ISO 11794:2017
(Main)Copper, lead, zinc and nickel concentrates — Sampling of slurries
Copper, lead, zinc and nickel concentrates — Sampling of slurries
ISO 11794:2017 sets out the basic methods for sampling particulate material that is mixed with a liquid, usually water, to form a slurry. In industry and in the mining and mineral processing literature, slurry is also referred to as pulp, but this term is not used in ISO 11794:2017. At very high ratios of fine particulate solids to liquids where material assumes a soft plastic form, the mixture is correctly termed as a paste. Sampling of pastes is not covered in ISO 11794:2017. The procedures described in ISO 11794:2017 apply to sampling of particulate materials that are transported in moving streams as slurries, but not pressurized slurries. These streams may fall freely or be confined in pipes, launders, flumes, sluices, spirals or similar channels. Sampling of slurries in stationary situations, such as a settled or even a well-stirred slurry in a holding vessel or dam, is not recommended and is not covered in ISO 11794:2017. ISO 11794:2017 describes procedures that are designed to provide samples representative of the slurry solids and particle-size distribution of the slurry under examination. After draining the slurry sample of fluid and measuring the fluid volume, damp samples of the contained particulate material in the slurry are available for drying (if required) and measurement of one or more characteristics in an unbiased manner and with a known degree of precision. The characteristics are measured by chemical analysis, physical testing or both. The sampling methods described are applicable to slurries that require inspection to verify compliance with product specifications, determination of the value of a characteristic as a basis for settlement between trading partners or estimation of a set of average characteristics and variances that describes a system or procedure. Provided that flow rates are not too high, the reference method against which other sampling procedures are compared is one where the entire stream is diverted into a vessel for a specified time or volume interval. This method corresponds to the stopped-belt method described in ISO 12743.
Concentrés de cuivre, de plomb, de zinc et de nickel — Échantillonnage des schlamms
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INTERNATIONAL ISO
STANDARD 11794
Second edition
2017-06
Copper, lead, zinc and nickel
concentrates — Sampling of slurries
Concentrés de cuivre, de plomb, de zinc et de nickel —
Échantillonnage des schlamms
Reference number
©
ISO 2017
© ISO 2017, Published in Switzerland
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ii © ISO 2017 – All rights reserved
Contents Page
Foreword .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 2
4 Principles of sampling slurries . 2
4.1 General . 2
4.2 Sampling errors . 3
4.2.1 General. 3
4.2.2 Preparation error, PE . 4
4.2.3 Delimitation and extraction errors, DE and EE . 4
4.2.4 Weighting error, WE . 6
4.2.5 Periodic quality-fluctuation error, QE .
3 6
4.3 Sampling and total variance . 6
4.3.1 Sampling variance. 6
4.3.2 Total variance . 6
4.3.3 Sampling-stage method of estimating sampling and total variance . 8
4.3.4 Simplified method of estimating sampling and total variance . 9
4.3.5 Interleaved sample method of measuring total variance .10
5 Establishing a sampling scheme .11
6 Minimization of bias and unbiased increment mass .16
6.1 Minimization of bias .16
6.2 Volume of increment for falling-stream samplers to avoid bias.17
7 Number of increments .17
7.1 General .17
7.2 Simplified method .18
8 Minimum mass of solids contained in lot and sub-lot samples .18
8.1 Minimum mass of solids in lot samples .18
8.2 Minimum mass of solids in sub-lot samples .18
8.3 Minimum mass of solids in lot and sub-lot samples after size reduction .18
9 Time-basis sampling .19
9.1 General .19
9.2 Sampling interval .19
9.3 Cutters .19
9.4 Taking of increments .19
9.5 Constitution of lot or sub-lot samples .19
9.6 Division of increments and sub-lot samples .20
9.7 Division of lot samples .20
9.8 Number of cuts for division .20
10 Stratified random sampling within fixed time intervals.20
11 Mechanical sampling from moving streams .21
11.1 General .21
11.2 Design of the sampling system .21
11.2.1 Safety of operators .21
11.2.2 Location of sample cutters .21
11.2.3 Provision for duplicate sampling .21
11.2.4 System for checking the precision and bias.21
11.2.5 Avoiding bias .21
11.3 Slurry sample cutters .22
11.3.1 General.22
11.3.2 Falling-stream cutters .22
11.3.3 Cutter velocities .23
11.4 Mass of solids in increments .23
11.5 Number of primary increments .23
11.6 Routine checking .23
12 Manual sampling from moving streams .23
12.1 General .23
12.2 Choosing the sampling location .24
12.3 Sampling implements .24
12.4 Mass of solids in increments .25
12.5 Number of primary increments .25
12.6 Sampling procedures .25
13 Sampling of stationary slurries .25
14 Sample preparation .25
14.1 General .25
14.2 Sample division.25
14.3 Sample grinding .26
14.4 Chemical analysis samples . .26
14.5 Physical test samples .26
15 Packing and marking of samples .26
Annex A (normative) Sampling-stage method for estimating sampling and total variance .27
Annex B (informative) Examples of correct slurry sampling devices .34
Annex C (informative) Examples of incorrect slurry sampling devices .37
Annex D (normative) Manual sampling implements .41
Bibliography .42
iv © ISO 2017 – All rights reserved
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
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/ patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO’s adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following
URL: w w w . i s o .org/ iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 183, Copper, lead, zinc and nickel ores and
concentrates.
This second edition cancels and replaces the 2010 edition (ISO 11794:2010), of which it constitutes
a minor revision. The main changes are the deletion of reference ISO 20212, which has not yet been
published, and the replacement of “should” with “shall” where the criteria and/or requirements are
mandatory.
INTERNATIONAL STANDARD ISO 11794:2017(E)
Copper, lead, zinc and nickel concentrates — Sampling of
slurries
WARNING — This document may involve hazardous materials, operations and equipment.
It is the responsibility of the user of this document to establish appropriate health and safety
practices and determine the applicability of any other limitations prior to use.
1 Scope
This document sets out the basic methods for sampling particulate material that is mixed with a liquid,
usually water, to form a slurry. In industry and in the mining and mineral processing literature, slurry is
also referred to as pulp, but this term is not used in this document. At very high ratios of fine particulate
solids to liquids where material assumes a soft plastic form, the mixture is correctly termed as a paste.
Sampling of pastes is not covered in this document.
The procedures described in this document apply to sampling of particulate materials that are
transported in moving streams as slurries, but not pressurized slurries. These streams may fall freely
or be confined in pipes, launders, flumes, sluices, spirals or similar channels. Sampling of slurries in
stationary situations, such as a settled or even a well-stirred slurry in a holding vessel or dam, is not
recommended and is not covered in this document.
This document describes procedures that are designed to provide samples representative of the slurry
solids and particle-size distribution of the slurry under examination. After draining the slurry sample of
fluid and measuring the fluid volume, damp samples of the contained particulate material in the slurry
are available for drying (if required) and measurement of one or more characteristics in an unbiased
manner and with a known degree of precision. The characteristics are measured by chemical analysis,
physical testing or both.
The sampling methods described are applicable to slurries that require inspection to verify compliance
with product specifications, determination of the value of a characteristic as a basis for settlement
between trading partners or estimation of a set of average characteristics and variances that describes
a system or procedure.
Provided that flow rates are not too high, the reference method against which other sampling
procedures are compared is one where the entire stream is diverted into a vessel for a specified time or
volume interval. This method corresponds to the stopped-belt method described in ISO 12743.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements 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 12743, Copper, lead, zinc and nickel concentrates — Sampling procedures for determination of metal
and moisture content
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 terms and definitions given in ISO 12743, ISO 12744 and
ISO 13292 apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http:// www .electropedia .org/
— ISO Online browsing platform: available at http:// www .iso .org/ obp
4 Principles of sampling slurries
4.1 General
In this document, a slurry is defined as “any fluid mixture of a solid of nominal top size < 1 mm that
is mixed with water, which is frequently used as a convenient form to handle solids in bulk”. Slurry
flows are found in many mineral processing plants, with the water and entrained solids mixture being
transported through the plant circuits by means of pumps and pipelines and under gravity in sluices,
flumes and launders. In a number of operations, ore is transported to the mill in slurry form, and in
others concentrates are transported long distances in slurry pipelines. Tailings from wet plants are also
discharged as slurries through pipelines to the tailings dam. In many of these operations, collection of
increments at selected sample points is required for evaluation of the particulate material in the slurry.
A lot sample is constituted from a set of unbiased primary increments from a lot. The sample container is
weighed immediately after collection and combination of increments to avoid water loss by evaporation
or spillage. Weighing is necessary to determine the percentage of solids by mass in the slurry sample.
The sample may then be filtered, dried and weighed. Alternatively, the sample may be sealed in plastic
bags after filtering for transport and drying at a later stage. The liquid removed during filtration should
be retained if it needs to be analysed.
Test samples are prepared from samples after filtering and drying. Test portions may then be taken
from the test sample and analysed using an appropriate and properly calibrated analytical method or
test procedure under prescribed conditions.
The objective of the measurement chain is to determine the characteristic of interest in an unbiased
manner with an acceptable and affordable degree of precision. The general sampling theory, which is
based on the additive property of variances, can be used to determine how the variances of sampling,
sample preparation and chemical analysis or physical testing propagate and hence determine the
total variance for the measurement chain. This sampling theory can also be used to optimize manual
sampling methods and mechanical sampling systems.
If a sampling scheme is to provide representative samples, all parts of the slurry in the lot must have
an equal opportunity of being selected and appearing in the sample for testing. Hence, slurries are to
be sampled in such a manner that all possible primary increments in the set into which the slurry can
be divided have the same probability of being selected. Any deviation from this basic requirement can
result in bias. A sampling scheme with incorrect selection techniques, i.e. with non-uniform selection
probabilities, cannot be relied upon to provide representative samples.
Sampling of slurries should preferably be carried out by systematic sampling on a time basis (see
Clause 9). If the slurry flow rate and the solids concentration vary with time, the slurry volume and the
mass of dry solids for each increment will vary accordingly. It needs to be shown that no systematic
error (bias) is introduced by periodic variation in quality or quantity, where the proposed sampling
interval is approximately equal to a multiple of the period of variation in quantity or quality. Otherwise,
stratified random sampling should be used (see Clause 10).
Best practice for sampling slurries is to cut freely falling streams mechanically (see Clause 11), with
a complete cross-section of the stream being taken during the traverse of the cutter. Access to freely
falling streams can sometimes be engineered at the end of pipes or, alternatively, a full-stream sample
2 © ISO 2017 – All rights reserved
by-line can be added to a pipe that diverts the slurry into a holding tank, or weirs can be incorporated in
launders, sluices and flumes. If samples are not collected in this manner, non-uniform concentration of
solids in the slurry due to segregation and stratification of the solids may lead to bias in the sample that
is collected. Slurry flow in pipes can be homogeneous with very fine particles, such as clays, dispersed
uniformly in turbulent suspension along the length and across the diameter of the pipe. However, more
commonly, the slurry in a pipe will have significant particle concentration gradients across the pipe and
there may be particle concentration fluctuations along the length of the pipe. These common conditions
are called heterogeneous flow. Examples of such flow are full-pipe flow of a heterogeneous suspension,
or partial-pipe flow of a fine particle suspension above a slower moving or even stationary bed of
coarser particles in the slurry.
For heterogeneous flow, bias is likely to occur where a tapping is made into the slurry pipe to locate
either a flush-fitting sample take-off pipe or a sample tube projecting into the slurry stream for
extraction of samples. The bias is caused by non-uniform radial concentration profiles in the pipe and
the different trajectories followed by particles of different masses due to their inertia, resulting in
larger or denser particles being preferentially rejected from, or included in, the sample.
In slurry channels such as launders, heterogeneous flow is almost always present, and this non-
uniformity in particle concentration is usually preserved in the discharge over a weir or step. However,
sampling at a weir or step allows complete access to the full width and breadth of the stream, thereby
enabling all parts of the slurry stream to be collected with equal probability.
Sampling of slurries in stationary situations, such as a settled or even a well-stirred slurry in a tank,
holding vessel or dam, is not recommended, because it is virtually impossible to ensure that all parts
of the slurry in the lot have an equal opportunity of being selected and appearing in the lot sample for
testing. Instead, sampling shall be carried out from moving streams, as the tank, vessel or dam is filled
or emptied.
4.2 Sampling errors
4.2.1 General
The processes of sampling, sample preparation and measurement are experimental procedures, and
each procedure has its own uncertainty appearing as variations in the final results. Where the average
of these variations is close to zero, they are called random errors. More serious variations contributing
to the uncertainty of results are systematic errors, which have averages biased away from zero. There
are also human errors that introduce variations due to departures from prescribed procedures for
which statistical analysis procedures are not applicable.
The characteristics of the solids component of a slurry can be determined by extracting samples from
the slurry stream, preparing test samples and measuring the required quality characteristics. The
total sampling error TSE can be expressed as the sum of a number of independent components (Gy,
1992; Pitard, 1993). Such a simple additive combination would not be possible if the components were
correlated. The sampling error, expressed as a sum of its components, is given by Formula (1):
TSEQ=+EQEQ++EWED++EEEP+ E (1)
12 3
where
QE is the short-range quality-fluctuation error associated with short-range variations in quality
of the solids component of the slurry;
QE is the long-range quality-fluctuation error associated with long-range variations in quality of
the solids component of the slurry;
QE is the periodic quality-fluctuation error associated with periodic variations in quality of the
solids component of the slurry;
WE is the weighting error associated with variations in the slurry flow rate;
DE is the increment delimitation error introduced by incorrect increment delimitation;
EE is the increment extraction error introduced by incorrect increment extraction from the slurry;
PE is the preparation error (also known as accessory error) introduced by departures (usually
unintentional) from correct practices, e.g. during constitution of the lot sample, draining and
filtering away the water, and transportation and drying of the sample.
The short-range quality-fluctuation error consists of two components, as shown by Formula (2):
QE =+FE GE (2)
where
FE is the fundamental error due to variation in quality between particles;
GE is the segregation and grouping error.
The fundamental error results from the composition heterogeneity of the lot, i.e. the heterogeneity that
is inherent to the composition of each particle making up the solids component of the lot. The greater
the differences in the compositions of particles, the greater the composition heterogeneity and the
higher the fundamental error variance. The fundamental error can never be completely eliminated.
It is an inherent error resulting from the variation in composition of the particles in the slurry being
sampled.
The segregation and grouping error results from the distribution heterogeneity of the sampled material
(Pitard, 1993). The distribution heterogeneity of a lot is the heterogeneity arising from the manner in
which particles are distributed in the slurry. It can be reduced by taking a greater number of smaller
increments, but it can never be completely eliminated.
A number of the components of the total sampling error, namely DE, EE and PE, can be minimized, or
reduced to an acceptable level, by correct design of the sampling procedure.
4.2.2 Preparation error, PE
In this context, the preparation error includes errors associated with non-selective sample-preparation
operations that should not change mass, such as sample transfer, draining and filtering, drying, crushing,
grinding or mixing. It does not include errors associated with sample division. Preparation errors, also
known as accessory errors, include sample contamination, loss of sample material, alteration of the
chemical or physical composition of the sample, operator mistakes, fraud or sabotage. These errors can
be made negligible by correct design of the sampling system and by staff training. For example, cross-
stream slurry cutters should have caps to prevent entry of splashes when the cutter is in the parked
position, and care needs to be taken during filtering to avoid loss of fines that are still suspended in the
water to be discarded.
4.2.3 Delimitation and extraction errors, DE and EE
Delimitation and extraction errors arise from incorrect sample cutter design and operation. The
increment delimitation error, DE, results from an incorrect shape of the volume delimiting the slurry
increment, and this can be due to both design and operation faults. Because of the incorrect shape of
the slurry increment volume, sampling with non-uniform selection probabilities results. The average
of DE is often non-zero, which makes it a source of sampling bias. The delimitation error can be made
negligible if all parts of the stream cross-section are diverted by the sample cutter for the same length
of time.
4 © ISO 2017 – All rights reserved
Sampling from moving slurry streams usually involves methods that fall into three broad operational
categories as follows (Pitard, 1993).
a) Taking the whole stream for part of the time with a cross-stream cutter as shown in Figure 1 a)
(after Pitard, 1993), usually where the slurry falls from a pipe or over a weir or step. Cuts 1 and
2 show correct sampling with the cutter diverting all parts of the stream for the same length of
time. Cuts 3 to 5 show incorrect sampling where the cutter diverts different parts of the stream for
different lengths of time.
b) Taking part of the stream all of the time as shown in Figure 1 b) (after Pitard, 1993) with an on-
stream point sampler or probe within a pipe or channel, which is always incorrect.
c) Taking part of the stream part of the time as shown in Figure 1 c) (after Pitard, 1993), also with an
on-stream point sampler or probe within a pipe or channel, which is always incorrect.
a) Taking all of the stream for part of the time
b) Taking part of the stream for all of the time (always incorrect)
c) Taking part of the stream for part of the time (always incorrect)
Key
a
Correct.
b
Incorrect.
Figure 1 — Plan view of volumes diverted by a slurry cutter
The increment extraction error, EE, results from incorrect extraction of the slurry increment. The
extraction is said to be correct if, and only if, all particles in the slurry that have their centre of gravity
inside the boundaries of the correctly delimited increment are extracted. The average of EE is often
non-zero, which makes it a source of sampling bias. The extraction error can be made negligible by
ensuring that the slurry increment is completely extracted from the stream without any particulate
material being lost from the cutter in splashes or slops. The depth and capacity of the cutter shall be
sufficient to avoid slurry reflux from the cutter aperture, resulting in loss of part of the extracted slurry
increment.
4.2.4 Weighting error, WE
The weighting error is an error component arising from the selection model underlying Formula (1).
In the model, the time-dependent flow rate of the solids in the slurry stream is a weighting function
applied to the corresponding time-dependent quality characteristic over time, which gives the
weighted average quality characteristic of the solids component of the lot. The weighting error results
from the application of incorrect weights to the quality characteristics. The best solution to reducing
the weighting error is to stabilize the flow rate. As a general rule, the weighting error is negligible for
variations in flow rate up to 10 %, and acceptable for variations in flow rate up to 20 %.
4.2.5 Periodic quality-fluctuation error, QE
Periodic quality-fluctuation errors result from periodic variations in quality generated by some
equipment used for slurry processing and transportation, e.g. grinding and screening circuits, splitters
and pumps. In such cases, stratified random sampling shall be carried out as discussed in Clause 10.
The alternative is to reduce the source of periodic variations in quality significantly, which may require
plant redesign.
4.3 Sampling and total variance
4.3.1 Sampling variance
Assume that the weighting error (WE), increment delimitation error (DE), increment extraction error
(EE) and preparation error (PE) described in 4.2 have been eliminated or reduced to insignificant
values by careful design and sampling practice. In addition, assume that periodic variations in quality
have been eliminated and that the flow rate has been regulated. The sampling error in Formula (1) then
reduces to the form:
TSEQ=+EQE (3)
Hence, the sampling variance s is given by:
()S
2 2 2
ss=+ s (4)
S QE1 QE2
The short-range quality-fluctuation variance, s , arises from the different internal composition of
QE1
increments taken at the shortest possible interval apart. This is a local or random variance due to the
particulate nature of the solids in the slurry.
The long-range quality-fluctuation variance, s , arises from the continuous trends in quality that
QE2
occur while sampling a slurry and is usually space- and time-dependent. This component is often the
combination of a number of trends generated by diverse causes.
4.3.2 Total variance
Assuming that sources of bias have either been eliminated or minimized, the next objective of a
sampling scheme is to provide one or more test portions, sufficiently representative of a lot, for
6 © ISO 2017 – All rights reserved
determination of the quality characteristics of the lot with good precision, i.e. low variance. The total
variance of the final result, denoted by s , consists of the variance of sampling (including sample
T
processing) plus the variance of analysis (chemical analysis, determination of particle-size distribution,
etc.) as follows:
22 2
ss=+ s (5)
TS A
where
s is the sampling variance (including sample processing);
S
s is the analytical variance.
A
In Formula (5), 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
A
determine 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=+ (6)
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.
The first method, which was developed by Gy, is to break up the sampling variance into its components
for each sampling stage, as specified in Annex A. The total variance is then given by:
s
22 22 A
ss=+.++ss + (7)
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 4.3.3) and provides very detailed information
on the variance components that 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 4.3.4), is to break up the total variance into
primary sampling, sample processing and analytical variances only as follows:
s
22 2 A
ss=+s + (8)
T P
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
P
the test portion;
is the analytical variance, including selection of the test portion [at stage u in Formula (7)].
s
A
The primary sampling variance is identical to the sampling variance for stage 1 in Formula (7), while
s is equal to the total sampling variance for the remaining sampling stages, except for selection of the
P
test portion which is included in the analytical variance. The relative magnitudes of the variance
components in Formula (8) 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 interleaved duplicate
T
samples (see 4.3.5). 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 duplicate samples (Merks, 1986). 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 is 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.
4.3.3 Sampling-stage method of estimating sampling and total variance
The sampling variance for stage i (see Annex A) is given by:
s
b
i
s = (9)
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
n
xx−
()
∑ j
j=1
2 2
s = − s (10)
PA
b
i
n −1
i
where
is the test result for increment j;
x
j
is the mean test result for all increments;
x
is the variance of subsequent sample processing and analysis.
s
PA
8 © ISO 2017 – All rights reserved
The variance of subsequent sample processing and analysis of each increment, s , has been taken
PA
into account in Formula (10) 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 , the total sampling variance is given by:
A
u−1
s
b
2 i
s = (11)
S ∑
n
i
i=1
Combining Formulae (6) and (11) gives the total variance s as follows:
T
u−1
s 2
b s
i A
s =+ (12)
T
∑
n r
i
i=1
For a three-stage sampling scheme (including selection of the test portion), Formula (12) reduces to:
2 2
s s 2
bb s
12 A
s =+ + (13)
T
n n r
1 2
The best way of reducing the value of s to an acceptable level is to reduce the largest terms in
T
Formula (12) first. Clearly, sn/ for a given sampling stage can be reduced by increasing the number
i
b
i
of increments n or reducing s by homogenizing the slurry prior to sampling. The last term can be
i
b
i
reduced by reducing the 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
i
iterations to obtain the required total variance s .
T
4.3.4 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 Formula (11), the primary sampling variance s is given by:
S
s
b
s = (14)
S
n
where
n is the number of primary increments;
s
is the variance between primary increments determined using Formula (10).
b
The primary sampling variance can be reduced by increasing the number of primary increments, n .
2 2
The sampling processing variance s and analytical variance s are determined experimentally by
P A
duplicate 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 Formulae (8) and (14) 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
s
b
2 1 2 A
s =+s + (15)
T P
n r
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
s s
b
2 1 PA
s =+ + (16)
T
n k 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 r
s = (17)
T
n
4.3.5 Interleaved sample method of measuring total variance
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