CEN/TR 16886:2016
(Main)Guidance on the application of statistical methods for determining the properties of masonry products
Guidance on the application of statistical methods for determining the properties of masonry products
In the masonry unit standards and in national legislation, some properties need to be declared based on a certain fractile and confidence level. To demonstrate compliance with that a statistical tool can be used.
The purpose of this Technical Report is to exemplify how a statistical tool can be used in practice. This document should not contradict nor extend the scope of the work and role of a Notified Body, nor impose additional burdens on the manufacturer, beyond those laid down in the Construction Products Regulation and the product standards.
Mechanical and other properties of building materials and components are in the report described by random variables with a certain type of probability distribution. The popular normal distribution (Laplace-Gauss distribution) is given in Annex A. Normal distribution may be used to approximate many actual symmetrical distributions. When a remarkable asymmetry is observed, then another type of distribution reflecting this asymmetry should be considered, leading to a more complex method to demonstrate compliance with the product standard. More information on the normality test of Shapiro-Wilk is given in Annex D.
Leitfaden für die Anwendung statistischer Methoden zur Bestimmung der Eigenschaften von Mauerwerk Produkten
Guide pour l'application de méthodes statistiques pour la détermination des propriétés des éléments de maçonnerie
Navodilo za uporabo statističnih metod za določanje lastnosti zidarskih proizvodov
V standardih za zidake in nacionalni zakonodaji morajo biti na podlagi določenega kvantila in stopnje zaupanja navedene nekatere lastnosti. Za dokazovanje skladnosti je mogoče uporabiti statistično orodje.
Namen tega tehničnega poročila je ponazoriti, kako je mogoče uporabiti statistično orodje v praksi. Ta dokument naj ne bi nasprotoval ali razširjal področja uporabe in vloge priglašenega organa niti nalagal dodatnih obremenitev proizvajalcu, razen tistih iz Uredbe o gradbenih proizvodih in standardov za proizvode.
Mehanske in druge lastnosti gradbenih materialov ter komponent so v poročilu opisane z naključnimi spremenljivkami z določeno vrsto verjetnostne porazdelitve. Priljubljena normalna porazdelitev (Gauss-Laplaceova porazdelitev) je podana v dodatku A. Normalna porazdelitev se lahko uporablja za določanje približka številnih dejanskih simetričnih porazdelitev. V primeru izredne nesimetričnosti je treba upoštevati drugo vrsto porazdelitve, ki odraža to nesimetričnost, kar vodi do bolj zapletene metode za dokazovanje skladnosti s standardom za proizvod. Več informacij o Shapiro-Wilkovem preskusu normalnosti je podanih v dodatku D.
General Information
Standards Content (Sample)
SLOVENSKI STANDARD
01-marec-2017
1DYRGLOR]DXSRUDERVWDWLVWLþQLKPHWRG]DGRORþDQMHODVWQRVWL]LGDUVNLK
SURL]YRGRY
Guidance on the application of statistical methods for determining the properties of
masonry products
Leitfaden für die Anwendung statistischer Methoden zur Bestimmung der Eigenschaften
von Mauerwerk Produkten
Guide pour l'application de méthodes statistiques pour la détermination des propriétés
des éléments de maçonnerie
Ta slovenski standard je istoveten z: CEN/TR 16886:2016
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
91.080.30 Zidane konstrukcije Masonry
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
CEN/TR 16886
TECHNICAL REPORT
RAPPORT TECHNIQUE
November 2016
TECHNISCHER BERICHT
ICS 03.120.30; 91.100.25
English Version
Guidance on the application of statistical methods for
determining the properties of masonry products
Guide pour l'application de méthodes statistiques pour Leitfaden für die Anwendung statistischer Methoden
la détermination des propriétés des éléments de zur Bestimmung der Eigenschaften von Mauerwerk
maçonnerie Produkten
This Technical Report was approved by CEN on 24 August 2015. It has been drawn up by the Technical Committee CEN/TC 125.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
CEN-CENELEC Management Centre: Avenue Marnix 17, B-1000 Brussels
© 2016 CEN All rights of exploitation in any form and by any means reserved Ref. No. CEN/TR 16886:2016 E
worldwide for CEN national Members.
Contents Page
European Foreword .3
Introduction .4
1 Scope .5
2 Normative references .5
3 Terms, definitions and symbols .5
3.1 Terms and definitions .5
3.2 Symbols .8
4 General .9
5 Statistical evaluation . 10
5.1 Factory production control . 10
5.2 Finished product testing . 10
5.2.1 General . 10
5.2.2 Inspection lot. 11
5.2.3 Spot sampling and sample sizes . 11
5.2.4 Production types . 12
5.2.5 Control method A: Batch control . 12
5.2.6 Control method B:'Rolling' inspection . 13
5.2.7 Evaluation of test results . 15
5.2.8 How to come from unknown to known standard deviation? . 18
5.2.9 Conformity . 18
5.2.10 Simple and conservative approach . 19
5.2.11 Non-conforming products . 19
5.2.12 Guidance . 19
6 Product type determination . 22
Annex A (informative) Normal distribution (Laplace-Gauss distribution) . 23
Annex B (informative) Tables for acceptance coefficient k depending on the used
n
fractile p and confidence level γ (taken from ISO 16269-6:2005) . 24
Annex C (informative) Examples of statistical evaluation . 40
C.1 Example 1 . 40
C.2 Example 2 . 42
C.3 Example 3 . 44
C.4 Example 4 . 45
C.5 Example 5 . 47
Annex D (informative) Normality test of SHAPIRO – WILK . 50
D.1 General . 50
D.2 Normality test of SHAPIRO-WILK . 50
Bibliography . 54
European Foreword
This document (CEN/TR 16886:2016) has been prepared by Technical Committee CEN/TC 125
“Masonry”, the secretariat of which is held by BSI.
Attention is drawn to the possibility that some of the elements of this document may be the
subject of patent rights. CEN shall not be held responsible for identifying any or all such patent
rights
This document has been prepared under a mandate given to CEN by the European Commission
and the European Free Trade Association.
Introduction
This document is informative for the guidance of manufacturers and Notified Bodies (NBs), who
want to use statistical methods for the evaluation of conformity and Factory Production Control
of masonry products. Its use is optional. Other statistical methods and non-statistical methods
may be used.
Quality control of building materials and components is an indispensable part of an overall
concept of structural reliability. As quality control is generally a time-consuming and expensive
task, various operational techniques and activities have been developed to fulfil safety
requirements in buildings. Properly employed statistical methods are one way to provide
efficient, economic and effective means of quality control.
Background: “The terms and definitions in EN 1990 (Eurocode: Basis of structural design) are
derived from ISO 2394 (General principles on reliability for structures). For the design of
structures, EN 1996-1-1 (Eurocode 6: Design of masonry structures — Part 1-1: General rules for
reinforced and unreinforced masonry structures) is intended to be used together with EN 1990.
ISO 12491 (Statistical methods for quality control of building materials and components) gives
general principles for the application of statistical methods for the quality control of building
materials and components, in compliance with the safety and serviceability requirements of
ISO 2394. ISO 12491 is applicable to all buildings and other civil engineering works, existing or
under construction, whatever nature or combination of materials used, e.g. concrete, steel, wood,
bricks. The EN 771 series specifies that one method of satisfying the conformity criterion laid
down in these product standards is to use the approach given in ISO 12491.”
This Technical Report gives guidance on how a statistical evaluation can be put into practice
based on the background of ISO 12491.
A simplified method is also given based on information obtained from practice about the
possible distribution in production for specific product characteristics.
The method may also be used for the evaluation of different properties at the different stages of
the factory production control (FPC) with the aim to minimize testing costs for the manufacturer
and to ensure that the requirements are fulfilled.
Detailed examples are given in Annex C. For other more sophisticated techniques and specific
problems, other international standards can be applied.
The initial draft of this document was prepared by the joint working group CEN/TC 125/TG 5
and the Sector Group 10 of Notified Bodies under the Construction Products Directive. The
CEN/TR is a tool available for manufacturers and Notified Bodies.
It is laid down in the hEN’s of masonry products that the manufacturer should demonstrate
compliance for his product with the requirements of the harmonized product standards.
The purpose of this Technical Report is to put statistical evaluation into practice. Detailed
examples are given in the annexes.
1 Scope
In the masonry unit standards and in national legislation, some properties need to be declared
based on a certain fractile and confidence level. To demonstrate compliance with that a
statistical tool can be used.
The purpose of this Technical Report is to exemplify how a statistical tool can be used in
practice. This document should not contradict nor extend the scope of the work and role of a
Notified Body, nor impose additional burdens on the manufacturer, beyond those laid down in
the Construction Products Regulation and the product standards.
Mechanical and other properties of building materials and components are in the report
described by random variables with a certain type of probability distribution. The popular
normal distribution (Laplace-Gauss distribution) is given in Annex A. Normal distribution may
be used to approximate many actual symmetrical distributions. When a remarkable asymmetry
is observed, then another type of distribution reflecting this asymmetry should be considered,
leading to a more complex method to demonstrate compliance with the product standard. More
information on the normality test of Shapiro-Wilk is given in Annex D.
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.
EN 1990, Eurocode - Basis of structural design
EN 1996 (all parts), Eurocode 6 — Design of masonry structures
3 Terms, definitions and symbols
For the purposes of this document, the following terms, definitions and symbols apply.
3.1 Terms and definitions
3.1.1
unit
defined quantity of building material, component or element that can be individually considered
and separately tested
3.1.2
population
totality of units under consideration
3.1.3
variable
X
variable which can take any of the values of a specified set of values and with which is associated
a probability distribution
3.1.4
probability distribution
function which gives the probability that a variable X takes any given value (in the case of a
discrete variable) or belongs to a given set of values (in the case of a continuous variable)
3.1.5
distribution function
Π(x)
function giving, for every value of x, the probability that the variable X is less than or equal to x:
P=(x) PX,(≤ x)
3.1.6
(probability) density function
f(x)
derivative (when it exists) of the distribution function
3.1.7
parameter (population)
quantity used in describing the distribution of a random variable in a population
3.1.8
fractile
x
if X is a continuous variable and p is a real number between 0 and 1, the p-fractile is the value of
a variable X for which the distribution function equals p
Note 1 to entry: Thus x is a fractile if P,(X ≤ x ) = p
p p
3.1.9
population mean
μ
for a continuous variable X having the probability density f(x), the mean, if it exists, is given by:
µ ×f ()x dx
∫
the integral being extended over the interval(s) of variation of the variable X
3.1.10
population variance
σ
for a continuous variable X having the probability density function f(x), the variance, if it exists,
is given by
σµ(x− ) f ()x dx
∫
the integral being extended over the interval(s) of variation of the variable X
3.1.11
population standard deviation
σ
positive square root of the population variance σ
=
=
3.1.12
normal distribution
probability distribution of a continuous variable X, the probability density function of which is
11 x−µ
fx( ) exp−
2 σ
σp2
3.1.13
random sample
one or more sampling units taken from a population in such a way that each unit of the
population has the same probability of being taken
3.1.14
sample size
n
number of sampling units in the sample
3.1.15
sample mean
x
m
sum of n values x of sampling units divided by the sample size n
i
Xi
∑
xm=
n
3.1.16
sample variance
s
sum of n squared deviations from the sample mean x divided by the sample size n minus 1
m
Σ−()X i xm
s =
n−1
3.1.17
sample standard deviation
s
positive square root of the sample variance s
3.1.18
estimation
operation of assigning, from observations on a sample, numerical values to the parameter of a
distribution chosen as the statistical model of the population from which this sample was taken
3.1.19
estimator
function of a set of the sample random variables used to estimate a population parameter
3.1.20
estimate
value of an estimator obtained as a result of an estimation
=
3.1.21
confidence level
γ
given value of the probability associated with a confidence interval
3.1.22
lot
definite quantity of units, manufactured or produced under the same conditions which are
presumed uniform
3.1.23
isolated lot
lot separated from the sequence of lots in which it was produced or collected, and not forming
part of a current sequence of inspection lots
3.1.24
conforming unit
unit which satisfies all the specified requirements
3.1.25
non-conforming unit
unit containing at least one non-conformity which causes the unit not to satisfy specified
requirements
3.1.26
sampling inspection
inspection in which decisions are made to accept or not accept a lot, based on results of a sample
selected from that lot
3.1.27
sampling plan
plan in accordance with which one or more samples are taken in order to obtain information
and the possibility of reaching a decision concerning the acceptance of the lot
3.2 Symbols
k is the acceptance coefficient
n
k is the acceptance coefficient one-sided tolerance interval
k is the acceptance coefficient two-sided tolerance interval
k is the corrected acceptance coefficient
c
k is the acceptance coefficient for known standard deviation
k
k is the acceptance coefficient for unknown standard deviation
u
n is the number of test samples within the spot sample
x is the mean test result
m
x is the test result for test sample i
i
i is the number of the individual test sample
x is the test result of the estimated normal distribution of the spot sample
est
s is the standard deviation of the test results
s is the standard deviation of the test results of a spot sample
s
σ is the known standard deviation
l is the number of inspection lots
λ is the thermal conductivity of the unit
10,dry,unit
p is the fractile
γ is the confidence level
4 General
It is specified in the product standards that the manufacturer should demonstrate compliance
for his product with the requirements of the relevant European standard and with the declared
values for the product properties by carrying out both:
a) product type determination, which can be type testing, type calculation, reference to
tabulated values or descriptive documentation of the product;
b) factory production control (FPC).
If a manufacturer of masonry elements intends to declare that the units are Category I units,
then the units shall fulfil the definition of Category I units which is 'Units with a declared
compressive strength with a probability of failure to reach it not exceeding 5 %', which means
that the manufacturer is declaring that the customer can be 95 % confident that the delivered
units fulfilled the declared compressive strength. To be able to demonstrate this, the
manufacturer can operate a FPC that includes a statistical evaluation.
The confidence level for a property shall be fixed depending on how important the property is in
a building. The higher the confidence level is the lower is the risk that the product does not fulfil
the declared values. When dealing with the safety of a building it is necessary to presuppose a
minimum confidence level fulfilled by the used products, otherwise the partial safety factors
cannot be fixed.
Confidence levels other than 95 % can be used, e.g. the safety system specified in EN 1990 to
which the Eurocode for masonry (EN 1996 series) refers to for safety aspects, is based on the
assumption that declared values for the used product properties fulfil a confidence level of 75 %.
For characteristics, where a certain minimum confidence level is not fixed in a technical
specification or in a contract to be fulfilled, the manufacturer is free to fix the confidence level he
will operate with, and the higher the chosen level is the lower the risk that the manufacturer is
running that the delivered products do not fulfil the declared values. The risk the manufacturer
is running is fixed by a combination of the actual variation in test results over time, the
frequencies of checking and testing, the way the FPC system is developed and how close the
declared value is to the tested values.
In the product standard the conformity criteria are related to a 'consignment', that is a delivery
to a building site. The product standard defines a declared value as a value that the
manufacturer is confident in achieving, bearing in mind the precision of the tests and the
variability of the production process, and when the declared values are accompanying the
product to the building site, they are valid for the delivered consignment. Since it is impractical
to test each consignment, the manufacturer should plan the FPC system in such a way that the
effect of the variations of product characteristics during the production is taken into account
when declaring the characteristics for the consignment. In some production processes products
are naturally separated into batches and a consignment is quite often only a part of a batch. If a
production is based on a continuous flow a consignment is only a part of the continuous
production.
5 Statistical evaluation
5.1 Factory production control
The FPC system can be developed in such a way that the checking procedures are:
— mainly related to the process only (full process control and consequently only a small
amount of finished product testing); or
— mainly related to the finished products only (and consequently limited process control); or
— a combination of both.
It can even be so that the amount of process control and finished product testing varies
depending on the property to be assessed. If the test for the property is low cost, e.g. a test of
dimensions, and if the property is less important in relation to the end use then it might be the
right solution to use finished product testing. But if the testing of the property is expensive, e.g.
frost resistance tests, then the solution might be to base the assessment on process control using
proxy tests.
The manufacturer defines the product groups. A product group consists of products from one
manufacturer having common values for one or more characteristics. That means that the
products belonging to a product group might differ according to the characteristics in question.
If a product group is defined, then the FPC system should ensure that all types of units within a
group are controlled and over time also in the finished product testing, if that is part of the FPC.
Depending on the way the FPC system is developed (mainly related to process control only,
mainly related to finished product testing only or a combination of both) a selection of these
should be considered.
Samples, taken during the process and finished product samples need to be representative for
the inspection lot. For that reason the sampling procedure is important and so should be
specified. When the frequency of testing is fixing the size of the inspection lot and thereby the
manufacturer’s risk the frequency should be carefully considered, decided and recorded. If test
results and FPC system give evidence of problems then the frequencies can be reconsidered and
reduced compared to the ones used.
5.2 Finished product testing
5.2.1 General
When testing the finished product in FPC, it is possible to use alternative test methods if a
correlation can be established between the alternative test method and the reference test
method or if a safe relationship can be demonstrated when using the alternative method
compared to the reference method.
It is also important to notice that a test result of a spot sample (see 5.2.3) is representing an
inspection lot (see 5.2.2). If an evaluated test result is not conforming, the whole production
since the last test should be looked upon as non-conforming. For that reason it can be
recommended that for properties where the reference test is time consuming and might be
costly, alternative tests or proxy tests that are less time consuming and costly are used. By doing
so the time span between the tests can be shortened and the amount of products covered by a
non-conforming test result will be less and thereby reduce the manufacturer’s risk.
The amount of products produced between two tests is an inspection lot. The frequency of
testing can vary from one property to another and thereby the inspection lot can vary from one
property to another.
5.2.2 Inspection lot
The production is divided into inspection lots.
An inspection lot shall consist of units produced under uniform conditions:
• same raw materials;
• same dimensions;
• same production process.
If a certain characteristic is the same for multiple units, where the dimension has no influence,
these units can belong to the same product family.
This means that an inspection lot for the characteristic in question can only consist of products
belonging to the same product group.
The manufacturer decides on the size of the inspection lot from:
• raw material mixing lots; or
• number/volume of units; or
• number of production days.
Independent of the way the size of the inspection lot is decided, it shall be possible to draw a
representative spot sample.
5.2.3 Spot sampling and sample sizes
When the inspection lot has been decided, the sampling procedure for a spot sample shall be
fixed in such a way that the spot sample is representative for the inspection lot as shown in the
example of Figure 1.
Figure 1 — An example of representative sampling
Sampling procedures for stacks and banded packs are given in the European product standard. It
is also possible to sample from the conveyer belt or, in the case of fired units, after the kiln.
The number of units in the spot sample is decided by the manufacturer. If a minimum number of
units has been fixed then this should be accepted.
By deciding on the size of the inspection lot the manufacturer is fixing the frequencies of tests to
be done. The size of the inspection lot should be decided based on:
• how close the declared value is to the test value;
• the deviation of the test values;
• how much process control is going on.
These decisions allow the manufacturer to manage their own risks.
5.2.4 Production types
A production, which is naturally separated into batches, is named a batch production. In the case
of the batch production the properties of the units may change batch by batch. A batch is
normally looked upon as a separate inspection lot. If the process control minimizes the changes
from one batch to another, an inspection lot can cover more than one batch. An example of a
batch production is shown in Figure 2.
Figure 2 — Example of batch production
A production, which is based on a continuous flow, is named a series production. An example of
a series production is given in Figure 3. In the case of series production the properties of the
units are the same within a series. A series production usually contains more than one
inspection lot.
Figure 3 — Example of series production
5.2.5 Control method A: Batch control
When a batch production is in operation, then the FPC system needs to be based on a batch
control, which means, that each batch is controlled separately as shown in Figure 4.
When dealing with the evaluation of test results, the acceptance coefficient k is given in Tables
n
1 and 2 (5.2.7). These tables show that there is a great difference in using k for three or for six
n
test results and for that reason it is recommended to operate with spot sample sizes of at least
six units.
Figure 4 — Example of Method A: Each inspection lot is evaluated individually
5.2.6 Control method B:'Rolling' inspection
In a series production there are a series of inspection lots, which should not exceed a total
number of five. In the example in Figure 5 four are used.
Figure 5 — Example with 4 inspection lots in a series
For the first inspection lot a spot sample size of three is taken and tested. For the second
inspection lot three new samples are taken, tested and evaluated together with the ones from
the first inspection lot and therefore the spot sample size will be six. For the third inspection lot
three new samples are taken, tested and evaluated together with the ones from the first and
second inspection lot and therefore the spot sample size will be nine. For the fourth inspection
lot three new samples are taken and tested and evaluated together with the ones from the first
three inspection lots and therefore the spot sample size will be 12. For the fifth inspection lot
three new samples are taken, tested and evaluated together with the ones from the second, third
and fourth inspection lots and therefore the spot sample size will be 12. The described rolling
system will continue for the following inspection lots. The rolling system is illustrated in
Figure 6. When dealing with the evaluation of test results the acceptance coefficient k is given in
n
Tables 1 and 2 (5.2.7). These tables show that there is a great difference for 6 and 12 test results,
and the number of tests to be done is half compared to the batch control when the size of the
inspection lot is the same. Another possibility is to half the size of the inspection lot and
therefore to reduce the number of units covered by non-conformity, if that occurs.
Figure 6 — Example of method B,'Rolling' inspection: series of four inspection lots
Another possibility is the so-called 'progressive' sampling procedure (see Figure 7). For each of
the first to fifth inspection lots a spot size of one sample is taken and tested. These lots are
evaluated together. For the sixth and following inspection lots one additional sample is taken,
tested and evaluated together with the ones from the previous inspection lots. The spot size is
gradually increased from 5 to 15 samples.
From then on, one additional sample is taken from each next inspection lot but the spot sample
is limited to the last 15 samples. The spot sample size continues to be 15.
Figure 7 — Example of method B,'Rolling' inspection 'Progressive' sampling: series of 15
inspection lots
5.2.7 Evaluation of test results
Where and when possible and applicable, the results of the checks and testing should be
interpreted by means of statistical techniques, by attributes or by variables to verify the product
characteristics and to determine if the production conforms to the compliance criteria and the
products conform to the declared values. One method of satisfying this conformity criterion is to
use the approach given in ISO 12491. This approach is shown in detail in this subclause.
When using the test results of a spot sample with a limited number of samples to estimate the
characteristics of the production there are some uncertainties. The deviation within the test
results is one uncertainty and how representative the spot sample is for the production is
another uncertainty. The first uncertainty is dealt with in the evaluation by taking into account
the standard deviation s of the test results of the spot sample. The second uncertainty is dealt
with by using an acceptance coefficient k . The acceptance coefficient k can be regarded as a
n n
factor minimizing the statistical uncertainties from spot sampling. k is dependent on several
n
factors:
— the number of samples in the inspection lot n;
— the confidence level γ;
(a)
— the fractile p ;
— the standard deviation is unknown. The symbol used is k ;
u
— the standard deviation is known. The symbol used is k ;
k
— one-sided limit evaluation. The symbol used is k ;
— two-sided limit evaluation. The symbol used is k .
When evaluating the test results from a spot sample, the following procedure should be used:
Calculate the mean value of the test results using Formula (1):
n
xx= (1)
mi∑
n
i=1
where
x is the mean test result
m
x is the test result for test sample i
i
n is the number of test samples within the spot sample
i is the number of the individual test sample
Calculate the standard deviation s for the test results of the spot sample using Formula (2):
s
n
xx−
( )
∑
i m
i=1
(2)
s=
n−1
where
s is the standard deviation for the test results
n is the number of test samples within the spot sample
i is the number of the individual test sample
x is the test result for test sample i
i
x is the mean test result
m
The mean test result x and the calculated standard deviation of the test results s are the specific
m
values of the corresponding estimator of the population mean μ and standard deviation σ.
Be aware that a 5 % characteristic value corresponds with a fractile P = 95 and a 95 %
characteristic value also corresponds with a fractile P = 95. A 50 % characteristic value
corresponds with a fractile P = 50.
If the standard deviation is unknown and if the test results shall be compared with a lower limit
value then calculate the test result of the estimated normal distribution x using Formula (3):
est
Xest=XKm− 1,u×Ss (3)
If the standard deviation is unknown and if the test results shall be compared with an upper
limit value then calculate the test result of the estimated normal distribution xest using
Formula (4):
Xest XKm+ 1,u×Ss (4)
If the standard deviation is unknown and if the test results shall be compared with a two-sided
limit value then calculate the test result of the estimated normal distribution x using
est
Formula (5):
(5)
Xest=XKm±×2,u Ss
If the standard deviation σ is known and if the test results shall be compared with a lower limit
value then calculate the test result of the estimated normal distribution x using Formula (6):
est
Xest=XKm−×1, k σ (6)
If the standard deviation σ is known and if the test results shall be compared with an upper limit
value then calculate the test result of the estimated normal distribution x using Formula (7):
est
Xest=XKm+×1, k σ (7)
=
If the standard deviation σ is known and if the test results shall be compared with a two-sided
upper limit value then calculate the test result of the estimated normal distribution x using
est
Formula (8):
(8)
Xest=XKm±×2, k σ
where
x is the test result of the estimated normal distribution of the spot sample
est
x is the mean test result
m
k is the acceptance coefficient for unknown standard deviation and one-sided limit
1,u
evaluation to be taken from Table 1 or 2 or the relevant tables in Annex B
k2,u is the acceptance coefficient for unknown standard deviation and two-sided limit
evaluation to be taken from the relevant tables in Annex B
s is the standard deviation for the test results of the spot sample
s
k is the acceptance coefficient for known standard deviation and one-sided limit
1,k
evaluation to be taken from Table 1 or 2 or the relevant tables in Annex B
k is the acceptance coefficient for known standard deviation and two-sided limit
2,k
evaluation to be taken from the relevant tables in Annex B
σ is the known standard deviation
Table 1 — k for 50 % characteristic value (50 % fractile) and 95 % confidence level
Standard n=3 4 5 6 7 8 9 10 11 12 14 15
deviation
Unknown 1,69 1,18 0,95 0,82 0,74 0,67 0,62 0,58 0,55 0,52 0,47 0,46
Known 0,95 0,82 0,74 0,67 0,62 0,58 0,55 0,52 0,50 0,48 0,44 0,43
Table 2 — k for 5 % characteristic value (95 % fractile) and 95 % confidence level
Standard n=3 4 5 6 7 8 9 10 11 12 14 15
deviation
Unknown 7,66 5,14 4,20 3,71 3,40 3,19 3,03 2,91 2,82 2,74 2,62 2,57
Known 2,60 2,47 2,38 2,32 2,27 2,23 2,19 2,17 2,14 2,12 2,09 2,07
More tables for the acceptance coefficients k and k , depending on the used fractile p (50 %,
1 2
75 %, 90 %, 95 %) and the used confidence level γ (50 %, 75 %, 90 %, 95 %) for known and
unknown standard deviation are given in Annex B.
The method of using the acceptance coefficient for known standard deviation k is only valid
k
when the standard deviation s of the spot sample corresponds to Formula (9):
s
0,63 σ ≤ s ≤ 1,37 σ (9)
s
If, as part of the evaluation, it turns out that s > 1,37 σ, the manufacturer shall either restart or
s
continue working with the unknown acceptance coefficient k . This means that the inspection
u
lots shall be treated separately.
If, as part of the evaluation, it turns out that s < 0,63 σ, the producer may decide:
s
- to restart, or
- to continue working with the unknown acceptance coefficient k , or
u
- to continue working with the known acceptance coefficients, knowing he is evaluating on
the safe side
5.2.8 How to come from unknown to known standard deviation?
In control method A (5.2.5) the standard deviation of the population is considered to be
unknown at least for the first 40 test samples and the acceptance coefficient k is taken from
u
tables for unknown standard deviation. For the next 80 test samples the standard deviation can
be considered to be known, but the used acceptance coefficient is corrected (k ). The acceptance
c
coefficient for the known standard deviation k is taken from tables for known standard
k
deviation. The corrected acceptance coefficient k is calculated by a linear interpolation between
c
the acceptance coefficient k and k . The known standard deviation σ is calculated based on at
u k
least the first 40 test results.
In control method B (5.2.6) the standard deviation of the population is considered to be
unknown at least for the first 20 test samples and the acceptance coefficient k is taken from
u
tables for unknown standard deviation. For the next 40 test samples the standard deviation can
be considered to be known, but the used acceptance coefficient is corrected (k ) as mentioned in
c
the previous section. The acceptance coefficient for the known standard deviation k is taken
k
from tables for known standard deviation. The known standard deviation σ is calculated based
on at least the first 20 test results.
If 'progressive sampling' is used the standard deviation of the population is considered to be
unknown for at least the first 30 test samples and the acceptance coefficient k is taken from
u
tables for unknown standard deviation. For the next 30 test samples the standard deviation can
be considered to be known, but the used acceptance coefficient is corrected (kc) as mentioned in
is
the first section of this clause. The acceptance coefficient for the known standard deviation kk
taken from tables for known standard deviation. The known standard deviation σ is calculated
based on at least the first 30 test results.
5.2.9 Conformity
After calculating x by testing the inspection lots the result shall be compared with either the
est
declared value (DV) or a lower (LL) or upper limit (UL) depending on the property.
For instance:
- for compressive strength it is the declared value or the lower limit. The declared value
needs to be equal to or lower than the lower limit value.
- for process control properties it can be the upper and lower declared value or the upper and
lower limit.
- for thermal values it is the declared value or the upper limit. The declared value needs to be
equal to or higher than the lower limit value.
5.2.10 Simple and conservative approach
A simple and conservative approach can be to evaluate single test results for at least one year for
a given property and calculate the mean value and the standard deviation, then fix a band in
which new test results shall fit in. The upper band limit and lower band limit can then be two
times the standard deviation away from the mean value. Then the declared value is
recommended to be 0,4 times the standard deviation away from the respective band limits. If
non-conformity occurs the evaluation of at least the last year of single test results, including the
non-conforming values, should be repeated and the band limit values adjusted accordingly. The
same should happen for the declared value. The non-conforming inspection lot can be treated as
described in 5.2.11 using control method A.
5.2.11 Non-conforming products
When an evaluation of the test results of the last spot sample is leading to non-conformity, it is
important to avoid the whole inspection lot being mixed up with the other inspection lots. The
non-conforming inspection lot shall be treated separately. It may be reclassified by the
manufacturer and given different declared values. If it is not segregated the whole stock shall be
treated as non-conforming. For that reason a procedure for dealing with non-conforming
products should be developed.
It should be in the interest of the manufacturer to avoid that the same non-conformity occurs
again. When non-conformity occurs, it is important to try to identify the reason why, otherwise
it is difficult to find out what to do to avoid it happening again. Testing can be part of the
identification.
To ensure that the personnel managing the production know what to do when check and
measuring values are passing the limit values, it is important to have the necessary instructions
documented.
Non-conformities will normally result in higher frequencies than the ones used. The background
for that is to reduce the size of the next batch that might also not comply.
5.2.12 Guidance
5.2.12.1 How to use the different possibilities?
A manufacturer is producing units in two different ways:
a) Product 1 is a special unit produced very rarely and only in small quantities. The
characteristics of the product can vary from production to production.
b) Product 2 is one of the core units of the production site. It is produced in a series of variable
length – sometimes only two days of production – but it is produced within short-time
intervals.
For product 1 it is obvious to use control method A (batch control). For product 2 both
control methods A and B can be used. For product 2 it is even possible to use control method
A for some properties and control method B for other properties. If using method B a re-
declaration in connection with a non-conformity is possible based on test results obtained
by testing a new spot sample taken at random from the inspection lot following control
method A. However it is necessary to keep the test results leading to the non-conformity in
the method B control system when evaluating the next spot sample.
The following details can be used when planning the setup of the FPC system:
Control method A:
1) verification of separate inspection lots.
2) inspection lots are defined to be the full prod
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