ISO/TR 5307:1991

TECHNICAL
REPORT TR 5307
First edition
1991-12-01
Solid fertilizers - Derivation of a sampling plan
for the evaluation of a large delivery
Ma tibes fertilisan tes solides - Fondements theoriques du plan
d%chantNonnage destinb A Mvaluation d ’une grosse livraison
Reference number
ISO/TR 5307: 199 1 (E)
Contents
Page
1 Scope .
2 References .
3 Notation and Symbols .
4 Preliminary hypotheses .
Principle of the sampling plan .
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Information .
............................. 8
5.3 What is determined by the proposed method
............................................................ 8
6 Theory of the sampling plan
6.1 Definitions .
.................................................................... 9
6.2 Determination of limits
........................................ 10
6.3 Use of two non-central t distributions
6.4 Determination of N and N’ .
........... 15
7 Practical procedure for the determination of N and N’
7.1 Basic information . 15
.......................................................................................... 15
7.2 Calculation
........................................... 16
7.3 Simplified calculation when N’ > 30
................................................................. 17
8 Examples of calculations
........................................... 17
8.1 Calculation by the complete process
8.2 Simplified calculation .
... 20
9 Effect of the values of the various Parameters on N and K
10 Evaluation of a delivery .
.
Annexes
A Table of values of the standardized normal variable u as
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
a function of P
B Table of values of (1 - a2)/a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 ISO 1991
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Permission in writing from the publisher.
International Organlzation for Standardization
Case Postale 56 l CH-121 1 Geneve 20 l Switzerland
Printed in Switzerland
ii
ISOITR 5307:1991 (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. Esch 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, govern-
mental 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 main task of technical committees is to prepare International Stan-
dards, but in exceptional circumstances a technical committee may
propose the publication of a Technical Report of one of the following
types:
- type 1, when the required support cannot be obtained for the publi-
cation of an International Standard, despite repeated efforts;
- type 2, when the subject is still under technical development or
where for any other reason there is the future but not immediate
possibility of an agreement on an International Standard;
- type 3, when a technical committee has collected data of a different
kind from that which is normally published as an International Stan-
dard ( “state of the art ”, for example).
Technical Reports of types 1 and 2 are subject to review within three
years of publication, to decide whether they tan be transformed into
International Standards. Technical Reports of type 3 do not necessarily
have to be reviewed until the data they provide are considered to be no
longer valid or useful.
lSO/TR 5307, which is a Technical Report of type 3, was prepared by
Technical Committee ISO/TC 134, Fertilizers and seil conditioners.
This document is a type 3 Technical Report. lt is not envisaged that it
will be published as an International Standard. lt gives the mathematical
derivation of the sampling plan specified in ISO 8634.
Annexes A and B are for information only.

Intrduction
Within the framework of its work on sampling, Technical Committee 134
"Fertilizers and Soil Conditioners" has, through its subcommittee 2,
carried out statistical studies on various sampling Plans which may be
used to assess large deliveries of fertilizers. This work complements
other Standards for fertilizers, currently under prebaration, and
provides the theoretical background necessary to appreciate fully the
requirements of those Standards. This technical report (type 3), which is
different from the international Standards usually produced by
ISO/TC 134/SC4, is intended to act as a complement to them, as a
basis for the sampling of fertilizer deliveries.
Esch country has its own regulations applicable to the fertilizer trade;
an official department is responsible for carrying out Checks regarding
application of the regulations. If these regulations are violated,
sanctions may be taken against those responsible for placing the
fertilizer on the market in that country. In the case of an imported
delivery, it is the representative of the manufacturing Company in the
or the importer who is considered by the relevant authorities to
country,
be responsible for the declared contents shown on labels or other
documentation accompanying the fertilizers.
ISO 8634 concerns the case of an importer who resells, on his own
responsibility, a large amount of fertilizer received from abroad. After
unloading, this delivery is resold in smaller lots to traders (dealers or
fartikr cooperatives) who will themselves be direct suppliers to farmers.
In the case in question, it is the importer whose name is associated with
the fertilizer; and it is therefore he who will be considered by the
retailers and users to be responsible for the declared contents.
ISO 8634 is designed for acceptance inspection. It determines the
rules for:
.
a) sampling (i.e. the sampling plan);
b) acceptance (the acceptance or rejection of the delivery);
and both apply to the bulk delivery imported.
iv
The location of the acceptance inspection, as defined in ISO 8634, in the
chain of transactions tan be represented by the following diagram:
Country of manufacture Importing country
Seller- Importer - Retailers -,Farmers
. . . . . .
. . .
. . . . . .
Official inspections of
lots resold in accordance
with national regulations
This page intentionaiiy left blank

TECHNICAL REPORT ISO/TR 5307:1991(E)
- Derivation of a sampling plan for the
Solid fertilizers
evaluation of a large delivery
1 Scope
This Technical Report presents the sampling theory which h s
resulted in
If
the definition of the sampling plan described in ISO 8634
. l
The sampling plan is applicable to a large delivery of more than
250 t of fertilizer supplied to another Party, for resale, on his own
responsibility, in small lots, each of which would be subject to
legislation.
By large amount is understood, for example, a full boat-load
(5,000 t, 10,000 t or more) thus corresponding to a relatively long
period of manufacture,
but the theory applies to any delivery of 250 t or
more.
2 References
ISO 8157: 1984, Fertilizers and soil conditioners - Vocabulary.
ISO 8634:.'1, Solid fertilizers - Sampling plan for the evaluation
of a large delivery.
3 Notation and symbols
The following Symbols appear in this Technical Report and have the
meanings assigned to them below.
Actual mean value and Standard
deviation between sampling units in the
delivery.
Mean value and Standard deviation
between sampling units in a delivery of just
acceptable quality.
. l) To be published.
Mean value and Standard deviation
between sampling units in a delivery of just
unacceptable quality.
Mean and Standard deviation,
respectively, of two lots which tan be
considered by the importer to be of the Same
quality.
Number of sampling units in the
u
delivery.
N Number of sampling units to be selected
during the sampling of the delivery.
(Increments).
r
Number of analyses to be carried out on
N
the N increments during the inspection of
the delivery.
N Number of sampling units contained in
R
the smallest lot presented for resale.
Number of increments to be combined
k
into each aggregate Sample for analysis.
Number of sampling units which will be
mandatorily selected during the official
sampling of a lot of NR sampling units.
-
Mean value found by analysis after the
x
selection of n sampling units from a lot of
sampling units.
Estimate of g/ with the aid of the
S’
J- k
Nr analyses, where CI is the Standard
deviation between the sampling units in the
delivery.
Analytical result obtained on the
x
i
Sample of rank i.
Estimate of the mean value of the
delivery with the aid of the N' analyses.
Declared value e.g. of a plant
nutrient in the fertilizer delivery.
Official inspection limit value which
depends on the declared value(l)). It may be
equal to D or less than D by a
prescribed tolerante which may depend on the
size of the lot Sold.
Probability that the mean value
ra
of n sampling units is lower than the
official limit value (L), just acceptable
by the importer.
Probability that the mean value
r
r
of n sampling units is lower than the
official limit value (L), just unacceptable
by the importer.
Probability of rejection of a delivery of
a
just acceptable quality (Sellers or producer's
risk).
Probability of acceptance of a delivery of
just unacceptable quality (importe& or
sk).
consumer's ff.
Value of the standardized normal variable
L-
such that Pr u > ulwr ] equals ra .
a
a
standardized normal variable
Value of the
L-
r such that Pr u > ulmr .] equals 2-r a
r
Value of the standardized normal
‘1-a
variable such that Pr[u > ulBa]
-
equals a.
-
Value of the standardized normal
u -
variable such that Pr[u >
1 Jg
equals ß.
Non-centrality Parameter.
Calculation coefficient which is dependent
on n, the risk levels a and ß and the
probability levels r, and r,T
Constant factor dependent on N' which
represents the uncertainty associated with the
estimate of the Standard deviation.
Value of the non-central Student ratio
tO
corresponding to the level of probability for a
non-centrality Parameter equal to
N UI-r
J
a
f n
Limit value of the estimate calculated
BO
from tO.
A,B . Calculation intermediates used during the
estimation of the lot after analysis.
F - Calculation intermediate used to facilitate
the calculation of k and N.
4 Preliminary hypotheses
Thesampling plan has been dra wn up on the assumpt ion that there is no
serial correlation between the success i ve units of the del ivery.
The N units inspected are selected at random from the delivery, each
unit having the Same Chance of being selected, and the N groups of
k units made up at random from the N. It is also understood that
the lots made up by the importer represent a random Sample from among the
U bags of the delivery and that the increments taken from a lot by
the authorities responsible for the inspection are taken at random from
the lot.
In the subsequent theory, it is assumed that a Single plant nutrient is
of interest or that, if this is not the case, each plant nutrient is
considered separately. It is also assumed that the fertilizer is
packaged. although similar arguments will also apply to products in
bulk.
The analytical error is considered to be negligible in relation to the
sampling error.
Finally, it is assumed once.and for all, as has been shown by the studies
of data from production and dispatch inspection carried out in various
countries:
a) that the mean concentration of a certain component or value in
the sampling units (e.g. bag) constituting a definite lot of
fertilizer shall be considered as a random quantity which obeys a
normal distribution;
b) that the distribution of this random quantity does not depend, at
least for sufficiently large lots, on their size.
5 Principle of the ~ampling plan
5.1 General
The sampling plan described in ISO 8634 defines a pair of numbers, N
and N', which depend on:
a) the legal requirements of the importing country (acceptable limit
for the value and the size of the smallest lot which tan be
inspected);
b) the risks which the importer accepts.
NOTE - It should be remembered that it is intended for the
inspection of the delivery received by the importer, and not
for the lots resold by the Same importer.
.
N is the number of increments which are to be taken from the delivery
and N' the number of analyses to be carried out on these N
increments.
The N increments are combined and mixed k by k (I( is a
whole number), thus resulting in N' aggregate samples
(N = MV') and an analysis is carried out on each of these N'
aggregate samples.
This procedure is explained by the relatively long and costly nature for
the analyses for determining the content of the various fertilizer
nutrients.
The sampling-plan adopted is based on the use of two non-central Student
distributions.
As the Standard deviation of the population is only known through N'
analyses and the corresponding estimate s, the confidence intervals
to be used should draw on Student's distribution and not Gaussian
distribution. Moreover, in the present case., the two central values of
the limit distributions which the buyer's and seller's risks should
cover, will be defined on the basis of a fixed value (L) by a shift
based on the Standard deviation cf of the population. In this case, the
reduced value of the interval between the value L and the confidence
interval limits obeys a non-central Student distribution, which has been
tabulated in particular by Neyman and Tokarska. It depends only on the
shift of the central value (in relative value) in r.elation to the
Standard deviation CI of the population.
-
Given that in each non-central Student test (one linked to the seller's
risk, and the other to the buyer's risk) the Same Standard deviation
(i.e. s or -r> arose in the non-centrality Parameter and in the
dispersion of the mean of the N Sample values; then the determination
of N and N' is independent of the value of the actual Standard
deviation of the lot.
5.2 Information
This is of two types. The first type is derived from the national
regulations of the importing country. That is:
n The number of sampling units from which, in accordance with
the regulations, partial samples are to be taken, in the case
of the smallest lot that tan be inspected.
L The official inspection limit; if the declared value
is D,
it tan be equal to D or less than D by a
permitted tolerante which may or may not be a function of the
number of lots inspected. (L = D - T,
'ifT= tolerante).
The second type tan be fixed by mutual agreement between the two
contracting Parties (the supplier and the importer), taking into account

ISOITR 5307:1991 (E)
the conditions of application of the regulations in the importing country
(frequency and stringency of inspections, punitive sanctions, etc.):
That is:
This is the fundamental Parameter as it defines the
ya
"level of quality" which shall be the minimum objective of the
manufacturer in production, in Order to giVe satisfaction to
the importer (see figure 1 and 7.1).
Production will normally be centred upon the declared
value D; but it is not sufficient for it to fulfil this
condition. What is required by the importer, and it should be
noted that he is not the User, is to be able to resell small
lots without being penalised by the official inspection
Service. He therefore wishes it to be impossible to draw from
the Overall delivery small lots which, after sampling, reveal
average contents less than L, under official inspection
conditions. The ideal would be for the production to contain no
small lot likely to appear on inspection to have a value less
than L; but this ideal is impossible to attain under
practical manufacturing conditions and would only be verifiable
by a full inspection, at a proh.ibitive tost. The importer
therefore accepts a certain percentage of incorrect units
if the bag is a sampling unit) defining the quality
(i.e. bags,
level of the Overall delivery which he considers acceptable;
this percentage'is expressed by the Parameter ra which
tan be defined as "the probability; which is just acceptable to
- the importer, that the average value of n sampling units is
less than the official limit L ”.
Distribution of
the delivery
(Sample average
of 22 increments)
Content
D
Minimum quality defining an acceptable
delivery
- The relationship between D, L and ra
Figure 1
NOTE - The regulations of the importing country may also
require that the complete delivery should,respect the declared
content D. In this case, the tolerante T is only
applicable-to small lots resold and their acceptance is
accompanied by a verification of the compensations between
recorded under-contents and over-contents. This aspect of the
question is not examined in this technical report.
a The seller's risk or 'probability of rejecting a delivery
-
of acceptable quality' (see also 7.1). *
This technical report defines a statistical test. The aim of
the sampling plan is to obtain sufficient information to be
able to say, With certainwell defined risks, whether or not
the delivery is indeed of the acceptable quality level as
defined above by r,: This actual quality of the delivery
in any case, be known with absolute certainty. Certain
cannot,
These risks are defined in relation
risks have to be accepted.
,to a hypothesis of what the delivery i& in fact. As far as
the seller's risk, a, is concerned, the Supposition is made
that the manufacturerhas in fact supplied a correct delivery,
i.e. corresponding to the quality level acceptable to the
importer, defined as a limit by r,. With this hypothesis
-. of a correct delivery, it may happen, by the misfortune of
the Seller, that the random drawing of sampling units results
in a Sample which gives a distorted image of the delivery,
making it to be declared defective whereas it is in fact
correct.
The manufacturer thus sees the delivery wrongly rejected,
.
whilst with this hypothesis his delivery is actually correct.
For the manufacturer, the sampling plan should be such that a
correct delivery is only wrongly rejected in less than
r, % of cases.
This risk, or the seller's risk, is the
maximum which the manufacturer agrees to bear.
This Parameter defines the 'quality level' of the
*r
delivery, which is too low to be acceptable to the importer
(see also 7.1).
It was found above, with regard to r,, that the importer
could not require an ideal which is impossible to obtain or
that the total absence of sampling units of content less
than L bad-to be checked. But he should require that the
sampling plan guarantees him against an excessive Proportion of
l s
small lots which on inspection are found to be deficient in
content. This is why he requires that he should define as
unacceptable a delivery which, on inspection, contains more
than rr % of small lots in which the mean value (based
on n sampling units) is less than the official
limit L.
The importer's risk
ß
-
This, for the importer,
complements the seller's risk a (see
also 7.1).
ISOITR 5307:1991 (E)
In the hypothesis in whichthe actual quality level of the delivery
is as 1OW as Yr, the importer wishes to be Sure that the
sampling plan will not lead him wrongly to accept the delivery as
correct, in more than ß % of cases.
5.3 Obj'ective of the determination
5.3.1 General
Two Points should be considered.
Firstly, the sampling plan itself, i.e. the increments to be taken, their
combinations
(applicable) the number of analyses etc.
in accordance with the results
Secondly, the rule of acceptance,
obtained.
5.3.2 The sampling plan
This will define the number of sampling units (bags), N, from which
the increments are to be taken.
Either each increment, obtained from a sampling unit, is analysed; or
they are grouped in twos, threes or fours etc. (k) and in this case
an analysis is only carried out for each of the groups of k aggregate
samples (N' analyses).
Thus: k = N/N’
If k = 1, it is sufficient to determine N; in other cases, the
plan determines N and k, and hence N'.
5.3.3 Rule of acceptance
The sampling plan leads to N' results being obtained for each content
to be determined. With these N' results, the Standard deviation of
the delivery, s, tan be estimated.
The Standard also gives the value of a coefficient K which, as will
be seen later on, depends on the preceding data: ra, Yr, E, ß
and n.
Taking x as the mean of the N' results, the delivery will be
accepted if:
2 2 L f Ks, or B 2 Bg
but if:
2 < L + Ks, or B < Bg
then the delivery will be rejected.
6 Theory of the sampling plan
6.1 Definitions
The delivery is made up of U sampling units (bags, etc.).

ISOITR 5307:1991 (E)
The mean content of the delivery is p, its Standard deviation CI and
-
its declared content D.
The importer does not know either p or CL On the other hand, he
knows D which is the declared compositi& under which the fertilizer
is to be Sold, and he is faced with the Problem of ensuring, by suitable
that the lots which he sells will conform to the
sampling and analysis,
specifications of local regulations.
It is assumed that local regulations generally require that an analysis,
carried out after sampling any lot (which, in extreme Gases, may consist
shall not fall below a limit L. This may be the
of a Single bag),
or it may be less by a permitted tolerante
declared composition D,
(which will generally dep6nd on the size of the lot).
In view of the fact that the importer will usually resell the fertilizer
the sampling scheme should be designed to
in sub-lots of varying size,
give suitable protection to the smallest lots which he intends to sell.
Assuming that the smallest lot intended for sale is NR bags then,
for inspection at this Stage (resold lots) a Sample of n bags will be
taken (n is thus fixed by local regulations).
Using these n bags, an aggregate Sample will be made up, the analysis
of which will lead to the (mean) value x.
Sample of n bags ; it is the
x is the (mean) value of the
lot
...