ISO 13909-7:2016

INTERNATIONAL ISO
STANDARD 13909-7
Second edition
2016-07-01
Hard coal and coke — Mechanical
sampling —
Part 7:
Methods for determining the
precision of sampling, sample
preparation and testing
Houille et coke — Échantillonnage mécanique —
Partie 7: Méthodes pour la détermination de la fidélité de
l’échantillonnage, de la préparation de l’échantillon et de l’essai
Reference number
ISO 13909-7:2016(E)
©
ISO 2016

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ISO 13909-7:2016(E)

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ISO 13909-7:2016(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 General . 1
5 Formulae relating to factors affecting precision . 2
5.1 General . 2
5.2 Sampling . 3
6 Estimation of primary increment variance . 4
6.1 Direct determination of individual primary increments . 4
6.2 Determination using the estimate of precision. 5
7 Methods for estimating precision . 5
7.1 General . 5
7.2 Duplicate sampling with twice the number of increments. 5
7.3 Duplicate sampling during routine sampling . 8
7.4 Alternatives to duplicate sampling . 9
7.5 Precision adjustment procedure . 9
8 Calculation of precision .10
8.1 Replicate sampling .10
8.2 Normal sampling scheme .11
9 Methods of checking sample preparation and testing errors .12
9.1 General .12
9.2 Target value for variance of sample preparation and analysis .12
9.2.1 General.12
9.2.2 Off-line preparation . .13
9.2.3 On-line preparation .13
9.3 Checking procedure as a whole .13
9.4 Checking stages separately .14
9.4.1 General.14
9.4.2 Procedure 1 .15
9.4.3 Procedure 2 .18
9.4.4 Interpretation of results .21
9.5 Procedure for obtaining two samples at each stage .22
9.5.1 With a riffle .22
9.5.2 With a mechanical sample divider .22
9.6 Example .22
Annex A (informative) Variogram method for determining variance .26
Annex B (informative) Grubbs’ estimators method for determining sampling precision .34
Bibliography .43
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ISO 13909-7:2016(E)

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.
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 meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
Barriers to Trade (TBT), see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 27, Solid mineral fuels, Subcommittee SC 4,
Sampling.
This second edition cancels and replaces the first edition (ISO 13909-7:2001), which has been technically
revised.
ISO 13909 consists of the following parts, under the general title Hard coal and coke — Mechanical
sampling:
— Part 1: General introduction
— Part 2: Coal — Sampling from moving streams
— Part 3: Coal — Sampling from stationary lots
— Part 4: Coal — Preparation of test samples
— Part 5: Coke — Sampling from moving streams
— Part 6: Coke — Preparation of test samples
— Part 7: Methods for determining the precision of sampling, sample preparation and testing
— Part 8: Methods of testing for bias
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ISO 13909-7:2016(E)

Introduction
Two different situations are considered when a measure of precision is required. In the first, an estimate
is made of the precision that can be expected from an existing sampling scheme and, if this is different
from that desired, adjustments are made to correct it. In the second, the precision that is achieved on a
particular lot is estimated from the experimental results actually obtained using a specifically designed
sampling scheme.
The formulae developed in this part of ISO 13909 are based on the assumption that the quality of the
fuel varies in a random manner throughout the mass being sampled and that the observations will
follow a normal distribution. Neither of these assumptions is strictly correct. Although the assumption
that observations will follow a normal distribution is not strictly correct for some fuel parameters, this
deviation from assumed conditions will not materially affect the validity of the formulae developed for
precision checking since the statistics used are not very sensitive to non-normality. Strictly speaking,
however, confidence limits will not always be symmetrically distributed about the mean. For most
practical uses of precision, however, the errors are not significant.
NOTE In the text, the term “fuel” is used where both coal and coke would be applicable in the context and
either “coal” or “coke” where that term is exclusively applicable.
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INTERNATIONAL STANDARD ISO 13909-7:2016(E)
Hard coal and coke — Mechanical sampling —
Part 7:
Methods for determining the precision of sampling, sample
preparation and testing
1 Scope
In this part of ISO 13909, formulae are developed which link the variables that contribute to overall
sampling precision. Methods are described for estimating overall precision and for deriving values for
primary increment variance which can be used to modify the sampling scheme to change the precision.
Methods for checking the variance of sample preparation and testing are also described.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 13909-1, Hard coal and coke — Mechanical sampling — Part 1: General introduction
ISO 13909-2, Hard coal and coke — Mechanical sampling — Part 2: Coal — Sampling from moving streams
ISO 13909-3, Hard coal and coke — Mechanical sampling — Part 3: Coal — Sampling from stationary lots
ISO 13909-4, Hard coal and coke — Mechanical sampling — Part 4: Coal — Preparation of test samples
ISO 13909-5, Hard coal and coke — Mechanical sampling — Part 5: Coke — Sampling from moving streams
ISO 13909-6, Hard coal and coke — Mechanical sampling — Part 6: Coke — Preparation of test samples
ISO 13909-8, Hard coal and coke — Mechanical sampling — Part 8: Methods of testing for bias
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 13909-1 apply.
4 General
When designing a sampling scheme in order to meet a required precision of results, formulae are
necessary that link certain fuel and sampling characteristics to that precision. The main factors to be
considered are the variability of primary increments, preparation and testing errors, the number of
increments and samples taken to represent the lot, and the mass of the samples. These formulae are
derived in Clause 5. Methods for estimating the parameters used in those formulae are given in Clause 6.
Once a sampling system has been designed and installed, the precision which is being achieved on
a routine basis should be checked. An estimate of the precision can be obtained from the primary
increment variance, V , the numbers of increments, n, and sub-lots, m, (see Clause 5) and the preparation
I
and testing variance, V . The preparation component of V is made up of on-line sample processing
PT PT
and off-line sample preparation.
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ISO 13909-7:2016(E)

Sampling variance is a function of product variability, so the same number of increments, sub-lots, and
preparation and testing errors will yield different precision with fuels that exhibit different product
variability.
Depending on the extent to which serial correlation exists and which method of estimating primary
increment variance is used, such an estimate could represent a considerable overestimate of the
numerical value of the precision (i.e. indicate that it is worse than is really the case). In addition, in
order for the results to be meaningful, large numbers of increments (in duplicate) would need to be
prepared and analysed for the estimation of V and V .
I PT
Quality variations obtained in the form of primary increment variances on existing systems are not
absolute and therefore designers should exercise caution when using such results in a different
situation. The estimated value of the primary increment variance, V , should be derived experimentally
I
for each fuel and at each sampling location.
Whenever a sampling scheme is used for determining increment variance, the operating conditions
should be as similar as possible to the conditions known, or anticipated, to prevail during the sampling
for which the increment variance is needed, whether it be carried out by the same or by a different
sampling system.
An estimate of the precision actually achieved can be obtained by taking the sample in a number of
parts and comparing the results obtained from these parts. There are several methods of doing this,
depending on
a) the purpose of the test, and
b) the practical limitations imposed by the available sampling procedures and equipment.
Where a sampling system is in existence, the purpose of the test is to check that the scheme is in fact
achieving the desired precision (see Clause 7). If it is not, it may need to be modified and rechecked until
it meets the precision required. In order to do this, a special check scheme should be devised which may
be different from the regular scheme but which measures the precision of the regular scheme.
For regular sampling schemes, the most rigorous approach is that of duplicate sampling of sub-lots.
In many existing mechanical sampling systems, however, the capacity of individual components and
the interval between increments in the regular scheme is insufficient to allow the taking of extra
increments. In such cases, duplicate samples can be constituted from the normal number of increments
and the result adjusted for the smaller number of increments in each sample (see 7.3).
The need may arise to sample a particular lot and to know the precision of the result obtained
(see Clause 8). Once again, a special check scheme needs to be devised, but in this case, it is the precision
achieved by that scheme on that lot which is required. For the measurement of the precision achieved
for a particular lot, replicate sampling is the best method.
Methods for detailed checking of preparation and testing errors are given in Clause 9. The results may
also be used to provide data for the formulae used in Clause 5.
5 Formulae relating to factors affecting precision
5.1 General
Precision is a measure of the closeness of agreement between the results obtained by repeating a
measurement procedure several times under specified conditions and is a characteristic of the method
used. The smaller the random errors of a method, the more precise the method is. A commonly accepted
index of precision is two times the sample estimate of the population standard deviation and this index
of precision is used throughout this part of ISO 13909.
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ISO 13909-7:2016(E)

If a large number of replicate samples, j, are taken from a sub-lot of fuel and are prepared and analysed
separately, the estimated precision, P, of a single observation is given by Formula (1):
Ps22 V (1)
SPT
where
s is the sample estimate of the population standard deviation;
V is the total variance.
SPT
The total variance, V , in Formula (1) is a function of the primary increment variance, the number of
SPT
increments, and the errors associated with sample preparation and testing.
NOTE The components of primary increment variance are the variance of sample extraction and the
variance contributed by product variability. The variance contributed by product variability is generally, but not
always, the largest source of variance in sampling.
For a single sample, this relationship is expressed by Formula (2):
V
I
V =+V (2)
SPT PT
n
where
V is the primary increment variance;
l
V is the preparation and testing variance;
PT
n is the number of primary increments in the sample.
5.2 Sampling
Where the test result is the arithmetic mean of a number of samples, resulting from dividing the lot into
a series of sub-lots and taking a sample from each, Formula (2) becomes:
V V
IPT
V =+ (3)
SPT
mn m
where
m is the number of sample results used to obtain the mean.
Since a sample is equivalent to one member of a set of replicate samples, by combining Formulae (1) and
(3), it can be shown that:
V V
IPT
P =+2 (4)
mn m
Formula (4) gives an estimate of the precision that can be expected to be achieved when a given
sampling scheme is used for testing a given fuel, the variability of which is known or can somehow
be estimated. In addition, Formula (4) enables the designer of the sampling scheme to determine, for
the desired precision and with fuel of known or estimated variability, the combination of numbers of
increments and samples, respectively, which will be most favourable considering the relative merits
of the sampling equipment and the laboratory facilities in question. For the latter purpose, however,
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ISO 13909-7:2016(E)

it is more convenient to use either or both Formulae (5) and (6), both of which have been derived by
rearranging Formula (4).
4V
I
n = (5)
2
mP −4V
PT
4 Vn+ V
()
IPT
m = (6)
2
nP
NOTE Results obtained from solid mineral fuels flowing in a stream will frequently display serial
correlation, i.e. immediately adjoining fuel tends to be of similar composition, while fuel further apart tends to
be of dissimilar composition. When this is so, the estimates of precision of the result of a single sample based on
primary increment variance and the variance of preparation and testing would indicate precision that is worse,
i.e. numerically higher, than the precision actually achieved. The effect of serial correlation can be taken into
account using the variographic method of determining variance given in informative Annex A.
6 Estimation of primary increment variance
6.1 Direct determination of individual primary increments
The direct estimation of primary increment variance can be accomplished with a duplicate sampling
scheme comprised of several hierarchical levels which allows both the overall variance and the variance
of preparation and testing to be estimated. The estimated variance of primary increments can then be
obtained by subtraction of the variance of preparation and testing from the estimated overall variance.
A number of primary increments is taken systematically and either divided into two parts or prepared
so that duplicate samples are taken at the first division stage. Each part is prepared and tested for
the quality characteristic of interest, using the same methods that are expected to be used in routine
operations. The mean of the two results and the difference between the two results are calculated for
each pair.
It is recommended that at least 30 increments be taken, spread if possible over an entire lot or even
over several lots of the same type of fuel.
The procedure is as follows.
a) Calculate the preparation and testing variance, V .
PT
2
d
∑
V = (7)
PT
2n
p
where
d is the difference between pair members;
n is the number of pairs.
p
b) Calculate the primary increment variance, V .
l
2
1
2
x − x
()
∑ ∑
n
V
p
PT
V = − (8)
I
2
n −1
()
p
where
x is the mean of the two measurements for each increment.
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ISO 13909-7:2016(E)

An alternative method for estimating primary increment variance, V is as follows:
l,
2
D
V
∑
PT
V =− (9)
I
22h
where
D is the difference between the means of successive pairs;
h is the number of successive pairs.
This method avoids the overestimation of sampling variance when there is serial correlation (see Note
in 5.2) but can only be used if the primary-increment sampling interval at which the increments are
taken is more than or approximately equal to the primary-increment sampling interval used when the
scheme is implemented in routine sampling operations.
The most rigorous treatment of serial correlation is to use the variographic method given in Annex A.
This takes into account both serial correlation and sampling interval effects, thereby avoiding
overestimation of sampling variance and number of primary increments due to these factors.
6.2 Determination using the estimate of precision
The primary increment variance can be calculated from the estimate of precision obtained either using
the method of duplicate sampling given in 7.2 or replicate sampling given in Clause 8 according to
Formula (10) which is derived by rearranging the terms of Formula (4).
2
mnP
V =−nV (10)
IPT
4
This value can then be used to adjust the sampling scheme if necessary.
7 Methods for estimating precision
7.1 General
For all the methods given in this Clause, the following symbols and definitions apply:
— n is the number of increments in a sub-lot for the regular scheme;
0
— m is the number of sub-lots in a lot for the regular scheme;
0
— P is the desired precision for the regular scheme;
0
— P is the worst (highest absolute value) precision to be permitted.
W
In all cases, the same methods of sample preparation shall be used as for the regular scheme.
7.2 Duplicate sampling with twice the number of increments
Twice the normal number of increments (2n ) are taken from each sub-lot and combined as duplicate
0
samples (see Figure 1), each containing n increments. This process is repeated, if necessary, over
0
several lots of the same fuel, until at least 10 pairs of duplicate samples have been taken.
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ISO 13909-7:2016(E)

A parameter of the fuel is chosen to be analysed, e.g. ash content (dry basis) for coal, or Micum 40 index
for coke. The standard deviation within duplicate samples for the test parameter is then calculated
using Formula (11):
2
d
∑
s= (11)
2n
p
where
d is the difference between duplicates;
n is the number of pairs of duplicates being examined.
p
An example of results for coal ash is given in Table 1.
Table 1 — Results of duplicate sampling, % ash, dry basis
Duplicate values
Difference between duplicates
%
Sample pair no.
2
A B |A − B| = d d
1 11,1 10,5 0,6 0,36
2 12,4 11,9 0,5 0,25
3 12,2 12,5 0,3 0,09
4 10,6 10,3 0,3 0,09
5 11,6 12,5 0,9 0,81
6 11,8 12,0 0,2 0,04
7 11,8 12,2 0,4 0,16
8 10,8 10,0 0,8 0,64
9 7,9 8,2 0,3 0,09
10 10,8 10,3 0,5 0,25
Total 2,78
The number of pairs, n is 10. The variance of the ash content is therefore
p
2
d
∑
2
s =
2n
p
27, 8
==0,139
20
and the standard deviation is:
s 0,,13900 373
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ISO 13909-7:2016(E)

Key
increment from regular scheme
extra increment for precision check scheme
Figure 1 — Example of a plan of duplicate sampling
The precision of the result for a single sub-lot is therefore:
Ps= 2
= 20,,373 = 075 % ash
()
The precision achieved for the mean ash of a normal lot sampled as m sub-lots is given by 2sm . For
example, if m = 10, then:
20,373
()
P = = 0,235 9 %
10
These values of P have been calculated using point estimates for the standard deviation and represent
the best estimate for precision.
If an interval estimate for the standard deviation is used, then on a 95 % confidence level, the precision
is within an interval with upper and lower limits. These limits are calculated from the point estimate
of precision and factors which depend on the degrees of freedom (f) used in calculating the standard
deviation (see Table 2).
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ISO 13909-7:2016(E)

Table 2 — Factors used for calculation of precision intervals
ƒ (number of observations) 5 6 7 8 9 10 15 20 25 50
Lower limit 0,62 0,64 0,66 0,68 0,69 0,70 0,74 0,77 0,78 0,84
Upper limit 2,45 2,20 2,04 1,92 1,83 1,75 1,55 1,44 1,38 1,24
2
NOTE The factors in Table 2 are derived from the estimate of s obtained from the squared differences of n
2
pairs of observations. Since there is no constraint in this case, the estimate as well as d will have n degrees of
freedom. The values in Table 2 are derived from the relationship:
2 2
ns ns
2
< 2 2
χχ
nn,,0025 ,,00975
2
The body of Table 2 gives the values for n/χ , which are multiplied by s to obtain the confidence limits.
For example, for the lot with 10 sub-lots used in the example above:
Upper limit = 1,75 (0,235 9) = 0,41 %
Lower limit = 0,70 (0,235 9) = 0,17 %
where the factors are obtained from Table 2 using ƒ = n , i.e. 10. The true precision lies between 0,17 %
P
and 0,41 % ash at the 95 % confidence level.
7.3 Duplicate sampling during routine sampling
If operational conditions do not allow the taking of 2n increments from each normal sub-lot or
0
precision is to be determined during normal sampling, then, provided that all increments can be kept
separate, adopt the following procedure for estimating precision.
Take the normal number of increments, n , from each sub-lot and combine them as duplicate samples
0
each comprising n /2 increments (see Figure 2). Repeat this process, if necessary, over several lots of
0
the same fuel until at least 10 pairs of duplicate samples have been obtained. In this case, the precision
obtained using the procedure in 7.2 will be for n
...