags.
or
reservoirs,
of
letters
.
were
a
distribution
of
normalized
helium
iso
-
topic
ratios
one
might
conclude
that
the
popula-
tion
that
was
being
sampled
by
four-letter
words
did
not
contain
the
extreme
values
that
one-
and
two-letter
words
were
sampling.
The
lower
curve
in
Figure
12.1
approximates
the
helium
isotope
histogram
for
MORB;
the
other
curves
look
more
like
OIB
statistics.
SU
M
A-
sam
pl
ing
upo
n
meltin
g
a
nd
averaging
In
dealing
with
the
mantle,
one
is
dealing
with
a
heterogenous
system
-
heterogenous
in
prop-
erties
and
processes.
In
geochemical
box
mod-
els
the
mantle
is
viewed
as
a
collection
of
large,
discrete
and
isolated
reservoirs,
with
definite
loca-
tions
and
compositions.
In
petrological
models
there
are
distinct
components
that
can
be
inti-
mately
mixed
.
The
samp
ling
process
usually
envis-
aged
is
similar
to
using
a
dipper
in
a
bucket
of
water.
In
models
based
on
sampling
theory
the
vari-
ous
products
of
mantle
differentiation
are
viewed
as
averages,
different
kind
of
samples
from
a
heterogenous
population;
a
homogenous
product
does
not
imply
a
homogenous
or
well-mixed
source.
In
the
real
world,
sampling
at
volcanoes
involves
melting,
recrystallization
and
incom
-
plete
extraction,
as
well
as
statistics.
Fast
spreading
oceanic
ridges
process
lar
ge
volumes
of
the
mantle
and
involve
large
degrees
of
melting.
A
consequence
of
the
central
limit
theorem
is
that
the
variance
of
samples
drawn
from
a
heterogenous
population
(reser-
voir)
depends
inversely
on
the
sampled
volume
(Anderson
,
2000b;
Meibom
&
Anderson,
2003).
The
homogeneity
of
a
sample
population
(e.g.
all
MORB
samples)
can
thus
simply
reflect
the
integration
effect
of
large
volume
sampling.
The
presumed
homogeneity
of
the
MORB
source
may
thus
be
an
illusion.
The
'
MORB
reservoir
' is
thought
to
be
homo
ge-
nous
because
some
isotopic
ratios
show
less
scatter
in
MORB
than
in
ocean-island
basalts
(OIB).
The
common
explanation
is
that
MORB
are
derived
from
a well-stirred,
convecting
part
of
the
mantle
while
OIB
are
derived
from
a
dif-
ferent,
deeper
reservoir
.
Alternatively
,
the
homo-
geneity
of
MORB
can
be
explained
as
a
consequence
of
the
sampling
process.
The
standard,
two-reservoir
model
of
geochem-
istry
is
reinforced
by
questionable
-
from
a
sta-
tistical
point
of
view
-
data
filtering
practices.
Samples
that
are
judged
to
be
contaminated
by
plumes
(i.e.
OIB-like
samples)
are
often
removed
from
the
dataset
prior
to
statistical
analysis.
Sometimes
the
definition
of
plume
influence
is
arbitrary.
For
example,
isotopic
ratios
that
exceed
an
arbitrary
cutoff
may
be
eliminated
from
the
dataset
.
In
this
way
,
the
MORB
dataset
is
forced
to
appear
more
homogenous
than
it
really
is.
This
method
is
commonly
applied
to
3
HerHe.
Despite
this
,
various
ridges
still
have
different
means
and
variances,
and
these
depend
on
spreading
rate
and
ridge
maturity.
The
upper
mantle,
sub-continental
lith
osphere,
and
lower
mantle
are
usually
treated
as
distinct
and
accessible
geochemical
reservoirs.
There
is
evidence,
however
, for
ubiquitous
small-
to
moderate-scale
heterogeneity
in
the
upper
mantle,
referred
to
as
the
statistical
upper
mantle
assem-
blage
(
SUMA)
.
This
heterogeneity
is
the
result
of
processes
such
as
inefficient
melt
extrac-
tion
and
long-term
plate
tectonic
recycling
of
sedimentary
and
crustal
components.
The
SUMA
146
STATISTICS
AND
OTHER
DAMN
ED
LIES
concept
derives
from
the
CLT
and
contrasts
with
the
idea
of
a
convective
ly
homogenized
MORB
mantle
reservoir,
and
different
reservoirs
for
OIB
where
homogenization
of
the
source
is
achieved
by
convective
stirring
and
mixing.
In
contrast,
in
the
SUMA
model,
the
isotopic
compositions
of
MORB
and
OIB
are
the
outcome
of
homog-
enization
during
sampling,
by
partial
melting
and
magma
mixing.
The
primary
homogeniza-
tion
process
sampling
upon
melting
and
averaging,
SUMA
,
does
not
require
the
participation
of
dis-
tinct
(e.g.
lower
mantle)
reservoirs
to
explain
OIB
compositions.
Statistical
distributions
of
litho-
logic
components
and
sampling
theory
replace
the
concept
of
distinct
,
isolated
geoc
hemical
reservoirs,
and
extensive
solid-state
convective
stirring
prior
to
sampling.
In
sampling
theory
terms,
SUMA
is
the
heterogenous
population
to
be
sampled
-
the
probability
density
function.
MORB
represents
a
large
scale
sample,
or
average,
from
this
population
;
near-ridge
seamounts
are
a
smaller-scale
sample;
grain
boundaries
, fluid-
inclusions
and
melt-inclusions
are
very
small-
scale
samples.
MORB
is
uniform
and
does
not
have
the
extremes
of
composition
b
eca
use
it
is
large-scale
average
of
the
sampled
mantle.
Math-
ematical
and
statistical
treatments
of
isotopic
heterogeneity
of
basalts
and
upper
mantle
assemblages
are
starting
to
replace
the
static
reservoir
and
convective
stir-
ring
concepts.
Bayesian
statistics
The
use
of
prior
probabilities
and
subjective
constraints
external
to
the
dataset
is
known
as
bayesian
statistics
[Harold
Jeffreys
baye-
sian
statistics].
Bayesian
reasoning
is
com-
mon
in
geophysical
inversion
problems
and
in
the
statistical
treatments
of
isotope
data.
For
example,
it
is
often
assumed
that
there
are
two
populations
in
isotopic
data
-
the
MORB
reser-
voir,
corresponding
to
the
'co
nvectin
g
de
gasse
d
upper
mantle',
and
the
OIB
reservoir,
assumed
to
be
an
isolated,
more
primitive,
less-degassed,
more
variable
reservoir
in
the
lower
mantle.
Data
are
corrected
or
filtered
to
remove
the
influ-
ence
of
contamination
or
pollution
by
materials
from
the
'wrong'
reservoir,
and
then
statistics
is
applied.
In
seismology,
a
prior
or
reference
model
is
adopted
and
perturbations
are
made
to
this
to
satisfY
new
datasets
.
Which
kind
of
statistical
approach
is
prefer-
able
in
these
situations?
Bayesian
meth-
ods
have
a
long
and
controversial
his-
tory.
Bayesian
reasoning
has
emerged
from
an
intuitive
to
a
formal
level
in
many
fields
of
science.
Subjective
probability
was
developed
to
quantifY
the
plausibility
of
events
under
circum-
stances
of
uncertainty.
Bayes'
theorem
is
a
natu-
ral
way
of
reasoning,
e.g.
www.ipac.caltech.edu/
level5/March01/Dagostini/Dagostini2.html.
There
is
an
apparent
contradiction
between
rigorous
normal
statistics
and
the
intuition
of
geologists.
Geologists'
intuition
resembles
the
less
familiar
bayesian
approach;
conclusions
should
not
deviate
too
much
from
prior
beliefs.
On
the
other
hand,
a
common
objection
to
bayesian
statistics
is
that
science
must
be
objec-
tive
-
there
is
no
room
for
belief
or
prejudice
.
Science
is
not
a
matter
of
religion.
But
scien-
tific
beliefs
and
assumptions
are
always
there,
but
often
well
hidden.
There
are
good
reasons
for
applying
bayesian
methods
to
geo
lo
g ical
problems
and
geochemi-
cal
datasets
. First
of
all,
geoc
hemical
data
are
often
ratios,
they
cannot
be
negative.
Conclu-
sions
should
not
depend
on
whether
a
ratio
or
its
inverse
is
analysed.
In
the
absence
of
infor-
mation
to
the
contrary
it
can
be
assumed
that
all
values
of
0
to
infinity
are
equally
probable
in
the
underlyin
g
distribution
(i.e.
the
magma
source-
or
mantle-
prior
to
sampling
and
averag-
in
g).
Ruling
out
negative
values
already
is
a
prior
co
nstraint.
Sampling
of
a
heterogenous
source,
according
to
the
central
limit
theorem
(CLT)
will
yield
a
peaked
distribution
that,
in
the
limit
of
a
large
sample
volume,
is
nOI'mal
or
lo
g
-n
ormal.
Man
y
geochemical
samples
can
be
considered
to
have
sampled
fairly
large
volumes.
These
consid-
erations
are
more
critical
for
3
He
j4
He
than
for
heavier
isotopes
since
the
spread
of
values
about
the
mean
is
lar
ger,
and
median
values
are
not
far
from
0 .
Similarly,
histograms
of
seismic
wave
travel-time
residuals,
which
ca
n
be
negative,
or
he
at
flow
values,
depend
on
the
distance
the
rays
travel
or
the
number
of
samples
averaged
used
to
characterize
a
heat
flow
province.
Large
volcanoes
and
global
tomography
average
large
volumes
of
the
mantle.
Distributions
are
commonly
asymmetric
and
skewed.
Medians
of
isotope
data
are
more
robust
measures
than
the
arithmetic
means,
with
which
they
commonly
disagree
.
Log-normal
distribu-
tions
are
more
appropriate
than
linear
Gaussian
distributions
for
many
geophysical
and
geochem-
ical
datasets.
These
are
relatively
mild
applica-
tions
of
bayesian
reasoning
.
Stronger
bayesian
priors
would
involve
placing
a
low
prior
probabil-
ity
on
certain
ranges
of
values
of
the
parameter
being
estimated,
or
on
external
parameters.
Basaltic
volcanism
is
by
nature
an
integrator
of
the
underlying
source.
All
volcanoes
average,
to
a
greater
or
lesser
extent,
the
underlying
het-
erogeneities.
To
determine
the
true
heterogeneity
of
the
mantle,
samples
from
a
large
variety
of
environments
are
required,
including
fast
and
slow
spreading
ridges,
small
off-axis
seamounts,
fracture
zones,
new
and
dying
ridges,
various
ridge
depths
,
overlapping
spreading
centers,
melt-starved
regions,
unstable
ridge
systems
such
as
back-arc
ridges,
and
so
on
.
These
regions
are
often
avoided,
as
being
anomalous.
Various
materials
enter
subduction
zones,
including
sed-
iments
,
altered
oceanic
crust
and
peridotite,
and
some
of
these
are
incorporated
into
the
upper
mantle
. To
the
extent
that
anomalous
mate-
rials
are
excluded,
or
anomalous
regions
left
unsampled
,
the
degree
of
true
intrinsic
het-
erogeneity
of
the
mantle
will
be
unknown.
In
essence,
one
must
sample
widely,
collecting
speci-
mens
that
represent
different
degrees
of
melting
and
different
source
volumes
.
The
data
can
be
weighted
or
given
low
probability
in
the
bayesian
approach.
The
main
distinguishing
feature
of
the
bayesian
approach
is
that
it
makes
use
of
more
information
than
the
standard
statistical
approach
.
Whereas
the
latter
is
based
on
analy-
sis
of
'hard
data
', i.e.
data
derived
from
a well-
defined
observation
process,
bayesian
statistics
accommodates
'prior
information'
which
is
usu-
ally
less
well
specified
and
may
even
be
subjec-
tive.
This
makes
bayesian
methods
potentially
more
powerful,
but
also
imposes
the
require-
ment
for
extra
care
in
their
use
.
In
particular,
FALLACIES
147
we
are
no
longer
approaching
an
analysis
in
an
'open-minded'
manner,
allowing
the
data
to
determine
the
result.
Instead,
we
input
'prior
information'
about
what
we
think
the
answer
is
before
we
analyse
the
data!
The
danger
of
subjec-
tive
bayesian
priors,
if
improperly
applied,
is
that
prior
beliefs
become
immune
to
data.
Fallacies
Logic,
argument
,
rhetoric
and
fallacies
are
branches
of
philosophy
;
science
started
out
as
a
branch
of
philosophy.
Scientific
truth
is
now
treated
differently
from
logical
truth,
or
mathemat-
ical
truth,
and
statistics.
But
in
the
search
for
sci-
entific
truth
it
is
important
not
to
make
logical
errors,
and
not
to
make
arguments
based
on
log-
ical
fallacies.
The
rules
of
logical
inference
form
the
foundation
of
good
science.
A
fallacy
is
an
error
in
reasoning
and
dif-
fers
from
a
factual
or
statistical
error
. A
fal-
lacy
is
an
'argument'
in
which
the
premises
given
for
the
conclusion
do
not
provide
the
necessary
support.
Many
of
the
paradoxes
in
Earth
science,
as
discussed
in
the
following
chapters,
are
the
result
of
poor
assumptions
or
erroneous
reasoning
(l
o
gic
a l
parad
ox
e s )
[mantleplumes
falla
c
ie
s
].
The
following
chapters
discuss
some
of
the
controversies
and
paradoxes
in
Earth
science
.
Some
of
these
can
be
traced
to
assumptions
and
fallacies.
The
following
are
a
few
examples
of
logical
fallacies;
Midocean-ridge
basalts
are
homogenous;
therefore
their
mantle
source
is
homogenous
;
therefore
it
is
vigorously
convecting.
therefore
it
is
well
stirred.
(There
are
four
logical
fallacies
in
that
sentence,
and
at
least
one
factual
error.)
Midocean-ridge
basalts
are
derived
fi·om
the
upp
e r
mantle
;
oce
a
n-island
basalts
are
not
midocean-ridge
basalts
;
they
therefore
are
derived
from
the
lower
mantle.
Seismic
velocities
and
density
decrease
with
temperature
;
therefore
regions
of
the
mantle
that
have
low
seismic
velocities
are
hotter
and
less
dense
than
other
regions
.
Some
other
well
known
fallacies
are
catego-
rized
below,
with
examples.
The
examples
are