ISO 12743:2006

Copper, lead, zinc and nickel concentrates - Sampling procedures for determination of metal and moisture content

ISO 12743:2006 sets out the basic methods for sampling copper, lead, zinc and nickel concentrates from moving streams and stationary lots, including stopped-belt sampling, to provide samples for chemical analysis, physical testing and determination of moisture content, in accordance with the relevant International Standards.

Concentrés de cuivre, de plomb, de zinc et de nickel — Procédures d'échantillonnage pour la détermination de la teneur en métal et de l'humidité

General Information

• Status: Withdrawn
• Publication Date: 18-Jun-2006
• Withdrawal Date: 18-Jun-2006
• ICS: 73.060.99 - Other metalliferous minerals
• Technical Committee: ISO/TC 183 - Copper, lead, zinc and nickel ores and concentrates
• Drafting Committee: ISO/TC 183/WG 9 - Sampling of copper, lead and zinc concentrates
• Current Stage: 9599 - Withdrawal of International Standard
• Start Date: 08-Jun-2018
• Completion Date: 13-Dec-2025

Relations

• Revised: ISO 12743:2018 - Copper, lead, zinc and nickel concentrates - Sampling procedures for determination of metal and moisture content
• Effective Date: 04-Nov-2015
• Revises: ISO 12743:1998 - Copper, lead and zinc sulfide concentrates - Sampling procedures for determination of metal and moisture content
• Effective Date: 15-Apr-2008

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 12743
Second edition
2006-06-15
Copper, lead, zinc and nickel
concentrates — Sampling procedures
for determination of metal and moisture
content
Concentrés de cuivre, de plomb, de zinc et de nickel — Procédures
d'échantillonnage pour la détermination de la teneur en métal et de
l'humidité
Reference number
©
ISO 2006
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©  ISO 2006
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ii © ISO 2006 – All rights reserved

Contents Page
Foreword. vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions. 2
4 Sampling theory. 4
4.1 General. 4
4.2 Total variance. 5
4.3 Sampling-stage method of estimating sampling and total variance. 6
4.4 Simplified method of estimating sampling and total variance . 9
4.5 Interleaved sample method of measuring total variance . 10
5 Establishing a sampling scheme . 12
6 Mass of increment . 17
6.1 General. 17
6.2 Mass of increment for falling-stream samplers. 17
6.3 Mass of increment for cross-belt samplers . 17
6.4 Mass of increment for manual sampling from stationary lots . 18
6.4.1 Primary increments . 18
6.4.2 Mass of secondary and subsequent increments . 18
6.5 Mass of increment for stopped-belt reference sampling. 18
7 Methods of sampling from concentrate streams . 18
7.1 General. 18
7.2 Mass-basis systematic sampling. 18
7.2.1 General. 18
7.2.2 Sampling interval. 19
7.2.3 Sample cutter . 19
7.2.4 Taking of primary increments . 19
7.2.5 Constitution of subsamples and lot samples . 19
7.2.6 Types of division. 20
7.2.7 Division of increments . 20
7.2.8 Division of subsamples. 20
7.2.9 Division of lot samples. 20
7.3 Time-basis systematic sampling. 21
7.3.1 General. 21
7.3.2 Sampling interval. 21
7.3.3 Sample cutter . 21
7.3.4 Taking of primary increments . 22
7.3.5 Constitution of subsamples and lot samples . 22
7.3.6 Types of division. 22
7.3.7 Division of increments and subsamples. 22
7.3.8 Division of lot samples. 22
7.4 Stratified random sampling . 22
7.4.1 Fixed mass intervals. 22
7.4.2 Fixed time intervals . 23
8 Mechanical sampling of concentrate streams. 23
8.1 General. 23
8.2 Design of the sampling system. 23
8.2.1 Safety of operators . 23
8.2.2 Location of sample cutters . 23
8.2.3 Provision for interleaved sampling . 23
8.2.4 Provision for stratified random sampling. 24
8.2.5 Checking precision and bias . 24
8.2.6 Avoiding bias. 24
8.2.7 Minimizing bias . 24
8.2.8 Configuration of the sampling system . 24
8.3 Sample cutters. 24
8.3.1 General . 24
8.3.2 Design criteria . 25
8.3.3 Cutter speed . 26
8.4 Mass of increments. 27
8.5 Number of increments . 27
8.6 Sampling interval . 27
8.7 Routine checking . 27
9 Manual sampling of concentrate streams . 28
9.1 General . 28
9.2 Choosing the sampling location. 28
9.3 Sampling implements . 28
9.4 Mass of increments. 28
9.5 Number of increments . 28
9.6 Sampling interval . 28
9.7 Sampling procedures . 29
9.7.1 General . 29
9.7.2 Full stream cut from a falling stream . 29
9.7.3 Partial stream cuts from a falling stream . 29
9.7.4 Sampling from moving conveyor belts.30
10 Stopped-belt reference sampling . 30
11 Sampling from grabs . 31
11.1 General . 31
11.2 Mass of primary increments . 31
11.3 Number of primary increments. 31
11.4 Method of sampling . 31
11.5 Constitution of subsamples and lot samples . 31
12 Sampling from trucks, railway wagons and sampling hoppers. 32
12.1 General . 32
12.2 Mass of primary increments . 32
12.3 Number of primary increments. 32
12.4 Method of sampling . 32
12.5 Constitution of subsamples and lot samples . 32
13 Sampling of concentrate in bags or drums. 35
13.1 General . 35
13.2 Mass of primary increments . 35
13.3 Number of primary increments. 35
13.4 Method of sampling . 36
13.4.1 General . 36
13.4.2 Sampling during filling or emptying . 36
13.4.3 Spear sampling . 36
13.5 Constitution of subsamples and lot samples . 36
14 Sampling of stockpiles . 37
15 Methods of comminution, mixing and division. 37
15.1 General . 37
15.2 Comminution . 37
15.2.1 General . 37
15.2.2 Mills . 37
15.3 Mixing. 38
15.3.1 General . 38
iv © ISO 2006 – All rights reserved

15.3.2 Methods of mixing . 38
15.4 Division . 40
15.4.1 Chemical analysis samples . 40
15.4.2 Moisture samples. 40
15.4.3 Number of increments for division . 40
15.4.4 Minimum mass of divided sample . 40
15.4.5 Rotary sample division. 41
15.4.6 Cutter-type division . 42
15.4.7 Manual increment division. 43
15.4.8 Spear division . 43
15.4.9 Fractional shovelling. 44
15.4.10 Ribbon division. 45
15.4.11 Riffle division . 47
16 Sample requirements . 49
16.1 Moisture samples. 49
16.1.1 Mass of test portion. 49
16.1.2 Processing of samples. 49
16.2 Chemical analysis samples . 49
16.3 Physical test samples. 50
17 Packing and marking of samples. 50
Annex A (normative) Sampling stage method of estimating sampling and total variance . 51
Annex B (informative) Estimation of total variance — Barge unloading using a grab . 58
Annex C (informative) Mechanical sample cutters . 62
Annex D (informative) Checklist for mechanical sampling systems . 67
Annex E (normative) Manual sampling devices. 71
Annex F (informative) Apparatus for manual sampling of concentrates from stopped belts. 73
Annex G (informative) Sampling of stockpiles. 74
Annex H (normative) Increment division scoops for conducting manual increment division . 76
Bibliography . 77

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 12743 was prepared by Technical Committee ISO/TC 183, Copper, lead, zinc and nickel ores and
concentrates.
This second edition cancels and replaces the first edition (ISO 12743:1998), which has been technically
revised. The principal change is the extension of the scope to cover the sampling of nickel concentrates.

vi © ISO 2006 – All rights reserved

INTERNATIONAL STANDARD ISO 12743:2006(E)

Copper, lead, zinc and nickel concentrates — Sampling
procedures for determination of metal and moisture content
WARNING — This International Standard may involve hazardous materials, operations and
equipment. It is the responsibility of the user of this International Standard to establish
appropriate health and safety practices and determine the applicability of regulatory
limitations prior to use.
1 Scope
This International Standard sets out the basic methods for sampling copper, lead, zinc and nickel
concentrates from moving streams and stationary lots, including stopped-belt sampling, to provide samples for
chemical analysis, physical testing and determination of moisture content, in accordance with the relevant
International Standards. Where the concentrates are susceptible to significant oxidation or decomposition, it is
necessary to use a common sample for moisture determination and chemical analysis to eliminate bias (see
ISO 10251). In such cases, the common sample must be sufficiently representative, i.e. unbiased and
sufficiently precise, for chemical analysis and determination of moisture content. Any large agglomerates
(> 10 mm) present in the primary sample should be crushed prior to further sample processing. Sampling of
concentrates in slurry form is specifically excluded from this International Standard.
Stopped-belt sampling is the reference method for collecting concentrate samples against which mechanical
and manual-sampling procedures may be compared. Sampling from moving streams is the preferred method.
Both falling-stream and cross-belt samplers are described.
Sampling from stationary lots is used only where sampling from moving streams is not possible. The
procedures described in this International Standard, for sampling from stationary lots, only minimize some of
the systematic sampling errors.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 10251, Copper, lead, zinc and nickel concentrates — Determination of mass loss of bulk material on
drying
ISO 12744, Copper, lead, zinc and nickel concentrates — Experimental methods for checking the precision of
sampling
ISO 13292, Copper, lead, zinc and nickel concentrates — Experimental methods for checking the bias of
sampling
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.1
representative sample
quantity of concentrate representing a larger mass of concentrate with both precision and bias within
acceptable limits
3.2
lot
quantity of concentrate to be sampled
3.3
lot sample
quantity of concentrate representative of the lot
3.4
sub-lot
subdivided parts of a lot which are processed separately, each of them producing a subsample which is
analysed separately, e.g. for moisture determination
3.5
subsample
quantity of concentrate representative of the sub-lot
3.6
sampling
sequence of operations aimed at obtaining a sample representative of a lot
NOTE It comprises a series of sampling stages, each stage usually comprising operations of selection and
preparation
3.7
selection
operation by which a smaller quantity of concentrate is taken from a larger quantity of concentrate
3.8
increment
quantity of concentrate selected by a sampling device in one operation
3.9
increment selection
selection process that consists of extracting from the lot, or from an intermediate sample, successive
increments which can be combined to constitute a sample
3.10
division
operation of decreasing sample mass, without change of particle size, where a representative part of the
sample is retained
3.11
constant-mass division
method of division in which the retained portions from individual increments or subsamples are of uniform
mass
2 © ISO 2006 – All rights reserved

3.12
proportional division
method of division in which the retained portions from individual increments or subsamples are a constant
proportion of their original mass
3.13
preparation
nonselective operation without division such as sample transfer, drying, comminution or homogenization
3.14
sample processing
whole sequence of selection and preparation operations which transforms a stage i sample into a test sample
3.15
comminution
operation of reducing particle size by crushing, grinding or pulverisation
3.16
stage i sample
sample obtained at the ith stage of the sampling scheme
3.17
moisture sample
representative quantity of concentrate from which test portions are taken for moisture determination
NOTE Alternatively, the whole moisture sample may be dried to determine its moisture content
3.18
laboratory sample
sample that is processed so that it can be sent to the laboratory and used for further processing and selection
of one or more test samples for analysis
3.19
common sample
representative quantity of concentrate which is dried to determine its mass loss and subsequently used for
further processing and selection of one or more test samples for chemical analysis
3.20
test sample
representative quantity of concentrate obtained from a laboratory sample when additional preparation, such as
drying or hygroscopic moisture determination, is needed prior to the selection of one or more test portions
3.21
test portion
representative quantity of concentrate taken from a moisture sample, a laboratory sample or a test sample
which is submitted to moisture determination or analysis in its entirety
3.22
systematic sampling
selection of increments in which the concentrate being sampled is divided into equal strata and the first
increment is taken at random within the first stratum, the interval between subsequent increments being equal
to the stratum size
3.23
stratified random sampling
selection of increments in which the concentrate being sampled is divided into equal strata, each increment
being taken at random within each stratum
3.24
homogenisation
preparation operation which reduces the distribution heterogeneity of the concentrate
3.25
agglomerate
cluster of particles that are held together by chemical or physical phenomena
3.26
nominal top size
aperture size of a test sieve that retains 5 % of the mass of concentrate
3.27
moisture determination
quantitative measurement of the mass loss of the moisture test portion under the conditions of drying specified
in ISO 10251
3.28
chemical analysis
quantitative determination of the required chemical constituents of the analysis test portion
3.29
error
in any quantitative measurement, the difference between the true value and the value obtained for an
individual measurement
3.30
bias
statistically significant difference between the mean of the test results and an accepted reference value
See also ISO 13292.
3.31
precision
closeness of agreement between independent test results obtained under stipulated conditions
See also ISO 12744.
3.32
interleaved samples
samples constituted by placing consecutive primary increments alternately into two separate sample
containers
4 Sampling theory
4.1 General
The basic rule for a correct sampling method is that all possible increments from the concentrate stream or
stratum shall have the same probability of being selected and appearing in the sample. Any deviation from this
basic requirement can result in a bias. An incorrect sampling scheme cannot be relied on to provide
representative samples.
Sampling should preferably be carried out on a systematic basis, either on a mass basis (see 7.2) or on a time
basis (see 7.3), but only where it can be shown that no systematic error (or bias) could be introduced due to
any periodic variation in quality or quantity that may coincide with, or approximate to, any multiples of the
proposed sampling interval. In such cases, it is recommended that stratified random sampling within fixed time
or mass intervals be carried out (see 7.4).
4 © ISO 2006 – All rights reserved

The methods for sampling, including sample processing, depend on the final choice of the sampling scheme
and on the steps necessary to minimize possible systematic errors. The aim is always to reduce the total
variance to an acceptable level, while at the same time eliminating any significant biases, e.g. minimizing
degradation of samples used for determination of size distribution.
Moisture samples shall be processed as soon as possible and test portions shall be weighed immediately. If
this is not possible, samples shall be stored in impervious airtight containers with a minimum of free air space
to minimize any change in moisture content, but should be prepared without delay.
4.2 Total variance
The general aim of a sampling scheme is to provide one or several test portions, sufficiently representative of
a lot, for determination of the quality characteristics of the lot. The total variance of the final result, denoted by
s , consists of the variance of sampling (including sample processing) plus the variance of analysis (chemical
T
analysis, moisture determination, determination of particle size distribution, etc.) as follows:
22 2
s=+ss …(1)
TS A
where
s is the sampling variance (including sample processing);
S
s is the analytical variance.
A
In Equation 1, the sampling variance includes the variances due to all sampling (and sample processing)
steps, except selection of the test portion. The variance due to selection of the test portion is included in the
analytical variance, s , which is determined in accordance with ISO 12744, because it is difficult to determine
A
separately the “true” analytical variance.
Often replicate analyses of quality characteristics are carried out, which reduces the total variance. In this
case, if r replicate analyses are made:
s
22 A
ss=+ …(2)
TS
r
The estimation or measurement of the total variance can be carried out in several ways, depending on the
purpose of the exercise. In many respects, the different approaches are complementary.
[3, 4]
The first method, which was developed by Gy , is to break up the sampling variance into its components
for each sampling stage (see Annex A). The total variance is then given by:
s
22 2 2 A
ss=+ .+s+ .+s + …(3)
T
SS S
11iu−
r
where
s is the sampling variance for stage 1, i.e. the primary sampling variance;
S
s is the sampling variance for stage i;
S
i
s is the sampling variance for stage u − 1, the second last stage;
S
u−1
u is the number of sampling stages, stage u corresponding to selection of the test portion.
This is referred to as the “sampling stage” method (see 4.3) and provides very detailed information on the
variance components, which is particularly useful for designing and assessing sampling schemes. However,
to obtain maximum benefit, it is necessary to collect data at each sampling stage.
The second method, called the “simplified” method (see 4.4), is to break up the total variance into primary
sampling, sample processing and analytical variances only as follows:
s
22 2 A
ss=+s+ …(4)
TP
S
r
where
s is the primary sampling variance;
S
s is the variance due to all subsequent sampling steps, i.e. sample processing, except selection of the
P
test portion;
s is the analytical variance, including selection of the test portion (at stage u in Equation 3).
A
The primary sampling variance is identical to the sampling variance for stage 1 in Equation 3, while s is
P
equal to the total sampling variance for the remaining sampling stages, except for selection of the test portion
which is included in the analytical variance. The relative magnitudes of the variance components in Equation 4
indicate where additional effort is required to reduce the total variance. However, it is not possible to separate the
variances of the separate sample-processing stages. This method is suitable for estimating the total variance
for new sampling schemes based on the same sample-processing procedures, where the numbers of primary
increments, sample processings and analyses are varied.
Finally, the total variance s can be estimated experimentally by collecting interpenetrating duplicate samples
T
(see 4.5). This is called the “interleaved sample” method and gives valuable information on the total variance
actually achieved for a given sampling scheme with no extra effort, provided facilities are available for collecting
[5]
duplicate samples (Merks ). It gives no information on variance components, but the total variance can be
compared with the analytical variance to ascertain whether the sampling scheme used was optimized or not. It
is therefore of limited use for designing sampling schemes, but it can be used to monitor whether a sampling
scheme is in control.
4.3 Sampling-stage method of estimating sampling and total variance
The sampling variance for stage i is given by (see Annex A):
s
b
i
s = …(5)
S
i
n
i
where
s is the variance between increments for stage i;
b
i
n is the number of increments for stage i.
i
The variance between increments for stage i, s , can be estimated using the following equation:
b
i
6 © ISO 2006 – All rights reserved

n
()xx−
j
∑
j=1
s=− s …(6)
PA
b
i
n −1
i
where
x is the test result for increment j;
j
x is the mean test result for all increments;
s is the variance of subsequent sample processing and analysis.
PA
The variance of subsequent sample processing and analysis of each increment, s , has been taken into
PA
account in Equation 6 to obtain an unbiased estimate of s .
b
i
NOTE Care is needed in subtracting variances. The difference is significant only when the F ratio of the variances
being subtracted is statistically significant.
Remembering that the variance due to selection of the test portion is included in the analytical variance s ,
A
the total sampling variance is given by:
u−1
s
b
2 i
s = …(7)
S
∑
n
i
i=1
Combining Equations 2 and 7 gives the total variance s as follows:
T
u−1 2
s
b s
2 i
A
s=+ …(8)
T
∑
nr
i
i=1
For a three-stage sampling scheme (including selection of the test portion), Equation 8 reduces to:
2 2
s s 2
s
b b
i 2 A
s=+ + …(9)
T
nn r
The best way of reducing the value of s to an acceptable level is to reduce the largest terms in Equation 8 first.
T
Clearly s / n for a given sampling stage can be reduced by increasing the number of increments n or
i i
b
i
reducing s by homogenizing the concentrate prior to sampling. The last term can be reduced by reducing the
b
i
particle size prior to selection of the test portion, or performing replicate analyses. Selecting the optimum number
of increments n for each sampling stage may require several iterations to obtain the required total variance
i
s .
T
EXAMPLE Consider a four-stage sampling scheme for determining the metal content of a copper concentrate
containing 31,2 % Cu. Assume that the concentrate is being conveyed at 500 t/h on a conveyor belt, that the lot size is
500 t, and that the following parameters have been determined using Equation 6 where appropriate:
s = 0,3 % Cu
b
s = 0,2 % Cu
b
s = 0,1 % Cu
b
s = 0,05 % Cu
A
NOTE Many measurements may be required to obtain good estimates of s , s , s and s .
b b b A
1 2 3
Stage 1
Assume that the primary cutter takes increments of 12 kg mass at 2 min intervals. Thus:
n = 30
Primary sample mass = 360 kg
Equation 5 gives:
s = (0,3) /30 = 0,003 0
S
Stage 2
The primary increments are collected in a hopper, and then fed to the secondary cutter at a rate of 360 kg/h.
Secondary increments of 0,01 kg are taken at 30 s intervals. Thus:
n = 120
Divided sample mass = 1,2 kg
s = (0,2) /120 = 0,000 333
S
Stage 3
The 1,2 kg sample is transported to the sample-processing laboratory and fed through a rotary sample divider
−1
with a sample-collection canister divided into 8 equal sectors rotating at 30 rev/min (0,5 s ). Sample division
takes 2 min. Thus:
n = 60
Divided sample mass = 150 g
s = (0,1) /60 = 0,000 167
S
Stage 4
Dry the sample and then pulverize it to 150 µm. Select a 1 g test portion, by taking 10 increments of 0,1 g
with a spatula, and conduct a single analysis. Thus:
s = 0,05 % Cu
A
Total variance
The total variance is given by:
22 2 2 2
s=+ss+s+s
TA
SS S
12 3
= 0,003 0 + 0,000 333 + 0,000 167 + 0,002 5
= 0,006
8 © ISO 2006 – All rights reserved

Hence:
s = 0,077 % Cu
T
In this example, the largest components of variance are due to primary sampling and analysis. Consequently,
the total variance can be reduced by increasing the number of primary increments and conducting replicate
analyses.
An example of the application of the sampling-stage method of estimating total variance to sampling from
grabs is given in Annex B.
4.4 Simplified method of estimating sampling and total variance
While it is not possible to partition, i.e. separate, the variances of the individual sample-processing stages, the
simplified method is suitable for estimating the total variance for new sampling schemes based on the same
sample-processing procedures, where the numbers of primary increments, sample processings and analyses
are varied.
Using Equation 5, the primary sampling variance s is given by:
S
s
b
2 1
s = …(10)
S
n
where
n is the number of primary increments;
s is the variance between primary increments determined using Equation 6.
b
The primary sampling variance can be reduced by increasing the number of primary increments n .
2 2
The sample-processing variance s and analytical variance s are determined experimentally by duplicate
P A
sample processing and determination of quality characteristics in accordance with ISO 12744. The analytical
variance s can also be obtained by carrying out duplicate analyses on test samples.
A
Multiple sample processings and analyses are often carried out to reduce the total variance. In this case,
combining Equations 4 and 10 gives the following.
a) Where a single sample is constituted for the lot and r replicate analyses are carried out on the test
sample:
s
b s
1 A
ss=++ …(11)
TP
nr
b) Where the lot is divided into k sub-lots, a subsample is constituted for each sub-lot, and r replicate
analyses are carried out on each resultant test sample:
s
b s s
1 PA
s=+ + …(12)
T
nk rk
c) Where sample processing and analysis is carried out on each increment taken from the lot and r replicate
analyses are carried out:
s
22 A
ss++
P
b
2 1
r
s = …(13)
T
n
EXAMPLE Assume that 50 primary increments are taken from a zinc concentrate lot that has been divided into two
sub-lots. The resultant two subsamples are processed separately and analysed in duplicate. Assume that the primary
increment, sample processing and analytical standard deviations have been determined experimentally as follows:
s = 0,3 % Zn
b
s = 0,1 % Zn
b
s = 0,05 % Zn
b
Using Equation 12, the total variance is given by:
2 2 2
s = (0,3) /50 + (0,1) /2 + (0,05) /(2 × 2)
T
= 0,001 8 + 0,005 0 + 0,000 625
= 0,007 43
Hence:
s = 0,086 % Zn
T
In this example, the major component of variance is sample processing. This component could be reduced by
dividing the lot into a larger number of sub-lots, and constituting a subsample for each sub-lot.
4.5 Interleaved sample method of measuring total variance
The total variance s achieved for a given sampling operation can be estimated experimentally by collecting
T
interleaved duplicate samples as shown in Figure 1. If the number of primary increments for routine sampling
is n , then 2n primary increments are taken from each lot and the odd- and even-numbered increments are
1 1
separately combined to give samples A and B for the lot. Samples A and B are then separately submitted to
sample processing and analysis. This procedure is repeated until sampling has been completed. The total
variance for a single lot is then given by:
N
⎡⎤
⎢⎥xx−
∑AB
ii
⎢⎥
π
i=1
s = …(14)
⎢⎥
T
4 N
⎢⎥
⎢⎥
⎣⎦
where
x and x are the analyses for each pair of samples A and
...