Geoscience Reference
In-Depth Information
If the experiment of counting 100 points would be repeated many times, the
number of times (
K
) that the mineral A is counted would describe a binomial
distribution with mean 12 and standard deviation 3.25. According to the central-
limit theorem to be discussed in Sect.
2.3.1
, a binomial distribution approaches
a normal distribution when
n
increases. Hence it already can be said for a single
experiment that
P
(
¼
95 %. The resulting value of
K
is between 5.6 and 18.4 with a probability
of 95 %. Precision can be increased by counting more points. For example, if
n
μ
˃ <
K
<μ
˃
¼
0.95 or
P
(
5.
6
<
K
<
1.96
+1.96
)
18.4)
¼1,000, then
μ
¼120 and
˃
¼10.3, and
P
(99.8
<
K
<
140.2) ¼95 %. This result
also can be written as
k
¼120
20.2. In order to compare the experiment of
counting 1,000 points to that for 100 points, the result must be divided by
10, giving
k
0
¼
2.02. This number represents the estimate of volume percent
for the mineral A. It is 3.25/(10.3/10)
12
3.2 times as precise as the first estimate
that was based on 100 points only. This increase in precision is in agreement
wit
h
the basic statistical result that if
X
is the mean of a sample of size
n
,
˃
¼
2
X
¼
˃
2
X
ðÞ=
n
. Since ten times as many points were counted, the result has
become 10
0.5
3.2 times more precise.
Chayes (
1956
) discusses petrographic modal analysis in more detail. A second
example of usefulness of the binomial distribution model is taken from quantitative
stratigraphy. In general, most biostratigraphic correlation is based on biozonations
derived from range charts using observed oldest and youngest occurrences of
microfossil taxa. For example, in exploratory drilling for hydrocarbon deposits in
a sedimentary basin, a sequence of borehole samples along a well drilled in the
stratigraphically downward direction is systematically checked for first occurrences
of new species. When the samples are cuttings taken at a regular interval, there is
the possibility that younger material drops down the well so that highest or last
occurrences cannot be observed in that situation. The probability of detecting a
species in a single sample depends primarily on its abundance. As a measure,
relative abundance (to be written as
p
) of a species in a population of microfossils
is commonly used. Together with sample size (
n
),
p
specifies the binomial distri-
bution that
k
microfossils of the taxon will be observed in a single sample. If
p
is
very small, the binomial probability can be approximated by the probability of the
Poisson distribution:
¼
k
e
ʻ
ʻ
PK
ð
¼
k
Þ
¼
ð
k
¼
0, 1,
...
,
n
Þ
k
!
2
(
K
)
which is determined by its single parameter
. The Poisson
distribution can be derived from the binomial distribution by keeping
ʻ
with
E
(
K
)
¼
˃
¼
ʻ
np
constant and letting
n
tend to infinity while
p
tends to zero. Like many other
frequency distribution models, the approach can be extended to more than a single
variable.
ʻ
¼
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