GNB-CPD
SG10
NB-CPD/SG10/12/091
Guidance from the Group of Notified Bodies
for the Construction Products Directive
89/106/EEC
Issued: 9 March 2012
APPROVED GUIDANCE
GNB-CPD position paper from SG10 - EN 771-1 to EN 771-6
Evaluation of conformity for masonry units
General scope, limitations and aim of this guidance for notified bodies
This position paper contains guidance for notified bodies (NBs) involved in the attestation of
conformity of FPC of masonry units according to EN 771-1 to EN 771-6. The purpose is to help
NBs work equivalently and come to common judgments. This guidance contains informative
material (which NBs should or may follow) and/or normative guidance (which NBs shall follow or at
least work equivalently to as circumstances demand).
The primary document for NBs is the edition of the relevant harmonized standard that is currently
cited in the Official Journal of the EU to which the manufacturer works. This guidance is thought
necessary to provide clarity and completeness for NBs so that they can work equivalently. It
supplements and makes practical for NBs the harmonized standards EN 771-1 to EN 771-6,
approved AG guidance, and Standing Committee guidance in the form of GPs, which also apply unless otherwise explicitly stated in this guidance. This position paper should not contradict nor
extend the scope of the work and role of a NB, nor impose additional burdens on the manufacturer,
beyond those laid down in the CPD and EN 771-1 to EN 771-6.
This guidance should be considered valid until the relevant standards are amended to include the
guidance (as thought fit by the CEN/TC); or until guidance from Commission, SCC or AG has
changed on relevant matters. Whereupon, the paper should be considered for withdrawal/revision
and be replaced by new guidance as necessary.
This position paper was considered approved by SG10 on 30 September 2011 and by Advisory
Group on 22 February 2012.
This position paper was developed primarily by CEN/TC 125 ‘Masonry’, to offer a statistical method
for the evaluation of conformity of masonry units. It has been published as a SG10 position paper to
enable it to be made available as soon as possible. It is expected that it will subsequently be
published as a CEN Technical Report.
NB-CPD/SG10/03/006r2
Page 1 of 74
Contents
1 Foreword ..................................................................................................................................................... 3 2 Symbols ...................................................................................................................................................... 3 3 Reference list .............................................................................................................................................. 4 4 General ....................................................................................................................................................... 4 5 Factory production control .......................................................................................................................... 5 5.1 5.2 5.3 5.4 5.5 5.6 General ................................................................................................................................................................. 5 Testing and measuring equipment ....................................................................................................................... 6 Production equipment ........................................................................................................................................... 6 Raw materials ....................................................................................................................................................... 6 Production process ............................................................................................................................................... 7 Finished product testing ....................................................................................................................................... 8 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5 5.6.6 5.6.7 5.6.8 5.6.9 5.6.10 5.6.11 5.6.12 6 Inspection lot ............................................................................................................................................................................ 9 Spot sampling and sample sizes .............................................................................................................................................. 9 Production types ..................................................................................................................................................................... 11 Method A: Batch control ......................................................................................................................................................... 11 Method B: ”Rolling” inspection ............................................................................................................................................... 12 Evaluation of test results ........................................................................................................................................................ 14 How to come from unknown to known standard deviation? ................................................................................................... 18 Conformity .............................................................................................................................................................................. 19 A simple and conservative approach ..................................................................................................................................... 24 Non-conforming products ....................................................................................................................................................... 24 Guidance ................................................................................................................................................................................ 25 Records .................................................................................................................................................................................. 28 Initial type tests ......................................................................................................................................... 28 Annex A Tables for acceptance coefficient kn depending on the used fractile p and confidence level γ
(taken from ISO 16269-6 (2005)) ................................................................................................... 30 Annex B Examples of statistical evaluation ................................................................................................... 46 NB-CPD/SG10/03/006r2
Page 2 of 74
1
Foreword
By agreement with CEN/ TC 125, SG10 has prepared this position paper to have a tool available for
Notified Bodies (NBs) and manufacturers.
It is laid down in the hENs that the manufacturer shall demonstrate compliance for his product with
the requirements of the harmonised standards EN 771-1 to EN 771-6.
The purpose of this guidance document is to put statistical evaluation into practice. It can be used
for the evaluation of different properties at the different stages of the FPC with the aim to minimise
testing costs for the manufacturer and to ensure that the requirements are fulfilled. Detailed
examples are given in the Annexes.
To maintain equivalent use and interpretation of this document, notified bodies are strongly invited
to raise any questions, remarks or problems related to the use of this document with the secretariat
of the NB-CPD/SG10. The address of the secretariat can be found in GNB-CPD Monitoring report
NB-CPD/M02 ‘Officials of the GNB-CPD’. At the time of writing, this can be found on the CIRCA
website in folder http://circa.europa.eu/Members/irc/nbg/cdpgnb/library?l=/monitoring_gnbcpd&vm=detailed&sb=Title, but GNB-CPD information is expected to be transferred to an area of
the CIRCABC website.
2
Symbols
kn
is the acceptance coefficient
k1
is the acceptance coefficient one-sided tolerance interval
k2
is the acceptance coefficient two-sided tolerance interval
kc
is the corrected acceptance coefficient
kk
is the acceptance coefficient for known standard deviation
ku
is the acceptance coefficient for unknown standard deviation
n
is the number of test samples within the spot sample
xm
is the mean test result
xi
is the test result for test sample i
i
is the number of the individual test sample
xest
is the estimated test result of the spot sample
s
is the standard deviation of the test results
ss
is the standard deviation of the test results of a spot sample
σ
is the known standard deviation
l
is the number of inspection lots
λ10,dry,unit
is the thermal conductivity of the unit
p
is the fractile
γ
is the confidence level
NB-CPD/SG10/03/006r2
Page 3 of 74
3
Reference list
EN 771-1:2011
Specification for masonry units - Part :1 Clay masonry units
EN 771-2:2011
Specification for masonry units - Part 2: Calcium silicate masonry units
EN 771-3:2011
Specification for masonry units - Part 3: Aggregate concrete masonry
units (dense and lightweight aggregates)
EN 771-4:2011
Specification for masonry units - Part 4: Autoclaved aerated concrete
masonry units
EN 771-5:2011
Specification for masonry units – Part 5: Manufactured stone masonry
units
EN 771-6:2011
Specification for masonry units – Part 6: Natural stone masonry units
EN 1990:2002/A1:2005
Eurocode - Basis of structural design
EN 1996-1-1:2005
Eurocode 6: Design of masonry structures - Part 1-1: General rules for
reinforced and unreinforced masonry structures
EN 1996-1-2:2005
Eurocode 6: Design of masonry structures - Part 1-2: General rules Structural fire design
EN 1996-2:2006
Eurocode 6: Design of masonry structures - Part 2: Design
considerations, selection of materials and execution of masonry
EN 1996-3:2006
Eurocode 6: Design of masonry structures - Part 3: Simplified calculation
methods for unreinforced masonry structures
4
General
It is specified in the EN 771 series that the manufacturer shall 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:
• initial type testing of the product (ITT);
• factory production control (FPC).
If the manufacturer intends to declare that the units are Category I units, then the units have to 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 it is necessary for the manufacturer to operate a FPC that
includes a statistical evaluation.
The confidence level for a property has to 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.
NB-CPD/SG10/03/006r2
Page 4 of 74
It is not possible to operate with a 100 % confidence level for a property to be tested by a
destructive test, and for properties tested by a non-destructive test it will be too expensive to
operate with a 100 % confidence level. A confidence level of 95 % is very high and considered
more acceptable.
Confidence levels other than 95 % can be used, e.g. the safety system specified in the Eurocode,
EN 1990, to which the Eurocode for masonry (EN 1996) refers 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 is 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 test 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 has to 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
Factory production control
5.1
General
The factory production control (FPC) system may 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;
• any combination of both.
It may 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. test of dimensions, and
if the property is less important in relation to the end use then it may 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 may be to base the assessment on process control using proxy tests.
In some companies responsibility for the production is placed only on one person, and if this person
is not available, the responsibility for taking decisions is unclear. This can result in unnecessary and
costly stops of the production or the manufacture of non-conforming products. It should be in the
interest of the manufacturer to avoid this by establishing the responsibility, authority and interrelation
NB-CPD/SG10/03/006r2
Page 5 of 74
of all personnel who manage, perform and verify the work affecting the quality of masonry unit
products and the evaluation of conformity.
The procedures to be followed when controlling the production are of course of great importance as
the quality of the products is directly linked to that. It should be in the interest of the manufacturer to
obtain the best quality of the products and therefore to have an interest in clear procedures. The
best way of achieving this is to have them in a written form. Procedures for what to do, when
control and check parameters during the production are not obtained or fulfilled, are of the same or
may be of greater importance. Therefore the need for having them in a written form is crucial.
The manufacturer may define 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 may differ according to the characteristics in question. If a product
group is defined, then the FPC system shall 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 (process control only, finished product testing
only or a combination of both) a selection of these may be considered.
5.2
Testing and measuring equipment
The accuracy of the testing or measuring equipment used in the control procedures are to be in
accordance with the test standard. If it is not defined there, then a ‘rule of thumb’ can be 1/5 – 1/10
times of the accuracy of the value to be declared. Testing or measuring data are not helpful in itself,
unless you know that the data are accurate. It should be in the interest of the manufacturer to know
that testing and measuring data are reliable. To obtain that, all relevant weighing, measuring and
testing equipment that have an influence on the declared values, need to be verified and regularly
inspected.
A verification of testing and measuring equipment needs only to be done in the measuring area
used. If the length of a unit is 300 mm, then the measuring area for the length is approximately 290
– 310 mm and can be verified using a fixed measuring length, e.g. iron prism, iron block or iron bar
with a length of 300 mm. Weighing equipment can be verified by the manufacturer using fixed
weights covering the weighing area used.
5.3
Production equipment
Most production equipment contains moving parts, which need adjustment from time to time.
During production wear and tear can also happen. For that reason, it is recommended that all parts
of production equipment that have an influence on the declared values are to be controlled and
regularly inspected.
5.4
Raw materials
The product properties depend on the constituents used and variations in their quality. To eliminate
this influence as much as possible the manufacturer has to define his own acceptance criteria of
raw materials and the procedures with which to operate to ensure that these are met. This is
independent of the way the constituents are received in the factory – bought from a supplier or
delivered from the manufacturer’s own sources. If the constituents or some of them are bought from
a supplier, the manufacturer is advised to be sure that the control system for the constituents carried
out by the supplier is sufficient. Normally it is acceptable if the control system of the supplier is
NB-CPD/SG10/03/006r2
Page 6 of 74
supervised by a third party, and then the manufacturer has only to check the delivery notes and
make a visual inspection to ensure that the delivery is in line with the order. If the raw materials are
delivered from the manufacturer’s own sources, for example the manufacturer’s own clay pit, then a
procedure to check, if the grain size distribution of the clay is kept constant, could be to measure
regularly the amount of clay in a test sample passing a 90 μm test sieve. An example of control
data is given in Figure 1 along with the acceptance criteria fixed by the manufacturer, the upper limit
(UL) and lower limit (LL).
Figure 1. Example of variation in the amount of clay particles passing a 90 μm test sieve
5.5
Production process
The production process and the controlling of production are of great importance for the properties
of the products and variation in the properties. It should be in the interest of the manufacturer to
obtain the best quality of the products and therefore to want to have the best handling of the
production. The best way of achieving this is to identify relevant measuring and check parameters
in the process, and then to fix for each parameter requirements to be fulfilled or limits (upper and
lower limits, UL and LL) between which the parameter is allowed to vary. These limits and the
frequency of measuring or checking the parameter have to be based on the manufacturer’s
experience and on the importance and the variation of the parameter. The manufacturer should
also specify what should be done, when control and check parameters during the production are not
fulfilling the requirement or passing the limit value.
In the following example, Figure 2, the length of the green clay masonry units is measured to control
the wear and tear of the mould in which the units are produced. In the following part of the
NB-CPD/SG10/03/006r2
Page 7 of 74
production process the units will shrink 0,1 mm, and the intension is to declare a length of 228,5 mm
and a tolerance of ± 0,5 mm. Both aspects need to be taken into account when fixing the control
limits. The reason for the dramatic drop is a renewal of the mould. The renewal of the mould
should have taken place at spot sample 11 as it was leading to a situation where all units in the
inspection lot produced between spot sample 11 and 12 did not conform to the fixed upper limit.
Figure 2. Example of variation in the length of green clay masonry units over time
It is possible to operate with two sets of control limits, a narrow and a wider range. If the parameter
is passing the control limit of the narrow range, it can be looked upon as a warning, and a small
correction of the process may be made, but when the parameter passes the control limit for the
wider range, a more radical correction of the process will be needed.
5.6
Finished product testing
When testing the finished product, it is possible to use alternative test methods if a correlation can
be established between the alternative test method and the reference test method.
It is also important to notice that a test result of a spot sample (see clause 5.6.2) is representing an
inspection lot (see clause 5.6.1). If an evaluated test result is not conforming, the whole production
since the last test has to be looked upon as non-conforming. For that reason it can be
recommended, that for properties where the reference test is time consuming and may 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 nonconforming test result will be less and thereby reduce the manufacturer’s risk.
NB-CPD/SG10/03/006r2
Page 8 of 74
The amount of products produced between 2 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.
5.6.1
Inspection lot
The production is divided into inspection lots.
An inspection lot must 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.
That 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 has been decided, it must be possible to draw
a representative spot sample.
5.6.2
Spot sampling and sample sizes
When the inspection lot has been decided, the sampling procedure for a spot sample has to be fixed
in such a way that the spot sample is representative for the inspection lot.
Figure 3. An example of representative sampling
NB-CPD/SG10/03/006r2
Page 9 of 74
In the European Product Standard sampling procedures for stacks and banded packs are given. 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 somewhere a minimum
number of units has been fixed then this must 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 his own risks.
In the following Figure 4 the variation over time for the mean compressive strength is given.
Figure 4. Example of variation in mean compressive strength over time
On the basis of the test results from testing the spot sample it has to be decided whether the
inspection lot is accepted or not, see clause 5.6.8. In this respect the test results can be dealt with
separately or treated together with the previous results. It depends on the type of production (batch
production or series production).
NB-CPD/SG10/03/006r2
Page 10 of 74
5.6.3
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 minimises the changes from one
batch to another, an inspection lot can cover more than one batch
A production, which is based on a continuous flow, is named a series production. In the case of
series production the properties of the units are the same within a series. A series production
contains normally more than one inspection lot.
5.6.4
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.
In clause 5.6.6 when dealing with the evaluation of test results the acceptance coefficient kn is given
in Tables 1 and 2. These tables show that there is a great difference in using kn for 3 or for 6 test
results and for that reason it is recommended to operate with spot sample sizes of at least 6 units.
NB-CPD/SG10/03/006r2
Page 11 of 74
Figure 5. Example of Method A: Each inspection lot is evaluated individually
5.6.5
Method B: ”Rolling” inspection
In a series production there are a series of inspection lots, which should not exceed a total number
of 5. In the following 4 are used.
Figure 6. Example with 4 inspection lots in a series
For the 1st inspection lot a spot sample size of 3 is taken and tested. For the 2nd inspection lot 3
new samples are taken and tested and evaluated together with the ones from the 1st inspection lot
and therefore the spot sample size will be 6. For the 3rd inspection lot 3 new samples are taken and
tested and evaluated together with the ones from the 1st and the 2nd inspection lot and therefore the
spot sample size will be 9. For the 4th inspection lot 3 new samples are taken and tested and
evaluated together with the ones from the 1st, 2nd and 3rd inspection lot and therefore the spot
sample size will be 12. For the 5th inspection lot 3 new samples are taken and tested and evaluated
NB-CPD/SG10/03/006r2
Page 12 of 74
together with the ones from the 2nd, 3rd and 4th inspection lot 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 the following Figure. In clause 5.6.6 when dealing with the evaluation of test
results the acceptance coefficient kn is given in Tables 1 and 2. These tables show that there is a
great difference for 6 and for 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 7. Example of Method B, “Rolling” inspection: series of 4 inspection lots
Another possibility is the so-called “progressive” sampling procedure. For each of the 1st to 5th
inspection lots a spot size of one sample is taken and tested. These lots are evaluated together.
For the 6th and following inspection lots 1 additional sample is taken and 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, 1 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.
NB-CPD/SG10/03/006r2
Page 13 of 74
Figure 8. Example of Method B, ”Rolling” inspection “Progressive” sampling: series of 15
inspection lots
5.6.6
Evaluation of test results
Where and when possible and applicable, the results of checks and testing shall 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 section.
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 kn. The acceptance coefficient kn can be regarded as a factor minimising the
statistic uncertainties from spot sampling. kn is dependent on several factors:
• The number of samples in the inspection lot n
• The confidence level γ
• The fractile p *)
NB-CPD/SG10/03/006r2
Page 14 of 74
• The standard deviation is unknown. The symbol used is ku
• The standard deviation is known. The symbol used is kk
• One-sided limit evaluation. The symbol used is k1
• Two-sided limit evaluation. The symbol used is k2
*) Be aware that a 5 % characteristic value corresponds with a fractile p = 95 and a 95 % characteristic value
corresponds also with a fractile p = 95. 50 % characteristic value corresponds with a fractile p = 50.
When evaluating the test results from a spot sample, then use the following procedure:
Calculate the mean value of the test results using the following equation:
xm =
1
n
n
∑x
(1)
i
i=1
where
• xm is the mean test result
• xi is the test result for test sample 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 ss for the test results of the spot sample using the following
equation:
n
s=
∑ (x
i =1
i
− xm )
2
(2)
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
• xi is the test result for test sample i
• xm
is the mean test result
If the standard deviation is unknown and if the test results have to be compared with a lower limit
value then calculate the estimated test result xest using the following equation:
xest = xm – k1,u × ss
(3)
If the standard deviation is unknown and if the test results have to be compared with an upper limit
value then calculate the estimated test result xest using the following equation:
xest = xm + k1,u × ss
NB-CPD/SG10/03/006r2
(4)
Page 15 of 74
If the standard deviation is unknown and if the test results have to be compared with a two-sided
limit value then calculate the estimated test result xest using the following equation:
xest = xm ± k2,u × ss
5
If the standard deviation σ is known and if the test results have to be compared with a lower limit
value then calculate the estimated test result xest using the following equation:
xest = xm – k1,k × σ
6
If the standard deviation σ is known and if the test results have to be compared with an upper limit
value then calculate the estimated test result xest using the following equation:
7
xest = xm + k1,k × σ
If the standard deviation σ is known and if the test results have to be compared with a two-sided
upper limit value then calculate the estimated test result xest using the following equation:
xest = xm ± k2,k × σ
(8)
where
• xest
is the estimated test result of the spot sample
• xm
is the mean test result
• k1,u
is the acceptance coefficient for unknown standard deviation and one-sided
evaluation to be taken from Table 1 or 2 or relevant tables in Annex A
• k2,u
is the acceptance coefficient for unknown standard deviation and two-sided limit
evaluation to be taken from relevant tables in Annex A
• ss
is the standard deviation for the test results of the spot sample
• k1,k
is the acceptance coefficient for known standard deviation and one-sided limit
evaluation to be taken from Table 1 or 2 or relevant tables in Annex A
• k2,k
is the acceptance coefficient for known standard deviation and two-sided limit
evaluation to be taken from relevant tables in Annex A
• σ
is the known standard deviation
limit
Standard
deviation
n=3
4
5
6
7
8
9
10
11
12
14
15
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 1. kn for 50 % characteristic value (50 % fractile) and 95 % confidence level
NB-CPD/SG10/03/006r2
Page 16 of 74
Standard
deviation
n=3
4
5
6
7
8
9
10
11
12
14
15
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
Table 2. kn for 5 % characteristic value (95 % fractile) and 95 % confidence level
More tables are given in Annex A.
The method of using the acceptance coefficient for known standard deviation kk is only valid when
the standard deviation ss of the spot sample corresponds to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
(9)
If as part of the evaluation it turns out not to be the case, the manufacturer has to restart or he
decides to continue working with the unknown acceptance coefficient ku. This means that the
inspection lots have to be treated separately.
The effect of the size of the spot sample and the standard deviation of the test results of the sample
on the acceptance coefficient kn and the estimated compressive strength are shown in Table 3.
In the first example of Table 3 the spot sample representing an inspection lot consists of 6 units and
the results of the compressive strength are given on each unit. The mean value and the standard
deviation are calculated. From the table for “kn for 50 % fractile and 95 % confidence level” the
acceptance coefficient kn for unknown standard deviation and n = 6 are taken and the estimated
compressive strength for the inspection lot is calculated.
In the second example the spot sample size and the mean value are kept the same, but there is a
greater variation in the test results leading to a higher standard deviation, which again is leading to a
lower estimated compressive strength. A higher standard deviation is demonstrating less control
compared to the first example. When keeping the confidence level the estimated compressive
strength for the inspection lot needs to be lower.
In the third example the two previous spot samples are looked upon as one spot sample consisting
of 12 units. The mean value and the standard deviation are calculated. From Table 1 the
acceptance coefficient kn for unknown standard deviation and n = 12 are taken and the estimated
compressive strength for the lot is calculated. By enlarging the number of units to be tested of the
spot sample the estimated value is more certain leading to a higher estimated compressive strength
of the inspection lot compared to the second example, where the mean value and the standard
deviation are about the same.
NB-CPD/SG10/03/006r2
Page 17 of 74
Spot
Mean Std.
sample value deviation
size
in MPa in MPa
Coefficient
95 %,
unknown
kn
Estimated
comp.
strength in
MPa
6
20
1,3
0,82
19
6
20
3,2
0,82
17
12
20
3,0
0,52
18
Table 3. Example showing the effect of spot sample size and deviation
As you see, when reducing the variation in the test results by operating a better process control the
estimated value for the tested property will be higher. The same will be achieved by increasing the
number of units of the spot sample.
5.6.7
How to come from unknown to known standard deviation?
Looking at the tables for kn, Tables 1 and 2, it is clear, that there is a considerable effect in going
from an unknown to known standard deviation. In control method A (clause 5.6.4) the standard
deviation of the population is considered to be unknown at least for the first 40 test samples and the
acceptance coefficient ku has to be taken from 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 (kc). The acceptance coefficient for the known standard deviation kk is taken
from tables for known standard deviation. The corrected acceptance coefficient kc is calculated by a
linear interpolation between the acceptance coefficient ku and kk. The known standard deviation σ is
calculated based on the first at least 40 test results.
In control method B (clause 5.6.5) the standard deviation of the population is considered to be
unknown at least for the first 20 test samples and the acceptance coefficient ku has to be taken from
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 (kc) as above. The
acceptance coefficient for the known standard deviation kk is taken from tables for known standard
deviation. The known standard deviation σ is calculated based on the first at least 20 test results.
NB-CPD/SG10/03/006r2
Page 18 of 74
If “progressive sampling” is used the standard deviation of the population is considered to be
unknown at least for the first 30 test samples and the acceptance coefficient ku has to be taken from
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 above. The
acceptance coefficient for the known standard deviation kk is taken from tables for known standard
deviation. The known standard deviation σ is calculated based on the first at least 30 test results.
5.6.8
Conformity
After calculating xest by testing the inspection lots the result has to be compared with either the
declared value or a lower or upper limit depending on the property. For compressive strength it is
the declared value or the lower limit and for dimension it is the upper and lower declared value or
the upper and lower limit. In Figure 8 the estimated mean compressive strength is based on 95 %
confidence level for the different spot samples using the calculations of the test data shown in
Figure 4. In Figure 9 the estimated 5 % characteristic compressive strength based on 95 %
confidence level is shown using the same test data.
UL
LL
DV
Figure 9. Example of variation in the estimated mean compressive strength over time
NB-CPD/SG10/03/006r2
Page 19 of 74
Figure 10. Example of variation in the estimated 5 % characteristic compressive strength
over time
In Figure 9 and 10 the estimated compressive strength is varying between the upper and lower limit
and therefore conforming to the fixed limit values. The declared value needs to be equal to or lower
than the lower limit value.
In Figure 11 the variation in the length of the units over time is given. The units are from the same
production as the ones checked as green units, see Figure 2.
NB-CPD/SG10/03/006r2
Page 20 of 74
Figure 11. Example of variation in the length of the finished units over time
As mentioned before the intention is to declare a length of 228,5 mm and a tolerance of ± 0,5 mm,
which means, that the upper and lower declared value is fixed by the tolerance. When the renewal
of the mould did not take place at the production spot sample 11, see Figure 2, then the length of
the units of spot sample 13 does not comply with the declared value and the belonging tolerance.
A manufacturer of units with a shape shown in Figure 12 would like to declare the thermal
conductivity, λ10,dry,unit, of the unit.
Figure 12. Example of a shape of a masonry unit
NB-CPD/SG10/03/006r2
Page 21 of 74
By carrying out tests for masonry made of specific units it is possible for these units to establish a
relationship between the thermal conductivity, λ10,dry,unit, and the gross dry density of the units as
shown in Figure 13.
Figure 13. Example of a relationship between the gross dry density and the thermal
conductivity of a unit
By testing and controlling the gross dry density it is possible to declare the thermal conductivity,
λ10,dry,unit, of the unit. The gross dry density is used as a proxy property for the thermal conductivity.
In Figure 14 the variation in the gross dry density over time is shown. The variation in the gross dry
density is coming from 2 contributions, variation in the shape and variation in the net dry density of
the material. When a dramatic drop occurs periodically the probable reason for the variation in the
gross dry density is a renewal of the mould and therefore the variation in the shape and not a
variation in the net dry density.
NB-CPD/SG10/03/006r2
Page 22 of 74
Figure 14. Example of variation in the gross dry density of the units over time
If the variation in the gross dry density is as shown in Figure 15 the reason seems to be the
variation in the shape as well as the variation in the net dry density.
If the declared thermal conductivity value has to be a 50 % fractile with a confidence level of 50 %
the test results of the spot samples have to be evaluated, e.g. by the calculation procedures
described in clause 5.6.6 using Table A1 or A5 in Annex A.
NB-CPD/SG10/03/006r2
Page 23 of 74
Figure 15. Example of variation in the gross dry density of the units over time
5.6.9
A simple and conservative approach
A simple and conservative approach can be to evaluate single test results of at least 1 year for a
given property and calculate the mean value and the standard deviation and fix then a band in
which new test results have to fit in. The upper band limit and lower band limit then can be 2 times
of 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 shall
be repeated and the band limit values adjusted accordingly. The same shall happen for the
declared value. The non-conforming inspection lot can be treated as described in the next clause
using control method A.
5.6.10 Non-conforming products
When an evaluation of the test results of the last spot sample is leading to non-conformity, e.g. as
shown in Figure 11, it is important to avoid that the whole inspection lot is mixed up with the other
inspection lots. The non-conforming inspection lot has to be treated separately. It may be
reclassified by the manufacturer and given different declared values. If it is not segregated the
whole stock has to be treated as non-conforming. For that reason a procedure for dealing with nonconforming products should be developed.
NB-CPD/SG10/03/006r2
Page 24 of 74
It should be in the interest of the manufacturer to avoid that the same non-conformity occurs again.
When non-conformity occurs, then it is important to try to identify the reason why, otherwise it is
difficult to find out, what to do to avoid that it occurs again. Testing can be part of the identification.
To ensure that the personnel managing the production knows 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.
5.6.11 Guidance
How to use the different possibilities?
A manufacturer is producing units in two different ways:
• Product 1 is a special unit produced very rarely and only in small quantities. The
characteristics of the product may vary from production to production.
• Product 2 is one of the core units of the production site. It is produced in series of variable
length – sometimes only 2 days of production – but it is produced within short-time intervals.
For product 1 it will be 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 for some properties control method B. If using method B a redeclaration 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,
but 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 may be used when planning the setup of the FPC system:
Control method A:
• Verification of separate inspection lots.
• Inspection lots are defined to be the full production series.
• The minimum sample size of the spot sample is 6 units (n ≥ 6).
• Level of confidence for compressive strength for Category I units is required to be 95 %.
For net dry density and dimension 75 % may be chosen. For gross dry density or net dry
density used as a proxy property to thermal conductivity a confidence level of 50 % or 90 %
may be chosen.
• If the spot sample size is 6 units, the acceptance constant kn for mean compressive
strength at a 95 % confidence level is k1,u = 0,82 for unknown standard deviation and k1,k =
0,67 for known standard deviation.
• If the spot sample size is 6 units, the acceptance constant kn for 5 % characteristic
compressive strength at a 95 % confidence level is k1,u = 3,71 for unknown standard
deviation and k1,k = 2,32 for known standard deviation.
• If the spot sample size is 6 units, the acceptance constant kn for mean compressive
strength at a 75 % confidence level (Category II units) is k1,u = 0,30 for unknown standard
deviation and k1,k = 0,28 for known standard deviation.
NB-CPD/SG10/03/006r2
Page 25 of 74
Control method B:
• Verification of series of inspection lots.
• Inspection lot can be defined to be the units produced within 1 production week / 5 days.
• The minimum sample size of the spot sample is 3 units (n ≥ 3).
• Size of series are 4 inspection lots (l = 4).
• In case of n = 3, the sample size used for evaluation of each inspection lot is 12.
• Level of confidence for compressive strength for Category I units is required to be 95 %.
For net dry density and dimension 75 % may be chosen. For gross dry density used as a
proxy property to thermal conductivity a confidence level of 50 % or 90 % may be chosen.
• If the spot sample size is 3 units, the acceptance constant kn for mean compressive
strength at a 95 % confidence level is k1,u = 0,52 for unknown standard deviation and k1,k =
0,47 for known standard deviation. If a sample size of a spot sample is raised to 6 units
instead of 3 then the acceptance constant kn for mean compressive strength is k1,u = 0,35
for unknown standard deviation and k1,k = 0,34 for known standard deviation.
• If the spot sample size is 3 units, the acceptance constant kn for 5 % characteristic
compressive strength at a 95 % confidence level is k1,u = 2,74 for unknown standard
deviation and k1,k = 2,12 for known standard deviation. If a sample size of a spot sample is
raised to 6 units instead of 3 then the acceptance constant kn for mean compressive
strength is k1,u = 2,31 for unknown standard deviation and k1,k = 1,98 for known standard
deviation.
• If the spot sample size is 3 units, the acceptance constant kn for mean compressive
strength at a 75 % confidence level (Category II units) is k1,u = 0,20 for unknown standard
deviation and k1,k = 0,19 for known standard deviation. If a sample size of a spot sample is
raised to 6 units instead of 3 then the acceptance constant kn for mean compressive
strength is k1,u = 0,14 for unknown standard deviation and k1,k = 0,14 for known standard
deviation.
What to do with an inspection lot where the evaluated test results for one or more properties
are leading to non-conformity?
Control method A:
• Discard the inspection lot, or;
• Sample a new and larger spot sample (e.g. 12 instead of 6), test the sample for the
properties leading to a non-conformity and evaluate the test results using a reduced
acceptance constant (e.g. 0,52 instead of 0,82) according to the higher number of units in
the test sample, or;
• Change the declaration of the units based on ITT.
Control method B:
• Discard the inspection lot, or;
• Sample a new larger spot sample (e.g. ≥ 6 instead of 3 units) using control method A and
evaluate the test results using a reduced acceptance constant, according to the number of
the units in the test sample and change eventually the declaration accordingly. *)
*) Always keep the results of the inspection lot within the system when evaluating the next inspection lot
or start from the very beginning.
NB-CPD/SG10/03/006r2
Page 26 of 74
When a non-conformity is identified in the finished product testing it is not possible to take any
corrective actions for the tested inspection lot. It can only be discarded or re-declared. The longer
the production process of the units lasts, the larger is the number of units produced before it is
possible to correct the process, leading again to a larger number of units to be discarded or redeclared. The example mentioned about measurement of the length of the green units, see Figure
2, demonstrates that it is possible to detect a problem (wear and tear of the mould) early in the
process, which leads to a non-conformity of the finished product in the process. Checking
dimensions, weights and temperatures are quite simple, but done at the right places in the process
they will give a lot of information valid for the control of the process and the properties of the finished
products. It may even be possible to counteract a detected problem later on in the process.
Consideration should be given to identifying the most economical way to arrange the control by the
right mix of process control and finished product testing, and to consider also the possibility of using
internal proxy tests in the process control.
The manufacturer may define 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 may differ according to the characteristics in question. If a product
group is defined, then the FPC system shall ensure that all types of units within a group are
controlled and over time also by the finished product testing, if this is part of the FPC.
For process control the evaluation procedure described in clause 5.6.6 may be used, when
appropriate.
Traceability in the process
The clause deals with the traceability in the process from raw materials to finished products. It is
not dealing with the traceability on the market.
As mentioned earlier it should be in the interest of the manufacturer to avoid that the same nonconformity occurring again. It is therefore important to try to identify the reason why, when it
occurred, otherwise it is difficult to find out, what to do to avoid it occurring again.
The better knowledge the manufacturer has about the variation in the raw materials, variation in the
different parts of the process and their influence on the variation in the properties of the finished
product the better he will be able to identify the reason for non-conformity. To be able to obtain that
knowledge it can be recommended that the manufacturer follows the same units all through the
process from time to time if not on every occasion and to evaluate all the checks and measurements
together and to compare the results with other similar evaluations done. Based on such an exercise
it may be possible to establish traceability in the process.
Marking and stock control of products
The more variations there are in the production in relation to the type of products and properties the
higher is the need for instructions dealing with the marking procedure and how to handle and to
control the stock. It is important that 2 types of units with the same shape but not the same
properties are marked in such a way that they will not be mixed up. Inspection lots of products
should be identifiable and traceable.
NB-CPD/SG10/03/006r2
Page 27 of 74
5.6.12 Records
Many years of experience have shown that it can be dangerous to have only one person who knows
all the information required for the production of masonry units and how to control it. The more this
information is in a written form the more it is available for others, and at the same time it is easier to
establish an overview in writing. It can be recommended to describe step by step what needs to be
done in the whole process from the raw material to the finished product leaving the gate of the
factory in order to be able to produce a high quality product. This can include specifying the position
of each check and observation points and control procedures. It is really valuable for the machine
operator to have information of corrective actions available when control parameters are passing the
control limits.
Experiences show also that on a busy day it is easy to forget important observations made during
the production control if these observations are not recorded. To make it easier to record
observations it can be recommended to use tables.
Samples are taken during the process and from finished products and these 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 may be
reconsidered and reduced compared to the ones used.
6
Initial type tests
It is important for a manufacturer to produce what is possible to sell and not to try to sell what is
possible to produce. A manufacturer would like to fulfil the market needs and therefore intends to
develop and to produce units with specific properties. To ensure that these properties are available
it is necessary after completion of the development of a new product type and before
commencement of the manufacture and offering for sale, that appropriate initial type tests had been
carried out to confirm that the properties predicted from the development meet the requirements of
the product standard and the values to be declared for the unit.
If the manufacturer is trying to sell what is possible to produce and nothing else then the full finished
product test done as part of the control method A can act as an initial type test if the reference test
methods are used and the sampling procedure for ITT. In that respect the declared values, which
may vary from batch to batch have to be determined batch by batch and have to be based on an
evaluation of the same test results (see clause 5.6.6). It will not be possible to sell the units before
the test results are available.
If in control method B non-conformity occurs and the inspection lot is re-declared following control
method A using the reference test methods and the sampling procedure for ITT, then the test can
be regarded as an initial type test.
Whenever a major change in the source, blend, or nature of raw materials occurs, or when there is
a change in processing conditions, leading to what the manufacturer considers will constitute a new
product type being produced, the appropriate initial type test shall be repeated. If the manufacturer
has doubts it can be recommended to check whether some of the characteristics have changed or
not by using the FPC test procedures.
The manufacturer may define product groups. The products belonging to a product group may
differ according to the characteristics in question.
NB-CPD/SG10/03/006r2
Page 28 of 74
In the ITT process a manufacturer may take into consideration already existing test results.
A manufacturer may use the ITT results obtained by someone else (e.g. another manufacturer or
an association) to justify his own declaration of conformity regarding a product that is manufactured
according to the same design and with raw materials, constituents and manufacturing methods of
the same kind, provided that permission is given, and the test is valid for both products.
NB-CPD/SG10/03/006r2
Page 29 of 74
Annex A
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Tables for acceptance coefficient kn depending on the used
fractile p and confidence level γ (taken from ISO 16269-6
(2005))
fractile : p
0,50
0,75
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
0,000
0,675
n
0,90
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
0,95
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,75
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,675
0,90
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
1,282
0,95
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
1,645
Table A1. k1 for one-sided statistical tolerance, standard deviation: known and confidence
level γ = 50 %
NB-CPD/SG10/03/006r2
Page 30 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,477
1,152
0,390
1,064
0,388
1,012
0,302
0,977
0,276
0,950
0,255
0,930
0,239
0,913
0,225
0,900
0,214
0,888
0,204
0,878
0,195
0,870
0,188
0,862
0,181
0,855
0,175
0,849
0,169
0,844
0,164
0,839
0,159
0,834
0,155
0,830
0,151
0,826
0,148
0,823
0,144
0,819
0,141
0,816
0,138
0,813
0,136
0,810
0,133
0,807
0,131
0,805
0,128
0,802
0,126
0,800
0,124
0,798
0,122
0,796
0,120
0,794
0,119
0,793
0,117
0,791
0,115
0,789
0,113
0,788
0,112
0,786
0,110
0,785
0,109
0,783
0,107
0,782
0,106
0,781
0,105
0,780
0,103
0,778
0,102
0,777
0,101
0,776
0,100
0,775
0,099
0,774
0,098
0,772
0,097
0,771
0,096
0,770
n
0,90
1,759
0,671
0,619
1,584
1,557
1,537
1,521
1,507
1,495
1,485
1,477
1,469
1,462
1,456
1,451
1,446
1,441
1,437
1,433
1,430
1,426
1,423
1,420
1,417
1,414
1,412
1,410
1,408
1,405
1,403
1,401
1,400
1,398
1,396
1,395
1,393
1,392
1,390
1,389
1,388
1,387
1,385
1,384
1,383
1,382
1,381
1,379
1,378
1,377
0,95
2,122
2,035
1,983
1,947
1,921
1,900
1,884
1,870
1,859
1,849
1,840
1,832
1,826
1,820
1,814
1,809
1,804
1,800
1,796
1,793
1,789
1,786
1,783
1,781
1,778
1,776
1,773
1,771
1,768
1,766
1,764
1,763
1,761
1,759
1,758
1,756
1,755
1,753
1,752
1,751
1,750
1,748
1,747
1,746
1,745
1,744
1,743
1,742
1,741
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,095
0,094
0,094
0,093
0,092
0,091
0,090
0,090
0,089
0,088
0,087
0,087
0,086
0,085
0,085
0,084
0,083
0,082
0,082
0,081
0,081
0,080
0,080
0,079
0,079
0,078
0,078
0,077
0,077
0,076
0,076
0,075
0,075
0,074
0,074
0,074
0,073
0,073
0,072
0,072
0,072
0,071
0,071
0,070
0,070
0,070
0,069
0,069
0,068
0,068
0,75
0,769
0,768
0,768
0,767
0,766
0,765
0,764
0,764
0,763
0,762
0,761
0,761
0,760
0,760
0,759
0,758
0,758
0,757
0,757
0,756
0,755
0,755
0,754
0,754
0,753
0,752
0,752
0,751
0,751
0,75
0,750
0,749
0,749
0,748
0,748
0,748
0,747
0,747
0,746
0,746
0,746
0,745
0,745
0,744
0,744
0,744
0,743
0,743
0,742
0,742
0,90
1,376
1,375
1,375
1,374
1,373
1,372
1,371
1,371
1,370
1,369
1,368
1,368
1,367
1,367
1,366
1,365
1,365
1,364
1,364
1,363
1,362
1,362
1,361
1,361
1,360
1,359
1,359
1,358
1,358
1,357
1,357
1,356
1,356
1,355
1,355
1,355
1,354
1,354
1,353
1,353
1,353
1,352
1,352
1,352
1,352
1,351
1,351
1,351
1,350
1,35
0,95
1,740
1,739
1,738
1,737
1,737
1,736
1,735
1,734
1,733
1,732
1,731
1,731
1,730
1,730
1,729
1,728
1,728
1,727
1,727
1,726
1,726
1,725
1,725
1,724
1,724
1,723
1,723
1,722
1,722
1,721
1,721
1,720
1,720
1,719
1,719
1,718
1,718
1,717
1,717
1,716
1,716
1,715
1,715
1,715
1,715
1,714
1,714
1,714
1,713
1,713
Table A2. k1 for one-sided statistical tolerance, standard deviation: known and confidence
level γ = 75 %
NB-CPD/SG10/03/006r2
Page 31 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,907
1,581
0,740
1,415
0,641
1,316
0,574
1,248
0,524
1,198
0,485
1,159
0,454
1,128
0,428
1,102
0,406
1,080
0,387
1,061
0,370
1,045
0,356
1,030
0,343
1,017
0,331
1,006
0,321
0,995
0,311
0,986
0,303
0,977
0,295
0,969
0,287
0,962
0,281
0,955
0,274
0,948
0,268
0,943
0,262
0,937
0,257
0,932
0,252
0,926
0,248
0,922
0,243
0,917
0,239
0,913
0,234
0,909
0,231
0,906
0,227
0,902
0,224
0,899
0,220
0,895
0,217
0,892
0,214
0,889
0,211
0,886
0,209
0,884
0,206
0,881
0,203
0,878
0,201
0,876
0,199
0,873
0,196
0,871
0,194
0,868
0,192
0,866
0,190
0,864
0,188
0,862
0,186
0,860
0,184
0,858
0,182
0,856
n
0,90
2,188
2,022
1,923
1,855
1,805
1,766
1,735
1,709
1,687
1,668
1,652
1,637
1,625
1,613
1,602
1,593
1,584
1,576
1,569
1,562
1,555
1,550
1,544
1,539
1,533
1,529
1,524
1,520
1,516
1,513
1,509
1,506
1,502
1,499
1,496
1,493
1,491
1,488
1,485
1,483
1,480
1,478
1,475
1,473
1,471
1,469
1,467
1,465
1,463
0,95
2,552
2,385
2,286
2,218
2,169
2,130
2,098
2,073
2,051
2,032
2,015
2,001
1,998
1,976
1,966
1,956
1,947
1,939
1,932
1,926
1,919
1,913
1,907
1,902
1,897
1,893
1,888
1,884
1,879
1,876
1,872
1,869
1,865
1,862
1,859
1,856
1,854
1,851
1,848
1,846
1,843
1,841
1,838
1,836
1,834
1,832
1,831
1,829
1,827
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,180
0,179
0,177
0,176
0,174
0,172
0,171
0,169
0,168
0,166
0,165
0,164
0,162
0,161
0,160
0,159
0,158
0,156
0,155
0,154
0,153
0,152
0,151
0,150
0,149
0,148
0,147
0,146
0,145
0,144
0,143
0,142
0,142
0,141
0,140
0,139
0,138
0,138
0,137
0,136
0,135
0,135
0,134
0,133
0,133
0,132
0,131
0,130
0,130
0,129
0,75
0,854
0,853
0,851
0,850
0,848
0,846
0,845
0,843
0,842
0,840
0,839
0,838
0,836
0,835
0,834
0,833
0,832
0,830
0,829
0,828
0,827
0,826
0,825
0,824
0,823
0,822
0,821
0,820
0,819
0,818
0,817
0,816
0,816
0,815
0,814
0,813
0,812
0,812
0,811
0,810
0,809
0,809
0,808
0,807
0,807
0,806
0,805
0,804
0,804
0,803
0,90
1,461
1,460
1,458
1,457
1,455
1,453
1,452
1,450
1,449
1,447
1,446
1,445
1,443
1,442
1,441
1,440
1,439
1,437
1,436
1,435
1,434
1,433
1,432
1,431
1,430
1,429
1,428
1,427
1,426
1,425
1,424
1,423
1,423
1,422
1,421
1,420
1,419
1,419
1,418
1,417
1,416
1,416
1,415
1,414
1,414
1,413
1,412
1,411
1,411
1,410
0,95
1,825
1,824
1,822
1,821
1,819
1,817
1,816
1,814
1,813
1,811
1,810
1,809
1,807
1,806
1,805
1,804
1,803
1,801
1,800
1,799
1,798
1,797
1,796
1,795
1,794
1,793
1,792
1,791
1,790
1,789
1,788
1,787
1,786
1,785
1,785
1,784
1,783
1,782
1,781
1,780
1,779
1,779
1,778
1,778
1,777
1,776
1,776
1,775
1,775
1,774
Table A3. k1 for one-sided statistical tolerance, standard deviation: known and confidence
level γ = 90 %
NB-CPD/SG10/03/006r2
Page 32 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
1,164
0,838
0,950
0,625
0,823
1,497
0,736
1,411
0,672
1,346
0,622
1,297
0,582
1,257
0,549
1,223
0,521
1,195
0,496
1,171
0,475
1,150
0,457
1,131
0,440
1,115
0,425
1,100
0,412
1,086
0,399
1,074
0,388
1,063
0,378
1,052
0,368
1,043
0,360
1,035
0,351
1,026
0,344
1,019
0,336
1,011
0,330
1,005
0,323
0,998
0,317
0,992
0,311
0,986
0,306
0,981
0,301
0,975
0,297
0,971
0,292
0,966
0,288
0,962
0,283
0,957
0,279
0,953
0,275
0,949
0,272
0,946
0,268
0,942
0,265
0,939
0,261
0,935
0,258
0,932
0,255
0,929
0,252
0,926
0,249
0,923
0,246
0,920
0,243
0,918
0,241
0,915
0,238
0,913
0,236
0,910
0,233
0,908
n
0,90
2,445
2,232
2,104
2,018
1,954
1,904
1,864
1,830
1,802
1,778
1,757
1,738
1,722
1,707
1,693
1,691
1,670
1,659
1,650
1,642
1,633
1,626
1,618
1,612
1,605
1,599
1,593
1,588
1,582
1,578
1,573
1,569
1,564
1,560
1,556
1,553
1,549
1,546
1,542
1,539
1,536
1,533
1,530
1,527
1,525
1,522
1,520
1,517
1,515
0,95
2,828
2,595
2,468
2,381
2,317
2,267
2,227
2,194
2,166
2,141
2,120
2,102
2,085
2,070
2,057
2,044
2,033
2,023
2,013
2,005
1,996
1,989
1,981
1,975
1,968
1,962
1,956
1,951
1,946
1,941
1,937
1,932
1,928
1,923
1,919
1,916
1,912
1,909
1,905
1,902
1,899
1,897
1,894
1,891
1,888
1,886
1,883
1,881
1,878
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,231
0,229
0,227
0,225
0,223
0,221
0,219
0,217
0,215
0,213
0,211
0,210
0,208
0,207
0,205
0,203
0,202
0,200
0,199
0,197
0,196
0,194
0,193
0,192
0,191
0,189
0,188
0,187
0,185
0,184
0,183
0,182
0,181
0,180
0,179
0,178
0,177
0,176
0,175
0,174
0,173
0,172
0,171
0,170
0,170
0,169
0,168
0,167
0,166
0,165
0,75
0,906
0,904
0,902
0,900
0,898
0,895
0,893
0,891
0,889
0,887
0,886
0,884
0,883
0,881
0,880
0,878
0,877
0,875
0,874
0,872
0,871
0,869
0,868
0,867
0,866
0,864
0,863
0,862
0,860
0,859
0,858
0,857
0,856
0,855
0,854
0,852
0,851
0,850
0,849
0,848
0,847
0,846
0,845
0,844
0,844
0,843
0,842
0,841
0,840
0,839
0,90
1,513
1,511
1,509
1,507
1,505
1,502
1,500
1,498
1,496
1,494
1,493
1,491
1,490
1,488
1,487
1,485
1,484
1,482
1,481
1,479
1,478
1,476
1,475
1,474
1,473
1,471
1,470
1,469
1,467
1,466
1,465
1,464
1,463
1,462
1,461
1,459
1,458
1,457
1,456
1,455
1,454
1,453
1,453
1,452
1,451
1,450
1,449
1,449
1,448
1,447
0,95
1,876
1,874
1,872
1,870
1,868
1,866
1,864
1,862
1,860
1,858
1,856
1,855
1,853
1,852
1,850
1,848
1,847
1,845
1,844
1,842
1,841
1,839
1,838
1,837
1,836
1,834
1,833
1,832
1,830
1,829
1,828
1,827
1,826
1,825
1,824
1,823
1,822
1,821
1,820
1,819
1,818
1,817
1,816
1,815
1,815
1,814
1,813
1,812
1,811
1,810
Table A4. k1 for one-sided statistical tolerance, standard deviation: known and confidence
level γ = 95 %
NB-CPD/SG10/03/006r2
Page 33 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,000
0,888
0,000
0,774
0,000
0,739
0,000
0,722
0,000
0,712
0,000
0,706
0,000
0,701
0,000
0,698
0,000
0,695
0,000
0,693
0,000
0,692
0,000
0,690
0,000
0,689
0,000
0,688
0,000
0,678
0,000
0,686
0,000
0,686
0,000
0,685
0,000
0,685
0,000
0,685
0,000
0,684
0,000
0,684
0,000
0,683
0,000
0,683
0,000
0,682
0,000
0,682
0,000
0,682
0,000
0,682
0,000
0,681
0,000
0,681
0,000
0,681
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,680
0,000
0,679
0,000
0,679
0,000
0,679
0,000
0,679
0,000
0,679
0,000
0,679
0,000
0,679
0,000
0,679
n
0,90
1,785
1,499
1,419
1,382
1,361
1,347
1,337
1,330
1,325
1,320
1,317
1,314
1,311
1,309
1,307
1,306
1,304
1,303
1,302
1,301
1,300
1,299
1,298
1,298
1,297
1,297
1,296
1,296
1,295
1,295
1,294
1,294
1,293
1,293
1,293
1,293
1,292
1,292
1,292
1,292
1,291
1,291
1,290
1,290
1,290
1,290
1,290
1,290
1,290
0,95
2,339
1,939
1,830
1,780
1,751
1,732
1,719
1,710
1,702
1,696
1,691
1,687
1,684
1,681
1,679
1,677
1,675
1,673
1,672
1,671
1,669
1,668
1,667
1,666
1,665
1,665
1,664
1,663
1,662
1,662
1,661
1,661
1,660
1,660
1,660
1,659
1,659
1,658
1,658
1,658
1,658
1,657
1,657
1,657
1,657
1,656
1,656
1,655
1,655
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,75
0,679
0,679
0,679
0,679
0,679
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,90
1,290
1,290
1,289
1,289
1,289
1,289
1,289
1,288
1,288
1,288
1,288
1,288
1,288
1,288
1,288
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,287
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
1,286
0,95
1,655
1,655
1,655
1,655
1,655
1,654
1,654
1,654
1,654
1,654
1,654
1,654
1,653
1,653
1,653
1,653
1,653
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,650
1,650
1,650
1,650
1,650
Table A5. k1 for one-sided statistical tolerance, standard deviation: unknown and confidence
level γ = 50 %
NB-CPD/SG10/03/006r2
Page 34 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,708
2,225
0,472
1,465
0,383
1,256
0,332
1,152
0,297
1,088
0,272
1,044
0,252
0,011
0,236
0,985
0,223
0,964
0,212
0,947
0,202
0,933
0,193
0,920
0,186
0,909
0,179
0,900
0,173
0,891
0,168
0,884
0,163
0,877
0,158
0,870
0,154
0,865
0,151
0,860
0,147
0,854
0,144
0,850
0,140
0,846
0,138
0,842
0,135
0,838
0,133
0,835
0,130
0,831
0,128
0,828
0,125
0,825
0,123
0,823
0,121
0,820
0,120
0,818
0,118
0,815
0,116
0,813
0,114
0,811
0,113
0,809
0,111
0,807
0,110
0,805
0,108
0,803
0,107
0,801
0,106
0,800
0,104
0,798
0,103
0,797
0,102
0,795
0,101
0,794
0,100
0,793
0,099
0,791
0,098
0,790
0,097
0,789
n
0,90
3,993
2,502
2,134
1,962
1,860
1,791
1,740
1,702
1,671
1,646
1,625
1,607
1,591
1,578
1,566
1,555
1,545
1,536
1,529
1,522
1,514
1,509
1,503
1,498
1,492
1,488
1,483
1,479
1,475
1,472
1,468
1,465
1,461
1,458
1,455
1,453
1,450
1,448
1,445
1,443
1,441
1,439
1,437
1,435
1,433
1,431
1,430
1,428
1,426
0,95
5,122
3,152
2,681
2,464
2,336
2,251
2,189
2,142
2,104
2,074
2,048
2,026
2,008
1,991
1,977
1,964
1,952
1,942
1,932
1,924
1,916
1,909
1,902
1,896
1,889
1,884
1,879
1,874
1,869
1,865
1,861
1,858
1,854
1,850
1,847
1,844
1,840
1,837
1,834
1,832
1,829
1,827
1,824
1,822
1,820
1,818
1,815
1,813
1,811
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,096
0,095
0,094
0,093
0,093
0,092
0,091
0,090
0,089
0,088
0,087
0,087
0,086
0,086
0,085
0,084
0,084
0,083
0,083
0,082
0,081
0,081
0,080
0,080
0,079
0,078
0,078
0,077
0,077
0,076
0,076
0,075
0,075
0,074
0,074
0,074
0,073
0,073
0,072
0,072
0,072
0,071
0,071
0,070
0,070
0,070
0,069
0,069
0,068
0,068
0,75
0,788
0,787
0,786
0,785
0,784
0,782
0,781
0,780
0,779
0,778
0,777
0,776
0,776
0,775
0,774
0,773
0,772
0,772
0,771
0,770
0,769
0,769
0,768
0,767
0,767
0,766
0,765
0,764
0,764
0,763
0,763
0,762
0,762
0,761
0,761
0,760
0,760
0,759
0,759
0,758
0,758
0,757
0,757
0,756
0,756
0,755
0,755
0,754
0,754
0,753
0,90
1,425
1,423
1,422
1,420
1,419
1,418
1,416
1,415
1,413
1,412
1,411
1,410
1,409
1,408
1,407
1,405
1,404
1,403
1,402
1,401
1,400
1,399
1,399
1,398
1,397
1,396
1,395
1,395
1,394
1,393
1,392
1,392
1,391
1,390
1,390
1,389
1,388
1,387
1,387
1,386
1,385
1,385
1,384
1,384
1,383
1,382
1,382
1,381
1,381
1,380
0,95
1,809
1,808
1,806
1,805
1,803
1,801
1,800
1,798
1,797
1,795
1,794
1,793
1,791
1,790
1,789
1,788
1,787
1,785
1,784
1,783
1,782
1,781
1,780
1,779
1,778
1,777
1,776
1,775
1,774
1,773
1,772
1,771
1,771
1,770
1,769
1,768
1,767
1,767
1,766
1,765
1,764
1,764
1,763
1,762
1,762
1,761
1,760
1,759
1,759
1,758
Table A6. k1 for one-sided statistical tolerance, standard deviation: unknown and confidence
level γ = 75 %
NB-CPD/SG10/03/006r2
Page 35 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
2,177
5,843
1,089
2,603
0,819
1,973
0,686
1,698
0,603
1,540
0,545
1,436
0,501
1,360
0,466
1,303
0,438
1,257
0,414
1,220
0,394
1,189
0,377
1,162
0,361
1,139
0,348
1,119
0,336
1,101
0,325
1,085
0,315
1,071
0,306
1,058
0,297
1,046
0,290
1,036
0,283
1,026
0,277
1,017
0,270
1,008
0,265
1,001
0,259
0,993
0,254
0,986
0,249
0,979
0,245
0,973
0,240
0,967
0,236
1,162
0,232
1,357
0,229
1,553
0,225
1,748
0,221
1,943
0,218
1,739
0,215
1,535
0,213
1,331
0,210
1,127
0,207
0,923
0,204
0,920
0,202
0,917
0,199
0,913
0,197
0,910
0,194
0,907
0,192
0,904
0,190
0,902
0,188
0,899
0,186
0,897
0,184
0,894
n
0,90
10,253
4,259
3,188
2,743
2,494
2,333
2,219
2,133
2,066
2,012
1,967
1,929
1,896
1,867
1,842
1,820
1,800
1,782
1,766
1,752
1,737
1,725
1,713
1,703
1,692
1,683
1,674
1,666
1,658
1,651
1,644
1,638
1,631
1,624
1,619
1,614
1,608
1,603
1,598
1,594
1,590
1,585
1,581
1,577
1,574
1,570
1,567
1,563
1,560
0,95
13,090
5,312
3,957
3,400
3,092
2,894
2,755
2,650
2,569
2,503
2,449
2,403
2,364
2,329
2,299
2,273
2,249
2,228
2,208
2,191
2,174
2,160
2,146
2,134
2,121
2,110
2,099
2,090
2,080
2,072
2,064
2,057
2,049
2,041
2,035
2,029
2,023
2,017
2,011
2,006
2,001
1,996
1,991
1,986
1,982
1,978
1,974
1,970
1,966
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,182
0,181
0,179
0,178
0,176
0,174
0,173
0,171
0,170
0,168
0,167
0,165
0,164
0,163
0,162
0,160
0,159
0,158
0,156
0,155
0,154
0,153
0,152
0,151
0,150
0,149
0,148
0,147
0,146
0,145
0,144
0,143
0,143
0,142
0,141
0,140
0,139
0,139
0,138
0,137
0,136
0,136
0,135
0,134
0,134
0,133
0,132
0,131
0,131
0,130
0,75
0,892
0,890
0,888
0,886
0,884
0,881
0,879
0,877
0,875
0,873
0,871
0,870
0,868
0,867
0,865
0,863
0,862
0,860
0,859
0,857
0,856
0,855
0,853
0,852
0,851
0,850
0,849
0,847
0,846
0,845
0,844
0,843
0,842
0,841
0,840
0,838
0,837
0,836
0,835
0,834
0,833
0,832
0,831
0,830
0,830
0,829
0,828
0,827
0,826
0,825
0,90
1,557
1,555
1,552
1,549
1,547
1,544
1,541
1,538
1,536
1,533
1,531
1,529
1,527
1,525
1,523
1,520
1,518
1,516
1,514
1,512
1,510
1,509
1,507
1,505
1,504
1,502
1,500
1,498
1,497
1,495
1,494
1,492
1,491
1,490
1,489
1,487
1,486
1,485
1,483
1,482
1,481
1,480
1,479
1,478
1,477
1,475
1,474
1,473
1,472
1,471
0,95
1,963
1,960
1,956
1,953
1,950
1,947
1,944
1,940
1,937
1,934
1,932
1,929
1,927
1,924
1,922
1,920
1,917
1,915
1,912
1,910
1,908
1,906
1,904
1,902
1,900
1,898
1,896
1,894
1,892
1,890
1,889
1,887
1,886
1,884
1,883
1,881
1,880
1,878
1,877
1,875
1,874
1,872
1,871
1,870
1,869
1,867
1,866
1,865
1,863
1,862
Table A7. k1 for one-sided statistical tolerance, standard deviation: unknown and confidence
level γ = 90 %
NB-CPD/SG10/03/006r2
Page 36 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
4,465
11,763
1,686
3,807
1,177
2,618
0,954
2,150
0,823
1,896
0,735
1,733
0,670
1,618
0,620
1,533
0,580
1,466
0,547
1,412
0,519
1,367
0,495
1,329
0,474
1,296
0,455
1,268
0,439
1,243
0,424
1,221
0,411
1,201
0,398
1,183
0,387
1,167
0,377
1,153
0,367
1,138
0,359
1,126
0,350
1,114
0,343
1,104
0,335
1,093
0,329
1,084
0,322
1,075
0,317
1,067
0,311
1,059
0,306
1,052
0,301
1,046
0,296
1,039
0,291
1,033
0,286
1,026
0,282
1,021
0,278
1,016
0,275
1,010
0,271
1,005
0,267
1,000
0,264
0,996
0,261
0,991
0,257
0,987
0,254
0,982
0,251
0,978
0,248
0,975
0,246
0,971
0,243
0,968
0,241
0,964
0,238
0,961
n
0,90
20,582
6,156
4,162
3,407
3,007
2,756
2,582
2,454
2,355
2,276
2,211
2,156
2,109
2,069
2,033
2,002
1,974
1,949
1,926
1,907
1,887
1,870
1,853
1,839
1,825
1,813
1,800
1,789
1,778
1,769
1,760
1,751
1,742
1,733
1,726
1,719
1,712
1,705
1,698
1,692
1,686
1,681
1,675
1,669
1,664
1,660
1,655
1,651
1,646
0,95
26,260
7,656
5,144
4,203
3,708
3,400
3,188
3,032
2,911
2,815
2,737
2,671
2,615
2,567
2,524
2,487
2,453
2,424
2,397
2,373
2,349
2,330
2,310
2,293
2,276
2,261
2,246
2,233
2,220
2,209
2,199
2,188
2,178
2,167
2,159
2,151
2,142
2,134
2,126
2,119
2,113
2,106
2,100
2,093
2,087
2,082
2,076
2,071
2,065
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,236
0,234
0,231
0,229
0,227
0,225
0,223
0,220
0,218
0,216
0,214
0,213
0,211
0,210
0,208
0,206
0,205
0,203
0,202
0,200
0,199
0,197
0,196
0,195
0,194
0,192
0,191
0,190
0,188
0,187
0,186
0,185
0,184
0,183
0,182
0,180
0,179
0,178
0,177
0,176
0,175
0,174
0,173
0,172
0,172
0,171
0,170
0,169
0,168
0,167
0,75
0,958
0,955
0,953
0,950
0,947
0,944
0,941
0,939
0,936
0,933
0,931
0,929
0,927
0,925
0,923
0,920
0,918
0,916
0,914
0,912
0,910
0,909
0,907
0,905
0,904
0,902
0,900
0,898
0,897
0,895
0,894
0,892
0,891
0,890
0,889
0,887
0,886
0,885
0,883
0,882
0,881
0,880
0,878
0,877
0,876
0,875
0,874
0,872
0,871
0,870
0,90
1,642
1,639
1,635
1,631
1,628
1,624
1,620
1,616
1,613
1,609
1,606
1,604
1,601
1,598
1,596
1,593
1,590
1,587
1,585
1,582
1,580
1,578
1,575
1,573
1,571
1,569
1,567
1,564
1,562
1,560
1,558
1,556
1,555
1,553
1,551
1,549
1,547
1,546
1,544
1,542
1,541
1,539
1,538
1,536
1,535
1,533
1,532
1,530
1,529
1,527
0,95
2,061
2,057
2,052
2,048
2,044
2,040
2,036
2,031
2,027
2,023
2,020
2,016
2,013
2,010
2,007
2,003
2,000
1,997
1,993
1,990
1,988
1,985
1,983
1,980
1,978
1,975
1,973
1,970
1,968
1,965
1,963
1,961
1,959
1,957
1,955
1,952
1,950
1,948
1,946
1,944
1,942
1,941
1,939
1,937
1,936
1,934
1,932
1,930
1,929
1,927
Table A8. k1 for one-sided statistical tolerance, standard deviation: unknown and confidence
level γ = 95 %
NB-CPD/SG10/03/006r2
Page 37 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,755
0,282
0,727
1,238
0,714
1,216
0,706
1,203
0,701
1,195
0,697
1,188
0,694
1,184
0,692
1,180
0,690
1,177
0,689
1,175
0,688
1,173
0,687
1,171
0,686
1,170
0,685
1,168
0,685
1,167
0,684
1,166
0,684
1,165
0,683
1,165
0,683
1,164
0,683
1,164
0,682
1,163
0,682
1,163
0,681
1,162
0,681
1,162
0,681
1,161
0,681
1,161
0,680
1,160
0,680
1,160
0,680
1,160
0,680
1,160
0,680
1,159
0,679
1,159
0,679
1,158
0,679
1,158
0,679
1,158
0,679
1,158
0,679
1,157
0,679
1,157
0,679
1,157
0,679
1,157
0,679
1,157
0,678
1,157
0,678
1,157
0,678
1,157
0,678
1,157
0,678
1,157
0,678
1,156
0,678
1,156
0,678
1,156
n
0,90
1,823
1,766
1,737
1,719
1,707
1,698
1,692
1,686
1,682
1,679
1,676
1,674
1,672
1,670
1,669
1,667
1,666
1,665
1,664
1,663
1,662
1,662
1,661
1,661
1,660
1,660
1,659
1,659
1,658
1,658
1,657
1,657
1,656
1,656
1,656
1,656
1,655
1,655
1,655
1,655
1,655
1,654
1,654
1,654
1,654
1,654
1,653
1,653
1,653
0,95
2,164
2,100
2,067
2,046
2,033
2,023
2,015
2,009
2,004
2,000
1,997
1,994
1,992
1,990
1,988
1,986
1,985
1,984
1,983
1,982
1,981
1,980
1,979
1,978
1,977
1,977
1,976
1,976
1,975
1,975
1,974
1,974
1,973
1,973
1,973
1,973
1,972
1,972
1,972
1,972
1,971
1,971
1,970
1,970
1,970
1,970
1,969
1,969
1,969
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,678
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,677
0,75
1,156
1,156
1,156
1,156
1,156
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,155
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,154
1,153
1,153
1,153
1,153
1,153
0,90
1,653
1,653
1,653
1,653
1,653
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,652
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,651
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,650
1,649
1,649
1,649
1,649
1,649
0,95
1,969
1,969
1,969
1,969
1,969
1,968
1,968
1,968
1,968
1,968
1,968
1,968
1,968
1,968
1,968
1,967
1,967
1,967
1,967
1,967
1,967
1,967
1,967
1,967
1,967
1,966
1,966
1,966
1,966
1,966
1,966
1,966
1,966
1,966
1,966
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
1,965
Table A9. k2 for two-sided statistical tolerance, standard deviation: known and confidence
level γ = 50 %
NB-CPD/SG10/03/006r2
Page 38 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
0,919
1,520
0,834
1,402
0,792
1,340
0,768
1,303
0,752
1,278
0,741
1,260
0,732
1,246
0,726
1,236
0,721
1,227
0,716
1,220
0,713
1,214
0,710
1,209
0,707
1,205
0,705
1,202
0,703
1,198
0,702
1,196
0,700
1,193
0,699
1,191
0,698
1,189
0,697
1,187
0,695
1,185
0,695
1,184
0,694
1,183
0,693
1,182
0,692
1,180
0,692
1,179
0,691
1,178
0,691
1,177
0,690
1,176
0,690
1,175
0,689
1,175
0,689
1,174
0,688
1,174
0,688
1,173
0,688
1,172
0,687
1,172
0,687
1,171
0,686
1,171
0,686
1,170
0,686
1,170
0,686
1,169
0,685
1,169
0,685
1,168
0,685
1,168
0,685
1,168
0,685
1,167
0,684
1,167
0,684
1,166
0,684
1,166
n
0,90
2,106
1,971
1,897
1,850
1,818
1,794
1,776
1,762
1,751
1,742
1,734
1,727
1,722
1,717
1,712
1,708
1,705
1,702
1,699
1,697
1,694
1,692
1,690
1,684
1,678
1,681
1,684
1,683
1,681
1,680
1,679
1,678
1,677
1,676
1,675
1,674
1,674
1,673
1,672
1,671
1,671
1,670
1,670
1,669
1,669
1,668
1,668
1,667
1,667
0,95
2,464
2,323
2,244
2,194
2,158
2,132
2,112
2,096
2,083
2,073
2,064
2,056
2,050
2,044
2,039
2,034
2,030
2,027
2,024
2,021
2,018
2,016
2,013
2,011
2,009
2,008
2,006
2,005
2,003
2,002
2,001
1,999
1,998
1,997
1,996
1,995
1,994
1,993
1,992
1,991
1,991
1,990
1,990
1,989
1,988
1,988
1,987
1,987
1,986
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,684
0,684
0,683
0,683
0,683
0,683
0,683
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,680
0,680
0,680
0,680
0,68
0,680
0,680
0,680
0,680
0,680
0,679
0,679
0,679
0,679
0,679
0,75
1,166
1,166
1,165
1,165
1,165
1,165
1,165
1,164
1,164
1,164
1,164
1,164
1,163
1,163
1,163
1,163
1,163
1,162
1,162
1,162
1,162
1,162
1,161
1,161
1,161
1,161
1,161
1,160
1,160
1,16
1,160
1,160
1,160
1,160
1,160
1,159
1,159
1,159
1,159
1,159
1,159
1,159
1,159
1,159
1,159
1,158
1,158
1,158
1,158
1,158
0,90
1,667
1,666
1,666
1,665
1,665
1,665
1,664
1,664
1,663
1,663
1,663
1,663
1,662
1,662
1,662
1,662
1,662
1,661
1,661
1,661
1,661
1,661
1,660
1,660
1,660
1,660
1,660
1,659
1,659
1,659
1,659
1,659
1,658
1,658
1,658
1,658
1,658
1,657
1,657
1,657
1,657
1,657
1,657
1,657
1,657
1,656
1,656
1,656
1,656
1,656
0,95
1,986
1,985
1,985
1,984
1,984
1,984
1,983
1,983
1,982
1,982
1,982
1,981
1,981
1,981
1,981
1,980
1,980
1,980
1,979
1,979
1,979
1,979
1,978
1,978
1,978
1,978
1,978
1,977
1,977
1,977
1,977
1,977
1,976
1,976
1,976
1,976
1,976
1,975
1,975
1,975
1,975
1,975
1,974
1,974
1,974
1,974
1,974
1,973
1,973
1,973
Table A10. k2 for two-sided statistical tolerance, standard deviation: known and confidence
level γ = 75 %
NB-CPD/SG10/03/006r2
Page 39 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
1,187
1,842
1,013
1,640
0,924
1,527
0,872
1,456
0,837
1,407
0,813
1,371
0,795
1,344
0,781
1,323
0,770
1,306
0,761
1,292
0,754
1,281
0,758
1,271
0,742
1,262
0,738
1,255
0,734
1,248
0,730
1,243
0,727
1,237
0,724
1,233
0,722
1,229
0,720
1,226
0,717
1,222
0,716
1,219
0,714
1,216
0,713
1,214
0,711
1,211
0,710
1,209
0,708
1,207
0,707
1,205
0,706
1,203
0,705
1,201
0,704
1,200
0,703
1,198
0,702
1,197
0,701
1,195
0,700
1,194
0,700
1,193
0,699
1,192
0,699
1,191
0,698
1,190
0,697
1,189
0,697
1,188
0,696
1,187
0,696
1,186
0,695
1,185
0,695
1,184
0,694
1,184
0,694
1,183
0,693
1,183
0,693
1,182
n
0,90
2,446
1,236
2,114
2,034
1,977
1,935
1,902
1,875
1,854
1,836
1,821
1,809
1,797
1,788
1,779
1,772
1,765
1,759
1,753
1,749
1,744
1,740
1,736
1,733
1,729
1,726
1,723
1,721
1,718
1,716
1,714
1,712
1,710
1,708
1,706
1,705
1,703
1,702
1,700
1,699
1,698
1,696
1,695
1,694
1,693
1,692
1,691
1,690
1,689
0,95
2,809
2,597
2,473
2,390
2,330
2,285
2,250
2,222
2,198
2,179
2,162
2,148
2,136
2,125
2,115
2,107
2,099
2,092
2,086
2,081
2,075
2,071
2,066
2,062
2,058
2,055
2,052
2,049
2,046
2,044
2,041
2,039
2,036
2,034
2,032
2,030
2,029
2,027
2,025
2,024
2,022
2,021
2,019
2,018
2,017
2,016
2,014
2,013
2,012
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,693
0,692
0,692
0,692
0,692
0,691
0,691
0,691
0,690
0,690
0,690
0,690
0,689
0,689
0,689
0,689
0,689
0,688
0,688
0,688
0,688
0,688
0,687
0,687
0,687
0,687
0,687
0,686
0,686
0,686
0,686
0,686
0,686
0,686
0,686
0,685
0,685
0,685
0,685
0,685
0,685
0,685
0,685
0,685
0,685
0,684
0,684
0,684
0,684
0,684
0,75
1,182
1,181
1,181
1,180
1,180
1,179
1,179
1,178
1,178
1,177
1,177
1,176
1,176
1,175
1,175
1,175
1,174
1,174
1,173
1,173
1,173
1,172
1,172
1,172
1,172
1,171
1,171
1,171
1,170
1,170
1,170
1,170
1,169
1,169
1,169
1,169
1,169
1,168
1,168
1,168
1,168
1,168
1,167
1,167
1,167
1,167
1,167
1,166
1,166
1,166
0,90
1,688
1,688
1,687
1,686
1,686
1,685
1,684
1,683
1,683
1,682
1,682
1,681
1,681
1,680
1,680
1,679
1,679
1,678
1,678
1,677
1,677
1,676
1,676
1,675
1,675
1,675
1,674
1,674
1,673
1,673
1,673
1,672
1,672
1,672
1,672
1,671
1,671
1,671
1,670
1,670
1,670
1,669
1,669
1,669
1,669
1,668
1,668
1,668
1,667
1,667
0,95
2,011
2,010
2,010
2,009
2,008
2,007
2,006
2,006
2,005
2,004
2,003
2,003
2,002
2,002
2,001
2,000
2,000
1,999
1,999
1,998
1,998
1,997
1,997
1,996
1,996
1,995
1,995
1,994
1,994
1,993
1,993
1,992
1,992
1,992
1,992
1,991
1,991
1,991
1,990
1,990
1,990
1,989
1,989
1,989
1,989
1,988
1,988
1,988
1,987
1,987
Table A11. k2 for two-sided statistical tolerance, standard deviation: known and confidence
level γ = 90 %
NB-CPD/SG10/03/006r2
Page 40 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
1,393
2,062
1,160
1,812
1,036
1,668
0,960
1,574
0,910
1,509
0,875
1,460
0,894
0,423
0,828
1,394
0,812
1,370
0,799
1,351
0,788
1,334
0,779
1,320
0,772
1,308
0,765
1,298
0,759
1,289
0,754
1,281
0,749
1,274
0,745
1,267
0,742
1,261
0,739
1,256
0,736
1,251
0,733
1,247
0,730
1,243
0,728
1,240
0,726
1,236
0,724
1,233
0,722
1,230
0,721
1,228
0,719
1,225
0,718
1,223
0,717
1,221
0,715
1,218
0,714
1,216
0,713
1,214
0,712
1,212
0,711
1,211
0,710
1,209
0,709
1,208
0,708
1,206
0,707
1,205
0,706
1,204
0,706
1,202
0,705
1,201
0,704
1,200
0,703
1,199
0,703
1,198
0,702
1,197
0,702
1,196
0,701
1,195
n
0,90
2,668
2,415
1,265
2,165
2,093
2,039
1,996
1,961
1,933
1,909
1,889
1,872
1,857
1,844
1,832
1,822
1,812
1,804
1,797
1,790
1,783
1,778
1,772
1,768
1,763
1,759
1,755
1,752
1,748
1,745
1,742
1,739
1,736
1,733
1,731
1,729
1,727
1,725
1,723
1,721
1,719
1,718
1,716
1,714
1,713
1,712
1,710
1,709
1,708
0,95
3,031
2,777
2,627
2,525
2,451
2,395
2,350
2,313
2,283
2,258
2,236
2,218
2,201
2,187
2,174
2,163
2,152
2,143
2,135
2,128
2,120
2,114
2,108
2,103
2,097
2,093
2,088
2,084
2,080
2,077
2,073
2,070
2,066
2,063
2,061
2,058
2,056
2,053
2,051
2,049
2,047
2,045
2,043
2,041
2,039
2,038
2,036
2,035
2,033
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,701
0,700
0,700
0,699
0,699
0,699
0,698
0,698
0,697
0,697
0,697
0,696
0,696
0,696
0,696
0,695
0,695
0,695
0,694
0,694
0,694
0,693
0,693
0,693
0,693
0,692
0,692
0,692
0,691
0,691
0,691
0,691
0,690
0,690
0,690
0,690
0,690
0,689
0,689
0,689
0,689
0,689
0,689
0,689
0,689
0,688
0,688
0,688
0,688
0,688
0,75
1,194
1,194
1,193
1,192
1,192
1,191
1,190
1,189
1,189
1,188
1,187
1,187
1,186
1,186
1,185
1,184
1,184
1,183
1,183
1,182
1,182
1,181
1,181
1,180
1,180
1,180
1,179
1,179
1,178
1,178
1,178
1,177
1,177
1,177
1,177
1,176
1,176
1,176
1,175
1,175
1,175
1,175
1,174
1,174
1,174
1,174
1,174
1,173
1,173
1,173
0,90
1,707
1,706
1,705
1,704
1,703
1,701
1,700
1,699
1,698
1,697
1,696
1,696
1,695
1,694
1,694
1,693
1,692
1,691
1,691
1,690
1,689
1,689
1,688
1,688
1,687
1,686
1,686
1,685
1,685
1,684
1,684
1,683
1,683
1,682
1,682
1,682
1,681
1,681
1,680
1,680
1,680
1,679
1,679
1,679
1,679
1,678
1,678
1,678
1,677
1,677
0,95
2,032
2,031
2,030
2,029
2,028
2,026
2,025
2,024
2,023
2,022
2,021
2,020
2,019
2,018
2,018
2,017
2,016
2,015
2,014
2,013
2,012
2,012
2,011
2,011
2,010
2,009
2,009
2,008
2,008
2,007
2,007
2,006
2,006
2,005
2,005
2,004
2,004
2,003
2,003
2,002
2,002
2,001
2,001
2,000
2,000
2,000
1,999
1,999
1,998
1,998
Table A12. k2 for two-sided statistical tolerance, standard deviation: known and confidence
level γ = 95 %
NB-CPD/SG10/03/006r2
Page 41 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
1,243
2,057
0,943
1,582
0,853
1,441
0,809
1,370
0,782
1,328
0,765
1,300
0,752
1,279
0,743
1,264
0,735
1,252
0,730
1,242
0,725
1,234
0,721
1,227
0,717
1,222
0,714
1,217
0,712
1,212
0,709
1,209
0,707
1,205
0,706
1,202
0,704
1,200
0,703
1,198
0,701
1,195
0,700
1,193
0,699
1,191
0,698
1,190
0,697
1,188
0,697
1,187
0,696
1,186
0,695
1,185
0,694
1,183
0,693
1,182
0,693
1,181
0,692
1,181
0,692
1,180
0,691
1,179
0,691
1,178
0,690
1,177
0,690
1,177
0,689
1,176
0,689
1,175
0,689
1,174
0,689
1,174
0,688
1,173
0,688
1,173
0,688
1,172
0,688
1,172
0,687
1,171
0,687
1,171
0,686
1,170
0,686
1,170
n
0,90
2,870
2,229
2,040
1,946
1,889
1,851
1,823
1,802
1,786
1,772
1,761
1,752
1,744
1,738
1,732
1,727
1,722
1,718
1,714
1,711
1,708
1,706
1,703
1,701
1,698
1,696
1,694
1,693
1,691
1,690
1,689
1,687
1,686
1,685
1,684
1,683
1,682
1,681
1,680
1,679
1,678
1,678
1,677
1,676
1,675
1,675
1,674
1,674
1,673
0,95
3,376
2,635
2,416
2,308
2,243
2,199
2,168
2,143
2,124
2,109
2,096
2,086
2,077
2,069
2,062
2,056
2,051
2,046
2,042
2,038
2,034
2,031
2,028
2,026
2,023
2,021
2,018
2,016
2,014
2,013
2,011
2,010
2,008
2,007
2,006
2,005
2,003
2,002
2,001
2,000
1,999
1,999
1,998
1,997
1,996
1,995
1,995
1,994
1,993
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,686
0,686
0,685
0,685
0,685
0,685
0,685
0,684
0,684
0,684
0,684
0,684
0,684
0,684
0,684
0,683
0,683
0,683
0,683
0,683
0,683
0,683
0,683
0,683
0,683
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,682
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,681
0,75
1,170
1,169
1,169
1,169
1,169
1,168
1,168
1,168
1,167
1,167
1,167
1,167
1,166
1,166
1,166
1,166
1,166
1,165
1,165
1,165
1,165
1,165
1,164
1,164
1,164
1,164
1,164
1,163
1,163
1,163
1,163
1,163
1,163
1,163
1,163
1,162
1,162
1,162
1,162
1,162
1,162
1,162
1,161
1,161
1,161
1,161
1,161
1,160
1,160
1,160
0,90
1,673
1,672
1,672
1,671
1,671
1,670
1,670
1,669
1,669
1,668
1,668
1,667
1,667
1,667
1,667
1,666
1,666
1,666
1,665
1,665
1,665
1,664
1,664
1,664
1,664
1,663
1,663
1,663
1,662
1,662
1,662
1,662
1,662
1,662
1,662
1,661
1,661
1,661
1,661
1,661
1,661
1,661
1,660
1,660
1,660
1,660
1,660
1,659
1,659
1,659
0,95
1,993
1,992
1,992
1,991
1,991
1,990
1,990
1,989
1,989
1,988
1,988
1,987
1,987
1,986
1,986
1,986
1,985
1,985
1,984
1,984
1,984
1,983
1,983
1,983
1,983
1,982
1,982
1,982
1,981
1,981
1,981
1,981
1,980
1,980
1,980
1,980
1,980
1,979
1,979
1,979
1,979
1,979
1,978
1,978
1,978
1,978
1,978
1,977
1,977
1,977
Table A13. k2 for two-sided statistical tolerance, standard deviation: unknown and
confidence level γ = 50 %
NB-CPD/SG10/03/006r2
Page 42 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
2,674
4,394
1,492
2,487
1,211
2,036
1,083
1,829
1,009
1,709
0,961
1,630
1,926
1,573
0,900
1,530
0,880
1,497
0,864
1,469
0,850
1,447
0,839
1,428
0,829
1,412
0,821
1,398
0,814
1,386
0,807
1,375
0,802
1,366
0,797
1,357
0,792
1,349
0,788
1,343
0,784
1,336
0,781
1,331
0,777
1,325
0,774
1,320
0,771
1,315
0,769
1,311
0,766
1,306
0,764
1,303
0,762
1,299
0,760
1,296
0,758
1,293
0,757
1,290
0,755
1,287
0,753
1,284
0,752
1,282
0,751
1,280
0,749
1,277
0,748
1,275
0,747
1,273
0,746
1,271
0,745
1,269
0,743
1,267
0,742
1,265
0,741
1,263
0,740
1,262
0,739
1,260
0,739
1,259
0,738
1,257
0,737
1,256
n
0,90
6,109
3,489
2,872
2,590
2,425
2,316
2,238
2,179
2,133
2,095
2,064
2,038
2,015
1,996
1,979
1,964
1,950
1,938
1,927
1,918
1,908
1,900
1,892
1,886
1,879
1,873
1,867
1,862
1,857
1,853
1,848
1,844
1,839
1,835
1,832
1,829
1,825
1,822
1,819
1,816
1,814
1,811
1,809
1,806
1,804
1,802
1,799
1,797
1,795
0,95
7,178
4,117
3,397
3,069
2,877
2,750
2,659
2,590
2,536
2,492
2,456
2,425
2,399
2,376
2,356
2,338
2,322
2,308
2,295
2,284
2,273
2,264
2,254
2,246
2,238
2,231
2,224
2,218
2,211
2,206
2,201
2,196
2,191
2,186
2,182
2,178
2,175
2,171
2,167
2,164
2,161
2,158
2,155
2,152
2,149
2,147
2,144
2,142
2,139
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,736
0,736
0,735
0,734
0,734
0,733
0,732
0,731
0,731
0,730
0,730
0,729
0,729
0,728
0,728
0,727
0,727
0,726
0,726
0,725
0,725
0,724
0,724
0,723
0,723
0,723
0,722
0,722
0,721
0,721
0,721
0,720
0,720
0,720
0,720
0,719
0,719
0,719
0,718
0,718
0,718
0,717
0,717
0,717
0,717
0,716
0,716
0,716
0,715
0,715
0,75
1,255
1,254
1,252
1,251
1,250
1,249
1,248
1,246
1,245
1,244
1,243
1,242
1,242
1,241
1,240
1,239
1,238
1,238
1,237
1,236
1,235
1,235
1,234
1,233
1,233
1,232
1,231
1,230
1,230
1,229
1,228
1,228
1,227
1,227
1,226
1,225
1,225
1,224
1,224
1,223
1,223
1,222
1,222
1,221
1,221
1,221
1,220
1,220
1,219
1,219
0,90
1,793
1,792
1,790
1,789
1,787
1,785
1,784
1,782
1,781
1,779
1,778
1,776
1,775
1,774
1,773
1,771
1,770
1,769
1,767
1,766
1,765
1,764
1,763
1,762
1,762
1,761
1,760
1,759
1,758
1,757
1,756
1,755
1,755
1,754
1,753
1,752
1,751
1,751
1,750
1,749
1,748
1,748
1,747
1,746
1,746
1,745
1,744
1,743
1,743
1,742
0,95
2,137
2,135
2,133
2,131
2,129
2,127
2,125
2,123
2,121
2,119
2,118
2,116
2,115
2,113
2,112
2,111
2,109
2,108
2,106
2,105
2,104
2,103
2,101
2,100
2,099
2,098
2,097
2,095
2,094
2,093
2,092
2,091
2,090
2,089
2,089
2,088
2,087
2,086
2,085
2,084
2,083
2,082
2,082
2,081
2,080
2,079
2,078
2,078
2,077
2,076
Table A14. k2 for two-sided statistical tolerance, standard deviation: unknown and
confidence level γ = 75 %
NB-CPD/SG10/03/006r2
Page 43 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
6,809
11,166
2,492
4,135
1,766
2,954
1,473
2,478
1,314
2,218
1,213
2,053
1,144
1,939
1,093
1,854
1,053
1,789
1,022
1,737
0,996
1,694
0,975
1,659
0,957
1,628
0,941
1,602
0,928
1,580
0,916
1,560
0,905
1,542
0,896
1,526
0,887
1,512
0,880
1,500
0,873
1,487
0,867
1,477
0,861
1,466
0,856
1,458
0,850
1,449
0,846
1,442
0,841
1,434
0,837
1,427
0,833
1,420
0,830
1,415
0,827
1,409
0,823
1,404
0,820
1,398
0,817
1,393
0,815
1,389
0,812
1,385
0,810
1,380
0,807
1,376
0,805
1,372
0,803
1,369
0,801
1,366
0,799
1,362
0,797
1,359
0,795
1,356
0,793
1,353
0,792
1,350
0,790
1,348
0,789
1,345
0,787
1,342
n
0,90
15,513
5,789
4,158
3,500
3,141
2,913
2,755
2,637
2,546
2,474
2,414
2,365
2,322
2,286
2,254
2,226
2,201
2,179
2,159
2,142
2,124
2,110
2,095
2,083
2,070
2,059
2,048
2,039
2,029
2,021
2,014
2,006
1,999
1,991
1,985
1,979
1,974
1,968
1,962
1,957
1,952
1,948
1,943
1,938
1,934
1,930
1,927
1,923
1,919
0,95
18,221
6,824
4,913
4,143
3,723
3,456
3,270
3,133
3,026
2,941
2,871
2,813
2,763
2,720
2,683
2,650
2,620
2,594
2,570
2,550
2,529
2,512
2,494
2,480
2,465
2,452
2,439
2,428
2,417
2,408
2,399
2,390
2,381
2,372
2,365
2,358
2,351
2,344
2,337
2,331
2,326
2,320
2,315
2,309
2,304
2,300
2,295
2,291
2,286
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,786
0,785
0,783
0,782
0,781
0,780
0,779
0,777
0,776
0,775
0,774
0,773
0,772
0,771
0,771
0,770
0,769
0,768
0,767
0,766
0,765
0,765
0,764
0,763
0,763
0,762
0,761
0,760
0,760
0,759
0,758
0,758
0,757
0,757
0,756
0,755
0,755
0,754
0,754
0,753
0,753
0,752
0,752
0,751
0,751
0,750
0,750
0,749
0,749
0,748
0,75
1,340
1,338
1,336
1,334
1,332
1,329
1,327
1,325
1,323
1,321
1,320
1,318
1,317
1,315
1,314
1,312
1,311
1,309
1,308
1,306
1,305
1,304
1,302
1,301
1,300
1,299
1,298
1,296
1,295
1,294
1,293
1,292
1,291
1,290
1,289
1,288
1,287
1,286
1,285
1,284
1,283
1,282
1,282
1,281
1,280
1,279
1,278
1,278
1,277
1,276
0,90
1,916
1,913
1,910
1,907
1,904
1,901
1,898
1,895
1,892
1,889
1,887
1,885
1,882
1,880
1,878
1,876
1,874
1,871
1,869
1,867
1,865
1,863
1,862
1,860
1,858
1,856
1,854
1,853
1,851
1,849
1,848
1,846
1,845
1,843
1,842
1,841
1,839
1,838
1,836
1,835
1,834
1,833
1,832
1,831
1,830
1,828
1,827
1,826
1,825
1,824
0,95
2,282
2,279
2,275
2,272
2,268
2,264
2,261
2,257
2,254
2,250
2,247
2,245
2,242
2,240
2,237
2,234
2,232
2,229
2,227
2,224
2,222
2,220
2,218
2,216
2,214
2,211
2,209
2,207
2,205
2,203
2,201
2,200
2,198
2,197
2,195
2,193
2,192
2,190
2,189
2,187
2,186
2,184
2,183
2,181
2,180
2,179
2,177
2,176
2,174
2,173
Table A15. k2 for two-sided statistical tolerance, standard deviation: unknown and
confidence level γ = 90 %
NB-CPD/SG10/03/006r2
Page 44 of 74
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
fractile : p
0,50
0,75
13,652
22,383
3,585
5,938
2,288
3,819
1,812
3,041
1,566
2,639
1,416
2,392
1,314
2,224
1,240
2,101
1,183
2,008
1,139
1,935
1,103
1,875
1,074
1,825
1,049
1,784
1,027
1,748
1,009
1,717
0,992
1,689
0,978
1,665
0,965
1,644
0,954
1,625
0,944
1,608
0,934
1,591
0,926
1,577
0,918
1,563
0,911
1,552
0,904
1,540
0,898
1,530
0,892
1,519
0,887
1,511
0,881
1,502
0,877
1,495
0,873
1,488
0,868
1,480
0,864
1,473
0,860
1,466
0,857
1,460
0,854
1,455
0,850
1,449
0,847
1,444
0,844
1,438
0,841
1,434
0,839
1,430
0,836
1,425
0,834
1,421
0,831
1,417
0,829
1,413
0,827
1,410
0,825
1,406
0,823
1,403
0,821
1,399
n
0,90
31,093
8,306
5,369
4,291
3,733
3,390
3,157
2,987
2,857
2,754
2,671
2,602
2,543
2,493
2,449
2,411
2,377
2,347
2,319
2,296
2,272
2,253
2,233
2,217
2,200
2,186
2,171
2,159
2,146
2,136
2,126
2,115
2,105
2,095
2,087
2,079
2,072
2,064
2,056
2,050
2,044
2,037
2,031
2,025
2,020
2,015
2,010
2,005
2,000
0,95
36,520
9,789
6,342
5,077
4,423
4,020
3,746
3,546
3,394
3,273
3,175
3,094
3,025
2,965
2,914
2,869
2,829
2,793
2,761
2,733
2,705
2,682
2,659
2,639
2,619
2,602
2,585
2,570
2,555
2,543
2,531
2,519
2,507
2,495
2,486
2,477
2,467
2,458
2,449
2,442
2,434
2,427
2,419
2,412
2,406
2,400
2,394
2,388
2,382
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
fractile : p
0,50
0,819
0,818
0,816
0,814
0,813
0,811
0,809
0,807
0,806
0,804
0,803
0,802
0,800
0,799
0,798
0,797
0,796
0,794
0,793
0,792
0,791
0,790
0,789
0,788
0,788
0,787
0,786
0,785
0,784
0,783
0,782
0,782
0,781
0,780
0,780
0,779
0,778
0,777
0,777
0,776
0,775
0,775
0,774
0,773
0,773
0,772
0,771
0,770
0,770
0,769
0,75
1,396
1,393
1,391
1,388
1,385
1,382
1,379
1,377
1,374
1,371
1,369
1,367
1,365
1,363
1,361
1,359
1,357
1,355
1,353
1,351
1,349
1,348
1,346
1,345
1,343
1,341
1,340
1,338
1,337
1,335
1,334
1,332
1,331
1,330
1,329
1,327
1,326
1,325
1,323
1,322
1,321
1,320
1,319
1,318
1,317
1,316
1,315
1,314
1,313
1,312
0,90
1,996
1,992
1,988
1,984
1,980
1,976
1,972
1,968
1,964
1,960
1,957
1,954
1,951
1,948
1,946
1,943
1,940
1,937
1,934
1,931
1,929
1,927
1,924
1,922
1,920
1,918
1,916
1,913
1,911
1,909
1,907
1,905
1,903
1,901
1,900
1,898
1,896
1,894
1,892
1,890
1,889
1,887
1,886
1,884
1,883
1,881
1,880
1,878
1,877
1,875
0,95
2,377
2,373
2,368
2,364
2,359
2,354
2,350
2,345
2,341
2,336
2,333
2,329
2,326
2,322
2,319
2,315
2,312
2,308
2,305
2,301
2,298
2,296
2,293
2,290
2,288
2,285
2,282
2,279
2,277
2,274
2,272
2,270
2,267
2,265
2,263
2,261
2,259
2,256
2,254
2,252
2,250
2,248
2,247
2,245
2,243
2,241
2,239
2,238
2,236
2,234
Table A16. k2 for two-sided statistical tolerance, standard deviation: unknown and
confidence level γ = 95 %
NB-CPD/SG10/03/006r2
Page 45 of 74
Annex B
Examples of statistical evaluation
Example 1
Example of statistical analysis of compressive strength using batch control.
The fractile p = 50%
The confidence level γ = 95%
The number of series of inspection lots is l = 1.
One-sided tolerance interval, lower limit
The declared mean compressive strength is 15 N/mm²
For the first and the following inspection lots a sample size of 6 samples are taken and tested and
evaluated inspection lot by inspection lot (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and γ: 95
%)).
For the first 42 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A8 (p: 50 % and γ: 95 %) is 0,823. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 50 % and γ: 95 %) and is 0,672. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 42 test results.
From inspection lot 21 and so on 6 samples are taken from each next inspection lot and the test
results are evaluated inspection lot by inspection lot (xm (equation 1), known standard deviation σ
and xest (equation 6) according to clause 5.6.6 using k1,k taken from Annex A Table A4 (p: 50 % and
γ: 95 %)).
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 21). A non-conforming inspection lot has to be
treated separately as described in the text.
NB-CPD/SG10/03/006r2
Page 46 of 74
EXAMPLE 1
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
METHOD A : use at least 6 testresults per inspection lot
50
95
Start correction
End correction
7
20
Series of inspection lots
Declared Value
15
1,409
Inspection lot
test 1
test 2
test 3
test 4
test 5
test 6
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
18,1
16,1
17,7
21,4
19,5
19,9
17,2
20,2
21,0
20,2
20,0
23,8
21,1
21,0
18,8
19,7
20,4
20,5
19,8
19,7
16,5
21,3
19,3
18,5
17,9
18,4
19,1
20,8
20,8
19,3
19,2
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
14,2
21,4
17,5
21,8
18,3
18,8
17,9
19,6
19,7
20,1
17,0
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
15,3
21,3
18,3
19,4
19,4
17,6
18,1
18,5
21,1
18,8
18,2
19,9
18,9
21,3
19,9
17,1
19,1
15,8
20,1
18,1
20,8
22,4
19,2
20,9
16,4
18,9
17,2
17,2
17,7
15,7
15,7
18,6
19,4
21,1
19,4
21,9
18,8
22,4
20,1
21
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
14,3
21,2
19,1
21,3
19,2
17
18,1
18,1
18,6
20,6
17,1
22,6
19,3
22,6
22,6
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18
16,4
20,8
16,7
17,4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
18,43
17,27
17,77
19,50
19,85
19,97
18,02
19,78
18,62
21,02
21,10
19,80
21,03
18,62
19,75
19,13
18,93
20,40
19,25
18,58
15,52
20,82
18,02
19,27
0,70
1,24
1,12
1,35
0,94
0,84
1,09
2,39
2,09
2,03
1,33
3,00
1,44
2,47
1,59
1,21
2,35
1,41
0,83
1,47
1,08
0,96
1,06
1,94
0,823
0,823
0,823
0,823
0,823
0,823
0,823
NB-CPD/SG10/03/006r2
1
kc
0,823
0,811
0,800
0,788
0,777
0,765
0,753
0,742
0,730
0,718
0,707
0,695
0,684
0,672
k1,k
σ
0,672
1,409
0,672
0,672
0,672
0,672
0,672
1,858
1,858
Xest
17,85
16,25
16,84
18,39
19,08
19,28
17,12
Xest
16,86
18,64
17,49
19,91
20,01
18,72
19,97
17,57
18,72
18,12
17,94
19,42
18,29
17,33
Xest
17,33
14,27
19,57
16,77
18,02
Equation OK?
OK
OK
OK
OK
OK
Page 47 of 74
Example 2
Example of statistical analysis of compressive strength using batch control.
This example is similar to example 1. The only difference is that the declared compressive strength
is a 5 % characteristic value.
The fractile p = 95%
The confidence level γ = 95%
The number of series of inspection lots is l = 1.
One-sided tolerance interval, lower limit
The declared 5 % characteristic compressive strength is 10 N/mm²
For the first and the following inspection lots a sample size of 6 samples are taken and tested and
evaluated inspection lot by inspection lot (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 95 % and γ: 95
%)).
For the first 42 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A8 (p: 95 % and γ: 95 %) is 3,708. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 95 % and γ: 95 %) and is 2,317. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 42 test results.
From inspection lot 21 and so on 6 samples are taken from each next inspection lot and the test
results are evaluated inspection lot by inspection lot (xm (equation 1), known standard deviation σ
and xest (equation 6) according to clause 5.6.6 using k1,k taken from Annex A Table A4 (p: 95 % and
γ: 95 %)).
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side. A non-conforming inspection lot has to be treated
separately as described in the text.
NB-CPD/SG10/03/006r2
Page 48 of 74
EXAMPLE 2
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
METHOD A : use at least 6 testresults per inspection lot
95
95
Start correction
End correction
7
20
Series of inspection lots
Declared Value
10
1,409
Inspection lot
test 1
test 2
test 3
test 4
test 5
test 6
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
18,1
16,1
17,7
21,4
19,5
19,9
17,2
20,2
21,0
20,2
20,0
23,8
21,1
21,0
18,8
19,7
20,4
20,5
19,8
19,7
16,5
21,3
19,3
18,5
17,9
18,4
19,1
20,8
20,8
19,3
19,2
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
14,2
21,4
17,5
21,8
18,3
18,8
17,9
19,6
19,7
20,1
17,0
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
15,3
21,3
18,3
19,4
19,4
17,6
18,1
18,5
21,1
18,8
18,2
19,9
18,9
21,3
19,9
17,1
19,1
15,8
20,1
18,1
20,8
22,4
19,2
20,9
16,4
18,9
17,2
17,2
17,7
15,7
15,7
18,6
19,4
21,1
19,4
21,9
18,8
22,4
20,1
21
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
14,3
21,2
19,1
21,3
19,2
17
18,1
18,1
18,6
20,6
17,1
22,6
19,3
22,6
22,6
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18
16,4
20,8
16,7
17,4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
18,43
17,27
17,77
19,50
19,85
19,97
18,02
19,78
18,62
21,02
21,10
19,80
21,03
18,62
19,75
19,13
18,93
20,40
19,25
18,58
15,52
20,82
18,02
19,27
0,70
1,24
1,12
1,35
0,94
0,84
1,09
2,39
2,09
2,03
1,33
3,00
1,44
2,47
1,59
1,21
2,35
1,41
0,83
1,47
1,08
0,96
1,06
1,94
3,708
3,708
3,708
3,708
3,708
3,708
3,708
NB-CPD/SG10/03/006r2
1
kc
3,708
3,601
3,494
3,387
3,280
3,173
3,066
2,959
2,852
2,745
2,638
2,531
2,424
2,317
k1,k
σ
2,317
1,409
2,317
2,317
2,317
2,317
2,317
1,858
1,858
Xest
15,83
12,69
13,61
14,50
16,38
16,86
13,99
Xest
12,79
14,71
13,69
16,24
16,48
15,33
16,71
14,45
15,73
15,26
15,22
16,83
15,83
14,28
Xest
14,28
11,21
16,51
13,71
14,96
Equation OK?
OK
OK
OK
OK
OK
Page 49 of 74
Example 3
Example of statistical analysis of compressive strength using batch control.
This example is similar to example 1. The only difference is, that the used confidence level is 75 %.
The fractile p = 50%
The confidence level γ = 75%
The number of series of inspection lots is l = 1.
One-sided tolerance interval, lower limit
The declared mean compressive strength is 15 N/mm²
For the first and the following inspection lots a sample size of 6 samples are taken and tested and
evaluated inspection lot by inspection lot (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A6 (p: 50 % and γ: 75
%)).
For the first 42 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A6 (p: 50 % and γ: 75 %) is 0,297. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A2 (p: 50 % and γ: 75 %) and is 0,276. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 42 test results.
From inspection lot 21 and so on 6 samples are taken from each next inspection lot and the test
results are evaluated inspection lot by inspection lot (xm (equation 1), known standard deviation σ
and xest (equation 6) according to clause 5.6.6 using k1,k taken from Annex A Table A2 (p: 50 % and
γ: 75 %)).
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side. A non-conforming inspection lot has to be treated
separately as described in the text.
NB-CPD/SG10/03/006r2
Page 50 of 74
EXAMPLE 3
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
METHOD A : use at least 6 testresults per inspection lot
50
75
Start correction
End correction
7
20
Series of inspection lots
Declared Value
15
1,409
Inspection lot
test 1
test 2
test 3
test 4
test 5
test 6
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
18,1
16,1
17,7
21,4
19,5
19,9
17,2
20,2
21,0
20,2
20,0
23,8
21,1
21,0
18,8
19,7
20,4
20,5
19,8
19,7
16,5
21,3
19,3
18,5
17,9
18,4
19,1
20,8
20,8
19,3
19,2
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
14,2
21,4
17,5
21,8
18,3
18,8
17,9
19,6
19,7
20,1
17,0
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
15,3
21,3
18,3
19,4
19,4
17,6
18,1
18,5
21,1
18,8
18,2
19,9
18,9
21,3
19,9
17,1
19,1
15,8
20,1
18,1
20,8
22,4
19,2
20,9
16,4
18,9
17,2
17,2
17,7
15,7
15,7
18,6
19,4
21,1
19,4
21,9
18,8
22,4
20,1
21
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
14,3
21,2
19,1
21,3
19,2
17
18,1
18,1
18,6
20,6
17,1
22,6
19,3
22,6
22,6
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18
16,4
20,8
16,7
17,4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
18,43
17,27
17,77
19,50
19,85
19,97
18,02
19,78
18,62
21,02
21,10
19,80
21,03
18,62
19,75
19,13
18,93
20,40
19,25
18,58
15,52
20,82
18,02
19,27
0,70
1,24
1,12
1,35
0,94
0,84
1,09
2,39
2,09
2,03
1,33
3,00
1,44
2,47
1,59
1,21
2,35
1,41
0,83
1,47
1,08
0,96
1,06
1,94
0,297
0,297
0,297
0,297
0,297
0,297
0,297
NB-CPD/SG10/03/006r2
1
kc
0,297
0,295
0,294
0,292
0,291
0,289
0,287
0,286
0,284
0,282
0,281
0,279
0,278
0,276
k1,k
σ
0,276
1,409
0,276
0,276
0,276
0,276
0,276
1,858
1,858
Xest
18,22
16,90
17,43
19,10
19,57
19,72
17,69
Xest
17,60
19,37
18,20
20,60
20,69
19,39
20,63
18,21
19,35
18,74
18,54
20,01
18,86
18,07
Xest
18,07
15,00
20,30
17,50
18,75
Equation OK?
OK
OK
OK
OK
OK
Page 51 of 74
Example 4
Example of statistical analysis of compressive strength using “Rolling“ inspection.
The fractile p = 50%
The confidence level γ = 95%
The number of series of inspection lots is l = 4.
One-sided tolerance interval, lower limit
The declared mean compressive strength is 15 N/mm²
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 3) according to clause 5.6.6 using
k1,u taken from Annex A Table A8 (p: 50 % and γ: 95 %)). For the next and the following 2
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 3)
according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and γ: 95 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 12 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and
γ: 95 %)) but the spot sample size is limited to the last 12 samples. The spot sample size continues
to be 12.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A8 (p: 50 % and γ: 95 %) is 0,519. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 50 % and γ: 95 %) and is 0,475. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A4 (p: 50 % and γ: 95 %)) and the spot sample size is still limited to the last 12
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 1). A non-conforming inspection lot has to be
treated separately as described in the text.
NB-CPD/SG10/03/006r2
Page 52 of 74
EXAMPLE4
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
50
95
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
Declared Value
15
2,325
Inspection lot
test 1
test 2
test 3
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
20,2
21,0
20,2
20,0
23,8
21,1
21,0
18,8
19,7
20,4
20,5
19,8
19,7
18,7
21,3
19,3
18,5
19,9
18,9
21,3
19,9
17,1
19,1
15,8
20,1
18,1
20,8
22,4
19,2
20,9
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
20,0
21,4
17,5
21,8
21,9
18,8
22,4
20,1
21,0
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
19,7
21,3
18,3
19,4
22,6
19,3
22,6
22,6
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18,0
3
6
9
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
18,10
18,17
18,76
19,40
20,31
20,93
20,51
20,14
19,64
19,47
20,10
20,00
19,58
19,44
19,58
19,31
19,77
20,27
19,68
20,62
20,86
19,96
20,54
19,77
19,46
19,63
18,75
19,01
18,86
19,00
1,97
2,39
2,43
2,45
2,45
1,63
2,02
1,91
1,47
1,35
0,65
0,79
1,35
1,34
1,49
1,55
1,41
1,66
1,59
1,62
1,53
2,21
2,37
2,54
2,57
2,27
2,32
2,21
1,94
1,93
1,686
0,823
0,620
0,519
0,519
0,519
0,519
NB-CPD/SG10/03/006r2
4
kc
0,519
0,516
0,512
0,509
0,505
0,502
0,499
0,495
0,492
0,489
0,485
0,482
0,478
0,475
k1,k
σ
0,475
2,325
0,475
0,475
0,475
0,475
0,475
0,475
0,475
0,475
0,475
0,475
0,475
1,762
1,762
Xest
14,78
16,20
17,25
18,13
19,03
20,09
19,46
Xest
19,30
18,94
18,45
18,28
18,93
18,83
18,42
18,29
18,44
18,17
18,64
19,15
18,57
19,78
Xest
19,78
20,02
19,12
19,70
18,93
18,62
18,79
17,91
18,17
18,02
18,16
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
Page 53 of 74
Example 5
Example of statistical analysis of compressive strength using "Rolling" inspection.
This example is similar to example 4. The only difference is the size of the series of inspection lots.
In example 4 the series is l = 4 and in example 5 the series is l = 5.
The fractile p = 50%
The confidence level γ = 95%
The number of series of inspection lots is l = 5.
One-sided tolerance interval, lower limit
The declared mean compressive strength is 15 N/mm²
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 3) according to clause 5.6.6 using
k1,u taken from Annex A Table A8 (p: 50 % and γ: 95 %)). For the next and the following 3
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 3)
according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and γ: 95 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 15 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and
γ: 95 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A8 (p: 50 % and γ: 95 %) is 0,455. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 50 % and γ: 95 %) and is 0,425. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A4 (p: 50 % and γ: 95 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
NB-CPD/SG10/03/006r2
Page 54 of 74
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 1 and 22). A non-conforming inspection lot has
to be treated separately as described in the text.
EXAMPLE 5
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
50
95
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
Declared Value
15
2,490
Inspection lot
test 1
test 2
test 3
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
20,2
21,0
20,2
20,0
23,8
21,1
15,4
18,8
19,7
20,4
20,5
19,8
19,7
18,7
14,7
19,3
20,8
15,2
14,1
14,4
19,9
14,9
16,0
15,8
20,1
18,1
20,8
22,4
19,2
20,9
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
20,0
15,8
17,5
15,3
16,1
16,1
15,5
16,1
16,0
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
19,7
16,2
14,4
19,4
14,5
15,4
15,7
14,3
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18,0
3
6
9
12
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
18,10
18,17
18,76
19,40
19,87
20,39
20,02
20,01
19,69
19,35
19,35
19,97
19,61
19,56
18,67
17,93
17,73
17,17
16,32
16,25
16,19
15,58
16,59
17,35
18,23
18,59
19,06
19,01
19,01
18,91
1,97
2,39
2,43
2,45
2,47
2,19
2,40
2,24
2,06
1,68
1,69
0,79
1,20
1,22
2,02
1,98
2,10
2,26
2,03
2,06
2,11
1,38
2,76
2,99
2,99
2,75
2,60
2,26
1,96
1,82
1,686
0,823
0,620
0,519
0,455
0,455
0,455
NB-CPD/SG10/03/006r2
5
kc
0,455
0,453
0,450
0,448
0,446
0,443
0,441
0,439
0,437
0,434
0,432
0,430
0,427
0,425
k1,k
σ
0,425
2,490
0,425
0,425
0,425
0,425
0,425
0,425
0,425
0,425
0,425
0,425
0,425
2,461
2,461
Xest
14,78
16,20
17,25
18,13
18,74
19,40
18,93
Xest
18,89
18,88
18,57
18,23
18,24
18,87
18,51
18,47
17,58
16,85
16,66
16,10
15,26
15,20
Xest
15,20
15,14
14,53
15,54
16,30
17,19
17,54
18,01
17,96
17,97
17,86
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
Page 55 of 74
Example 6
Example of statistical analysis of compressive strength using a special type of “Rolling” inspection
"Progressive Sampling”.
The fractile p = 50%
The confidence level γ = 95%
The number of series of inspection lots is l = 15.
One-sided tolerance interval, lower limit
The declared mean compressive strength is 6 N/mm²
For each of the 1st to 5th inspection lots a spot size of one sample is taken and tested. These
inspection lots are evaluated together (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and γ: 95
%)). For the 6th and following inspection lots 1 additional sample is taken and tested and evaluated
together with the ones from the previous inspection lots (xm (equation 1), standard deviation ss
(equation 2) and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8
(p: 50 % and γ: 95 %)). The spot size is gradually increased from 5 to 15 samples.
From then on, 1 additional sample is taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 50 % and
γ: 95 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 30 samples, the standard deviation of the population is considered to be unknown and
the k1,u factor taken from Annex A Table A8 (p: 50 % and γ: 95 %) is 0,455.
For the inspection lots 30 – 60 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 50 % and γ: 95 %) and is 0,425. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 61 and so on 1 additional sample is taken from each next inspection lot and the
test results are evaluated together with the ones from the previous inspection lots (xm (equation 1),
known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A4 (p: 50 % and γ: 95 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 2). A non-conforming inspection lot has to be
treated separately as described in the text.
NB-CPD/SG10/03/006r2
Page 56 of 74
EXAMPLE 6
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
50
95
test 1
test 2
6,78
8,36
8,64
8,41
8,25
7,44
8,57
6,25
7,83
7,40
8,57
8,27
7,67
8,13
6,53
7,85
7,20
7,95
7,39
5,96
6,26
5,60
7,15
6,01
7,88
6,01
5,91
7,05
5,51
5,87
7,59
8,64
6,72
8,11
8,21
8,48
7,87
7,36
8,11
6,50
7,38
7,78
8,11
8,70
7,46
7,76
6,46
7,78
6,73
7,47
6,98
6,35
6,35
5,53
6,68
5,86
7,40
6,42
5,80
6,57
5,71
5,78
10,46
12,26
8,90
NB-CPD/SG10/03/006r2
test 3
Start correction
End correction
METHOD B: progressive sampling : use only 1 testresult per inspection lot
30
60
n
Xm
Ss
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
6,78
7,57
7,92
8,05
8,09
7,98
8,06
7,84
7,84
7,79
7,86
7,90
7,88
7,90
7,81
7,88
7,80
7,75
7,69
7,53
7,46
7,26
7,32
7,20
7,23
7,06
6,90
6,86
6,69
6,64
6,62
6,72
6,64
6,69
6,84
6,98
7,14
7,15
7,29
7,20
7,29
7,41
7,48
7,70
7,80
7,81
7,67
7,74
7,65
7,60
7,50
7,40
7,33
7,16
7,17
7,07
7,04
6,93
6,74
6,68
6,54
6,49
6,67
7,04
7,14
#DEEL/0!
1,11
1,00
0,85
0,74
0,72
0,69
0,91
0,85
0,81
0,80
0,77
0,74
0,72
0,78
0,72
0,73
0,69
0,67
0,79
0,85
0,92
0,88
0,93
0,94
0,91
0,89
0,87
0,86
0,88
0,86
1,00
0,94
1,00
1,05
1,11
1,07
1,07
1,05
1,05
1,00
0,93
0,94
0,81
0,65
0,64
0,69
0,64
0,68
0,66
0,63
0,68
0,74
0,83
0,83
0,89
0,87
0,83
0,72
0,69
0,67
0,69
1,21
1,88
1,94
Series of inspection lots
15
Declared Value
6
1,009
k1,u
4,465
1,686
1,177
0,954
0,823
0,735
0,670
0,620
0,580
0,547
0,519
0,495
0,474
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
0,455
kc
0,455
0,454
0,453
0,452
0,451
0,450
0,449
0,448
0,447
0,446
0,445
0,444
0,443
0,442
0,441
0,440
0,439
0,438
0,437
0,436
0,435
0,434
0,433
0,432
0,431
0,430
0,429
0,428
0,427
0,426
0,425
k1,k
σ
0,425
1,009
0,425
0,425
0,425
0,425
0,425
0,425
0,935
0,935
Xest
2,59
6,24
7,04
7,38
7,39
7,56
7,23
7,31
7,32
7,42
7,49
7,51
7,56
7,45
7,55
7,47
7,44
7,38
7,17
7,07
6,84
6,92
6,78
6,80
6,64
6,50
6,47
6,30
6,24
Xest
6,18
6,17
6,26
6,18
6,23
6,38
6,53
6,68
6,70
6,84
6,75
6,84
6,97
7,04
7,25
7,36
7,37
7,23
7,30
7,21
7,16
7,06
6,96
6,89
6,72
6,73
6,63
6,61
6,50
6,31
6,28
Xest
6,28
6,14
6,10
6,28
6,64
6,74
Equation OK?
OK
OK
OK
OK
OK
OK
Page 57 of 74
Example 7
Example of statistical analysis of compressive strength using “Rolling” inspection.
This example is similar to example 4. The only difference is that the declared compressive strength
is a 5 % characteristic value.
The fractile p = 95%
The confidence level γ = 95%
The number of series of inspection lots is l = 4.
One-sided tolerance interval, lower limit
The declared 5 % characteristic compressive strength is 10 N/mm²
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 3) according to clause 5.6.6 using
k1,u taken from Annex A Table A8 (p: 95 % and γ: 95 %)). For the next and the following 3
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 3)
according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 95 % and γ: 95 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 12 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 95 % and
γ: 95 %)) but the spot sample size is limited to the last 12 samples. The spot sample size continues
to be 12.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A8 (p: 95 % and γ: 95 %) is 2,737. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 95 % and γ: 95 %) and is 2,120. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A4 (p: 95 % and γ: 95 %)) and the spot sample size is still limited to the last 12
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
NB-CPD/SG10/03/006r2
Page 58 of 74
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 1 and 2). A non-conforming inspection lot has to
be treated separately as described in the text.
EXAMPLE 7
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
95
95
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
Declared Value
10
2,490
Inspection lot
test 1
test 2
test 3
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
20,2
21,0
20,2
20,0
23,8
21,1
15,4
18,8
19,7
20,4
20,5
19,8
19,7
18,7
14,7
19,3
20,8
15,2
14,1
14,4
19,9
14,9
16,0
15,8
20,1
18,1
20,8
22,4
19,2
20,9
17,8
14,7
17,3
22,8
20,6
21,0
16,5
20,5
19,5
19,6
21,4
20,5
17,6
20,0
15,8
17,5
15,3
16,1
16,1
15,5
16,1
16,0
23,4
21,7
21,7
20,3
14,5
19,1
19,1
18,4
16,3
19,0
22,3
21,2
20,8
20,1
17,2
20,3
20,0
20,1
20,4
18,1
16,9
19,7
16,2
14,4
19,4
14,5
15,4
15,7
14,3
15,5
21,5
19,5
17,1
17,2
18,2
18,6
18,8
18,0
3
6
9
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
18,10
18,17
18,76
19,40
20,31
20,93
20,04
19,68
19,18
19,00
20,10
20,00
19,58
19,44
18,14
17,54
17,65
16,60
16,51
16,04
15,61
15,66
16,93
17,88
18,60
19,37
18,75
19,01
18,86
19,00
1,97
2,39
2,43
2,45
2,45
1,63
2,49
2,32
1,84
1,70
0,65
0,79
1,35
1,34
1,90
1,98
2,27
2,16
2,23
2,02
1,54
1,51
2,97
3,12
3,02
2,50
2,32
2,21
1,94
1,93
7,656
3,708
3,032
2,737
2,737
2,737
2,737
NB-CPD/SG10/03/006r2
4
kc
2,737
2,690
2,642
2,595
2,547
2,500
2,452
2,405
2,357
2,310
2,262
2,215
2,167
2,120
k1,k
σ
2,120
2,490
2,120
2,120
2,120
2,120
2,120
2,120
2,120
2,120
2,120
2,120
2,120
2,461
2,461
Xest
3,04
9,31
11,38
12,68
13,59
16,48
13,23
Xest
13,23
12,98
12,60
12,54
13,76
13,78
13,48
13,45
12,27
11,79
12,02
11,08
11,11
10,83
Xest
10,83
10,39
10,44
11,72
12,67
13,38
14,15
13,53
13,79
13,64
13,78
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
Page 59 of 74
Example 8
Example of statistical analysis of compressive strength using a special type of “Rolling” inspection:
“Progressive Sampling”
This example is similar to example 6. The only difference is that the declared compressive strength
is a 5 % characteristic value.
The fractile p = 95%
The confidence level γ = 95%
The number of series of inspection lots is l = 15.
One-sided tolerance interval, lower limit
The declared 5 % characteristic compressive strength is 4 N/mm²
For each of the 1st to 5th inspection lots a spot size of one sample is taken and tested. These
inspection lots are evaluated together (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 95 % and γ: 95
%)). For the 6th and following inspection lots 1 additional sample is taken and tested and evaluated
together with the ones from the previous inspection lots (xm (equation 1), standard deviation ss
(equation 2) and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8
(p: 95 % and γ: 95 %)). The spot size is gradually increased from 5 to 15 samples.
From then on, 1 additional sample is taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A8 (p: 95 % and
γ: 95 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 30 samples, the standard deviation of the population is considered to be unknown and
the k1,u factor taken from Annex A Table A8 (p: 95 % and γ: 95 %) is 2,567.
For the inspection lots 30 – 60 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A4 (p: 95 % and γ: 95 %) and is 2,070. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 61 and so on 1 additional sample is taken from each next inspection lot and the
test results are evaluated together with the ones from the previous inspection lots (xm (equation 1),
known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A4 (p: 95 % and γ: 95 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
NB-CPD/SG10/03/006r2
Page 60 of 74
In the last column it is indicated whether the mentioned equation fits or does not fit. At batch 68
there is a non-conformity. The manufacturer has to restart or he decides to continue working with
the acceptance coefficient k1,u. This means that the inspection lots have to be treated separately.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 2, 3, 4, 34 and 68). A non-conforming inspection
lot has to be treated separately as described in the text.
EXAMPLE 8
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
95
95
test 1
test 2
6,78
8,36
8,64
8,41
8,25
7,44
8,57
6,25
7,83
7,40
8,57
8,27
7,67
8,13
6,53
7,85
7,20
7,95
7,39
5,96
6,26
5,60
7,15
6,01
7,88
6,01
5,91
7,05
5,51
5,87
7,59
8,64
6,72
4,91
8,21
8,48
7,87
7,36
8,11
6,50
7,38
7,78
8,11
8,70
7,46
7,76
6,46
7,78
6,73
7,47
6,98
6,35
6,35
5,53
6,68
5,86
7,40
6,42
5,80
6,57
5,71
5,78
10,46
11,26
8,90
11,50
5,89
10,50
NB-CPD/SG10/03/006r2
test 3
Start correction
End correction
METHOD B: progressive sampling : use only 1 testresult per inspection lot
30
60
n
Xm
Ss
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
6,78
7,57
7,92
8,05
8,09
7,98
8,06
7,84
7,84
7,79
7,86
7,90
7,88
7,90
7,81
7,88
7,80
7,75
7,69
7,53
7,46
7,26
7,32
7,20
7,23
7,06
6,90
6,86
6,69
6,64
6,62
6,72
6,64
6,47
6,62
6,77
6,92
6,94
7,08
6,98
7,07
7,20
7,27
7,48
7,59
7,60
7,45
7,52
7,65
7,60
7,50
7,40
7,33
7,16
7,17
7,07
7,04
6,93
6,74
6,68
6,54
6,49
6,67
6,97
7,07
7,37
7,34
7,62
#DEEL/0!
1,11
1,00
0,85
0,74
0,72
0,69
0,91
0,85
0,81
0,80
0,77
0,74
0,72
0,78
0,72
0,73
0,69
0,67
0,79
0,85
0,92
0,88
0,93
0,94
0,91
0,89
0,87
0,86
0,88
0,86
1,00
0,94
1,01
1,09
1,19
1,17
1,18
1,18
1,17
1,14
1,11
1,13
1,08
0,98
0,98
0,98
0,96
0,68
0,66
0,63
0,68
0,74
0,83
0,83
0,89
0,87
0,83
0,72
0,69
0,67
0,69
1,21
1,69
1,76
2,10
2,12
2,25
Series of inspection lots
15
Declared Value
4
1,009
k1,u
26,260
7,656
5,144
4,203
3,708
3,400
3,188
3,032
2,911
2,815
2,737
2,671
2,615
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
2,567
kc
2,567
2,550
2,534
2,517
2,501
2,484
2,468
2,451
2,434
2,418
2,401
2,385
2,368
2,352
2,335
2,319
2,302
2,285
2,269
2,252
2,236
2,219
2,203
2,186
2,169
2,153
2,136
2,120
2,103
2,087
2,070
k1,k
σ
2,070
1,009
2,070
2,070
2,070
2,070
2,070
2,070
2,070
2,070
2,070
0,974
0,974
Xest
‐21,69
0,26
3,66
4,96
5,32
5,71
4,95
5,27
5,44
5,60
5,78
5,89
6,02
5,81
6,02
5,93
5,97
5,95
5,51
5,26
4,90
5,07
4,82
4,81
4,72
4,61
4,63
4,48
4,37
Xest
4,05
4,05
4,16
4,10
3,95
4,12
4,28
4,45
4,48
4,64
4,56
4,67
4,81
4,90
5,13
5,25
5,28
5,15
5,24
5,37
5,34
5,26
5,17
5,12
4,97
5,00
4,91
4,90
4,81
4,63
4,66
Xest
4,66
4,52
4,48
4,66
4,96
5,05
5,35
5,32
5,60
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
NOK
Page 61 of 74
Example 9
Example of statistical analysis of compressive strength using a special type of “Rolling” inspection:
“Progressive Sampling”
This example is similar to example 8. The only difference is that the confidence level in this
example is 75 %.
The fractile p = 95%
The confidence level γ = 75%
The number of series of inspection lots is l = 15.
One-sided tolerance interval, lower limit
The declared 5 % characteristic compressive strength is 4 N/mm²
For each of the 1st to 5th inspection lots a spot size of one sample is taken and tested. These
inspection lots are evaluated together (xm (equation 1), standard deviation ss (equation 2) and xest
(equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A6 (p: 95 % and γ: 75
%)). For the 6th and following inspection lots 1 additional sample is taken and tested and evaluated
together with the ones from the previous inspection lots (xm (equation 1), standard deviation ss
(equation 2) and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A6
(p: 95 % and γ: 75 %)). The spot size is gradually increased from 5 to 15 samples.
From then on, 1 additional sample is taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 3) according to clause 5.6.6 using k1,u taken from Annex A Table A6 (p: 95 % and
γ: 75 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 30 samples, the standard deviation of the population is considered to be unknown and
the k1,u factor taken from Annex A Table A6 (p: 95 % and γ: 75 %) is 1,991.
For the inspection lots 30 – 60 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A2 (p: 95 % and γ: 75 %) and is 1,820. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 61 and so on 1 additional sample is taken from each next inspection lot and the
test results are evaluated together with the ones from the previous inspection lots (xm (equation 1),
known standard deviation σ and xest (equation 6) according to clause 5.6.6 using k1,k taken from
Annex A Table A2 (p: 95 % and γ: 75 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
NB-CPD/SG10/03/006r2
Page 62 of 74
In the last column it is indicated whether the mentioned equation fits or does not fit. At batch 68
there is a non-conformity. The manufacturer has to restart or he decides to continue working with
the acceptance coefficient k1,u. This means that the inspection lots have to be treated separately.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 2 and 68). A non-conforming inspection lot has
to be treated separately as described in the text.
NB-CPD/SG10/03/006r2
Page 63 of 74
EXAMPLE 9
ONE SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
95
75
test 1
test 2
6,78
8,36
8,64
8,41
8,25
7,44
8,57
6,25
7,83
7,40
8,57
8,27
7,67
8,13
6,53
7,85
7,20
7,95
7,39
5,96
6,26
5,60
7,15
6,01
7,88
6,01
5,91
7,05
5,51
5,87
7,59
8,64
6,72
4,91
8,21
8,48
7,87
7,36
8,11
6,50
7,38
7,78
8,11
8,70
7,46
7,76
6,46
7,78
6,73
7,47
6,98
6,35
6,35
5,53
6,68
5,86
7,40
6,42
5,80
6,57
5,71
5,78
10,46
11,26
8,90
11,50
5,89
10,50
NB-CPD/SG10/03/006r2
test 3
Start correction
End correction
METHOD B: progressive sampling : use only 1 testresult per inspection lot
30
60
n
Xm
Ss
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
6,78
7,57
7,92
8,05
8,09
7,98
8,06
7,84
7,84
7,79
7,86
7,90
7,88
7,90
7,81
7,88
7,80
7,75
7,69
7,53
7,46
7,26
7,32
7,20
7,23
7,06
6,90
6,86
6,69
6,64
6,62
6,72
6,64
6,47
6,62
6,77
6,92
6,94
7,08
6,98
7,07
7,20
7,27
7,48
7,59
7,60
7,45
7,52
7,65
7,60
7,50
7,40
7,33
7,16
7,17
7,07
7,04
6,93
6,74
6,68
6,54
6,49
6,67
6,97
7,07
7,37
7,34
7,62
#DEEL/0!
1,11
1,00
0,85
0,74
0,72
0,69
0,91
0,85
0,81
0,80
0,77
0,74
0,72
0,78
0,72
0,73
0,69
0,67
0,79
0,85
0,92
0,88
0,93
0,94
0,91
0,89
0,87
0,86
0,88
0,86
1,00
0,94
1,01
1,09
1,19
1,17
1,18
1,18
1,17
1,14
1,11
1,13
1,08
0,98
0,98
0,98
0,96
0,68
0,66
0,63
0,68
0,74
0,83
0,83
0,89
0,87
0,83
0,72
0,69
0,67
0,69
1,21
1,69
1,76
2,10
2,12
2,25
Series of inspection lots
15
Declared Value
4
1,009
k1,u
5,122
3,152
2,681
2,464
2,336
2,251
2,189
2,142
2,104
2,074
2,048
2,026
2,008
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
1,991
kc
1,991
1,985
1,980
1,974
1,968
1,963
1,957
1,951
1,945
1,940
1,934
1,928
1,923
1,917
1,911
1,906
1,900
1,894
1,888
1,883
1,877
1,871
1,866
1,860
1,854
1,849
1,843
1,837
1,831
1,826
1,820
k1,k
σ
1,820
1,009
1,820
1,820
1,820
1,820
1,820
1,820
1,820
1,820
1,820
0,974
0,974
Xest
1,86
4,77
5,76
6,25
6,31
6,51
5,86
6,02
6,09
6,20
6,31
6,37
6,45
6,26
6,44
6,35
6,37
6,34
5,96
5,76
5,43
5,57
5,35
5,35
5,24
5,13
5,13
4,98
4,88
Xest
4,63
4,62
4,72
4,65
4,49
4,64
4,80
4,95
4,97
5,12
5,03
5,13
5,26
5,34
5,55
5,67
5,68
5,54
5,62
5,75
5,70
5,61
5,51
5,45
5,29
5,30
5,21
5,19
5,08
4,89
4,90
Xest
4,90
4,77
4,72
4,90
5,20
5,30
5,60
5,57
5,84
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
NOK
Page 64 of 74
Example10
Example of statistical analysis of gross dry density using “Rolling” inspection.
The fractile p = 50%
The confidence level γ = 90%
The number of series of inspection lots is l = 4.
One-sided tolerance interval, upper limit
The declared mean gross dry density is 700 kg/m3
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 4) according to clause 5.6.6 using
k1,u taken from Annex A Table A7 (p: 50 % and γ: 90 %)). For the next and the following 2
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 4)
according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 50 % and γ: 90 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 12 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 4) according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 50 % and
γ: 90 %)) but the spot sample size is limited to the last 12 samples. The spot sample size continues
to be 12.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A7 (p: 50 % and γ: 90 %) is 0,394. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A3 (p: 50 % and γ: 90 %) and is 0,370. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 7) according to clause 5.6.6 using k1,k taken from
Annex A Table A3 (p: 50 % and γ: 90 %)) and the spot sample size is still limited to the last 12
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side. A non-conforming inspection lot has to be treated
separately as described in the text.
NB-CPD/SG10/03/006r2
Page 65 of 74
EXAMPLE 10
ONE SIDED TOLERANCE INTERVAL-upper limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
50
90
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
test 3
n
Xm
Ss
k1,u
603,00
632,0
681,0
663,00
672,0
657,0
672,00
680,0
693,0
659,00
662,0
638,0
672,00
669,0
682,0
626,00
655,0
662,0
653,00
656,0
652,0
658,0
662,0
632,00
692,0
636,0
618,00
635,0
668,0
637,00
662,0
669,0
676,00
696,0
685,0
668,00
671,0
639,0
628,00
638,0
652,0
663,00
672,0
672,0
628,00
680,0
657,0
672,00
662,0
666,0
668,00
669,0
692,0
683,00
655,0
691,0
641,00
656,0
682,0
698,00
658,0
682,0
665,00
692,0
688,0
671,00
635,0
618,0
662,00
662,0
619,0
638,00
696,0
668,0
673,00
671,0
667,0
655,00
682,0
662,0
672,00
662,0
672,0
672,00
652,0
672,0
662,00
662,0
669,0
628,00
516,0
676,0
667,00
668,0
698,0
619,00
669,0
626,0
652,00
685,0
652,0
638,67
651,33
661,44
659,33
668,25
664,17
657,17
659,58
654,33
652,50
653,08
658,83
660,33
660,08
663,33
655,67
657,50
666,75
668,58
669,75
672,92
674,25
665,50
662,50
659,50
656,67
662,92
668,17
667,67
666,17
651,25
653,50
646,67
646,33
39,43
28,93
27,95
24,78
14,07
18,61
14,82
13,79
17,54
19,92
21,21
26,31
23,10
21,30
20,31
18,35
17,61
15,18
17,62
15,34
17,77
18,35
24,81
26,78
26,75
23,83
19,73
14,10
8,48
8,47
44,55
46,23
47,14
47,68
1,089
0,603
0,466
0,394
0,394
0,394
0,394
668,00
3
6
9
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
NB-CPD/SG10/03/006r2
Declared Value
700
20,92845
test 2
test 1
4
kc
0,394
0,392
0,390
0,388
0,387
0,385
0,383
0,381
0,379
0,377
0,376
0,374
0,372
0,370
k1,k
σ
0,370
20,928
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
20,217
20,217
Xest
681,60
668,78
674,47
669,10
673,79
671,50
663,01
Xest
665,41
667,79
662,50
660,63
661,17
666,89
668,35
668,06
671,27
663,56
665,36
674,57
676,37
677,23
Xest
677,23
680,40
681,73
672,98
669,98
666,98
664,15
670,40
675,65
675,15
673,65
658,73
660,98
654,15
653,81
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
Page 66 of 74
Example 11
Example of statistical analysis of net dry density using “Rolling” inspection.
The fractile p = 50%
The confidence level γ = 90%
The number of series of inspection lots is l = 4.
One-sided tolerance interval, upper limit
The declared mean net dry density is 1400 kg/m3
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 4) according to clause 5.6.6 using
k1,u taken from Annex A Table A7 (p: 50 % and γ: 90 %)). For the next and the following 2
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 4)
according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 50 % and γ: 90 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 12 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 4) according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 50 % and
γ: 90 %)) but the spot sample size is limited to the last 12 samples. The spot sample size continues
to be 12.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A7 (p: 50 % and γ: 90 %) is 0,394. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A3 (p: 50 % and γ: 90 %) and is 0,370. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 7) according to clause 5.6.6 using k1,k taken from
Annex A Table A3 (p: 50 % and γ: 90 %)) and the spot sample size is still limited to the last 12
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side. A non-conforming inspection lot has to be treated
separately as described in the text.
NB-CPD/SG10/03/006r2
Page 67 of 74
The last column makes the link to EN 1745 Annex A to calculate the λ10,dry (50/90) value of the
material.
EXAMPLE 11
ONE SIDED TOLERANCE INTERVAL-upper limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
50
90
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
test 3
n
Xm
Ss
k1,u
1206,00
1264,0
1362,0
1326,00
1344,0
1314,0
1344,00
1360,0
1386,0
1318,00
1324,0
1276,0
1344,00
1338,0
1364,0
1252,00
1310,0
1324,0
1306,00
1312,0
1304,0
1316,0
1324,0
1264,00
1384,0
1272,0
1236,00
1270,0
1336,0
1274,00
1324,0
1338,0
1352,00
1392,0
1370,0
1336,00
1342,0
1278,0
1256,00
1276,0
1304,0
1326,00
1344,0
1344,0
1256,00
1360,0
1314,0
1344,00
1324,0
1332,0
1336,00
1338,0
1384,0
1366,00
1310,0
1382,0
1282,00
1312,0
1364,0
1396,00
1316,0
1364,0
1330,00
1384,0
1376,0
1342,00
1270,0
1236,0
1324,00
1324,0
1238,0
1276,00
1392,0
1336,0
1346,00
1342,0
1334,0
1310,00
1364,0
1324,0
1344,00
1324,0
1344,0
1344,00
1304,0
1344,0
1324,00
1324,0
1338,0
1256,00
1032,0
1352,0
1334,00
1336,0
1396,0
1238,00
1338,0
1252,0
1277,33
1302,67
1322,89
1318,67
1336,50
1328,33
1314,33
1319,17
1308,67
1305,00
1306,17
1317,67
1320,67
1320,17
1326,67
1311,33
1315,00
1333,50
1337,17
1339,50
1345,83
1348,50
1331,00
1325,00
1319,00
1313,33
1325,83
1336,33
1335,33
1332,33
1302,50
1307,00
1293,33
78,85
57,86
55,90
49,55
28,13
37,21
29,64
27,59
35,08
39,83
42,42
52,61
46,20
42,59
40,63
36,69
35,22
30,37
35,25
30,68
35,55
36,70
49,63
53,56
53,51
47,67
39,46
28,20
16,96
16,95
89,10
92,46
94,29
1,089
0,603
0,466
0,394
0,394
0,394
0,394
1336,00
3
6
9
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
NB-CPD/SG10/03/006r2
Declared Value
1400
41,8569
test 2
test 1
4
kc
0,394
0,392
0,390
0,388
0,387
0,385
0,383
0,381
0,379
0,377
0,376
0,374
0,372
0,370
k1,k
σ
0,370
41,857
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
0,370
40,433
40,433
Xest
1363,20
1337,56
1348,94
1338,19
1347,58
1342,99
1326,01
Xest
1330,82
1335,58
1325,00
1321,26
1322,35
1333,77
1336,69
1336,12
1342,54
1327,13
1330,72
1349,14
1352,73
1354,46
Xest
1354,46
1360,79
1363,46
1345,96
1339,96
1333,96
1328,29
1340,79
1351,29
1350,29
1347,29
1317,46
1321,96
1308,29
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
λ 10,dry (50/90)‐
material concrete
0,383
0,357
0,368
0,357
0,367
0,362
0,345
0,355
0,344
0,340
0,341
0,353
0,356
0,355
0,362
0,346
0,350
0,369
0,372
0,374
0,381
0,384
0,365
0,359
0,353
0,347
0,360
0,371
0,370
0,367
0,336
0,340
0,326
Page 68 of 74
Example 12
Example of statistical analysis of net dry density using “Rolling” inspection.
This example is similar to example 11. The only difference is that the confidence level in this
example is 50 % and the numbers of series of inspection lots is 5.
The fractile p = 50%
The confidence level γ = 50%
The number of series of inspection lots is l = 5.
One-sided tolerance interval, upper limit
The declared mean net dry density is 1400 kg/m3
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 4) according to clause 5.6.6 using
k1,u taken from Annex A Table A5 (p: 50 % and γ: 50 %)). For the next and the following 2
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 4)
according to clause 5.6.6 using k1,u taken from Annex A Table A5 (p: 50 % and γ: 50 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 15 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 4) according to clause 5.6.6 using k1,u taken from Annex A Table A5 (p: 50 % and
γ: 50 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A5 (p: 50 % and γ: 50 %) is 0,000,
which means that xest = xm. For the inspection lots 8 – 20 the standard deviation can be considered
as known. The acceptance coefficient for the known standard deviation k1,k is taken from Annex A
Table A1 (p: 50 % and γ: 50 %) and is 0,000, which means that xest = xm. The known standard
deviation σ is calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 7) according to clause 5.6.6 using k1,k taken from
Annex A Table A1 (p: 50 % and γ: 50 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side. A non-conforming inspection lot has to be treated
separately as described in the text.
NB-CPD/SG10/03/006r2
Page 69 of 74
The last column makes the link to EN 1745 Annex A to calculate the λ10,dry (50/50) value of the
material.
EXAMPLE 12
ONE SIDED TOLERANCE INTERVAL-upper limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
50
50
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
test 3
n
Xm
Ss
k1,u
1206,00
1264,0
1362,0
1326,00
1344,0
1314,0
1344,00
1360,0
1386,0
1318,00
1324,0
1276,0
1344,00
1338,0
1364,0
1252,00
1310,0
1324,0
1306,00
1312,0
1304,0
1316,0
1324,0
1264,00
1384,0
1272,0
1236,00
1270,0
1336,0
1274,00
1324,0
1338,0
1352,00
1392,0
1370,0
1336,00
1342,0
1278,0
1256,00
1276,0
1304,0
1326,00
1344,0
1344,0
1256,00
1360,0
1314,0
1344,00
1324,0
1332,0
1336,00
1338,0
1384,0
1366,00
1310,0
1382,0
1282,00
1312,0
1364,0
1396,00
1316,0
1364,0
1330,00
1384,0
1376,0
1342,00
1270,0
1236,0
1324,00
1324,0
1238,0
1276,00
1392,0
1336,0
1346,00
1342,0
1334,0
1310,00
1364,0
1324,0
1344,00
1324,0
1344,0
1344,00
1304,0
1344,0
1324,00
1324,0
1338,0
1256,00
1032,0
1352,0
1334,00
1336,0
1396,0
1238,00
1338,0
1252,0
1304,00
1370,0
1304,0
1277,33
1302,67
1322,89
1318,67
1324,67
1328,27
1324,13
1316,53
1316,67
1303,07
1306,40
1319,20
1317,87
1312,27
1323,73
1323,33
1315,73
1322,53
1337,33
1333,60
1343,33
1349,33
1335,33
1323,87
1326,93
1323,33
1317,20
1328,13
1335,20
1334,00
1308,53
1313,07
1300,80
1299,87
78,85
57,86
55,90
49,55
45,94
33,47
34,15
26,93
35,61
38,35
37,64
46,90
48,51
45,41
38,67
41,62
33,99
36,37
31,49
35,73
32,16
34,16
47,11
50,05
52,45
48,32
44,29
35,57
26,58
15,58
80,66
83,02
85,44
85,88
0,000
0,000
0,000
0,000
0,000
0,000
0,000
1336,00
3
6
9
12
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
NB-CPD/SG10/03/006r2
Declared Value
1400
41,8569
test 2
test 1
5
kc
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
k1,k
σ
0,000
41,857
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
0,000
40,433
40,433
Xest
1277,33
1302,67
1322,89
1318,67
1324,67
1328,27
1324,13
Xest
1324,13
1316,53
1316,67
1303,07
1306,40
1319,20
1317,87
1312,27
1323,73
1323,33
1315,73
1322,53
1337,33
1333,60
Xest
1333,60
1343,33
1349,33
1335,33
1323,87
1326,93
1323,33
1317,20
1328,13
1335,20
1334,00
1308,53
1313,07
1300,80
1299,87
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
λ 10,dry (50/50)‐
material clay
0,291
0,301
0,309
0,308
0,310
0,312
0,310
0,307
0,307
0,301
0,303
0,308
0,307
0,305
0,310
0,310
0,306
0,309
0,315
0,314
0,318
0,320
0,314
0,310
0,311
0,310
0,307
0,312
0,314
0,314
0,304
0,305
0,300
0,300
Page 70 of 74
Example 13
Example of statistical analysis of net dry density using “Rolling” inspection.
This example is similar to example 11. The only difference is that the declared value is a 90 %
characteristic value and the number of series of inspection lots is l = 5.
The fractile p = 90%
The confidence level γ = 90%
The number of series of inspection lots is l = 5.
One-sided tolerance interval, upper limit
The declared 90 % characteristic net dry density is 1400 kg/m3
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 4) according to clause 5.6.6 using
k1,u taken from Annex A Table A7 (p: 90 % and γ: 90 %)). For the next and the following 3
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 4)
according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 90 % and γ: 90 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 15 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 4) according to clause 5.6.6 using k1,u taken from Annex A Table A7 (p: 90 % and
γ: 90 %)) but the spot sample size is limited to the last 15 samples. The spot sample size continues
to be 15.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k1,u factor taken from Annex A Table A7 (p: 90 % and γ: 90 %) is 1,867. For
the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k1,k is taken from Annex A Table A3 (p: 90 % and γ: 90 %) and is 1,613. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k1,u and k1,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 7) according to clause 5.6.6 using k1,k taken from
Annex A Table A3 (p: 90 % and γ: 90 %)) and the spot sample size is still limited to the last 15
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
NB-CPD/SG10/03/006r2
Page 71 of 74
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 1, 2, 3, 4, 5, 7, 19, 21, 22, 23 and 29). A nonconforming inspection lot has to be treated separately as described in the text.
The last column makes the link to EN 1745 Annex A to calculate the λ10,dry (90/90) value of the
material.
EXAMPLE 13
ONE SIDED TOLERANCE INTERVAL-upper limit
fractile p
confidence level
Inspection lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
90
90
Start correction
End correction
METHOD B: use at least 3 testresults per inspection lot
7
20
Series of inspection lots
test 3
n
Xm
Ss
k1,u
1206,00
1264,0
1362,0
1326,00
1344,0
1314,0
1344,00
1360,0
1386,0
1318,00
1324,0
1276,0
1344,00
1338,0
1364,0
1252,00
1310,0
1324,0
1306,00
1312,0
1304,0
1316,0
1324,0
1264,00
1384,0
1272,0
1236,00
1270,0
1336,0
1274,00
1324,0
1338,0
1352,00
1392,0
1370,0
1336,00
1342,0
1278,0
1256,00
1276,0
1304,0
1326,00
1344,0
1344,0
1256,00
1360,0
1314,0
1344,00
1324,0
1332,0
1336,00
1338,0
1384,0
1366,00
1310,0
1382,0
1282,00
1312,0
1364,0
1396,00
1316,0
1364,0
1330,00
1384,0
1376,0
1342,00
1270,0
1236,0
1324,00
1324,0
1238,0
1276,00
1392,0
1336,0
1346,00
1342,0
1334,0
1310,00
1364,0
1324,0
1344,00
1324,0
1344,0
1344,00
1304,0
1344,0
1324,00
1324,0
1338,0
1256,00
1032,0
1352,0
1334,00
1336,0
1396,0
1238,00
1338,0
1252,0
1277,33
1302,67
1322,89
1318,67
1324,67
1328,27
1324,13
1316,53
1316,67
1303,07
1306,40
1319,20
1317,87
1312,27
1323,73
1323,33
1315,73
1322,53
1337,33
1333,60
1343,33
1349,33
1335,33
1323,87
1326,93
1323,33
1317,20
1328,13
1335,20
1334,00
1308,53
1313,07
1300,80
78,85
57,86
55,90
49,55
45,94
33,47
34,15
26,93
35,61
38,35
37,64
46,90
48,51
45,41
38,67
41,62
33,99
36,37
31,49
35,73
32,16
34,16
47,11
50,05
52,45
48,32
44,29
35,57
26,58
15,58
80,66
83,02
85,44
4,259
2,494
2,133
1,967
1,867
1,867
1,867
1336,00
3
6
9
12
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
NB-CPD/SG10/03/006r2
Declared Value
1400
41,8569
test 2
test 1
5
kc
1,867
1,847
1,828
1,808
1,789
1,769
1,750
1,730
1,711
1,691
1,672
1,652
1,633
1,613
k1,k
σ
1,613
41,857
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
1,613
40,433
40,433
Xest
1613,16
1446,98
1442,13
1416,14
1410,43
1390,76
1387,88
Xest
1402,28
1393,86
1393,18
1378,76
1381,28
1393,26
1391,11
1384,69
1395,34
1394,12
1385,70
1391,68
1405,67
1398,82
Xest
1398,82
1408,55
1414,55
1400,55
1389,09
1392,15
1388,55
1382,42
1393,35
1400,42
1399,22
1373,75
1378,29
1366,02
Equation OK?
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
λ 10,dry (90/90)‐
material
0,427
0,360
0,358
0,347
0,345
0,337
0,336
0,338
0,338
0,332
0,333
0,338
0,337
0,335
0,339
0,338
0,335
0,337
0,343
0,340
0,344
0,347
0,341
0,336
0,338
0,336
0,334
0,338
0,341
0,340
0,330
0,332
0,327
Page 72 of 74
Example 14
Example of TWO-SIDED statistical analysis of dimension using “Rolling” inspection.
The fractile p = 50%
The confidence level γ = 75%
The number of series of inspection lots is l = 4.
Two-sided tolerance interval
The manufacturer wants to have the mean value of the length of the green units between 242
mm and 247 mm
For the first inspection lot a sample size of 3 samples are taken and tested and evaluated (xm
(equation 1), standard deviation ss (equation 2) and xest (equation 5) according to clause 5.6.6 using
k2,u taken from Annex A Table A14 (p: 50 % and γ: 75 %)). For the next and the following 3
inspection lots 3 additional samples are taken and tested and evaluated together with the ones from
the previous inspection lots (xm (equation 1), standard deviation ss (equation 2) and xest (equation 5)
according to clause 5.6.6 using k2,u taken from Annex A Table A14 (p: 50 % and γ: 75 %)). By doing
so the spot sample size evaluated together is gradually increased from 3 to 12 samples.
From then on, 3 additional samples are taken from each next inspection lot and evaluated together
with the ones from the previous inspection lots (xm (equation 1), standard deviation ss (equation 2)
and xest (equation 5) according to clause 5.6.6 using k2,u taken from Annex A Table A14 (p: 50 %
and γ: 75 %)) but the spot sample size is limited to the last 12 samples. The spot sample size
continues to be 12.
For the first 21 samples (1 – 7 inspection lot), the standard deviation of the population is considered
to be unknown and the k2,u factor taken from Annex A Table A14 (p: 50 % and γ: 75 %) is 0,821.
For the inspection lots 8 – 20 the standard deviation can be considered as known, but the used
acceptance coefficient is corrected (kc). The acceptance coefficient for the known standard
deviation k2,k is taken from Annex A Table A10 (p: 50 % and γ: 75 %) and is 0,705. The corrected
acceptance coefficient kc is calculated by a linear interpolation between the acceptance coefficient
k2,u and k2,k taking into account the considered inspection lot. The known standard deviation σ is
calculated based on the first 21 test results.
From inspection lot 21 and so on 3 additional samples are taken from each next inspection lot and
the test results are evaluated together with the ones from the previous inspection lots (xm (equation
1), known standard deviation σ and xest (equation 8) according to clause 5.6.6 using k2,k taken from
Annex A Table A10 (p: 50 % and γ: 75 %)) and the spot sample size is still limited to the last 12
samples.
After each evaluation the result has to be compared with the lower limit value (e.g. the declared
value) decided by the manufacturer.
Part of the evaluation is also to check that the standard deviation ss of the spot sample corresponds
to the following equation:
0,63 σ ≤ ss ≤ 1,37 σ
In the last column it is indicated whether the mentioned equation fits or does not fit.
If there is a non-conformity due to great differences between the test results, the estimated value is
highlighted by a red signal at the right side (batch 5, 6, 7, 14, 18, 19, 20, 23, 24 and 25). A nonconforming inspection lots give a warning to the manufacturer to take some corrective actions.
NB-CPD/SG10/03/006r2
Page 73 of 74
EXAMPLE 14
2,166
TWO SIDED TOLERANCE INTERVAL-lower limit
fractile p
confidence level
50
75
Start correction
End correction
7
20
Series of inspection lots
Inspection lot
test 1
test 2
test 3
n
Xm
Ss
k1,u
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
244
245
246
247
242
241
245
247
243
244
246
245
245
246
245
244
248
247
245
244
246
245
245
247
245
243
245
247
247
246
242
242
246
246
244
244
246
245
245
245
244
245
248
246
245
245
245
246
246
246
244
244
246
246
247
246
241
242
247
246
243
243
246
246
244
245
243
244
247
248
245
245
248
248
247
247
244
245
3
6
9
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
245,00
245,50
245,89
246,00
245,17
244,08
243,92
243,92
244,33
244,83
244,83
244,58
244,92
245,33
244,83
244,58
245,33
245,75
246,00
246,08
245,75
245,58
245,83
246,33
245,83
245,25
1,00
1,05
1,05
0,95
2,21
2,57
2,43
2,43
2,10
1,53
1,47
1,24
1,00
0,65
0,83
0,79
1,61
1,82
1,54
1,44
1,29
1,24
1,27
1,07
1,27
1,36
1,492
1,009
0,9
0,85
0,85
0,85
0,85
NB-CPD/SG10/03/006r2
kc
0,850
0,839
0,829
0,818
0,808
0,797
0,787
0,776
0,766
0,755
0,745
0,734
0,724
0,713
4
k1,k
Declared Value lower limit
Declared Value upper limit
σ
0,713
2,166
0,713
0,713
0,713
0,713
0,713
0,713
0,713
1,677
Xest lower limit
243,508
244,442
244,940
245,190
243,289
241,895
241,852
Xest upper limit
246,492
246,5582
246,8376
246,8104
247,0441
246,2718
245,9816
242
247
Xest lower limit
242,076
242,099
242,538
243,061
243,084
242,857
243,213
243,652
243,175
242,948
243,721
244,160
244,433
244,888
1,677
Xest upper limit
245,7576
245,7347
246,1286
246,6058
246,5829
246,3101
246,6206
247,0145
246,4916
246,2188
246,946
247,3398
247,567
247,2791
Xest lower Xest upper limit
limit
244,888
244,554
244,388
244,638
245,138
244,638
244,054
247,279
246,946
246,779
247,029
247,529
247,029
246,446
Equation OK?
OK
OK
OK
OK
OK
OK
OK
Page 74 of 74