Practical applications of sampling theory
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Practical applications of sampling theory

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of the materials
could be improved. The producers of the RMs usually
carry out homogeneity tests that provide data that could
be reported in a compressed form as sampling constants,
but unfortunately at the moment, these data are not fully
Below, some examples are given on how Gy\u2019s funda-
mental sampling error model can be used in practice to
design and audit analytical procedures. Some further
examples can be found in Ref. [8]. As mentioned above,
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 87
If the material to be sampled consists of particles
having different shapes and size distributions, it is difficult
to estimate the number of critical particles in the sample.
Gy has derived an equation that can also be used in this
case to estimate the relative variance of the fundamental
sampling error:
r2r ¼ Cd3
\ufffd 1
\ufffd \ufffd
rr ¼ ra
¼ Relative standard deviation of the fundamental
sampling error ð4Þ
where ra, absolute standard deviation (in concentration
units); aL, average concentration of the lot; d, characteristic
particle size = 95% limit of the size distribution; MS,
sample size; ML, lot size; and C is the sampling constant
that depends on the properties of the material sampled.
C is the product of four parameters:
C ¼ fgbc ð5Þ
where f is the shape factor (see Fig. 2). Shape factor is the
Sampling Expert: Well, let\u2019s consider the problem. We
could use the Poisson distribution to estimate the required
sample size. Let\u2019s see:
The maximum relative standard deviation sr = 20%= 0.2.
From Eq. (2), we can estimate how many coarse particles
there should be in the sample to have this standard deviation
n ¼ 1
¼ 1
¼ 25
If 1 tonne contains 5 coarse particles, this result means that
the primary sample should be 5 tonne. This is a good
example of an impossible sampling problem. Although you
could take a 5-tonne sample, there is no feasible technology
to separate and count the coarse particles from it. You should
not try the traditional analytical approach in controlling the
quality of your product. Instead, if the specification is really
sensible, you forget the particle size analyzers and maintain
the quality of your product by process technological means;
that is, you take care that all the equipment are regularly
serviced and their high performance maintained so that the
product quality is always maintained.
Plant Manager: Thank you. In light of what you said, it
seems that the expensive laser diffraction particle size
analyzer recommended to us will not solve our problem.
3.2. Applications of Gy\u2019s fundamental sampling error
equation for designing sample preparation procedures
ratio of the volume of the sampled particles having the
characteristic dimension d to the volume of the cube
having the same dimension. For spheroidal particles
fc 0.5, which often can be used as the default value for
this parameter. g is the size distribution factor ( g = 0.25 for
wide-size distribution, and g = 1 for uniform particle sizes),
and b is the liberation factor (see Fig. 2). Liberation factor
is an empirical correction for materials, where the critical
particles are found as inclusions in the matrix particles.
Liberation size L is defined as the size of the opening of a
screen below which 95% of the material has to be crushed
in order to liberate at least 85% of the critical particles;
bmax = 1 (liberated materials and materials ground below
the liberation size L), bmin = 0.03 (materials where the
critical particles are very small in comparison to d; note
that because b is dependent on the particle size d for a
given material, the sampling constant C changes when the
material is ground or crushed). c is the constitution factor
and can be estimated by using Eq. (6) if the necessary
material properties are available.
c ¼
1\ufffd aL
\ufffd \ufffd2
qc þ 1\ufffd
\ufffd \ufffd
qm ð6Þ
Here, aL is the average concentration of the lot; a, the
concentration of the analyte in the critical particles; qc, the
density of the critical particles; and qm, the density of the
matrix or diluent particles.
If the material properties are not available and they are
difficult to estimate, the sampling constant C can always
be estimated experimentally. International reference mate-
rials (RMs), for example, are a special group of materials
for which the sampling constant should always be esti-
mated and reported. Unfortunately, this is seldom done. If
the particle size distribution and sampling constants were
Fig. 2. Estimation of particle shape factor and liberation factor for
unliberated and liberated critical particles. L is the particle size of the
critical particles.
the fundamental sampling error is the minimum theoretical
error achievable in a sampling step. Therefore, the funda-
mental sampling error calculations give realistic estimates
for the global sampling error, only if the material is well-
mixed before sampling, and all sampling and subsampling
contains as an average 0.05% of an enzyme powder that has
a density of 1.08 g/cm3. The size distribution of the enzyme
was available, and from it was estimated that the
characteristic particle size d = 1.00 mm and the size factor
g = 0.5. Estimate the fundamental sampling error for the
following analytical procedure.
The actual concentration of a 25-kg bag is estimated by
taking first a 500-g sample from it. This material is ground
to particle size \ufffd 0.5 mm. Then, the enzyme is extracted
from a 2-g sample by using a proper solvent, and the
concentration is determined by using liquid chromatogra-
phy. The relative standard deviation of the chromatographic
measurement is 5%.
To estimate the errors of the two sampling steps, we have
the following material properties:
The total relative standard deviation can now be
estimated by applying the rule of propagation of errors:
st ¼
¼ 0:143 ¼ 14:3%
The largest error is generated in preparing the 2-g
sample for the extraction of the enzyme. To improve the
overall precision, this step should first be modified. The
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139488
These values give for constitution factor (Eq. (6)) the value
c = 2160 g/cm3, and for the sampling constants (Eq. (5)), C
C1 ¼ 540 g=cm3 and C2 ¼ 270 g=cm3
Eq. (3) gives now the standard deviation estimates for
sampling steps.
M1 = 500 g
ML1 = 25000 g
d1 = 0.1 cm
g1 = 0.5
aL= 0.05%
a= 100%
qc = 1.08 g/cm
qm= 0.67 g/cm
f = 0.5 . . .default value for spheroidal particles
b= 1 . . .liberated particles
M2 = 2.0 g . . .sample sizes
ML2 = 500 g . . .lot sizes
d2 = 0.05 cm . . .particle sizes
g2 = 0.25 . . .estimated size distribution factors
sr1 = 0.033 = 3.3% . . .primary sample
sr2 = 0.13 = 13% . . .secondary sample
procedures are carried out with equipment and methods
that follow the rules of sampling correctness defined in
Gy\u2019s sampling theory. Therefore, if large lots are sampled,
the uncertainty of the primary samples has to be estimated
in a different way, e.g., by using Gy\u2019s variographic
Example 2. A certain cattle feed (density = 0.67 g/cm3)
sr3 = 0.05 = 5% . . .analytical determination
recommendation from this exercise is that either a larger
sample should be used for the extraction, or the primary
sample should be pulverized to a finer particle size before
secondary sampling, whichever is more economic in
Example 3. Evaluate the feasibility of the following
procedure for calibrating an IR spectrometer for the
determination of quartz in mineral mixtures. To prepare
the calibration standards, pure minerals (d= 1 mm) were
ground individually for 2 min in a swing mill. Then 30
mg\u20132.95 g of each mineral was carefully weighed to
obtain the designed composition. The material was
carefully mixed for 3 min in a Retsch Spectro Mill, and
20 mg of the mineral mixture was carefully weighed into
4.98 g of KBr and mixed for 3