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# Practical applications of sampling theory

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```barrels
for shipment. For the analysis, these 50-g pieces are taken
as primary samples, and from them a 1-g sample is
dissolved for the cobalt determination. The cost of one
analytical determination is 12 o, that of taking one
primary sample (50-g piece from a given plate) is 2 o.
The standard deviations have been estimated as between-
plates standard deviation s1 = 35 g/tonne, within-plate
standard deviation (standard deviation of 50-g pieces taken
from a single plate) s2 = 15 g/tonne, and the standard
deviation between 1-g samples taken from a single 50-g
piece, s3 = 3.3 g/tonne. Optimize the analytical procedure
so that the standard deviation of the lot mean does not
exceed the value of 5 g/tonne.. Sampling is carried out at
three different error-generating levels. Following values
apply to these:
level 1: N1 = 25000 kg/50 kg = 500, s1 = 35 g/tonne,
c1 = 0 o, n1=?
level 2: N2 = 50 kg/50 g = 1000, s2 = 15 g/tonne, c2 =
2 o, n2=?
level 3: N3 = 50 g/1 g = 50, s3 = 3.3 g/tonne, c3 = 12o,
n3=?
At level 1, the unit cost is practically zero. At this
level, the only sampling procedure is the decision made
from which plates the 50-g primary samples should be
taken.
Solution. Eq. 10 gives the results: n3 = 0.09, by apply-
ing the constraints we have to select n3 = 1; n2 = 0, we have
to select n2 = 1.
For the level 1, the following results are obtained: n1=
53.3 (Eq. (12a)) or n1= 58.4 (Eq. (12b)). The approximate
solution slightly overestimates the required number of
plates to be sampled, but as long as the size of the sample
at any level is, say 10% or less from the lot size, the
difference is small.
In this case, the following sampling protocol could be
used.
A 50-g piece is taken from every ninth cathode plate
at the packing stage. This gives a total of 55 samples
per a lot of 25 tonne. From each 50-g piece, one cobalt
x¯ i ni
Total cost of the investigation in general case is
ct ¼
Xk
i¼1
nici ð16aÞ
Usually in practice, the costs of sampling and analysis are
independent from the strata from which the samples are
taken and this equation simplifies to
ct ¼ ntc*; if c1 ¼ c2 ¼; . . . ;¼ ck ¼ c* ð16bÞ
Optimization of investigation involves the optimal allo-
cation of the total number of samples that can be analyzed
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 91
determination is made. The total cost of this inspection
scheme is 55(0 o+ 2 o + 12 o) = 770 o. The ex-
pected standard deviation of the lot mean (from Eq. (8a))
is 4.9 g/tonne.
Fig. 5 shows the Operation Characteristics curve of this
inspection scheme. It shows both the producer\u2019s and buyer\u2019s
risk by using this inspection scheme.
4.2. Optimization of sampling plan, when the lot consists of
strata of different sizes and heterogeneities
Sometimes the lot to be investigated consists of well-
defined strata of different sizes and heterogeneities; for
example, when the batch to be processed is prepared by
mixing raw materials of different qualities to achieve the
required average composition. This kind of a lot is shown in
Fig. 6.
Fig. 5. Operation Characteristics of the optimized inspection protocol for
cobalt determination. In this figure x-axis shows the true mean value of the
lot, and y-axis, the probability that the mean value of the inspection exceeds
the specification 150 g/tonne (producer\u2019s risk) or the probability that the
inspection value is below the specification (buyer\u2019s risk).
The lot consists of k different strata. Quantities needed to
design a cost optimal sampling plan are
Wi ¼ MLiP
MLi
¼ Relative size of the stratum i ð14Þ
where MLi, sizes of strata (e.g., as mass or volume; i = 1,2,
. . ., k); Ni, relative size of stratum i expressed as the
number of potential samples that could be taken from the
strata =Ni=MLi/MSi; MSi is the size of samples taken from
stratum i; ri, standard deviation of one sample taken from
stratum i; ci, cost of one sample analyzed from stratum i;
ct, total cost of the estimation of the grand mean of the lot;
ni, number of samples taken and analyzed from stratum i; nt,
total number of samples analyzed =Sni; xij, analytical results
on samples from stratum i (i = 1,2,. . .,k; j = 1,2, . . ., ni);
x¯i ¼
Pni
j
xij
ni
=Mean of stratum i; x¯¯ ¼Pki¼1Wix¯i= grand mean
of the lot.
Variance of the lot mean
r2x¯¯ ¼
X
W 2i
Ni \ufffd ni
Ni \ufffd 1
r2i
ni
ð15aÞ
If the samples taken are small in comparison to the stratum
size (as is usually the case), this equation simplifies to
r2¯c
X
W 2
r2i ; if in all strata nibNi and NiH1 ð15bÞ
Fig. 6. Lot consisting of k strata of different sizes, and the quantities needed
to optimize the sampling plan.
between the strata in an optimal way. Mathematical opti-
Example 5 (Optimal design for estimating sulfur balance of
a pulp mill). Estimation of sulfur balance for a pulp mill is a
difficult task Fig. 7. Sulfur enters the mill in raw materials
(water, wood) and in chemicals. The outflowing streams
consist of products, wastewater, solid wastes, and atmo-
spheric emissions. Initial calculations showed that the mean
values of all other streams, except the emissions into
atmosphere, could be estimated reliably. Atmospheric
emissions comprised about one quarter of the total sulfur.
Estimation of atmospheric sulfur emissions in an old pulp
mill, like the one where this study was carried out, is
difficult and therefore optimization of the emission
measurement plan is a challenging task. This is due to the
fact that there is a large number of gaseous outlets into
atmosphere. These have different mass-flows, concentra-
tions of the sulfur compounds are highly variable and sulfur
is found in dust and in many gaseous compounds [SO2,
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139492
mum can again be derived by assuming that ni is continu-
ous. The results given below have been derived assuming
that investigation costs are independent from strata (Eq.
(16b) valid). Optimization strategy depends on how much
information is available. If only the sizes of strata are known
and the total cost ct of the investigation is fixed, then the
best strategy is to allocate the samples proportionally to the
sizes of strata:
ni ¼ Wint ð17Þ
and
nt ¼ ct
c*
ð18Þ
Both nt and ni have to be rounded to integers so that the total
cost will not be exceeded.
If the unit costs and standard deviations are available,
then even better plans can be designed. Laboratories usually
follow their costs and, consequently, good cost estimates are
available. The standard deviations can be estimated either
by using Gy\u2019s sampling theory, if the material properties
needed are available, or experimentally from a pilot study. If
the quality control of the analytical laboratory is well
planned, it also provides data that can be used for optimi-
zation of sampling and analytical procedures.
Cost optimal plan for the investigation can be derived in
two ways. Either the total cost of the investigation is fixed
and the variance of the grand mean of the lot is minimized
or the target value for the variance is given and the total cost
is minimized. By assuming that Eqs. (15b) and (16b) are
valid, that is, in all strata the samples are small in compar-
ison to the sizes of strata, and the cost of investigation is
independent of strata, the following results can be derived.
4.2.1. Maximum value, cmax, given to the total cost,
variance of the lot mean minimized
ni ¼ WiriXk
i¼1
Wiri
cmax
c*
ð19Þ
Here, ni has to be rounded to integers so that the target cost
is not exceeded.
4.2.2. Target value, rt, given to the standard deviation of the
lot mean, total cost minimized
ni ¼ Wirir2T
Xk
i¼1
Wiri ð20Þ
Again, ni has to be rounded to integers so that the required
standard deviation of the lot mean will not be exceeded,
i.e., r2x¯¯Vr
2
T.
H2S, CH3S, (CH3)2S, and (CH3)2S2]. In optimization, all
different emission sources and sulfur-containing compounds
analyzed```