EN ISO 4259:1995

SLOVENSKI STANDARD
01-maj-1998
1DIWQLSURL]YRGL'RORþDQMHLQXSRUDEDVWRSHQMQDWDQþQRVWLSULSUHVNXVQLK
PHWRGDK,62LQ3RSUDYHN
Petroleum products - Determination and application of precision data in relation to
methods of test (ISO 4259:1992/Cor 1:1993)
Mineralölerzeugnisse - Bestimmung und Anwendung der Werte für die Präzision von
Prüfverfahren (ISO 4259:1992/Cor 1:1993)
Produits pétroliers - Détermination et application des valeurs de fidélité relatives aux
méthodes d'essai (ISO 4259:1992/Cor 1:1993)
Ta slovenski standard je istoveten z: EN ISO 4259:1995
ICS:
75.080 Naftni proizvodi na splošno Petroleum products in
general
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

INTERNATIONAL Is0
STANDARD 4259
Second edition
1992-12-15
Petroleum products - Determination and
application of precision data in relation to
methods of test
Produits p6 troliers - D6termina tion et application des valeurs de fidblit6
relatives aux m6 thodes d ‘essai
Reference number
IS0 42597 992(E)
IS0 4259:1992(E)
Page
CONTENTS
Introduction .
Scope .
Normative Reference .
Definitions . 2
Stages in planning of an inter-laboratory test programme for the determination of the precision of a test method . 3
4.1 Preparing a draft method of test . 3
4.2 Planning a pilot programme with at least two laboratories . 3
4.3 Planning the inter-laboratory programme . 4
4.4 Executing the inter-laboratory programme . 4
5 Inspection of inter-laboratory results for uniformity and for outliers . 4
5.1 Transformation of data . 4
5.2 Tests for outliers . 5
5.3 Rejection of complete data from a sample . 6
5.4 Estimating missing or rejected values . 7
5.5 Rejection test for outlying laboratories . 8
5.6 Confirmation of selected transformation .
6 Analysis of variance and calculation of precision estimates .
6.1 Analysis of variance .
Expectation of mean squares and calculation of precision estimates . 10
6.2
Precision clause of a method of test . 12
6.3
7 . 12
Significance of repeatability r and reproducibility R as discussed in earlier clauses
7.1 Repeatability r .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.2 Reproducibility R
8 Specifications . 13
8.1 Aim of specifications . 13
8.2 Construction of specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
9 Quality control against specifications . 14
9.1 Testing margin at the supplier .
Testing margin at the recipient . 14
9.2
................................................................................................................ 14
10 Acceptance and rejection rules in case of dispute
Annexes
A (Normative) Determination of number of samples required . 16
B (Informative) Derivation of formula for calculating the number of samples required . 17
C (Normative) Notation and tests . 18
Example results of test for determination of bromine number and statistical tables . 22
D (Normative)
Types of dependence and corresponding transformations . 29
E (Normative)
F (Normative) Weighted linear regression analysis .
G (Normative) Rules for rounding off results .
H (Informative) Explanation of formulae in clause 7 .
Specifications which relate to a specified degree of criticality . 40
J (Informative)
,,.,.,.,.,.,.,.,.,.,. 42
Bibliography
0 IS0 1992
All rights reserved. No part of this publication may be reproduced or utilized in any form or
by any means, electronic or mechanical, including photocopying and microfilm, without per-
mission in writing from the publisher.
International Organization for Standardization
Case Postale 56 l CH-1211 Geneve 20 l Switzerland
Printed in Switzerland
ii
IS0 4259: 1992(E)
Foreword
IS0 (the International Organization for Standardization) is a worldwide
federation of national standards bodies (IS0 member bodies). The work
of preparing International Standards is normally carried out through IS0
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. IS0
collaborates closely with the International Electrotechnical Commission
(IEC) on all matters of electrotechnical standardization.
Draft International Standards adopted by the technical committees are
circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting
a vote.
International Standard IS0 4259 was prepared by Technical Committee
ISOTTC 28, Petroleum products and lubricants.
This second edition cancels and replaces the first edition
(IS0 4259:1979), of which it constitutes a technical revision.
Significant changes from the first edition include:
Independence of repeated results (subclause 3.17)
Blind coding in relation to repeated results (subclauses 3.4, 4.4)
Probability level for precision (subclauses 3.17, 3.19, 6.2.3)
Transformation procedure (subclause 5.1, annex E, annex F)
Hawkins’ outlier test (subclauses 5.2.2, 5.5, clause C.5)
Rejection criteria for complete data from a sample (subclause 5.3)
Variance ratio test/bias between laboratories (subclauses 6.1.4, 6.2.3.2,
clause C.6)
Formula for acceptability of results/confidence limits (clause 7, annex H)
Specifications which include a stated degree of criticality (subclause 8. ‘l,
annex J)
IS0 4259 makes reference to IS0 3534, Statistics - Vocabulary and
symbols, which gives a different definition of “true value” (see subclause
3.24). IS0 4259 also refers to IS0 5725, Precision of test methods -
Determination of repeatability and reproducibility for a standard test
method by inter-laboratory tests. The latter will be required in particular
and unusual circumstances (see subclause 5.1) for the purpose of esti-
mating precision.
Annexes form an integral part of this International
A, C, D, E, F and G
r information only.
Standard. Annexes B, H and J are fo

This page intentionally left blank

IS0 4259:1992(E)
INTERNATIONAL STANDARD
Petroleum products - Determination and application
of precision data in relation to methods of test
INTRODUCTION
For purposes of quality control and to check compliance with specifications, the properties of
commercial petroleum products are assessed by standard laboratory test methods. Two or more
measurements of the same property of a specific sample by any given test method do not usually
give exactly the same result. It is therefore necessary to take proper account of this fact, by arriving
at statistically based estimates of the precision for a method, i.e. an objective measure of the degree
of agreement to be expected between two or more results obtained in specified circumstances.

1 SCOPE 3.5 check sample : A sample taken at the place where the
product is exchanged, i.e. where the responsibility for the
This International Standard covers the calculation of product quality passes from the supplier to the recipient.
precision estimates and their application to specifications.
In particular, it contains definitions of relevant statistical
3.6 degrees of freedom : The divisor used in the
terms ( clause 3), the procedures to be adopted in the
calculation of variance; one less than the number of
planning of an inter-laboratory test programme to determine
independent results.
the precision of a test method (clause 4), the method of
calculating the precision from the results of such a
NOTE - The definition applies strictly only in the simplest
programme (clauses 5 and 6), and the procedure to be
cases. Complete deLfiniti0n.s are beyond the scope of this
followed in the interpretation of laboratory results in relation
International Standard.
both to precision of the methods and to the limits laid down
in specifications (clauses 7 to 10).
3.7 determination : The process of carrying out the series
of operations specified in the test method, whereby a single
It is emphasised that the procedures in this International
value is obtained.
Standard are designed to cover methods of test for petroleum
products only. The latter are, in general, homogeneous
products with which serious sampling problems do not 3.8 known val ue : The actual quantitative value implied by
normally arise. It would not be appropriate, therefore, to the preparation of the sample.
consider the procedures to be necessarily of wider
application, for example to heterogeneous solids. NOTE - The known value does not always exist, for example
for empirical tests such asjlash point.
2 NORMATIVE REFERENCE
3.9 mean; arithmetic mean; average : For a given set of
The following standard contains provisions which, through
results, the sum of the results divided by their number.
reference in this text, constitutes provisions of this
International Standard. At the time of publication, the
3.10 mean square : The sum of squares divided by the
edition indicated was valid. All standards are subject to
degrees of freedom.
revision, and parties to agreements based on this
International Standard are encouraged to investigate the
possibility of applying the most recent edition of the standard 3.11 normal distribution : The probability distribution of
listed below. Members of IEC and IS0 maintain registers a continuous random variable X such that, if x is any real
of currently valid International Standards. number, the probability density is
IS0 5725: “Precision of Test Methods - Determination of
-Lp[ -;( q] . .(l)
Repeatability and Reproducibility for a Standard Test
AX) diF
Method by Interlaboratory Tests ”.
-= NOTE - ~1 is the true value and o is the standard deviation
3 DEFINITIONS of the normal distribution (CJ > 0).
For the purposes of this International Standard, the following
3.12 operator : A person who normally and regularly
definitions apply :
carries out a particular test.
3.1 analysis of variance : A technique which enables the
3.13 outlier : A result far enough in magnitude from other
total variance of a method to be broken down into its
results to be considered not a part of the set.
component factors.
3.2 between-laboratory variance : When results obtained 3.14 precision : The closeness of agreement between the
by more than one laboratory are compared, the scatter is results obtained by applying the experimental procedure
usually wider than when the same number of tests are carried several times on identical materials and under prescribed
out by a single laboratory, and there is some variation conditions. The smaller the random part of the experimental
between means obtained by different laboratories. These error, the more precise is the procedure.
give rise to the between-laboratory variance which is that
component of the overall variance due to the difference in
3.15 random error : The chance variation encountered in
the mean values obtained by different laboratories. (There
all test work despite the closest control of variables.
is a corresponding definition for between-operator variance.)
3.16 recipient : Any individual or organization who
The term “between-laboratory” is often shortened to
receives or accepts the product delivered by the supplier.
“laboratory” when used to qualify representative parameters
of the d&&ion of the popilatkn of results, for example as
“laboratoAq variance ”. * * 3.17 repeatability :
a) Qualitatively
3.3 bias : The difference between the true value (related to
the method of test) (see 3.24) and the known value (see 3X),
The closeness of agreement between independent results
where this is available.
obtained in the normal and correct operation of the same
method on identical test material, in a short interval of
3.4 blind coding : The assignment of a different number to
time, and under the same test conditions (same operator,
each sample. No other identification or information on any
sample is given to the operator. same apparatus, same laboratory).

IS0 4259:1992(E)
me representative parameters of the dispersion of the 3.23 supplier : Any individual or organization responsible
population which may be associated with the results are for the quality of a product just before it is taken over by the
qualified by the term “repeatability ”, for example recipient.
repeatability standard deviation, repeatability variance.
The term “repeatability” shall not be confused with the 3.24 true value : For practical purposes, the value towards
terms “between repeats” or “repeats” when used in this which the average of single results obtained by n laboratories
tends, as n tends towards infinity; consequently, such a true
way (see 3.18). Repeatability refers to the state of
minimum random variability of results. The period of value is associated with the particular method of test.
time during which repeated results are to be obtained
shall therefore be short enough to exclude time - NOTE - A d@ierent and idealized definition is given in
dependent errors, for example, environmental and IS0 3534, Statistics - Vocabulary and symbols.
calibration errors.
3.25 variance : The mean of the squares of the deviation of
b) Quuntitatively a random variable from its mean, estimated by the mean
square.
The value equal to or below which the absolute difference
between two single test results obtained in the above
conditions may be expected to lie with a probability of 4 STAGES IN PLANNING OF AN INTER-
LABORATORY TEST PROGRAMME FOR THE
95 %.
DETERMINATION OF THE PRECISION OF A
TEST METHOD
3.18 replication : The execution of a test method more than
once so as to improve precision and to obtain a better
The stages in planning an inter-laboratory test programme
estimation of testing error. Replication shall be
are as follows :
distinguished from repetition in that the former implies that
repeated experiments are carried out at one place and, as far
as possible, one period of time. The representative
a) Preparing a draft method of test.
parameters of the dispersion of the population which may be
associated with repeated experiments are qualified by the
b) Planning a pilot programme with at least two
term “between repeats ”, or in shortened form “repeats ”, for
laboratories.
example “repeats standard deviation ”.
c) Planning the inter-laboratory programme.
3.19 reproducibility :
d) Executing the inter-laboratory programme.
a) Qualitatively
The four stages are described in turn.
The closeness of agreement between individual results
obtained in the normal and correct operation of the same
4.1 Preparing a draft method of test
method on identical test material but under different test
conditions (different operators, different apparatus and
This shall contain all the necessary details for carrying out
different laboratories).
the test and reporting the results. Any condition which could
alter the results shall be specified.
The representative parameters of the dispersion of the
population which may be associated with the results are
The clause on precision will be included at this stage only
qualified by the term “reproducibility ”, for example
as a heading.
reproducibility standard deviation, reproducibility
variance.
4.2 Planning a pilot programme with at least two
laboratories
b) Quantitatively
A pilot programme is necessary for the following reasons :
The value equal to or below which the absolute difference
between two single test results on identical material
obtained by opera;ors in different laboratories, using the
a) to verify the details in the operation of the test;
standardized test method, may be expected to lie with a
probability of 95 %.
b) to find out how well operators can follow the
instructions of the method;
3.20 result : The final value obtained by following the
complete set of instructions in the test method; it may be
c) to check the precautions regarding samples;
obtained from a single determination or from several
determinations depending on the instructions in the method.
d) to estimate roughly the precision of the test.
(It is assumed that the result is rounded off according to the
procedure specified in annex G.)
At least two samples are required, covering the range of
results to which the test is intended to apply; however, at
3.21 standard deviation : A measure of the dispersion of
least 12 laboratory/sample combinations shall be included.
a series of results around their mean, equal to the positive
Each sample is tested twice by each laboratory under
square root of the variance and estimated by the positive
repeatability conditions. If any omissions or inaccuracies in
square root of the mean square.
the draft method are revealed, they shall now be corrected.
The results shall be analysed for bias and precision : if either
3.22 sum of squares : The sum of squares of the is considered to be too large, then alterations to the method
differences between a series of results and their mean. shall be considered.
IS0 4259:1992(E)
4.3 Planning the inter-laboratory programme f) a blank form for reporting the results. For each
sample, there shall be space for the date of testing, the
two results, and any unusual occurrences. The unit of
There shall be at least five participating laboratories, but it
is preferable to exceed this number in order to reduce the accuracy for reporting the results shall be specified;
number of samples required.
g) a statement that the test shall be carried out under
The number of samples shall be sufficient to cover the range normal conditions, using operators with good experience
but not exceptional knowledge; and that the duration of
of the property measured, and to give reliability to the
precision estimates. If any variation of precision with level the test shall be the same as normal.
was observed in the results of the pilot programme, then at
least five samples shall be used in the inter-laboratory The pilot programme operators may take part in the
inter-laboratory programme. If their extra experience in
programme. In any case, it is necessary to obtain at least 30
degrees of freedom in both repeatability and reproducibility. testing a few more samples produces a noticeable effect,
For repeatability, this means obtaining a total of at least 30 it will serve as a warning that the method is not
pairs of results in the programme. satisfactory. ‘Ihey shall be identified in the report of the
results so that any effect may be noted.
For reproducibility, table 11 (annex A) gives the minimum
number of samples required in terms of L, P and Q, where
L is the number of participating laboratories and P and Q are
5 INSPECTION OF INTER-LABORATORY
the ratios of variance component estimates obtained from
RESULTS FOR UNIFORMITY AND FOR
Specifically, P is the ratio of the
the pilot programme.
OUTLlERS
interaction component to the repeats component, and Q is
the ratio of the laboratories component to the repeats
This clause specifies procedures for examining the results
component. Annex B gives the derivation of the formula
reported in a statistically designed inter-laboratory
used. If Q is much larger than P, then 30 degrees of freedom
programme (see clause 4) to establish
cannot be achieved; the blank entries in table 11 correspond
to this situation or the approach of it (i.e. when more than
a) the independence or dependence of precision and the
20 samples are required). For these cases, there is likely to
level of results;
be a significant bias between laboratories.
b) the uniformity of precision from laboratory to
4.4 Executing the inter-laboratory programme
laboratory, and to detect the presence of outliers.
One person shall be responsible for the entire programme,
The procedures are described in mathematical terms based
from the distribution of the texts and samples, to the final
on the notation of annex C and illustrated with reference to
appraisal of the results. He shall be familiar with the method,
the example data (calculation of bromine number) set out in
but shall not personally take part in the tests.
annex D.
The text of the method shall be distributed to all the
Throughout this clause (and clause 6), the procedures to be
laboratories in time to raise any queries before the tests begin.
used are first specified and then illustrated by a worked
If any laboratory wants to practice the method in advance,
example using data given in annex D.
this shall be done with samples other than those used in the
programme.
It is assumed throughout this clause that all the results are
either from a single\ormal distribution or capable of being
The samples shall be accumulated, subdivided and
transformed into-such a distribution (see 5.1). Other cases
distributed by the organizer, who shall also keep a reserve
(which are rare) would require different treatment which is
of each sample for emergencies. It is most important that
beyond the scope of this International Standard. See
the individual laboratory portions be homogeneous. They
reference [8] for a statistical test on normality.
shall be blind coded before distribution, and the following
instructions shall be sent with them :
Although the procedures shown here are in a form suitable
for hand calculation, it is strongly advised that an electronic
a) the agreed draft method of test;
computer be used to store and analyse inter-laboratory test
results, based on the procedures of this standard.
requirements for the
b) the handling and storage
samples;
5.1 Transformation of data
be tested
c) the order in which the samples are to
(a
In many test methods the precision depends on the 1 eve1 of
different random order for each laboratory)
the test result, and thus the variability of the reported results
is different from sample to sample. -? ‘he method of analysis
d) the statement that two results are to be obtained
outlined in this International Standard requires that this shall
consecutively on each sample by the same operator with
not be so and the position is rectified, if necessary, by a
thesameapparatus. Forstatisticalreasonsitisimperative
transformation.
that the two results are obtained independently of each
other, that is that the second result is not biased by
knowledge of the first. If this is regarded as impossible The laboratories standard deviations Dj, and the repeats
to achieve with the operator concerned, then the pairs of standard deviations ! ‘j (see annex C) are calculated and
results shall be obtained in a blind fashion, but ensuring plotted separately agamst the sample means mj. If the points
that they are carried out in a short period of time; so plotted may be considered as lying about a pair of lines
parallel to the m-axis, then no transformation is necessary.
If, however, the plotted points describe non-horizontal
e) the period of time during which repeated results are
to be obtained and the period of time during which all straight lines or curves of the form D = fi(m) and d = f2(m),
then a transformation will be necessary.
the samples are to be tested;
IS0 4259:1992(E)
Inspection of the figures in table 1 shows that both D and d
The relationships D = fi(m) and d --h(m) will not in general
be identical. The statistical procedures of this International increase with m, the rate of increase diminishing as m
increases. A plot of these figures on log-log paper (i.e. a
Standard require, however, that the same transformation be
graph of log D and log d against log m) shows that the points
applicable both for repeatability and for reproducibility. For
may reasonably be considered as lying about two straight
this reason the two relationships are combined into a single
lines (see figure F. 1 in annex F). From the example
dependency relationship D = f(m) (where D now includes
calculations given in annex F.4, the gradients of these lines
d) by including a dummy variable T. This will take account
are shown to be the same, with an estimated value of 0,638.
of the difference between the relationships, if one exists, and
will provide a means of testing for this difference (see Bearing in mind the errors in this estimated value, the
annex F.l). gradient may for convenience be taken as 2/s.
Hence, the same transformation is appropriate both for
The single relationship D = f(m) is best estimated by
repeatability and reproducibility, and is given by the formula
weighted linear regression analysis. Strictly speaking, an
iteratively weighted regression should be used, but in most
cases even an unweighted regression will give a satisfactory
x-$& = 3x’
. . .
(3)
The derivation of weights is described in
approximation.
I
annex F.2, and the computational procedure for the
regression analysis is described in annex F.3. Typical forms
Since the constant multiplier may be ignored, the
of dependence D = f(m) are given in annex E. 1. These are
transformation thus reduces to that of taking the cube roots
of a single transformation
all expressed in terms
of the reported results (bromine numbers). This yields the
parameter B.
transformed data shown in table 16 (annex D), in which the
cube roots are quoted correct to three decimal places.
The estimation of B. and the transformation nrocedure which
follows, are summarised in annex E.2. a This includes
5.2 Tests for outliers
statistical tests for the significance of the regression (i.e. is
the relationship D = f(m) parallel to the m-axis), and for the
The reported data, or if it has been decided that a
difference between the repeatability and reproducibility
transformation is necessary, the transformed results shall be
relationshms. based at the 5% significance level. If such a
inspected for outliers. These are the values which are so
difference ‘Is found to exist, or ifko suitable transformation
different from the remainder that it can only be concluded
exists, then the alternative methods of IS0 5725 shall be
that they have arisen from some fault in the application of
used. In such an event it will not be possible to test for
the method or from testing a wrong sample. Many possible
laboratory bias over all samples (clause 5.5) or separately
tests may be used and the associated significance levels
estimate the interaction component of variance (clause 6.1).
varied, but those that are specified in the following
sub-clauses have been found to be appropriate in this
If it has been shown at the 5% significance level that there
International Standard. These outlier tests all assume a
is a significant regression of the form D = f(m), then the
normal distribution of errors (see 5.).
appropriate transformation y = F(X), where x is the reported
result, is given by the formula
5.2.1 Uniformity of repeatability
F(x)=K = . . .
(2)
The first outlier test is concerned with detecting a discordant
x
I A)
result in a pair of repeat results. This tesP involves
where K is a constant. In that event all results shall be
calculating the e; over all the laboratory/sample
transformed accordingly and the remainder of the analysis
carried out in terms of the transformed results. Typical combinations. Co&ran ’s criterion at the 1 % significance
transformations are given in annex E. 1.
level is then used to test the ratio of the largest of these values
over their sum (see annex C, clause C.4). If its value exceeds
The choice of transformation is difficult to make the subject the value given in table 17 (annex D), corresponding to one
of formalized rules. Qualified statistical assistance may be
degree of freedom, n being the number of pairs available for
required in particular cases. The presence of outliers may comparison, then the member of the pair farthest from the
affect judgement as to the type of transformation required, sample mean shall be rejected and the process repeated,
if any (see 5.6).
reducing n by 1, until no more rejections are called for. In
certain cases, this test “snowballs” and leads to an
5.1.1 Worked example unacceptably large proportion of rejections, (say more than
10%). If this is so, this rejection test shall be abandoned and
some or all of the rejected results shall be retained. An
table 1 lists the values of m, D, and d for the eight samples
in the example given in annex D, correct to three significant arbitrary decision based on judgement will be necessary in
digits. Corresponding degrees of freedom are in parentheses. this case.
TABLE 1
Sample
Number 3 8 1 4 5 6 2 7
m 0,756 122 2,15 3964 10,9 4892 65,4 114
D 0,0669 (14) 0,159 (9) 0,729 (8) 0211 (11) 0291 (9) 1950 (9) 222 0 2993 0
0,935 (9)
d 0,0500(9) 0,0572(9) 0,127 (9) 0,116 (9) 0,0943(9) 0,527 (9) 0,818 (9)
l
5.2.1.1 WORKED EXAMPLE 5.2.2.1 WORKED EXAMPLE
In the case of the example given in annex D, the absolute The application of Hawkins’ test to cell means within
differences (ranges) between transformed repeat results, i.e. samples is shown below.
of the pairs of numbers in table 16, in units of the third
decimal place, are shown in table 2.
The first step is to calculate the deviations of cell means from
respective sample means over the whole array. These are
TABLE 2
shown in table 3, in units of the third decimal place.
Sample
I
The sum of squares of the deviations are then calculated for
Laboratory I 11 21 31 41
each sample. These are also shown in table 3 in units of the
third decimal place.
A
B
C TABLE 3
D 14 6 0 13 0 9 32
San 1
pie
E 65 4 0 0 14 5 7 28
Laboratory 1 2 3 4 5 6 7 8
F 23 20 34 29 20 30 43 0
G 62 4 78 0 0 16 18 56 15 10 48 3
A 20 8 14 6
H 29 44 0 27 4 32
44 20 B 75 7 20 9 10 47 6
J 0 59 0 40 0 30 26 0
C 64 35 3 20 30 4 22
D 314 33 18 42 7
39 80 50
E 32 32 30 9 7 18 18 39
The largest range is 0,078 for laboratory G on sample 3. The
F 75 97 31 20 30 8 74 53
sum of squares of all the ranges is
G 10 34 32 20 20 61 9 62
0,042* + 0,02 1* + . . . + 0,026* +d = 0,0439.
H 13 4 42 13
42 21 8 50
J 1 28 22 29 14 8
10 53
Thus, the ratio to be compared with Co&ran ’s criterion is
Sum of 117 15 4 6 3 11 13 17
0,078*
Squares
- = 0,138.
0,0439
The cell to be tested is the one with the most extreme
There are 72 ranges and, as from table 17 (annex D), the
deviation. This was obtained by laboratory D from sample 1.
criterion for 80 ranges is 0,1709, this ratio is not significant.
The appropriate Hawkins’ test ratio is therefore :
5.2.2 Uniformity of reproducibility
0,3 14
B*=
= 0,728 1.
The following outlier tests are concerned with establishing
Jo,1 17+0,015 . . . + 0,017
uniformity in the reproducibility estimate, and are designed
to detect either a discordant pair of results from a laboratory
The critical value, corresponding to n = 9 cells in sample 1
on a particular sample or a discordant set of results from a
and v = 56 extra degrees of freedom from the other samples,
laboratory on all samples. For both purposes, the Hawkins’
is interpolated from table 18 (annex D) as 0,3729. The test
test 121 is appropriate.
value is greater than the critical value, and so the results from
laboratory D on sample 1 are rejected.
This involves forming for each sample, and finally for the
overall laboratory averages (see 5.5), the ratio of the largest
As there has been a rejection, the mean value, deviations and
absolute deviation of laboratory mean from sample (or
sum of squares are recalculated for sample 1, and the
overall) mean to the square root of certain sums of squares
procedure is repeated. The next cell to be tested will be that
(see annex C.5).
obtained by laboratory F from sample 2. The Hawkins’ test
ratio for this cell is :
The ratio corresponding to the largest absolute deviation
shall be compared with the critical 1% values given in
0,097
table 18 (annex D), where n is the number of
B*=
= 0,3542.
laboratory/sample cells in the sample (or the number of
J 0,006+0,015 + . . . + 0,017
overall laboratory means) concerned and where v is the
degrees of freedom for the sum of squares which is additional
The critical value corresponding to n = 9 cells in sample 2
to that corresponding to the sample in question. In the test
and v = 55 extra degrees of freedom is interpolated from
for laboratory/sample cells v will refer to other samples, but
table 18 (annex D) as 0,3756. As the test ratio is less than
will be zero in the test for overall laboratory averages.
the critical value there will be no further rejections.
If a significant value is encountered for individual samples,
5.3 Rejection of complete data from a sample
the corresponding extreme values shall be omitted and the
process repeated. If any extreme values are found in the
The laboratories standard deviation and repeats standard
laboratory totals, then all the results from that laboratory
deviation shall be examined for any outlying samples. If a
shall be rejected.
transformation has been carried out or any rejection made,
new standard deviations shall be calculated.
If the test “snowballs ”, leading to an unacceptably large
proportion of rejections (say more than lo%), then this
rejection test shall be abandoned and some or all of the If the standard deviation for any sample is excessively large,
rejected results shall be retained. An arbitrary decision based it shall be examined with a view to rejecting the results from
on judgement will be necessary in this case. that sample.

IS0 4259:1992(E)
The variance ratio is then calculated as
Cochan ’s criterion at the 1% level can be used when the
standard deviations are based on the same number of degrees
(15,26)* / 19,96 = 11,66.
of freedom. This involves calculating the ratio of the largest
of the corresponding sums of squares (laboratories or
From table 20 (annex D) the critical value corresponding to
repeats, as appropriate) to their total (see annex C,
a significance level of 0,01/8 = 0,00125, on 8 and 63 degrees
clause C.4). If the ratio exceeds the critical value given in
of freedom, is approximately 4. This is less than the test
table 17 (annex D), with n as the number of samples and v
ratio, and results from sample 93 shall therefore be rejected.
the degrees of freedom, then all the results from the sample
in question shall be rejected. In such an event care should
Turning to repeats standard deviations, it is noted that
be taken that the extreme standard deviation is not due to the
degrees of freedom are identical for each sample and that
application of an inappropriate transformation (see 5.1), or
undetected outliers. Co&ran ’s test can therefore be applied. Cochran ’s criterion
will be the ratio of the largest sum of squares (sample 93) to
There is no optimal test when standard deviations are based the sum of all the sums of squares, that is
on different degrees of freedom. However the ratio of the
largest variance to that pooled from the remaining samples
2,97*/(1,13* +0,992 + . . . + 1,36*) = 0,510.
follows an F-distribution with v, and v2 degrees of freedom,
(see annex C, clause C.6). Here v1 is the degrees of freedom
This is greater than the critical value of 0,352 corresponding
of the variance in question and v2 is the degrees of freedom
to n = 8 and v = 8 (see table 17, annex D), and confirms that
from the remaining samples. If the ratio is greater than the
results from sample 93 shall be rejected.
critical value given in table 20 (annex D), corresponding to
a significance level of 0,01/S where S is the number of
samples, then results from the sample in question shall be 5.4 Estimating missing or rejected values
rejected.
5.4.1 One of the two repeat values missing or rejected
53.1 Worked example
If one of a pair of repeats (yijl or yijz) is missing or rejected,
The standard deviations of the transformed results, after the this shall be considered to have the same value as the other
rejection of the pair of results by laboratory D on sample 1,
repeat in accordance with the least squares method.
are given in table 4 in ascending order of sample mean,
.
correct to three significant digits. Corresponding degrees of
5.4.2 Both repeat values missing or rejected
freedom are in parentheses.
If both the repeat values are missing, estimates of aii
Inspection shows that there is no outlying sample amongst
(= yGl + yijJ shall be made by forming the laboratories x
these. It will be noted that the standard deviations are now
samples interaction sum of squares, including the missing
independent of the sample means, which was the purpose of
values of the totals of the laboratories/samples pairs of results
transforming the results.
as unknown variables. Any laboratory or sample from which
all the results were rejected shall be ignored and new values
The figures in table 5, taken from a test programme on
of L and S used. The estimates of the missing or rejected
bromine numbers over 100, will illustrate the case of a
values shall then be found by forming the partial derivatives
sample rejection.
of this sum of squares with respect to each variable in turn
and equating these to zero to solve as a set of simultaneous
It is clear, by inspection, that the laboratories standard
equations.
deviation of sample 93 at 1526 is far greater than the others.
It is noted that the repeats standard deviation in this sample
Formula (4) may be used where only one pair sum has to be
is correspondingly large.
estimated. If more estimates are to be made, the technique
of successive approximation can be used. In this, each pair
Since laboratory degrees of freedom are not the same over
sum is estimated in turn from formula (4), using L,, S, and
all samples, the variance ratio test is used. The variance
T, values which contain the latest estimates of the other
pooled from all samples excluding sample 93 is the sum of
missing pairs. Initial values for estimates can be based on
the sums of squares divided by the total degrees of freedom,
the appropriate sample mean, and the process usually
that is
converges to the required level of accuracy within three
(8 x 5,ld + 9 x 4,2d + . . . + 8 x 3,85*) = 19 96
complete iterations. See, for instance, reference [5] for
9 l
(8 + 9 + . . . + 8)
details.
TABLE 4
Sample number 3 8 1 4 5 6 2 7 ’
Sample mean 0,910 0 1,066 1,240 1,538 2,2 17 3,639 4,028 4,85 1
Laboratories standard deviation 0,0278( 14) 0,0473(9) 0,0354( 13) 0,0297( 11) 0,O 197(9) 0,0378(9) 0,0450(9) 0,04 16(9)
r
. Repeats standard
deviation .0,02 14(9) . 0,O 182(9) .0,028 l(8) . o,o w9) .0,0063(9) . 0,O 132(9) . 0,O 166(9) . 0,O 130(9) .
TABLE 5
Sample number 90 89 93 9
...