Geoscience Reference
In-Depth Information
The
Critical Content
,
a
, is the proportion of a critical
component that is to be estimated. The critical component
of a lot L is denoted a
L
, the critical content of a sample S is
denoted a
S
, etc.
non-zero, so bias can be introduced to the sampling pro-
cedure.
Accidental errors that occur during sampling or prepara-
tion cannot be analysed using statistical methods as they are
non-random events. Prevention of accidental errors is crucial
for reliable sampling, and have little to do with Sampling
Theory, more to do with good sampling practices.
Since the errors are random variables and are independent
the following relationships are true:
Weight of a critical component in the lot L
Critical content a
=
L
Weight all components in the lot L
Heterogeneity
is the condition of a lot where not all ele-
ments are identical. There are two types of heterogeneity that
we are concerned with: a) the constitution heterogeneity and
b) the distribution heterogeneity.
The
Constitution Heterogeneity
,
CH
, represents the dif-
ferences between the composition of each fragment within
the lot. Contributing factors are the fragment shape, size,
density, chemical composition, mineralogical composition,
etc. Constitution heterogeneity generates the fundamental
sampling error.
The
Distribution Heterogeneity
,
DH
, represents the dif-
ferences from one group to another within the lot. Contribut-
ing factors are the constitution heterogeneity, spatial distri-
bution, shape of the lot due to gravity, etc.
The
Sampling Protocol
is a set of steps for sample
taking and preparation meant to minimize errors and to
provide a representative sample that is within acceptable
standards.
• Total sampling error:
TE FEDEEE
=++
…
• Average error:
Ε
{TE}
=Ε
{FE}
+Ε
{DE}
+Ε
{EE}
+
…
•
Total error variance:
σ
2
{TE}
=
σ
2
{FE}
+
σ
2
{DE}
+
σ
2
{EE}
+
…
Thus, individual errors do not cancel out, but are compound-
ed. This compounding effect emphasises the care and atten-
tion that sampling requires.
When the mean of the sampling error, E{SE}, approaches
zero the sample is accurate, or non-biased. A sampling selec-
tion is said to be precise when the variance of the sampling
error, σ
2
(SE), is less than the standard required for a given
purpose. It is not related to the sample average or accuracy
of the sample. Accuracy and precision, the two measures of
sample quality, can be combined, leading to the notion of
representativeness:
5.2.2
Error Basics and Their Effects
on Sample Results
Errors may be introduced during the stages required for sam-
pling and sample preparation. They can be random with a
mean of zero, random with a non-zero mean, or accidental
(ocassional or non-systematic).
The “fundamental error (FE)” is the only error that can-
not be eliminated using proper sampling procedures. It will
be present even if the sampling operation is perfect. Funda-
mental error is a function of the constitution heterogeneity
of the material being sampled and it can be quantified before
sampling. The errors it generates are random with a mean
of zero.
The “increment delimitation error (DE)” and “increment
extraction error (EE)” are random errors but their mean is
typically non-zero. Unlike the fundamental error delimita-
tion and extraction error can be eliminated through proper
sampling procedures.
Delimitation error occurs when the shape of the volume
for the increment extracted is not correct; for example not
taking the entire cross section of a conveyor belt. Extrac-
tion error occurs when all of the fragments that belong in
the correctly delimited volume for the increment do not end
up in that volume. The mean of these errors is typically
r
2
{SE}
=
m {SE}
2
+ s
2
{SE}
£
r
2
0
When the mean square of the sampling error, r
2
(SE), is less
than a standard threshold,
r,
the sample is considered
representative.
5.2.3
Heterogeneity and the Fundamental Error
Sampling theory differentiates between two types of hetero-
geneity: (1) the distribution heterogeneity and (2) the consti-
tution heterogeneity.
The constitution heterogeneity could be considered in
two different ways: the heterogeneity between the fragments
making up a sample, or the heterogeneity within the frag-
ments of the sample. For our purposes, and sampling in gen-
eral, the heterogeneity between the fragments is more con-
sequential.
The constitution heterogeneity is defined based on the
number of fragments within the lot and still requiring the
