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
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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
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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
...