UE-Nr_8

GNB-CPD

SG10 Guidance from the Group of Notified Bodies

for the Construction Products Directive 89/106/EEC

NB-CPD/SG10/12/091 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.

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Contents

1 Foreword ..................................................................................................................................................... 3

2 Symbols ...................................................................................................................................................... 3

3 Reference list .............................................................................................................................................. 4

4 General ....................................................................................................................................................... 4

5 Factory production control .......................................................................................................................... 5 5.1 General ................................................................................................................................................................. 5 5.2 Testing and measuring equipment ....................................................................................................................... 6 5.3 Production equipment ........................................................................................................................................... 6 5.4 Raw materials ....................................................................................................................................................... 6 5.5 Production process ............................................................................................................................................... 7 5.6 Finished product testing ....................................................................................................................................... 8

5.6.1 Inspection lot ............................................................................................................................................................................ 9 5.6.2 Spot sampling and sample sizes .............................................................................................................................................. 9 5.6.3 Production types ..................................................................................................................................................................... 11 5.6.4 Method A: Batch control ......................................................................................................................................................... 11 5.6.5 Method B: ”Rolling” inspection ............................................................................................................................................... 12 5.6.6 Evaluation of test results ........................................................................................................................................................ 14 5.6.7 How to come from unknown to known standard deviation? ................................................................................................... 18 5.6.8 Conformity .............................................................................................................................................................................. 19 5.6.9 A simple and conservative approach ..................................................................................................................................... 24 5.6.10 Non-conforming products ....................................................................................................................................................... 24 5.6.11 Guidance ................................................................................................................................................................................ 25 5.6.12 Records .................................................................................................................................................................................. 28

6 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

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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_gnb-cpd&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

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http://circa.europa.eu/Members/irc/nbg/cdpgnb/library?l=/monitoring_gnb-cpd&vm=detailed&sb=Title

http://circa.europa.eu/Members/irc/nbg/cdpgnb/library?l=/monitoring_gnb-cpd&vm=detailed&sb=Title

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.

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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

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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

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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

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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 non-conforming test result will be less and thereby reduce the manufacturer’s risk.

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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

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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).

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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.

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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

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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.

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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 *)

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• 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 =1n

xii=1

n

∑

(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:

( )

11

2

−

−=∑=

n

xxs

n

imi

(2)

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


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 (4)

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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:

xest = xm + k1,k × σ 7

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


• k1,u is the acceptance coefficient for unknown standard deviation and one-sided limit 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

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

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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.

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Spot sample size

Mean value in MPa

Std. deviation 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.

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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

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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.

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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

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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.

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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.

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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 non-conforming products should be developed.

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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 re-declaration in connection with a non-conformity is possible based on test results obtained by testing a new spot sample taken at random from the inspection lot following control method A, 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.

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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.

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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 re-declared. 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 non-conformity 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.

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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.

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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.

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Annex A Tables for acceptance coefficient kn depending on the used fractile p and confidence level γ (taken from ISO 16269-6 (2005))

n fractile : p n fractile : p 0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95

2 0,000 0,675 1,282 1,645 51 0,000 0,675 1,282 1,645 3 0,000 0,675 1,282 1,645 52 0,000 0,675 1,282 1,645 4 0,000 0,675 1,282 1,645 53 0,000 0,675 1,282 1,645 5 0,000 0,675 1,282 1,645 54 0,000 0,675 1,282 1,645 6 0,000 0,675 1,282 1,645 55 0,000 0,675 1,282 1,645 7 0,000 0,675 1,282 1,645 56 0,000 0,675 1,282 1,645 8 0,000 0,675 1,282 1,645 57 0,000 0,675 1,282 1,645 9 0,000 0,675 1,282 1,645 58 0,000 0,675 1,282 1,645 10 0,000 0,675 1,282 1,645 59 0,000 0,675 1,282 1,645 11 0,000 0,675 1,282 1,645 60 0,000 0,675 1,282 1,645 12 0,000 0,675 1,282 1,645 61 0,000 0,675 1,282 1,645 13 0,000 0,675 1,282 1,645 62 0,000 0,675 1,282 1,645 14 0,000 0,675 1,282 1,645 63 0,000 0,675 1,282 1,645 15 0,000 0,675 1,282 1,645 64 0,000 0,675 1,282 1,645 16 0,000 0,675 1,282 1,645 65 0,000 0,675 1,282 1,645 17 0,000 0,675 1,282 1,645 66 0,000 0,675 1,282 1,645 18 0,000 0,675 1,282 1,645 67 0,000 0,675 1,282 1,645 19 0,000 0,675 1,282 1,645 68 0,000 0,675 1,282 1,645 20 0,000 0,675 1,282 1,645 69 0,000 0,675 1,282 1,645 21 0,000 0,675 1,282 1,645 70 0,000 0,675 1,282 1,645 22 0,000 0,675 1,282 1,645 71 0,000 0,675 1,282 1,645 23 0,000 0,675 1,282 1,645 72 0,000 0,675 1,282 1,645 24 0,000 0,675 1,282 1,645 73 0,000 0,675 1,282 1,645 25 0,000 0,675 1,282 1,645 74 0,000 0,675 1,282 1,645 26 0,000 0,675 1,282 1,645 75 0,000 0,675 1,282 1,645 27 0,000 0,675 1,282 1,645 76 0,000 0,675 1,282 1,645 28 0,000 0,675 1,282 1,645 77 0,000 0,675 1,282 1,645 29 0,000 0,675 1,282 1,645 78 0,000 0,675 1,282 1,645 30 0,000 0,675 1,282 1,645 79 0,000 0,675 1,282 1,645 31 0,000 0,675 1,282 1,645 80 0,000 0,675 1,282 1,645 32 0,000 0,675 1,282 1,645 81 0,000 0,675 1,282 1,645 33 0,000 0,675 1,282 1,645 82 0,000 0,675 1,282 1,645 34 0,000 0,675 1,282 1,645 83 0,000 0,675 1,282 1,645 35 0,000 0,675 1,282 1,645 84 0,000 0,675 1,282 1,645 36 0,000 0,675 1,282 1,645 85 0,000 0,675 1,282 1,645 37 0,000 0,675 1,282 1,645 86 0,000 0,675 1,282 1,645 38 0,000 0,675 1,282 1,645 87 0,000 0,675 1,282 1,645 39 0,000 0,675 1,282 1,645 88 0,000 0,675 1,282 1,645 40 0,000 0,675 1,282 1,645 89 0,000 0,675 1,282 1,645 41 0,000 0,675 1,282 1,645 90 0,000 0,675 1,282 1,645 42 0,000 0,675 1,282 1,645 91 0,000 0,675 1,282 1,645 43 0,000 0,675 1,282 1,645 92 0,000 0,675 1,282 1,645 44 0,000 0,675 1,282 1,645 93 0,000 0,675 1,282 1,645 45 0,000 0,675 1,282 1,645 94 0,000 0,675 1,282 1,645 46 0,000 0,675 1,282 1,645 95 0,000 0,675 1,282 1,645 47 0,000 0,675 1,282 1,645 96 0,000 0,675 1,282 1,645 48 0,000 0,675 1,282 1,645 97 0,000 0,675 1,282 1,645 49 0,000 0,675 1,282 1,645 98 0,000 0,675 1,282 1,645 50 0,000 0,675 1,282 1,645 99 0,000 0,675 1,282 1,645 100 0,000 0,675 1,282 1,645

Table A1. k1 for one-sided statistical tolerance, standard deviation: known and confidence level γ = 50 %

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n fractile : p n fractile : p 0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95

2 0,477 1,152 1,759 2,122 51 0,095 0,769 1,376 1,740 3 0,390 1,064 0,671 2,035 52 0,094 0,768 1,375 1,739 4 0,388 1,012 0,619 1,983 53 0,094 0,768 1,375 1,738 5 0,302 0,977 1,584 1,947 54 0,093 0,767 1,374 1,737 6 0,276 0,950 1,557 1,921 55 0,092 0,766 1,373 1,737 7 0,255 0,930 1,537 1,900 56 0,091 0,765 1,372 1,736 8 0,239 0,913 1,521 1,884 57 0,090 0,764 1,371 1,735 9 0,225 0,900 1,507 1,870 58 0,090 0,764 1,371 1,734 10 0,214 0,888 1,495 1,859 59 0,089 0,763 1,370 1,733 11 0,204 0,878 1,485 1,849 60 0,088 0,762 1,369 1,732 12 0,195 0,870 1,477 1,840 61 0,087 0,761 1,368 1,731 13 0,188 0,862 1,469 1,832 62 0,087 0,761 1,368 1,731 14 0,181 0,855 1,462 1,826 63 0,086 0,760 1,367 1,730 15 0,175 0,849 1,456 1,820 64 0,085 0,760 1,367 1,730 16 0,169 0,844 1,451 1,814 65 0,085 0,759 1,366 1,729 17 0,164 0,839 1,446 1,809 66 0,084 0,758 1,365 1,728 18 0,159 0,834 1,441 1,804 67 0,083 0,758 1,365 1,728 19 0,155 0,830 1,437 1,800 68 0,082 0,757 1,364 1,727 20 0,151 0,826 1,433 1,796 69 0,082 0,757 1,364 1,727 21 0,148 0,823 1,430 1,793 70 0,081 0,756 1,363 1,726 22 0,144 0,819 1,426 1,789 71 0,081 0,755 1,362 1,726 23 0,141 0,816 1,423 1,786 72 0,080 0,755 1,362 1,725 24 0,138 0,813 1,420 1,783 73 0,080 0,754 1,361 1,725 25 0,136 0,810 1,417 1,781 74 0,079 0,754 1,361 1,724 26 0,133 0,807 1,414 1,778 75 0,079 0,753 1,360 1,724 27 0,131 0,805 1,412 1,776 76 0,078 0,752 1,359 1,723 28 0,128 0,802 1,410 1,773 77 0,078 0,752 1,359 1,723 29 0,126 0,800 1,408 1,771 78 0,077 0,751 1,358 1,722 30 0,124 0,798 1,405 1,768 79 0,077 0,751 1,358 1,722 31 0,122 0,796 1,403 1,766 80 0,076 0,75 1,357 1,721 32 0,120 0,794 1,401 1,764 81 0,076 0,750 1,357 1,721 33 0,119 0,793 1,400 1,763 82 0,075 0,749 1,356 1,720 34 0,117 0,791 1,398 1,761 83 0,075 0,749 1,356 1,720 35 0,115 0,789 1,396 1,759 84 0,074 0,748 1,355 1,719 36 0,113 0,788 1,395 1,758 85 0,074 0,748 1,355 1,719 37 0,112 0,786 1,393 1,756 86 0,074 0,748 1,355 1,718 38 0,110 0,785 1,392 1,755 87 0,073 0,747 1,354 1,718 39 0,109 0,783 1,390 1,753 88 0,073 0,747 1,354 1,717 40 0,107 0,782 1,389 1,752 89 0,072 0,746 1,353 1,717 41 0,106 0,781 1,388 1,751 90 0,072 0,746 1,353 1,716 42 0,105 0,780 1,387 1,750 91 0,072 0,746 1,353 1,716 43 0,103 0,778 1,385 1,748 92 0,071 0,745 1,352 1,715 44 0,102 0,777 1,384 1,747 93 0,071 0,745 1,352 1,715 45 0,101 0,776 1,383 1,746 94 0,070 0,744 1,352 1,715 46 0,100 0,775 1,382 1,745 95 0,070 0,744 1,352 1,715 47 0,099 0,774 1,381 1,744 96 0,070 0,744 1,351 1,714 48 0,098 0,772 1,379 1,743 97 0,069 0,743 1,351 1,714 49 0,097 0,771 1,378 1,742 98 0,069 0,743 1,351 1,714 50 0,096 0,770 1,377 1,741 99 0,068 0,742 1,350 1,713 100 0,068 0,742 1,35 1,713


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n fractile : p n fractile : p

0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 0,907 1,581 2,188 2,552 51 0,180 0,854 1,461 1,825 3 0,740 1,415 2,022 2,385 52 0,179 0,853 1,460 1,824 4 0,641 1,316 1,923 2,286 53 0,177 0,851 1,458 1,822 5 0,574 1,248 1,855 2,218 54 0,176 0,850 1,457 1,821 6 0,524 1,198 1,805 2,169 55 0,174 0,848 1,455 1,819 7 0,485 1,159 1,766 2,130 56 0,172 0,846 1,453 1,817 8 0,454 1,128 1,735 2,098 57 0,171 0,845 1,452 1,816 9 0,428 1,102 1,709 2,073 58 0,169 0,843 1,450 1,814 10 0,406 1,080 1,687 2,051 59 0,168 0,842 1,449 1,813 11 0,387 1,061 1,668 2,032 60 0,166 0,840 1,447 1,811 12 0,370 1,045 1,652 2,015 61 0,165 0,839 1,446 1,810 13 0,356 1,030 1,637 2,001 62 0,164 0,838 1,445 1,809 14 0,343 1,017 1,625 1,998 63 0,162 0,836 1,443 1,807 15 0,331 1,006 1,613 1,976 64 0,161 0,835 1,442 1,806 16 0,321 0,995 1,602 1,966 65 0,160 0,834 1,441 1,805 17 0,311 0,986 1,593 1,956 66 0,159 0,833 1,440 1,804 18 0,303 0,977 1,584 1,947 67 0,158 0,832 1,439 1,803 19 0,295 0,969 1,576 1,939 68 0,156 0,830 1,437 1,801 20 0,287 0,962 1,569 1,932 69 0,155 0,829 1,436 1,800 21 0,281 0,955 1,562 1,926 70 0,154 0,828 1,435 1,799 22 0,274 0,948 1,555 1,919 71 0,153 0,827 1,434 1,798 23 0,268 0,943 1,550 1,913 72 0,152 0,826 1,433 1,797 24 0,262 0,937 1,544 1,907 73 0,151 0,825 1,432 1,796 25 0,257 0,932 1,539 1,902 74 0,150 0,824 1,431 1,795 26 0,252 0,926 1,533 1,897 75 0,149 0,823 1,430 1,794 27 0,248 0,922 1,529 1,893 76 0,148 0,822 1,429 1,793 28 0,243 0,917 1,524 1,888 77 0,147 0,821 1,428 1,792 29 0,239 0,913 1,520 1,884 78 0,146 0,820 1,427 1,791 30 0,234 0,909 1,516 1,879 79 0,145 0,819 1,426 1,790 31 0,231 0,906 1,513 1,876 80 0,144 0,818 1,425 1,789 32 0,227 0,902 1,509 1,872 81 0,143 0,817 1,424 1,788 33 0,224 0,899 1,506 1,869 82 0,142 0,816 1,423 1,787 34 0,220 0,895 1,502 1,865 83 0,142 0,816 1,423 1,786 35 0,217 0,892 1,499 1,862 84 0,141 0,815 1,422 1,785 36 0,214 0,889 1,496 1,859 85 0,140 0,814 1,421 1,785 37 0,211 0,886 1,493 1,856 86 0,139 0,813 1,420 1,784 38 0,209 0,884 1,491 1,854 87 0,138 0,812 1,419 1,783 39 0,206 0,881 1,488 1,851 88 0,138 0,812 1,419 1,782 40 0,203 0,878 1,485 1,848 89 0,137 0,811 1,418 1,781 41 0,201 0,876 1,483 1,846 90 0,136 0,810 1,417 1,780 42 0,199 0,873 1,480 1,843 91 0,135 0,809 1,416 1,779 43 0,196 0,871 1,478 1,841 92 0,135 0,809 1,416 1,779 44 0,194 0,868 1,475 1,838 93 0,134 0,808 1,415 1,778 45 0,192 0,866 1,473 1,836 94 0,133 0,807 1,414 1,778 46 0,190 0,864 1,471 1,834 95 0,133 0,807 1,414 1,777 47 0,188 0,862 1,469 1,832 96 0,132 0,806 1,413 1,776 48 0,186 0,860 1,467 1,831 97 0,131 0,805 1,412 1,776 49 0,184 0,858 1,465 1,829 98 0,130 0,804 1,411 1,775 50 0,182 0,856 1,463 1,827 99 0,130 0,804 1,411 1,775 100 0,129 0,803 1,410 1,774


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 1,164 0,838 2,445 2,828 51 0,231 0,906 1,513 1,876 3 0,950 0,625 2,232 2,595 52 0,229 0,904 1,511 1,874 4 0,823 1,497 2,104 2,468 53 0,227 0,902 1,509 1,872 5 0,736 1,411 2,018 2,381 54 0,225 0,900 1,507 1,870 6 0,672 1,346 1,954 2,317 55 0,223 0,898 1,505 1,868 7 0,622 1,297 1,904 2,267 56 0,221 0,895 1,502 1,866 8 0,582 1,257 1,864 2,227 57 0,219 0,893 1,500 1,864 9 0,549 1,223 1,830 2,194 58 0,217 0,891 1,498 1,862 10 0,521 1,195 1,802 2,166 59 0,215 0,889 1,496 1,860 11 0,496 1,171 1,778 2,141 60 0,213 0,887 1,494 1,858 12 0,475 1,150 1,757 2,120 61 0,211 0,886 1,493 1,856 13 0,457 1,131 1,738 2,102 62 0,210 0,884 1,491 1,855 14 0,440 1,115 1,722 2,085 63 0,208 0,883 1,490 1,853 15 0,425 1,100 1,707 2,070 64 0,207 0,881 1,488 1,852 16 0,412 1,086 1,693 2,057 65 0,205 0,880 1,487 1,850 17 0,399 1,074 1,691 2,044 66 0,203 0,878 1,485 1,848 18 0,388 1,063 1,670 2,033 67 0,202 0,877 1,484 1,847 19 0,378 1,052 1,659 2,023 68 0,200 0,875 1,482 1,845 20 0,368 1,043 1,650 2,013 69 0,199 0,874 1,481 1,844 21 0,360 1,035 1,642 2,005 70 0,197 0,872 1,479 1,842 22 0,351 1,026 1,633 1,996 71 0,196 0,871 1,478 1,841 23 0,344 1,019 1,626 1,989 72 0,194 0,869 1,476 1,839 24 0,336 1,011 1,618 1,981 73 0,193 0,868 1,475 1,838 25 0,330 1,005 1,612 1,975 74 0,192 0,867 1,474 1,837 26 0,323 0,998 1,605 1,968 75 0,191 0,866 1,473 1,836 27 0,317 0,992 1,599 1,962 76 0,189 0,864 1,471 1,834 28 0,311 0,986 1,593 1,956 77 0,188 0,863 1,470 1,833 29 0,306 0,981 1,588 1,951 78 0,187 0,862 1,469 1,832 30 0,301 0,975 1,582 1,946 79 0,185 0,860 1,467 1,830 31 0,297 0,971 1,578 1,941 80 0,184 0,859 1,466 1,829 32 0,292 0,966 1,573 1,937 81 0,183 0,858 1,465 1,828 33 0,288 0,962 1,569 1,932 82 0,182 0,857 1,464 1,827 34 0,283 0,957 1,564 1,928 83 0,181 0,856 1,463 1,826 35 0,279 0,953 1,560 1,923 84 0,180 0,855 1,462 1,825 36 0,275 0,949 1,556 1,919 85 0,179 0,854 1,461 1,824 37 0,272 0,946 1,553 1,916 86 0,178 0,852 1,459 1,823 38 0,268 0,942 1,549 1,912 87 0,177 0,851 1,458 1,822 39 0,265 0,939 1,546 1,909 88 0,176 0,850 1,457 1,821 40 0,261 0,935 1,542 1,905 89 0,175 0,849 1,456 1,820 41 0,258 0,932 1,539 1,902 90 0,174 0,848 1,455 1,819 42 0,255 0,929 1,536 1,899 91 0,173 0,847 1,454 1,818 43 0,252 0,926 1,533 1,897 92 0,172 0,846 1,453 1,817 44 0,249 0,923 1,530 1,894 93 0,171 0,845 1,453 1,816 45 0,246 0,920 1,527 1,891 94 0,170 0,844 1,452 1,815 46 0,243 0,918 1,525 1,888 95 0,170 0,844 1,451 1,815 47 0,241 0,915 1,522 1,886 96 0,169 0,843 1,450 1,814 48 0,238 0,913 1,520 1,883 97 0,168 0,842 1,449 1,813 49 0,236 0,910 1,517 1,881 98 0,167 0,841 1,449 1,812 50 0,233 0,908 1,515 1,878 99 0,166 0,840 1,448 1,811 100 0,165 0,839 1,447 1,810


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 0,000 0,888 1,785 2,339 51 0,000 0,679 1,290 1,655 3 0,000 0,774 1,499 1,939 52 0,000 0,679 1,290 1,655 4 0,000 0,739 1,419 1,830 53 0,000 0,679 1,289 1,655 5 0,000 0,722 1,382 1,780 54 0,000 0,679 1,289 1,655 6 0,000 0,712 1,361 1,751 55 0,000 0,679 1,289 1,655 7 0,000 0,706 1,347 1,732 56 0,000 0,678 1,289 1,654 8 0,000 0,701 1,337 1,719 57 0,000 0,678 1,289 1,654 9 0,000 0,698 1,330 1,710 58 0,000 0,678 1,288 1,654 10 0,000 0,695 1,325 1,702 59 0,000 0,678 1,288 1,654 11 0,000 0,693 1,320 1,696 60 0,000 0,678 1,288 1,654 12 0,000 0,692 1,317 1,691 61 0,000 0,678 1,288 1,654 13 0,000 0,690 1,314 1,687 62 0,000 0,678 1,288 1,654 14 0,000 0,689 1,311 1,684 63 0,000 0,678 1,288 1,653 15 0,000 0,688 1,309 1,681 64 0,000 0,678 1,288 1,653 16 0,000 0,678 1,307 1,679 65 0,000 0,678 1,288 1,653 17 0,000 0,686 1,306 1,677 66 0,000 0,678 1,287 1,653 18 0,000 0,686 1,304 1,675 67 0,000 0,678 1,287 1,653 19 0,000 0,685 1,303 1,673 68 0,000 0,678 1,287 1,652 20 0,000 0,685 1,302 1,672 69 0,000 0,678 1,287 1,652 21 0,000 0,685 1,301 1,671 70 0,000 0,678 1,287 1,652 22 0,000 0,684 1,300 1,669 71 0,000 0,678 1,287 1,652 23 0,000 0,684 1,299 1,668 72 0,000 0,678 1,287 1,652 24 0,000 0,683 1,298 1,667 73 0,000 0,678 1,287 1,652 25 0,000 0,683 1,298 1,666 74 0,000 0,678 1,287 1,652 26 0,000 0,682 1,297 1,665 75 0,000 0,678 1,287 1,652 27 0,000 0,682 1,297 1,665 76 0,000 0,677 1,287 1,652 28 0,000 0,682 1,296 1,664 77 0,000 0,677 1,287 1,652 29 0,000 0,682 1,296 1,663 78 0,000 0,677 1,287 1,652 30 0,000 0,681 1,295 1,662 79 0,000 0,677 1,287 1,652 31 0,000 0,681 1,295 1,662 80 0,000 0,677 1,287 1,652 32 0,000 0,681 1,294 1,661 81 0,000 0,677 1,287 1,652 33 0,000 0,680 1,294 1,661 82 0,000 0,677 1,287 1,652 34 0,000 0,680 1,293 1,660 83 0,000 0,677 1,287 1,652 35 0,000 0,680 1,293 1,660 84 0,000 0,677 1,287 1,652 36 0,000 0,680 1,293 1,660 85 0,000 0,677 1,287 1,652 37 0,000 0,680 1,293 1,659 86 0,000 0,677 1,286 1,651 38 0,000 0,680 1,292 1,659 87 0,000 0,677 1,286 1,651 39 0,000 0,680 1,292 1,658 88 0,000 0,677 1,286 1,651 40 0,000 0,680 1,292 1,658 89 0,000 0,677 1,286 1,651 41 0,000 0,680 1,292 1,658 90 0,000 0,677 1,286 1,651 42 0,000 0,680 1,291 1,658 91 0,000 0,677 1,286 1,651 43 0,000 0,679 1,291 1,657 92 0,000 0,677 1,286 1,651 44 0,000 0,679 1,290 1,657 93 0,000 0,677 1,286 1,651 45 0,000 0,679 1,290 1,657 94 0,000 0,677 1,286 1,651 46 0,000 0,679 1,290 1,657 95 0,000 0,677 1,286 1,651 47 0,000 0,679 1,290 1,656 96 0,000 0,677 1,286 1,650 48 0,000 0,679 1,290 1,656 97 0,000 0,677 1,286 1,650 49 0,000 0,679 1,290 1,655 98 0,000 0,677 1,286 1,650 50 0,000 0,679 1,290 1,655 99 0,000 0,677 1,286 1,650 100 0,000 0,677 1,286 1,650

Table A5. k1 for one-sided statistical tolerance, standard deviation: unknown and confidence level γ = 50 %

NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 0,708 2,225 3,993 5,122 51 0,096 0,788 1,425 1,809 3 0,472 1,465 2,502 3,152 52 0,095 0,787 1,423 1,808 4 0,383 1,256 2,134 2,681 53 0,094 0,786 1,422 1,806 5 0,332 1,152 1,962 2,464 54 0,093 0,785 1,420 1,805 6 0,297 1,088 1,860 2,336 55 0,093 0,784 1,419 1,803 7 0,272 1,044 1,791 2,251 56 0,092 0,782 1,418 1,801 8 0,252 0,011 1,740 2,189 57 0,091 0,781 1,416 1,800 9 0,236 0,985 1,702 2,142 58 0,090 0,780 1,415 1,798 10 0,223 0,964 1,671 2,104 59 0,089 0,779 1,413 1,797 11 0,212 0,947 1,646 2,074 60 0,088 0,778 1,412 1,795 12 0,202 0,933 1,625 2,048 61 0,087 0,777 1,411 1,794 13 0,193 0,920 1,607 2,026 62 0,087 0,776 1,410 1,793 14 0,186 0,909 1,591 2,008 63 0,086 0,776 1,409 1,791 15 0,179 0,900 1,578 1,991 64 0,086 0,775 1,408 1,790 16 0,173 0,891 1,566 1,977 65 0,085 0,774 1,407 1,789 17 0,168 0,884 1,555 1,964 66 0,084 0,773 1,405 1,788 18 0,163 0,877 1,545 1,952 67 0,084 0,772 1,404 1,787 19 0,158 0,870 1,536 1,942 68 0,083 0,772 1,403 1,785 20 0,154 0,865 1,529 1,932 69 0,083 0,771 1,402 1,784 21 0,151 0,860 1,522 1,924 70 0,082 0,770 1,401 1,783 22 0,147 0,854 1,514 1,916 71 0,081 0,769 1,400 1,782 23 0,144 0,850 1,509 1,909 72 0,081 0,769 1,399 1,781 24 0,140 0,846 1,503 1,902 73 0,080 0,768 1,399 1,780 25 0,138 0,842 1,498 1,896 74 0,080 0,767 1,398 1,779 26 0,135 0,838 1,492 1,889 75 0,079 0,767 1,397 1,778 27 0,133 0,835 1,488 1,884 76 0,078 0,766 1,396 1,777 28 0,130 0,831 1,483 1,879 77 0,078 0,765 1,395 1,776 29 0,128 0,828 1,479 1,874 78 0,077 0,764 1,395 1,775 30 0,125 0,825 1,475 1,869 79 0,077 0,764 1,394 1,774 31 0,123 0,823 1,472 1,865 80 0,076 0,763 1,393 1,773 32 0,121 0,820 1,468 1,861 81 0,076 0,763 1,392 1,772 33 0,120 0,818 1,465 1,858 82 0,075 0,762 1,392 1,771 34 0,118 0,815 1,461 1,854 83 0,075 0,762 1,391 1,771 35 0,116 0,813 1,458 1,850 84 0,074 0,761 1,390 1,770 36 0,114 0,811 1,455 1,847 85 0,074 0,761 1,390 1,769 37 0,113 0,809 1,453 1,844 86 0,074 0,760 1,389 1,768 38 0,111 0,807 1,450 1,840 87 0,073 0,760 1,388 1,767 39 0,110 0,805 1,448 1,837 88 0,073 0,759 1,387 1,767 40 0,108 0,803 1,445 1,834 89 0,072 0,759 1,387 1,766 41 0,107 0,801 1,443 1,832 90 0,072 0,758 1,386 1,765 42 0,106 0,800 1,441 1,829 91 0,072 0,758 1,385 1,764 43 0,104 0,798 1,439 1,827 92 0,071 0,757 1,385 1,764 44 0,103 0,797 1,437 1,824 93 0,071 0,757 1,384 1,763 45 0,102 0,795 1,435 1,822 94 0,070 0,756 1,384 1,762 46 0,101 0,794 1,433 1,820 95 0,070 0,756 1,383 1,762 47 0,100 0,793 1,431 1,818 96 0,070 0,755 1,382 1,761 48 0,099 0,791 1,430 1,815 97 0,069 0,755 1,382 1,760 49 0,098 0,790 1,428 1,813 98 0,069 0,754 1,381 1,759 50 0,097 0,789 1,426 1,811 99 0,068 0,754 1,381 1,759 100 0,068 0,753 1,380 1,758


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 2,177 5,843 10,253 13,090 51 0,182 0,892 1,557 1,963 3 1,089 2,603 4,259 5,312 52 0,181 0,890 1,555 1,960 4 0,819 1,973 3,188 3,957 53 0,179 0,888 1,552 1,956 5 0,686 1,698 2,743 3,400 54 0,178 0,886 1,549 1,953 6 0,603 1,540 2,494 3,092 55 0,176 0,884 1,547 1,950 7 0,545 1,436 2,333 2,894 56 0,174 0,881 1,544 1,947 8 0,501 1,360 2,219 2,755 57 0,173 0,879 1,541 1,944 9 0,466 1,303 2,133 2,650 58 0,171 0,877 1,538 1,940 10 0,438 1,257 2,066 2,569 59 0,170 0,875 1,536 1,937 11 0,414 1,220 2,012 2,503 60 0,168 0,873 1,533 1,934 12 0,394 1,189 1,967 2,449 61 0,167 0,871 1,531 1,932 13 0,377 1,162 1,929 2,403 62 0,165 0,870 1,529 1,929 14 0,361 1,139 1,896 2,364 63 0,164 0,868 1,527 1,927 15 0,348 1,119 1,867 2,329 64 0,163 0,867 1,525 1,924 16 0,336 1,101 1,842 2,299 65 0,162 0,865 1,523 1,922 17 0,325 1,085 1,820 2,273 66 0,160 0,863 1,520 1,920 18 0,315 1,071 1,800 2,249 67 0,159 0,862 1,518 1,917 19 0,306 1,058 1,782 2,228 68 0,158 0,860 1,516 1,915 20 0,297 1,046 1,766 2,208 69 0,156 0,859 1,514 1,912 21 0,290 1,036 1,752 2,191 70 0,155 0,857 1,512 1,910 22 0,283 1,026 1,737 2,174 71 0,154 0,856 1,510 1,908 23 0,277 1,017 1,725 2,160 72 0,153 0,855 1,509 1,906 24 0,270 1,008 1,713 2,146 73 0,152 0,853 1,507 1,904 25 0,265 1,001 1,703 2,134 74 0,151 0,852 1,505 1,902 26 0,259 0,993 1,692 2,121 75 0,150 0,851 1,504 1,900 27 0,254 0,986 1,683 2,110 76 0,149 0,850 1,502 1,898 28 0,249 0,979 1,674 2,099 77 0,148 0,849 1,500 1,896 29 0,245 0,973 1,666 2,090 78 0,147 0,847 1,498 1,894 30 0,240 0,967 1,658 2,080 79 0,146 0,846 1,497 1,892 31 0,236 1,162 1,651 2,072 80 0,145 0,845 1,495 1,890 32 0,232 1,357 1,644 2,064 81 0,144 0,844 1,494 1,889 33 0,229 1,553 1,638 2,057 82 0,143 0,843 1,492 1,887 34 0,225 1,748 1,631 2,049 83 0,143 0,842 1,491 1,886 35 0,221 1,943 1,624 2,041 84 0,142 0,841 1,490 1,884 36 0,218 1,739 1,619 2,035 85 0,141 0,840 1,489 1,883 37 0,215 1,535 1,614 2,029 86 0,140 0,838 1,487 1,881 38 0,213 1,331 1,608 2,023 87 0,139 0,837 1,486 1,880 39 0,210 1,127 1,603 2,017 88 0,139 0,836 1,485 1,878 40 0,207 0,923 1,598 2,011 89 0,138 0,835 1,483 1,877 41 0,204 0,920 1,594 2,006 90 0,137 0,834 1,482 1,875 42 0,202 0,917 1,590 2,001 91 0,136 0,833 1,481 1,874 43 0,199 0,913 1,585 1,996 92 0,136 0,832 1,480 1,872 44 0,197 0,910 1,581 1,991 93 0,135 0,831 1,479 1,871 45 0,194 0,907 1,577 1,986 94 0,134 0,830 1,478 1,870 46 0,192 0,904 1,574 1,982 95 0,134 0,830 1,477 1,869 47 0,190 0,902 1,570 1,978 96 0,133 0,829 1,475 1,867 48 0,188 0,899 1,567 1,974 97 0,132 0,828 1,474 1,866 49 0,186 0,897 1,563 1,970 98 0,131 0,827 1,473 1,865 50 0,184 0,894 1,560 1,966 99 0,131 0,826 1,472 1,863 100 0,130 0,825 1,471 1,862


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 4,465 11,763 20,582 26,260 51 0,236 0,958 1,642 2,061 3 1,686 3,807 6,156 7,656 52 0,234 0,955 1,639 2,057 4 1,177 2,618 4,162 5,144 53 0,231 0,953 1,635 2,052 5 0,954 2,150 3,407 4,203 54 0,229 0,950 1,631 2,048 6 0,823 1,896 3,007 3,708 55 0,227 0,947 1,628 2,044 7 0,735 1,733 2,756 3,400 56 0,225 0,944 1,624 2,040 8 0,670 1,618 2,582 3,188 57 0,223 0,941 1,620 2,036 9 0,620 1,533 2,454 3,032 58 0,220 0,939 1,616 2,031 10 0,580 1,466 2,355 2,911 59 0,218 0,936 1,613 2,027 11 0,547 1,412 2,276 2,815 60 0,216 0,933 1,609 2,023 12 0,519 1,367 2,211 2,737 61 0,214 0,931 1,606 2,020 13 0,495 1,329 2,156 2,671 62 0,213 0,929 1,604 2,016 14 0,474 1,296 2,109 2,615 63 0,211 0,927 1,601 2,013 15 0,455 1,268 2,069 2,567 64 0,210 0,925 1,598 2,010 16 0,439 1,243 2,033 2,524 65 0,208 0,923 1,596 2,007 17 0,424 1,221 2,002 2,487 66 0,206 0,920 1,593 2,003 18 0,411 1,201 1,974 2,453 67 0,205 0,918 1,590 2,000 19 0,398 1,183 1,949 2,424 68 0,203 0,916 1,587 1,997 20 0,387 1,167 1,926 2,397 69 0,202 0,914 1,585 1,993 21 0,377 1,153 1,907 2,373 70 0,200 0,912 1,582 1,990 22 0,367 1,138 1,887 2,349 71 0,199 0,910 1,580 1,988 23 0,359 1,126 1,870 2,330 72 0,197 0,909 1,578 1,985 24 0,350 1,114 1,853 2,310 73 0,196 0,907 1,575 1,983 25 0,343 1,104 1,839 2,293 74 0,195 0,905 1,573 1,980 26 0,335 1,093 1,825 2,276 75 0,194 0,904 1,571 1,978 27 0,329 1,084 1,813 2,261 76 0,192 0,902 1,569 1,975 28 0,322 1,075 1,800 2,246 77 0,191 0,900 1,567 1,973 29 0,317 1,067 1,789 2,233 78 0,190 0,898 1,564 1,970 30 0,311 1,059 1,778 2,220 79 0,188 0,897 1,562 1,968 31 0,306 1,052 1,769 2,209 80 0,187 0,895 1,560 1,965 32 0,301 1,046 1,760 2,199 81 0,186 0,894 1,558 1,963 33 0,296 1,039 1,751 2,188 82 0,185 0,892 1,556 1,961 34 0,291 1,033 1,742 2,178 83 0,184 0,891 1,555 1,959 35 0,286 1,026 1,733 2,167 84 0,183 0,890 1,553 1,957 36 0,282 1,021 1,726 2,159 85 0,182 0,889 1,551 1,955 37 0,278 1,016 1,719 2,151 86 0,180 0,887 1,549 1,952 38 0,275 1,010 1,712 2,142 87 0,179 0,886 1,547 1,950 39 0,271 1,005 1,705 2,134 88 0,178 0,885 1,546 1,948 40 0,267 1,000 1,698 2,126 89 0,177 0,883 1,544 1,946 41 0,264 0,996 1,692 2,119 90 0,176 0,882 1,542 1,944 42 0,261 0,991 1,686 2,113 91 0,175 0,881 1,541 1,942 43 0,257 0,987 1,681 2,106 92 0,174 0,880 1,539 1,941 44 0,254 0,982 1,675 2,100 93 0,173 0,878 1,538 1,939 45 0,251 0,978 1,669 2,093 94 0,172 0,877 1,536 1,937 46 0,248 0,975 1,664 2,087 95 0,172 0,876 1,535 1,936 47 0,246 0,971 1,660 2,082 96 0,171 0,875 1,533 1,934 48 0,243 0,968 1,655 2,076 97 0,170 0,874 1,532 1,932 49 0,241 0,964 1,651 2,071 98 0,169 0,872 1,530 1,930 50 0,238 0,961 1,646 2,065 99 0,168 0,871 1,529 1,929 100 0,167 0,870 1,527 1,927


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 0,755 0,282 1,823 2,164 51 0,678 1,156 1,653 1,969 3 0,727 1,238 1,766 2,100 52 0,678 1,156 1,653 1,969 4 0,714 1,216 1,737 2,067 53 0,678 1,156 1,653 1,969 5 0,706 1,203 1,719 2,046 54 0,678 1,156 1,653 1,969 6 0,701 1,195 1,707 2,033 55 0,678 1,156 1,653 1,969 7 0,697 1,188 1,698 2,023 56 0,678 1,155 1,652 1,968 8 0,694 1,184 1,692 2,015 57 0,678 1,155 1,652 1,968 9 0,692 1,180 1,686 2,009 58 0,678 1,155 1,652 1,968 10 0,690 1,177 1,682 2,004 59 0,678 1,155 1,652 1,968 11 0,689 1,175 1,679 2,000 60 0,678 1,155 1,652 1,968 12 0,688 1,173 1,676 1,997 61 0,678 1,155 1,652 1,968 13 0,687 1,171 1,674 1,994 62 0,678 1,155 1,652 1,968 14 0,686 1,170 1,672 1,992 63 0,678 1,155 1,652 1,968 15 0,685 1,168 1,670 1,990 64 0,678 1,155 1,652 1,968 16 0,685 1,167 1,669 1,988 65 0,678 1,155 1,652 1,968 17 0,684 1,166 1,667 1,986 66 0,677 1,155 1,651 1,967 18 0,684 1,165 1,666 1,985 67 0,677 1,155 1,651 1,967 19 0,683 1,165 1,665 1,984 68 0,677 1,155 1,651 1,967 20 0,683 1,164 1,664 1,983 69 0,677 1,155 1,651 1,967 21 0,683 1,164 1,663 1,982 70 0,677 1,155 1,651 1,967 22 0,682 1,163 1,662 1,981 71 0,677 1,155 1,651 1,967 23 0,682 1,163 1,662 1,980 72 0,677 1,155 1,651 1,967 24 0,681 1,162 1,661 1,979 73 0,677 1,155 1,651 1,967 25 0,681 1,162 1,661 1,978 74 0,677 1,155 1,651 1,967 26 0,681 1,161 1,660 1,977 75 0,677 1,155 1,651 1,967 27 0,681 1,161 1,660 1,977 76 0,677 1,154 1,650 1,966 28 0,680 1,160 1,659 1,976 77 0,677 1,154 1,650 1,966 29 0,680 1,160 1,659 1,976 78 0,677 1,154 1,650 1,966 30 0,680 1,160 1,658 1,975 79 0,677 1,154 1,650 1,966 31 0,680 1,160 1,658 1,975 80 0,677 1,154 1,650 1,966 32 0,680 1,159 1,657 1,974 81 0,677 1,154 1,650 1,966 33 0,679 1,159 1,657 1,974 82 0,677 1,154 1,650 1,966 34 0,679 1,158 1,656 1,973 83 0,677 1,154 1,650 1,966 35 0,679 1,158 1,656 1,973 84 0,677 1,154 1,650 1,966 36 0,679 1,158 1,656 1,973 85 0,677 1,154 1,650 1,966 37 0,679 1,158 1,656 1,973 86 0,677 1,154 1,650 1,965 38 0,679 1,157 1,655 1,972 87 0,677 1,154 1,650 1,965 39 0,679 1,157 1,655 1,972 88 0,677 1,154 1,650 1,965 40 0,679 1,157 1,655 1,972 89 0,677 1,154 1,650 1,965 41 0,679 1,157 1,655 1,972 90 0,677 1,154 1,650 1,965 42 0,679 1,157 1,655 1,971 91 0,677 1,154 1,650 1,965 43 0,678 1,157 1,654 1,971 92 0,677 1,154 1,650 1,965 44 0,678 1,157 1,654 1,970 93 0,677 1,154 1,650 1,965 45 0,678 1,157 1,654 1,970 94 0,677 1,154 1,650 1,965 46 0,678 1,157 1,654 1,970 95 0,677 1,154 1,650 1,965 47 0,678 1,157 1,654 1,970 96 0,677 1,153 1,649 1,965 48 0,678 1,156 1,653 1,969 97 0,677 1,153 1,649 1,965 49 0,678 1,156 1,653 1,969 98 0,677 1,153 1,649 1,965 50 0,678 1,156 1,653 1,969 99 0,677 1,153 1,649 1,965 100 0,677 1,153 1,649 1,965

Table A9. k2 for two-sided statistical tolerance, standard deviation: known and confidence level γ = 50 %

NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 0,919 1,520 2,106 2,464 51 0,684 1,166 1,667 1,986 3 0,834 1,402 1,971 2,323 52 0,684 1,166 1,666 1,985 4 0,792 1,340 1,897 2,244 53 0,683 1,165 1,666 1,985 5 0,768 1,303 1,850 2,194 54 0,683 1,165 1,665 1,984 6 0,752 1,278 1,818 2,158 55 0,683 1,165 1,665 1,984 7 0,741 1,260 1,794 2,132 56 0,683 1,165 1,665 1,984 8 0,732 1,246 1,776 2,112 57 0,683 1,165 1,664 1,983 9 0,726 1,236 1,762 2,096 58 0,682 1,164 1,664 1,983 10 0,721 1,227 1,751 2,083 59 0,682 1,164 1,663 1,982 11 0,716 1,220 1,742 2,073 60 0,682 1,164 1,663 1,982 12 0,713 1,214 1,734 2,064 61 0,682 1,164 1,663 1,982 13 0,710 1,209 1,727 2,056 62 0,682 1,164 1,663 1,981 14 0,707 1,205 1,722 2,050 63 0,682 1,163 1,662 1,981 15 0,705 1,202 1,717 2,044 64 0,682 1,163 1,662 1,981 16 0,703 1,198 1,712 2,039 65 0,682 1,163 1,662 1,981 17 0,702 1,196 1,708 2,034 66 0,681 1,163 1,662 1,980 18 0,700 1,193 1,705 2,030 67 0,681 1,163 1,662 1,980 19 0,699 1,191 1,702 2,027 68 0,681 1,162 1,661 1,980 20 0,698 1,189 1,699 2,024 69 0,681 1,162 1,661 1,979 21 0,697 1,187 1,697 2,021 70 0,681 1,162 1,661 1,979 22 0,695 1,185 1,694 2,018 71 0,681 1,162 1,661 1,979 23 0,695 1,184 1,692 2,016 72 0,681 1,162 1,661 1,979 24 0,694 1,183 1,690 2,013 73 0,681 1,161 1,660 1,978 25 0,693 1,182 1,684 2,011 74 0,681 1,161 1,660 1,978 26 0,692 1,180 1,678 2,009 75 0,681 1,161 1,660 1,978 27 0,692 1,179 1,681 2,008 76 0,681 1,161 1,660 1,978 28 0,691 1,178 1,684 2,006 77 0,681 1,161 1,660 1,978 29 0,691 1,177 1,683 2,005 78 0,681 1,160 1,659 1,977 30 0,690 1,176 1,681 2,003 79 0,681 1,160 1,659 1,977 31 0,690 1,175 1,680 2,002 80 0,681 1,16 1,659 1,977 32 0,689 1,175 1,679 2,001 81 0,681 1,160 1,659 1,977 33 0,689 1,174 1,678 1,999 82 0,681 1,160 1,659 1,977 34 0,688 1,174 1,677 1,998 83 0,681 1,160 1,658 1,976 35 0,688 1,173 1,676 1,997 84 0,681 1,160 1,658 1,976 36 0,688 1,172 1,675 1,996 85 0,681 1,160 1,658 1,976 37 0,687 1,172 1,674 1,995 86 0,680 1,159 1,658 1,976 38 0,687 1,171 1,674 1,994 87 0,680 1,159 1,658 1,976 39 0,686 1,171 1,673 1,993 88 0,680 1,159 1,657 1,975 40 0,686 1,170 1,672 1,992 89 0,680 1,159 1,657 1,975 41 0,686 1,170 1,671 1,991 90 0,68 1,159 1,657 1,975 42 0,686 1,169 1,671 1,991 91 0,680 1,159 1,657 1,975 43 0,685 1,169 1,670 1,990 92 0,680 1,159 1,657 1,975 44 0,685 1,168 1,670 1,990 93 0,680 1,159 1,657 1,974 45 0,685 1,168 1,669 1,989 94 0,680 1,159 1,657 1,974 46 0,685 1,168 1,669 1,988 95 0,680 1,159 1,657 1,974 47 0,685 1,167 1,668 1,988 96 0,679 1,158 1,656 1,974 48 0,684 1,167 1,668 1,987 97 0,679 1,158 1,656 1,974 49 0,684 1,166 1,667 1,987 98 0,679 1,158 1,656 1,973 50 0,684 1,166 1,667 1,986 99 0,679 1,158 1,656 1,973 100 0,679 1,158 1,656 1,973


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 1,187 1,842 2,446 2,809 51 0,693 1,182 1,688 2,011 3 1,013 1,640 1,236 2,597 52 0,692 1,181 1,688 2,010 4 0,924 1,527 2,114 2,473 53 0,692 1,181 1,687 2,010 5 0,872 1,456 2,034 2,390 54 0,692 1,180 1,686 2,009 6 0,837 1,407 1,977 2,330 55 0,692 1,180 1,686 2,008 7 0,813 1,371 1,935 2,285 56 0,691 1,179 1,685 2,007 8 0,795 1,344 1,902 2,250 57 0,691 1,179 1,684 2,006 9 0,781 1,323 1,875 2,222 58 0,691 1,178 1,683 2,006 10 0,770 1,306 1,854 2,198 59 0,690 1,178 1,683 2,005 11 0,761 1,292 1,836 2,179 60 0,690 1,177 1,682 2,004 12 0,754 1,281 1,821 2,162 61 0,690 1,177 1,682 2,003 13 0,758 1,271 1,809 2,148 62 0,690 1,176 1,681 2,003 14 0,742 1,262 1,797 2,136 63 0,689 1,176 1,681 2,002 15 0,738 1,255 1,788 2,125 64 0,689 1,175 1,680 2,002 16 0,734 1,248 1,779 2,115 65 0,689 1,175 1,680 2,001 17 0,730 1,243 1,772 2,107 66 0,689 1,175 1,679 2,000 18 0,727 1,237 1,765 2,099 67 0,689 1,174 1,679 2,000 19 0,724 1,233 1,759 2,092 68 0,688 1,174 1,678 1,999 20 0,722 1,229 1,753 2,086 69 0,688 1,173 1,678 1,999 21 0,720 1,226 1,749 2,081 70 0,688 1,173 1,677 1,998 22 0,717 1,222 1,744 2,075 71 0,688 1,173 1,677 1,998 23 0,716 1,219 1,740 2,071 72 0,688 1,172 1,676 1,997 24 0,714 1,216 1,736 2,066 73 0,687 1,172 1,676 1,997 25 0,713 1,214 1,733 2,062 74 0,687 1,172 1,675 1,996 26 0,711 1,211 1,729 2,058 75 0,687 1,172 1,675 1,996 27 0,710 1,209 1,726 2,055 76 0,687 1,171 1,675 1,995 28 0,708 1,207 1,723 2,052 77 0,687 1,171 1,674 1,995 29 0,707 1,205 1,721 2,049 78 0,686 1,171 1,674 1,994 30 0,706 1,203 1,718 2,046 79 0,686 1,170 1,673 1,994 31 0,705 1,201 1,716 2,044 80 0,686 1,170 1,673 1,993 32 0,704 1,200 1,714 2,041 81 0,686 1,170 1,673 1,993 33 0,703 1,198 1,712 2,039 82 0,686 1,170 1,672 1,992 34 0,702 1,197 1,710 2,036 83 0,686 1,169 1,672 1,992 35 0,701 1,195 1,708 2,034 84 0,686 1,169 1,672 1,992 36 0,700 1,194 1,706 2,032 85 0,686 1,169 1,672 1,992 37 0,700 1,193 1,705 2,030 86 0,685 1,169 1,671 1,991 38 0,699 1,192 1,703 2,029 87 0,685 1,169 1,671 1,991 39 0,699 1,191 1,702 2,027 88 0,685 1,168 1,671 1,991 40 0,698 1,190 1,700 2,025 89 0,685 1,168 1,670 1,990 41 0,697 1,189 1,699 2,024 90 0,685 1,168 1,670 1,990 42 0,697 1,188 1,698 2,022 91 0,685 1,168 1,670 1,990 43 0,696 1,187 1,696 2,021 92 0,685 1,168 1,669 1,989 44 0,696 1,186 1,695 2,019 93 0,685 1,167 1,669 1,989 45 0,695 1,185 1,694 2,018 94 0,685 1,167 1,669 1,989 46 0,695 1,184 1,693 2,017 95 0,685 1,167 1,669 1,989 47 0,694 1,184 1,692 2,016 96 0,684 1,167 1,668 1,988 48 0,694 1,183 1,691 2,014 97 0,684 1,167 1,668 1,988 49 0,693 1,183 1,690 2,013 98 0,684 1,166 1,668 1,988 50 0,693 1,182 1,689 2,012 99 0,684 1,166 1,667 1,987 100 0,684 1,166 1,667 1,987


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 1,393 2,062 2,668 3,031 51 0,701 1,194 1,707 2,032 3 1,160 1,812 2,415 2,777 52 0,700 1,194 1,706 2,031 4 1,036 1,668 1,265 2,627 53 0,700 1,193 1,705 2,030 5 0,960 1,574 2,165 2,525 54 0,699 1,192 1,704 2,029 6 0,910 1,509 2,093 2,451 55 0,699 1,192 1,703 2,028 7 0,875 1,460 2,039 2,395 56 0,699 1,191 1,701 2,026 8 0,894 0,423 1,996 2,350 57 0,698 1,190 1,700 2,025 9 0,828 1,394 1,961 2,313 58 0,698 1,189 1,699 2,024 10 0,812 1,370 1,933 2,283 59 0,697 1,189 1,698 2,023 11 0,799 1,351 1,909 2,258 60 0,697 1,188 1,697 2,022 12 0,788 1,334 1,889 2,236 61 0,697 1,187 1,696 2,021 13 0,779 1,320 1,872 2,218 62 0,696 1,187 1,696 2,020 14 0,772 1,308 1,857 2,201 63 0,696 1,186 1,695 2,019 15 0,765 1,298 1,844 2,187 64 0,696 1,186 1,694 2,018 16 0,759 1,289 1,832 2,174 65 0,696 1,185 1,694 2,018 17 0,754 1,281 1,822 2,163 66 0,695 1,184 1,693 2,017 18 0,749 1,274 1,812 2,152 67 0,695 1,184 1,692 2,016 19 0,745 1,267 1,804 2,143 68 0,695 1,183 1,691 2,015 20 0,742 1,261 1,797 2,135 69 0,694 1,183 1,691 2,014 21 0,739 1,256 1,790 2,128 70 0,694 1,182 1,690 2,013 22 0,736 1,251 1,783 2,120 71 0,694 1,182 1,689 2,012 23 0,733 1,247 1,778 2,114 72 0,693 1,181 1,689 2,012 24 0,730 1,243 1,772 2,108 73 0,693 1,181 1,688 2,011 25 0,728 1,240 1,768 2,103 74 0,693 1,180 1,688 2,011 26 0,726 1,236 1,763 2,097 75 0,693 1,180 1,687 2,010 27 0,724 1,233 1,759 2,093 76 0,692 1,180 1,686 2,009 28 0,722 1,230 1,755 2,088 77 0,692 1,179 1,686 2,009 29 0,721 1,228 1,752 2,084 78 0,692 1,179 1,685 2,008 30 0,719 1,225 1,748 2,080 79 0,691 1,178 1,685 2,008 31 0,718 1,223 1,745 2,077 80 0,691 1,178 1,684 2,007 32 0,717 1,221 1,742 2,073 81 0,691 1,178 1,684 2,007 33 0,715 1,218 1,739 2,070 82 0,691 1,177 1,683 2,006 34 0,714 1,216 1,736 2,066 83 0,690 1,177 1,683 2,006 35 0,713 1,214 1,733 2,063 84 0,690 1,177 1,682 2,005 36 0,712 1,212 1,731 2,061 85 0,690 1,177 1,682 2,005 37 0,711 1,211 1,729 2,058 86 0,690 1,176 1,682 2,004 38 0,710 1,209 1,727 2,056 87 0,690 1,176 1,681 2,004 39 0,709 1,208 1,725 2,053 88 0,689 1,176 1,681 2,003 40 0,708 1,206 1,723 2,051 89 0,689 1,175 1,680 2,003 41 0,707 1,205 1,721 2,049 90 0,689 1,175 1,680 2,002 42 0,706 1,204 1,719 2,047 91 0,689 1,175 1,680 2,002 43 0,706 1,202 1,718 2,045 92 0,689 1,175 1,679 2,001 44 0,705 1,201 1,716 2,043 93 0,689 1,174 1,679 2,001 45 0,704 1,200 1,714 2,041 94 0,689 1,174 1,679 2,000 46 0,703 1,199 1,713 2,039 95 0,689 1,174 1,679 2,000 47 0,703 1,198 1,712 2,038 96 0,688 1,174 1,678 2,000 48 0,702 1,197 1,710 2,036 97 0,688 1,174 1,678 1,999 49 0,702 1,196 1,709 2,035 98 0,688 1,173 1,678 1,999 50 0,701 1,195 1,708 2,033 99 0,688 1,173 1,677 1,998 100 0,688 1,173 1,677 1,998


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 1,243 2,057 2,870 3,376 51 0,686 1,170 1,673 1,993 3 0,943 1,582 2,229 2,635 52 0,686 1,169 1,672 1,992 4 0,853 1,441 2,040 2,416 53 0,685 1,169 1,672 1,992 5 0,809 1,370 1,946 2,308 54 0,685 1,169 1,671 1,991 6 0,782 1,328 1,889 2,243 55 0,685 1,169 1,671 1,991 7 0,765 1,300 1,851 2,199 56 0,685 1,168 1,670 1,990 8 0,752 1,279 1,823 2,168 57 0,685 1,168 1,670 1,990 9 0,743 1,264 1,802 2,143 58 0,684 1,168 1,669 1,989 10 0,735 1,252 1,786 2,124 59 0,684 1,167 1,669 1,989 11 0,730 1,242 1,772 2,109 60 0,684 1,167 1,668 1,988 12 0,725 1,234 1,761 2,096 61 0,684 1,167 1,668 1,988 13 0,721 1,227 1,752 2,086 62 0,684 1,167 1,667 1,987 14 0,717 1,222 1,744 2,077 63 0,684 1,166 1,667 1,987 15 0,714 1,217 1,738 2,069 64 0,684 1,166 1,667 1,986 16 0,712 1,212 1,732 2,062 65 0,684 1,166 1,667 1,986 17 0,709 1,209 1,727 2,056 66 0,683 1,166 1,666 1,986 18 0,707 1,205 1,722 2,051 67 0,683 1,166 1,666 1,985 19 0,706 1,202 1,718 2,046 68 0,683 1,165 1,666 1,985 20 0,704 1,200 1,714 2,042 69 0,683 1,165 1,665 1,984 21 0,703 1,198 1,711 2,038 70 0,683 1,165 1,665 1,984 22 0,701 1,195 1,708 2,034 71 0,683 1,165 1,665 1,984 23 0,700 1,193 1,706 2,031 72 0,683 1,165 1,664 1,983 24 0,699 1,191 1,703 2,028 73 0,683 1,164 1,664 1,983 25 0,698 1,190 1,701 2,026 74 0,683 1,164 1,664 1,983 26 0,697 1,188 1,698 2,023 75 0,683 1,164 1,664 1,983 27 0,697 1,187 1,696 2,021 76 0,682 1,164 1,663 1,982 28 0,696 1,186 1,694 2,018 77 0,682 1,164 1,663 1,982 29 0,695 1,185 1,693 2,016 78 0,682 1,163 1,663 1,982 30 0,694 1,183 1,691 2,014 79 0,682 1,163 1,662 1,981 31 0,693 1,182 1,690 2,013 80 0,682 1,163 1,662 1,981 32 0,693 1,181 1,689 2,011 81 0,682 1,163 1,662 1,981 33 0,692 1,181 1,687 2,010 82 0,682 1,163 1,662 1,981 34 0,692 1,180 1,686 2,008 83 0,682 1,163 1,662 1,980 35 0,691 1,179 1,685 2,007 84 0,682 1,163 1,662 1,980 36 0,691 1,178 1,684 2,006 85 0,682 1,163 1,662 1,980 37 0,690 1,177 1,683 2,005 86 0,681 1,162 1,661 1,980 38 0,690 1,177 1,682 2,003 87 0,681 1,162 1,661 1,980 39 0,689 1,176 1,681 2,002 88 0,681 1,162 1,661 1,979 40 0,689 1,175 1,680 2,001 89 0,681 1,162 1,661 1,979 41 0,689 1,174 1,679 2,000 90 0,681 1,162 1,661 1,979 42 0,689 1,174 1,678 1,999 91 0,681 1,162 1,661 1,979 43 0,688 1,173 1,678 1,999 92 0,681 1,162 1,661 1,979 44 0,688 1,173 1,677 1,998 93 0,681 1,161 1,660 1,978 45 0,688 1,172 1,676 1,997 94 0,681 1,161 1,660 1,978 46 0,688 1,172 1,675 1,996 95 0,681 1,161 1,660 1,978 47 0,687 1,171 1,675 1,995 96 0,681 1,161 1,660 1,978 48 0,687 1,171 1,674 1,995 97 0,681 1,161 1,660 1,978 49 0,686 1,170 1,674 1,994 98 0,681 1,160 1,659 1,977 50 0,686 1,170 1,673 1,993 99 0,681 1,160 1,659 1,977 100 0,681 1,160 1,659 1,977

Table A13. k2 for two-sided statistical tolerance, standard deviation: unknown and confidence level γ = 50 %

NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 2,674 4,394 6,109 7,178 51 0,736 1,255 1,793 2,137 3 1,492 2,487 3,489 4,117 52 0,736 1,254 1,792 2,135 4 1,211 2,036 2,872 3,397 53 0,735 1,252 1,790 2,133 5 1,083 1,829 2,590 3,069 54 0,734 1,251 1,789 2,131 6 1,009 1,709 2,425 2,877 55 0,734 1,250 1,787 2,129 7 0,961 1,630 2,316 2,750 56 0,733 1,249 1,785 2,127 8 1,926 1,573 2,238 2,659 57 0,732 1,248 1,784 2,125 9 0,900 1,530 2,179 2,590 58 0,731 1,246 1,782 2,123 10 0,880 1,497 2,133 2,536 59 0,731 1,245 1,781 2,121 11 0,864 1,469 2,095 2,492 60 0,730 1,244 1,779 2,119 12 0,850 1,447 2,064 2,456 61 0,730 1,243 1,778 2,118 13 0,839 1,428 2,038 2,425 62 0,729 1,242 1,776 2,116 14 0,829 1,412 2,015 2,399 63 0,729 1,242 1,775 2,115 15 0,821 1,398 1,996 2,376 64 0,728 1,241 1,774 2,113 16 0,814 1,386 1,979 2,356 65 0,728 1,240 1,773 2,112 17 0,807 1,375 1,964 2,338 66 0,727 1,239 1,771 2,111 18 0,802 1,366 1,950 2,322 67 0,727 1,238 1,770 2,109 19 0,797 1,357 1,938 2,308 68 0,726 1,238 1,769 2,108 20 0,792 1,349 1,927 2,295 69 0,726 1,237 1,767 2,106 21 0,788 1,343 1,918 2,284 70 0,725 1,236 1,766 2,105 22 0,784 1,336 1,908 2,273 71 0,725 1,235 1,765 2,104 23 0,781 1,331 1,900 2,264 72 0,724 1,235 1,764 2,103 24 0,777 1,325 1,892 2,254 73 0,724 1,234 1,763 2,101 25 0,774 1,320 1,886 2,246 74 0,723 1,233 1,762 2,100 26 0,771 1,315 1,879 2,238 75 0,723 1,233 1,762 2,099 27 0,769 1,311 1,873 2,231 76 0,723 1,232 1,761 2,098 28 0,766 1,306 1,867 2,224 77 0,722 1,231 1,760 2,097 29 0,764 1,303 1,862 2,218 78 0,722 1,230 1,759 2,095 30 0,762 1,299 1,857 2,211 79 0,721 1,230 1,758 2,094 31 0,760 1,296 1,853 2,206 80 0,721 1,229 1,757 2,093 32 0,758 1,293 1,848 2,201 81 0,721 1,228 1,756 2,092 33 0,757 1,290 1,844 2,196 82 0,720 1,228 1,755 2,091 34 0,755 1,287 1,839 2,191 83 0,720 1,227 1,755 2,090 35 0,753 1,284 1,835 2,186 84 0,720 1,227 1,754 2,089 36 0,752 1,282 1,832 2,182 85 0,720 1,226 1,753 2,089 37 0,751 1,280 1,829 2,178 86 0,719 1,225 1,752 2,088 38 0,749 1,277 1,825 2,175 87 0,719 1,225 1,751 2,087 39 0,748 1,275 1,822 2,171 88 0,719 1,224 1,751 2,086 40 0,747 1,273 1,819 2,167 89 0,718 1,224 1,750 2,085 41 0,746 1,271 1,816 2,164 90 0,718 1,223 1,749 2,084 42 0,745 1,269 1,814 2,161 91 0,718 1,223 1,748 2,083 43 0,743 1,267 1,811 2,158 92 0,717 1,222 1,748 2,082 44 0,742 1,265 1,809 2,155 93 0,717 1,222 1,747 2,082 45 0,741 1,263 1,806 2,152 94 0,717 1,221 1,746 2,081 46 0,740 1,262 1,804 2,149 95 0,717 1,221 1,746 2,080 47 0,739 1,260 1,802 2,147 96 0,716 1,221 1,745 2,079 48 0,739 1,259 1,799 2,144 97 0,716 1,220 1,744 2,078 49 0,738 1,257 1,797 2,142 98 0,716 1,220 1,743 2,078 50 0,737 1,256 1,795 2,139 99 0,715 1,219 1,743 2,077 100 0,715 1,219 1,742 2,076


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 6,809 11,166 15,513 18,221 51 0,786 1,340 1,916 2,282 3 2,492 4,135 5,789 6,824 52 0,785 1,338 1,913 2,279 4 1,766 2,954 4,158 4,913 53 0,783 1,336 1,910 2,275 5 1,473 2,478 3,500 4,143 54 0,782 1,334 1,907 2,272 6 1,314 2,218 3,141 3,723 55 0,781 1,332 1,904 2,268 7 1,213 2,053 2,913 3,456 56 0,780 1,329 1,901 2,264 8 1,144 1,939 2,755 3,270 57 0,779 1,327 1,898 2,261 9 1,093 1,854 2,637 3,133 58 0,777 1,325 1,895 2,257 10 1,053 1,789 2,546 3,026 59 0,776 1,323 1,892 2,254 11 1,022 1,737 2,474 2,941 60 0,775 1,321 1,889 2,250 12 0,996 1,694 2,414 2,871 61 0,774 1,320 1,887 2,247 13 0,975 1,659 2,365 2,813 62 0,773 1,318 1,885 2,245 14 0,957 1,628 2,322 2,763 63 0,772 1,317 1,882 2,242 15 0,941 1,602 2,286 2,720 64 0,771 1,315 1,880 2,240 16 0,928 1,580 2,254 2,683 65 0,771 1,314 1,878 2,237 17 0,916 1,560 2,226 2,650 66 0,770 1,312 1,876 2,234 18 0,905 1,542 2,201 2,620 67 0,769 1,311 1,874 2,232 19 0,896 1,526 2,179 2,594 68 0,768 1,309 1,871 2,229 20 0,887 1,512 2,159 2,570 69 0,767 1,308 1,869 2,227 21 0,880 1,500 2,142 2,550 70 0,766 1,306 1,867 2,224 22 0,873 1,487 2,124 2,529 71 0,765 1,305 1,865 2,222 23 0,867 1,477 2,110 2,512 72 0,765 1,304 1,863 2,220 24 0,861 1,466 2,095 2,494 73 0,764 1,302 1,862 2,218 25 0,856 1,458 2,083 2,480 74 0,763 1,301 1,860 2,216 26 0,850 1,449 2,070 2,465 75 0,763 1,300 1,858 2,214 27 0,846 1,442 2,059 2,452 76 0,762 1,299 1,856 2,211 28 0,841 1,434 2,048 2,439 77 0,761 1,298 1,854 2,209 29 0,837 1,427 2,039 2,428 78 0,760 1,296 1,853 2,207 30 0,833 1,420 2,029 2,417 79 0,760 1,295 1,851 2,205 31 0,830 1,415 2,021 2,408 80 0,759 1,294 1,849 2,203 32 0,827 1,409 2,014 2,399 81 0,758 1,293 1,848 2,201 33 0,823 1,404 2,006 2,390 82 0,758 1,292 1,846 2,200 34 0,820 1,398 1,999 2,381 83 0,757 1,291 1,845 2,198 35 0,817 1,393 1,991 2,372 84 0,757 1,290 1,843 2,197 36 0,815 1,389 1,985 2,365 85 0,756 1,289 1,842 2,195 37 0,812 1,385 1,979 2,358 86 0,755 1,288 1,841 2,193 38 0,810 1,380 1,974 2,351 87 0,755 1,287 1,839 2,192 39 0,807 1,376 1,968 2,344 88 0,754 1,286 1,838 2,190 40 0,805 1,372 1,962 2,337 89 0,754 1,285 1,836 2,189 41 0,803 1,369 1,957 2,331 90 0,753 1,284 1,835 2,187 42 0,801 1,366 1,952 2,326 91 0,753 1,283 1,834 2,186 43 0,799 1,362 1,948 2,320 92 0,752 1,282 1,833 2,184 44 0,797 1,359 1,943 2,315 93 0,752 1,282 1,832 2,183 45 0,795 1,356 1,938 2,309 94 0,751 1,281 1,831 2,181 46 0,793 1,353 1,934 2,304 95 0,751 1,280 1,830 2,180 47 0,792 1,350 1,930 2,300 96 0,750 1,279 1,828 2,179 48 0,790 1,348 1,927 2,295 97 0,750 1,278 1,827 2,177 49 0,789 1,345 1,923 2,291 98 0,749 1,278 1,826 2,176 50 0,787 1,342 1,919 2,286 99 0,749 1,277 1,825 2,174 100 0,748 1,276 1,824 2,173


NB-CPD/SG10/03/006r2 of 74


0,50 0,75 0,90 0,95 0,50 0,75 0,90 0,95 2 13,652 22,383 31,093 36,520 51 0,819 1,396 1,996 2,377 3 3,585 5,938 8,306 9,789 52 0,818 1,393 1,992 2,373 4 2,288 3,819 5,369 6,342 53 0,816 1,391 1,988 2,368 5 1,812 3,041 4,291 5,077 54 0,814 1,388 1,984 2,364 6 1,566 2,639 3,733 4,423 55 0,813 1,385 1,980 2,359 7 1,416 2,392 3,390 4,020 56 0,811 1,382 1,976 2,354 8 1,314 2,224 3,157 3,746 57 0,809 1,379 1,972 2,350 9 1,240 2,101 2,987 3,546 58 0,807 1,377 1,968 2,345 10 1,183 2,008 2,857 3,394 59 0,806 1,374 1,964 2,341 11 1,139 1,935 2,754 3,273 60 0,804 1,371 1,960 2,336 12 1,103 1,875 2,671 3,175 61 0,803 1,369 1,957 2,333 13 1,074 1,825 2,602 3,094 62 0,802 1,367 1,954 2,329 14 1,049 1,784 2,543 3,025 63 0,800 1,365 1,951 2,326 15 1,027 1,748 2,493 2,965 64 0,799 1,363 1,948 2,322 16 1,009 1,717 2,449 2,914 65 0,798 1,361 1,946 2,319 17 0,992 1,689 2,411 2,869 66 0,797 1,359 1,943 2,315 18 0,978 1,665 2,377 2,829 67 0,796 1,357 1,940 2,312 19 0,965 1,644 2,347 2,793 68 0,794 1,355 1,937 2,308 20 0,954 1,625 2,319 2,761 69 0,793 1,353 1,934 2,305 21 0,944 1,608 2,296 2,733 70 0,792 1,351 1,931 2,301 22 0,934 1,591 2,272 2,705 71 0,791 1,349 1,929 2,298 23 0,926 1,577 2,253 2,682 72 0,790 1,348 1,927 2,296 24 0,918 1,563 2,233 2,659 73 0,789 1,346 1,924 2,293 25 0,911 1,552 2,217 2,639 74 0,788 1,345 1,922 2,290 26 0,904 1,540 2,200 2,619 75 0,788 1,343 1,920 2,288 27 0,898 1,530 2,186 2,602 76 0,787 1,341 1,918 2,285 28 0,892 1,519 2,171 2,585 77 0,786 1,340 1,916 2,282 29 0,887 1,511 2,159 2,570 78 0,785 1,338 1,913 2,279 30 0,881 1,502 2,146 2,555 79 0,784 1,337 1,911 2,277 31 0,877 1,495 2,136 2,543 80 0,783 1,335 1,909 2,274 32 0,873 1,488 2,126 2,531 81 0,782 1,334 1,907 2,272 33 0,868 1,480 2,115 2,519 82 0,782 1,332 1,905 2,270 34 0,864 1,473 2,105 2,507 83 0,781 1,331 1,903 2,267 35 0,860 1,466 2,095 2,495 84 0,780 1,330 1,901 2,265 36 0,857 1,460 2,087 2,486 85 0,780 1,329 1,900 2,263 37 0,854 1,455 2,079 2,477 86 0,779 1,327 1,898 2,261 38 0,850 1,449 2,072 2,467 87 0,778 1,326 1,896 2,259 39 0,847 1,444 2,064 2,458 88 0,777 1,325 1,894 2,256 40 0,844 1,438 2,056 2,449 89 0,777 1,323 1,892 2,254 41 0,841 1,434 2,050 2,442 90 0,776 1,322 1,890 2,252 42 0,839 1,430 2,044 2,434 91 0,775 1,321 1,889 2,250 43 0,836 1,425 2,037 2,427 92 0,775 1,320 1,887 2,248 44 0,834 1,421 2,031 2,419 93 0,774 1,319 1,886 2,247 45 0,831 1,417 2,025 2,412 94 0,773 1,318 1,884 2,245 46 0,829 1,413 2,020 2,406 95 0,773 1,317 1,883 2,243 47 0,827 1,410 2,015 2,400 96 0,772 1,316 1,881 2,241 48 0,825 1,406 2,010 2,394 97 0,771 1,315 1,880 2,239 49 0,823 1,403 2,005 2,388 98 0,770 1,314 1,878 2,238 50 0,821 1,399 2,000 2,382 99 0,770 1,313 1,877 2,236 100 0,769 1,312 1,875 2,234


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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.

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EXAMPLE 1ONE SIDED TOLERANCE INTERVAL-lower limit METHOD A : use at least 6 testresults per inspection lot

50 Start correction 7 Series of inspection lots 1 Declared Value 15confidence level 95 End correction 20 1,409 1,858

Inspection lot test 1 test 2 test 3 test 4 test 5 test 6 n Xm Ss k1,u kc k1,k σ Xest Xest XestEquation

OK?

1 18,1 17,9 18,3 19,4 17,7 19,2 6 18,43 0,70 0,823 17,852 16,1 18,4 18,8 17,6 15,7 17 6 17,27 1,24 0,823 16,253 17,7 19,1 17,9 18,1 15,7 18,1 6 17,77 1,12 0,823 16,844 21,4 20,8 19,6 18,5 18,6 18,1 6 19,50 1,35 0,823 18,395 19,5 20,8 19,7 21,1 19,4 18,6 6 19,85 0,94 0,823 19,086 19,9 19,3 20,1 18,8 21,1 20,6 6 19,97 0,84 0,823 19,287 17,2 19,2 17,0 18,2 19,4 17,1 6 18,02 1,09 0,823 0,823 0,672 1,409 17,12 16,868 20,2 17,8 16,3 19,9 21,9 22,6 6 19,78 2,39 0,811 18,649 21,0 14,7 19,0 18,9 18,8 19,3 6 18,62 2,09 0,800 17,4910 20,2 17,3 22,3 21,3 22,4 22,6 6 21,02 2,03 0,788 19,9111 20,0 22,8 21,2 19,9 20,1 22,6 6 21,10 1,33 0,777 20,0112 23,8 20,6 20,8 17,1 21 15,5 6 19,80 3,00 0,765 18,7213 21,1 21,0 20,1 19,1 23,4 21,5 6 21,03 1,44 0,753 19,9714 21,0 16,5 17,2 15,8 21,7 19,5 6 18,62 2,47 0,742 17,5715 18,8 20,5 20,3 20,1 21,7 17,1 6 19,75 1,59 0,730 18,7216 19,7 19,5 20,0 18,1 20,3 17,2 6 19,13 1,21 0,718 18,1217 20,4 19,6 20,1 20,8 14,5 18,2 6 18,93 2,35 0,707 17,9418 20,5 21,4 20,4 22,4 19,1 18,6 6 20,40 1,41 0,695 19,4219 19,8 20,5 18,1 19,2 19,1 18,8 6 19,25 0,83 0,684 18,2920 19,7 17,6 16,9 20,9 18,4 18 6 18,58 1,47 0,672 0,672 1,858 17,33 17,33 OK21 16,5 14,2 15,3 16,4 14,3 16,4 6 15,52 1,08 0,672 14,27 OK22 21,3 21,4 21,3 18,9 21,2 20,8 6 20,82 0,96 0,672 19,57 OK23 19,3 17,5 18,3 17,2 19,1 16,7 6 18,02 1,06 0,672 16,77 OK24 18,5 21,8 19,4 17,2 21,3 17,4 6 19,27 1,94 0,672 18,02 OK

fractile p

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Example 2


This example is similar to example 1. The only difference is that the declared compressive strength is a 5 % characteristic value.





The declared 5 % characteristic compressive strength is 10 N/mm²





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.

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OK?

1 18,1 17,9 18,3 19,4 17,7 19,2 6 18,43 0,70 3,708 15,832 16,1 18,4 18,8 17,6 15,7 17 6 17,27 1,24 3,708 12,693 17,7 19,1 17,9 18,1 15,7 18,1 6 17,77 1,12 3,708 13,614 21,4 20,8 19,6 18,5 18,6 18,1 6 19,50 1,35 3,708 14,505 19,5 20,8 19,7 21,1 19,4 18,6 6 19,85 0,94 3,708 16,386 19,9 19,3 20,1 18,8 21,1 20,6 6 19,97 0,84 3,708 16,867 17,2 19,2 17,0 18,2 19,4 17,1 6 18,02 1,09 3,708 3,708 2,317 1,409 13,99 12,798 20,2 17,8 16,3 19,9 21,9 22,6 6 19,78 2,39 3,601 14,719 21,0 14,7 19,0 18,9 18,8 19,3 6 18,62 2,09 3,494 13,6910 20,2 17,3 22,3 21,3 22,4 22,6 6 21,02 2,03 3,387 16,2411 20,0 22,8 21,2 19,9 20,1 22,6 6 21,10 1,33 3,280 16,4812 23,8 20,6 20,8 17,1 21 15,5 6 19,80 3,00 3,173 15,3313 21,1 21,0 20,1 19,1 23,4 21,5 6 21,03 1,44 3,066 16,7114 21,0 16,5 17,2 15,8 21,7 19,5 6 18,62 2,47 2,959 14,4515 18,8 20,5 20,3 20,1 21,7 17,1 6 19,75 1,59 2,852 15,7316 19,7 19,5 20,0 18,1 20,3 17,2 6 19,13 1,21 2,745 15,2617 20,4 19,6 20,1 20,8 14,5 18,2 6 18,93 2,35 2,638 15,2218 20,5 21,4 20,4 22,4 19,1 18,6 6 20,40 1,41 2,531 16,8319 19,8 20,5 18,1 19,2 19,1 18,8 6 19,25 0,83 2,424 15,8320 19,7 17,6 16,9 20,9 18,4 18 6 18,58 1,47 2,317 2,317 1,858 14,28 14,28 OK21 16,5 14,2 15,3 16,4 14,3 16,4 6 15,52 1,08 2,317 11,21 OK22 21,3 21,4 21,3 18,9 21,2 20,8 6 20,82 0,96 2,317 16,51 OK23 19,3 17,5 18,3 17,2 19,1 16,7 6 18,02 1,06 2,317 13,71 OK24 18,5 21,8 19,4 17,2 21,3 17,4 6 19,27 1,94 2,317 14,96 OK

fractile p

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Example 3


This example is similar to example 1. The only difference is, that the used confidence level is 75 %.











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OK?

1 18,1 17,9 18,3 19,4 17,7 19,2 6 18,43 0,70 0,297 18,222 16,1 18,4 18,8 17,6 15,7 17 6 17,27 1,24 0,297 16,903 17,7 19,1 17,9 18,1 15,7 18,1 6 17,77 1,12 0,297 17,434 21,4 20,8 19,6 18,5 18,6 18,1 6 19,50 1,35 0,297 19,105 19,5 20,8 19,7 21,1 19,4 18,6 6 19,85 0,94 0,297 19,576 19,9 19,3 20,1 18,8 21,1 20,6 6 19,97 0,84 0,297 19,727 17,2 19,2 17,0 18,2 19,4 17,1 6 18,02 1,09 0,297 0,297 0,276 1,409 17,69 17,608 20,2 17,8 16,3 19,9 21,9 22,6 6 19,78 2,39 0,295 19,379 21,0 14,7 19,0 18,9 18,8 19,3 6 18,62 2,09 0,294 18,2010 20,2 17,3 22,3 21,3 22,4 22,6 6 21,02 2,03 0,292 20,6011 20,0 22,8 21,2 19,9 20,1 22,6 6 21,10 1,33 0,291 20,6912 23,8 20,6 20,8 17,1 21 15,5 6 19,80 3,00 0,289 19,3913 21,1 21,0 20,1 19,1 23,4 21,5 6 21,03 1,44 0,287 20,6314 21,0 16,5 17,2 15,8 21,7 19,5 6 18,62 2,47 0,286 18,2115 18,8 20,5 20,3 20,1 21,7 17,1 6 19,75 1,59 0,284 19,3516 19,7 19,5 20,0 18,1 20,3 17,2 6 19,13 1,21 0,282 18,7417 20,4 19,6 20,1 20,8 14,5 18,2 6 18,93 2,35 0,281 18,5418 20,5 21,4 20,4 22,4 19,1 18,6 6 20,40 1,41 0,279 20,0119 19,8 20,5 18,1 19,2 19,1 18,8 6 19,25 0,83 0,278 18,8620 19,7 17,6 16,9 20,9 18,4 18 6 18,58 1,47 0,276 0,276 1,858 18,07 18,07 OK21 16,5 14,2 15,3 16,4 14,3 16,4 6 15,52 1,08 0,276 15,00 OK22 21,3 21,4 21,3 18,9 21,2 20,8 6 20,82 0,96 0,276 20,30 OK23 19,3 17,5 18,3 17,2 19,1 16,7 6 18,02 1,06 0,276 17,50 OK24 18,5 21,8 19,4 17,2 21,3 17,4 6 19,27 1,94 0,276 18,75 OK

fractile p

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Example 4

Example of statistical analysis of compressive strength using “Rolling“ inspection.






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.


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.


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.


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EXAMPLE4ONE SIDED TOLERANCE INTERVAL-lower limit METHOD B: use at least 3 testresults per inspection lot


Inspection lot test 1 test 2 test 3 n Xm Ss k1,u kc k1,k σ Xest Xest XestEquation

OK?

1 20,2 17,8 16,3 3 18,10 1,97 1,686 14,782 21,0 14,7 19,0 6 18,17 2,39 0,823 16,203 20,2 17,3 22,3 9 18,76 2,43 0,620 17,254 20,0 22,8 21,2 12 19,40 2,45 0,519 18,135 23,8 20,6 20,8 12 20,31 2,45 0,519 19,036 21,1 21,0 20,1 12 20,93 1,63 0,519 20,097 21,0 16,5 17,2 12 20,51 2,02 0,519 0,519 0,475 2,325 19,46 19,308 18,8 20,5 20,3 12 20,14 1,91 0,516 18,949 19,7 19,5 20,0 12 19,64 1,47 0,512 18,4510 20,4 19,6 20,1 12 19,47 1,35 0,509 18,2811 20,5 21,4 20,4 12 20,10 0,65 0,505 18,9312 19,8 20,5 18,1 12 20,00 0,79 0,502 18,8313 19,7 17,6 16,9 12 19,58 1,35 0,499 18,4214 18,7 20,0 19,7 12 19,44 1,34 0,495 18,2915 21,3 21,4 21,3 12 19,58 1,49 0,492 18,4416 19,3 17,5 18,3 12 19,31 1,55 0,489 18,1717 18,5 21,8 19,4 12 19,77 1,41 0,485 18,6418 19,9 21,9 22,6 12 20,27 1,66 0,482 19,1519 18,9 18,8 19,3 12 19,68 1,59 0,478 18,5720 21,3 22,4 22,6 12 20,62 1,62 0,475 0,475 1,762 19,78 19,78 OK21 19,9 20,1 22,6 12 20,86 1,53 0,475 20,02 OK22 17,1 21,0 15,5 12 19,96 2,21 0,475 19,12 OK23 19,1 23,4 21,5 12 20,54 2,37 0,475 19,70 OK24 15,8 21,7 19,5 12 19,77 2,54 0,475 18,93 OK25 20,1 21,7 17,1 12 19,46 2,57 0,475 18,62 OK26 18,1 20,3 17,2 12 19,63 2,27 0,475 18,79 OK27 20,8 14,5 18,2 12 18,75 2,32 0,475 17,91 OK28 22,4 19,1 18,6 12 19,01 2,21 0,475 18,17 OK29 19,2 19,1 18,8 12 18,86 1,94 0,475 18,02 OK30 20,9 18,4 18,0 12 19,00 1,93 0,475 18,16 OK

fractile p

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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.












0,63 σ ≤ ss ≤ 1,37 σ


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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 5ONE SIDED TOLERANCE INTERVAL-lower limit METHOD B: use at least 3 testresults per inspection lot



OK?


fractile p

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Example 6

Example of statistical analysis of compressive strength using a special type of “Rolling” inspection "Progressive Sampling”.






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.



0,63 σ ≤ ss ≤ 1,37 σ



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EXAMPLE 6ONE SIDED TOLERANCE INTERVAL-lower limit METHOD B: progressive sampling : use only 1 testresult per inspection lot



OK?

1 6,78 1 6,78 #DEEL/0!2 8,36 2 7,57 1,11 4,465 2,593 8,64 3 7,92 1,00 1,686 6,244 8,41 4 8,05 0,85 1,177 7,045 8,25 5 8,09 0,74 0,954 7,386 7,44 6 7,98 0,72 0,823 7,397 8,57 7 8,06 0,69 0,735 7,568 6,25 8 7,84 0,91 0,670 7,239 7,83 9 7,84 0,85 0,620 7,3110 7,40 10 7,79 0,81 0,580 7,3211 8,57 11 7,86 0,80 0,547 7,4212 8,27 12 7,90 0,77 0,519 7,4913 7,67 13 7,88 0,74 0,495 7,5114 8,13 14 7,90 0,72 0,474 7,5615 6,53 15 7,81 0,78 0,455 7,4516 7,85 15 7,88 0,72 0,455 7,5517 7,20 15 7,80 0,73 0,455 7,4718 7,95 15 7,75 0,69 0,455 7,4419 7,39 15 7,69 0,67 0,455 7,3820 5,96 15 7,53 0,79 0,455 7,1721 6,26 15 7,46 0,85 0,455 7,0722 5,60 15 7,26 0,92 0,455 6,8423 7,15 15 7,32 0,88 0,455 6,9224 6,01 15 7,20 0,93 0,455 6,7825 7,88 15 7,23 0,94 0,455 6,8026 6,01 15 7,06 0,91 0,455 6,6427 5,91 15 6,90 0,89 0,455 6,5028 7,05 15 6,86 0,87 0,455 6,4729 5,51 15 6,69 0,86 0,455 6,3030 5,87 15 6,64 0,88 0,455 0,455 0,425 1,009 6,24 6,1831 7,59 15 6,62 0,86 0,454 6,1732 8,64 15 6,72 1,00 0,453 6,2633 6,72 15 6,64 0,94 0,452 6,1834 8,11 15 6,69 1,00 0,451 6,2335 8,21 15 6,84 1,05 0,450 6,3836 8,48 15 6,98 1,11 0,449 6,5337 7,87 15 7,14 1,07 0,448 6,6838 7,36 15 7,15 1,07 0,447 6,7039 8,11 15 7,29 1,05 0,446 6,8440 6,50 15 7,20 1,05 0,445 6,7541 7,38 15 7,29 1,00 0,444 6,8442 7,78 15 7,41 0,93 0,443 6,9743 8,11 15 7,48 0,94 0,442 7,0444 8,70 15 7,70 0,81 0,441 7,2545 7,46 15 7,80 0,65 0,440 7,3646 7,76 15 7,81 0,64 0,439 7,3747 6,46 15 7,67 0,69 0,438 7,2348 7,78 15 7,74 0,64 0,437 7,3049 6,73 15 7,65 0,68 0,436 7,2150 7,47 15 7,60 0,66 0,435 7,1651 6,98 15 7,50 0,63 0,434 7,0652 6,35 15 7,40 0,68 0,433 6,9653 6,35 15 7,33 0,74 0,432 6,8954 5,53 15 7,16 0,83 0,431 6,7255 6,68 15 7,17 0,83 0,430 6,7356 5,86 15 7,07 0,89 0,429 6,6357 7,40 15 7,04 0,87 0,428 6,6158 6,42 15 6,93 0,83 0,427 6,5059 5,80 15 6,74 0,72 0,426 6,3160 6,57 15 6,68 0,69 0,425 0,425 0,935 6,28 6,28 OK61 5,71 15 6,54 0,67 0,425 6,14 OK62 5,78 15 6,49 0,69 0,425 6,10 OK63 10,46 15 6,67 1,21 0,425 6,28 OK64 12,26 15 7,04 1,88 0,425 6,64 OK65 8,90 15 7,14 1,94 0,425 6,74 OK

fractile p

NB-CPD/SG10/03/006r2 of 74

Example 7

Example of statistical analysis of compressive strength using “Rolling” inspection.













0,63 σ ≤ ss ≤ 1,37 σ


NB-CPD/SG10/03/006r2 of 74


EXAMPLE 7ONE SIDED TOLERANCE INTERVAL-lower limit METHOD B: use at least 3 testresults per inspection lot



OK?


fractile p

NB-CPD/SG10/03/006r2 of 74

Example 8

Example of statistical analysis of compressive strength using a special type of “Rolling” inspection: “Progressive Sampling”














0,63 σ ≤ ss ≤ 1,37 σ

NB-CPD/SG10/03/006r2 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.




OK?

1 6,78 1 6,78 #DEEL/0!2 8,36 2 7,57 1,11 26,260 ‐21,693 8,64 3 7,92 1,00 7,656 0,264 8,41 4 8,05 0,85 5,144 3,665 8,25 5 8,09 0,74 4,203 4,966 7,44 6 7,98 0,72 3,708 5,327 8,57 7 8,06 0,69 3,400 5,718 6,25 8 7,84 0,91 3,188 4,959 7,83 9 7,84 0,85 3,032 5,2710 7,40 10 7,79 0,81 2,911 5,4411 8,57 11 7,86 0,80 2,815 5,6012 8,27 12 7,90 0,77 2,737 5,7813 7,67 13 7,88 0,74 2,671 5,8914 8,13 14 7,90 0,72 2,615 6,0215 6,53 15 7,81 0,78 2,567 5,8116 7,85 15 7,88 0,72 2,567 6,0217 7,20 15 7,80 0,73 2,567 5,9318 7,95 15 7,75 0,69 2,567 5,9719 7,39 15 7,69 0,67 2,567 5,9520 5,96 15 7,53 0,79 2,567 5,5121 6,26 15 7,46 0,85 2,567 5,2622 5,60 15 7,26 0,92 2,567 4,9023 7,15 15 7,32 0,88 2,567 5,0724 6,01 15 7,20 0,93 2,567 4,8225 7,88 15 7,23 0,94 2,567 4,8126 6,01 15 7,06 0,91 2,567 4,7227 5,91 15 6,90 0,89 2,567 4,6128 7,05 15 6,86 0,87 2,567 4,6329 5,51 15 6,69 0,86 2,567 4,4830 5,87 15 6,64 0,88 2,567 2,567 2,070 1,009 4,37 4,0531 7,59 15 6,62 0,86 2,550 4,0532 8,64 15 6,72 1,00 2,534 4,1633 6,72 15 6,64 0,94 2,517 4,1034 4,91 15 6,47 1,01 2,501 3,9535 8,21 15 6,62 1,09 2,484 4,1236 8,48 15 6,77 1,19 2,468 4,2837 7,87 15 6,92 1,17 2,451 4,4538 7,36 15 6,94 1,18 2,434 4,4839 8,11 15 7,08 1,18 2,418 4,6440 6,50 15 6,98 1,17 2,401 4,5641 7,38 15 7,07 1,14 2,385 4,6742 7,78 15 7,20 1,11 2,368 4,8143 8,11 15 7,27 1,13 2,352 4,9044 8,70 15 7,48 1,08 2,335 5,1345 7,46 15 7,59 0,98 2,319 5,2546 7,76 15 7,60 0,98 2,302 5,2847 6,46 15 7,45 0,98 2,285 5,1548 7,78 15 7,52 0,96 2,269 5,2449 6,73 15 7,65 0,68 2,252 5,3750 7,47 15 7,60 0,66 2,236 5,3451 6,98 15 7,50 0,63 2,219 5,2652 6,35 15 7,40 0,68 2,203 5,1753 6,35 15 7,33 0,74 2,186 5,1254 5,53 15 7,16 0,83 2,169 4,9755 6,68 15 7,17 0,83 2,153 5,0056 5,86 15 7,07 0,89 2,136 4,9157 7,40 15 7,04 0,87 2,120 4,9058 6,42 15 6,93 0,83 2,103 4,8159 5,80 15 6,74 0,72 2,087 4,6360 6,57 15 6,68 0,69 2,070 2,070 0,974 4,66 4,66 OK61 5,71 15 6,54 0,67 2,070 4,52 OK62 5,78 15 6,49 0,69 2,070 4,48 OK63 10,46 15 6,67 1,21 2,070 4,66 OK64 11,26 15 6,97 1,69 2,070 4,96 OK65 8,90 15 7,07 1,76 2,070 5,05 OK66 11,50 15 7,37 2,10 2,070 5,35 OK67 5,89 15 7,34 2,12 2,070 5,32 OK68 10,50 15 7,62 2,25 2,070 5,60 NOK

fractile p

NB-CPD/SG10/03/006r2 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 %.













0,63 σ ≤ ss ≤ 1,37 σ

NB-CPD/SG10/03/006r2 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.


NB-CPD/SG10/03/006r2 of 74




OK?

1 6,78 1 6,78 #DEEL/0!2 8,36 2 7,57 1,11 5,122 1,863 8,64 3 7,92 1,00 3,152 4,774 8,41 4 8,05 0,85 2,681 5,765 8,25 5 8,09 0,74 2,464 6,256 7,44 6 7,98 0,72 2,336 6,317 8,57 7 8,06 0,69 2,251 6,518 6,25 8 7,84 0,91 2,189 5,869 7,83 9 7,84 0,85 2,142 6,0210 7,40 10 7,79 0,81 2,104 6,0911 8,57 11 7,86 0,80 2,074 6,2012 8,27 12 7,90 0,77 2,048 6,3113 7,67 13 7,88 0,74 2,026 6,3714 8,13 14 7,90 0,72 2,008 6,4515 6,53 15 7,81 0,78 1,991 6,2616 7,85 15 7,88 0,72 1,991 6,4417 7,20 15 7,80 0,73 1,991 6,3518 7,95 15 7,75 0,69 1,991 6,3719 7,39 15 7,69 0,67 1,991 6,3420 5,96 15 7,53 0,79 1,991 5,9621 6,26 15 7,46 0,85 1,991 5,7622 5,60 15 7,26 0,92 1,991 5,4323 7,15 15 7,32 0,88 1,991 5,5724 6,01 15 7,20 0,93 1,991 5,3525 7,88 15 7,23 0,94 1,991 5,3526 6,01 15 7,06 0,91 1,991 5,2427 5,91 15 6,90 0,89 1,991 5,1328 7,05 15 6,86 0,87 1,991 5,1329 5,51 15 6,69 0,86 1,991 4,9830 5,87 15 6,64 0,88 1,991 1,991 1,820 1,009 4,88 4,6331 7,59 15 6,62 0,86 1,985 4,6232 8,64 15 6,72 1,00 1,980 4,7233 6,72 15 6,64 0,94 1,974 4,6534 4,91 15 6,47 1,01 1,968 4,4935 8,21 15 6,62 1,09 1,963 4,6436 8,48 15 6,77 1,19 1,957 4,8037 7,87 15 6,92 1,17 1,951 4,9538 7,36 15 6,94 1,18 1,945 4,9739 8,11 15 7,08 1,18 1,940 5,1240 6,50 15 6,98 1,17 1,934 5,0341 7,38 15 7,07 1,14 1,928 5,1342 7,78 15 7,20 1,11 1,923 5,2643 8,11 15 7,27 1,13 1,917 5,3444 8,70 15 7,48 1,08 1,911 5,5545 7,46 15 7,59 0,98 1,906 5,6746 7,76 15 7,60 0,98 1,900 5,6847 6,46 15 7,45 0,98 1,894 5,5448 7,78 15 7,52 0,96 1,888 5,6249 6,73 15 7,65 0,68 1,883 5,7550 7,47 15 7,60 0,66 1,877 5,7051 6,98 15 7,50 0,63 1,871 5,6152 6,35 15 7,40 0,68 1,866 5,5153 6,35 15 7,33 0,74 1,860 5,4554 5,53 15 7,16 0,83 1,854 5,2955 6,68 15 7,17 0,83 1,849 5,3056 5,86 15 7,07 0,89 1,843 5,2157 7,40 15 7,04 0,87 1,837 5,1958 6,42 15 6,93 0,83 1,831 5,0859 5,80 15 6,74 0,72 1,826 4,8960 6,57 15 6,68 0,69 1,820 1,820 0,974 4,90 4,90 OK61 5,71 15 6,54 0,67 1,820 4,77 OK62 5,78 15 6,49 0,69 1,820 4,72 OK63 10,46 15 6,67 1,21 1,820 4,90 OK64 11,26 15 6,97 1,69 1,820 5,20 OK65 8,90 15 7,07 1,76 1,820 5,30 OK66 11,50 15 7,37 2,10 1,820 5,60 OK67 5,89 15 7,34 2,12 1,820 5,57 OK68 10,50 15 7,62 2,25 1,820 5,84 NOK

fractile p

NB-CPD/SG10/03/006r2 of 74

Example10

Example of statistical analysis of gross dry density using “Rolling” inspection.




One-sided tolerance interval, upper limit

The declared mean gross dry density is 700 kg/m3







0,63 σ ≤ ss ≤ 1,37 σ



NB-CPD/SG10/03/006r2 of 74

EXAMPLE 10ONE SIDED TOLERANCE INTERVAL-upper limit METHOD B: use at least 3 testresults per inspection lot

fractile p 50 Start correction 7 Series of inspection lots 4 Declared Value 700confidence level 90 End correction 20 20,92845 20,217


OK?

1 603,00 632,0 681,0 3 638,67 39,43 1,089 681,602 663,00 672,0 657,0 6 651,33 28,93 0,603 668,783 672,00 680,0 693,0 9 661,44 27,95 0,466 674,474 659,00 662,0 638,0 12 659,33 24,78 0,394 669,105 672,00 669,0 682,0 12 668,25 14,07 0,394 673,796 626,00 655,0 662,0 12 664,17 18,61 0,394 671,507 653,00 656,0 652,0 12 657,17 14,82 0,394 0,394 0,370 20,928 663,01 665,418 668,00 658,0 662,0 12 659,58 13,79 0,392 667,799 632,00 692,0 636,0 12 654,33 17,54 0,390 662,5010 618,00 635,0 668,0 12 652,50 19,92 0,388 660,6311 637,00 662,0 669,0 12 653,08 21,21 0,387 661,1712 676,00 696,0 685,0 12 658,83 26,31 0,385 666,8913 668,00 671,0 639,0 12 660,33 23,10 0,383 668,3514 628,00 638,0 652,0 12 660,08 21,30 0,381 668,0615 663,00 672,0 672,0 12 663,33 20,31 0,379 671,2716 628,00 680,0 657,0 12 655,67 18,35 0,377 663,5617 672,00 662,0 666,0 12 657,50 17,61 0,376 665,3618 668,00 669,0 692,0 12 666,75 15,18 0,374 674,5719 683,00 655,0 691,0 12 668,58 17,62 0,372 676,3720 641,00 656,0 682,0 12 669,75 15,34 0,370 0,370 20,217 677,23 677,23 OK21 698,00 658,0 682,0 12 672,92 17,77 0,370 680,40 OK22 665,00 692,0 688,0 12 674,25 18,35 0,370 681,73 OK23 671,00 635,0 618,0 12 665,50 24,81 0,370 672,98 OK24 662,00 662,0 619,0 12 662,50 26,78 0,370 669,98 OK25 638,00 696,0 668,0 12 659,50 26,75 0,370 666,98 OK26 673,00 671,0 667,0 12 656,67 23,83 0,370 664,15 OK27 655,00 682,0 662,0 12 662,92 19,73 0,370 670,40 OK28 672,00 662,0 672,0 12 668,17 14,10 0,370 675,65 OK29 672,00 652,0 672,0 12 667,67 8,48 0,370 675,15 OK30 662,00 662,0 669,0 12 666,17 8,47 0,370 673,65 OK31 628,00 516,0 676,0 12 651,25 44,55 0,370 658,73 OK32 667,00 668,0 698,0 12 653,50 46,23 0,370 660,98 OK33 619,00 669,0 626,0 12 646,67 47,14 0,370 654,15 OK34 652,00 685,0 652,0 12 646,33 47,68 0,370 653,81 OK

NB-CPD/SG10/03/006r2 of 74

Example 11

Example of statistical analysis of net dry density using “Rolling” inspection.





The declared mean net dry density is 1400 kg/m3







0,63 σ ≤ ss ≤ 1,37 σ



NB-CPD/SG10/03/006r2 of 74

The last column makes the link to EN 1745 Annex A to calculate the λ10,dry (50/90) value of the material.




OK?

λ 10,dry (50/90)‐material concrete

1 1206,00 1264,0 1362,0 3 1277,33 78,85 1,089 1363,20 0,3832 1326,00 1344,0 1314,0 6 1302,67 57,86 0,603 1337,56 0,3573 1344,00 1360,0 1386,0 9 1322,89 55,90 0,466 1348,94 0,3684 1318,00 1324,0 1276,0 12 1318,67 49,55 0,394 1338,19 0,3575 1344,00 1338,0 1364,0 12 1336,50 28,13 0,394 1347,58 0,3676 1252,00 1310,0 1324,0 12 1328,33 37,21 0,394 1342,99 0,3627 1306,00 1312,0 1304,0 12 1314,33 29,64 0,394 0,394 0,370 41,857 1326,01 1330,82 0,3458 1336,00 1316,0 1324,0 12 1319,17 27,59 0,392 1335,58 0,3559 1264,00 1384,0 1272,0 12 1308,67 35,08 0,390 1325,00 0,34410 1236,00 1270,0 1336,0 12 1305,00 39,83 0,388 1321,26 0,34011 1274,00 1324,0 1338,0 12 1306,17 42,42 0,387 1322,35 0,34112 1352,00 1392,0 1370,0 12 1317,67 52,61 0,385 1333,77 0,35313 1336,00 1342,0 1278,0 12 1320,67 46,20 0,383 1336,69 0,35614 1256,00 1276,0 1304,0 12 1320,17 42,59 0,381 1336,12 0,35515 1326,00 1344,0 1344,0 12 1326,67 40,63 0,379 1342,54 0,36216 1256,00 1360,0 1314,0 12 1311,33 36,69 0,377 1327,13 0,34617 1344,00 1324,0 1332,0 12 1315,00 35,22 0,376 1330,72 0,35018 1336,00 1338,0 1384,0 12 1333,50 30,37 0,374 1349,14 0,36919 1366,00 1310,0 1382,0 12 1337,17 35,25 0,372 1352,73 0,37220 1282,00 1312,0 1364,0 12 1339,50 30,68 0,370 0,370 40,433 1354,46 1354,46 OK 0,37421 1396,00 1316,0 1364,0 12 1345,83 35,55 0,370 1360,79 OK 0,38122 1330,00 1384,0 1376,0 12 1348,50 36,70 0,370 1363,46 OK 0,38423 1342,00 1270,0 1236,0 12 1331,00 49,63 0,370 1345,96 OK 0,36524 1324,00 1324,0 1238,0 12 1325,00 53,56 0,370 1339,96 OK 0,35925 1276,00 1392,0 1336,0 12 1319,00 53,51 0,370 1333,96 OK 0,35326 1346,00 1342,0 1334,0 12 1313,33 47,67 0,370 1328,29 OK 0,34727 1310,00 1364,0 1324,0 12 1325,83 39,46 0,370 1340,79 OK 0,36028 1344,00 1324,0 1344,0 12 1336,33 28,20 0,370 1351,29 OK 0,37129 1344,00 1304,0 1344,0 12 1335,33 16,96 0,370 1350,29 OK 0,37030 1324,00 1324,0 1338,0 12 1332,33 16,95 0,370 1347,29 OK 0,36731 1256,00 1032,0 1352,0 12 1302,50 89,10 0,370 1317,46 OK 0,33632 1334,00 1336,0 1396,0 12 1307,00 92,46 0,370 1321,96 OK 0,34033 1238,00 1338,0 1252,0 12 1293,33 94,29 0,370 1308,29 OK 0,326

NB-CPD/SG10/03/006r2 of 74

Example 12


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 declared mean net dry density is 1400 kg/m3



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.




0,63 σ ≤ ss ≤ 1,37 σ



NB-CPD/SG10/03/006r2 of 74





OK?

λ 10,dry (50/50)‐material clay

1 1206,00 1264,0 1362,0 3 1277,33 78,85 0,000 1277,33 0,2912 1326,00 1344,0 1314,0 6 1302,67 57,86 0,000 1302,67 0,3013 1344,00 1360,0 1386,0 9 1322,89 55,90 0,000 1322,89 0,3094 1318,00 1324,0 1276,0 12 1318,67 49,55 0,000 1318,67 0,3085 1344,00 1338,0 1364,0 15 1324,67 45,94 0,000 1324,67 0,3106 1252,00 1310,0 1324,0 15 1328,27 33,47 0,000 1328,27 0,3127 1306,00 1312,0 1304,0 15 1324,13 34,15 0,000 0,000 0,000 41,857 1324,13 1324,13 0,3108 1336,00 1316,0 1324,0 15 1316,53 26,93 0,000 1316,53 0,3079 1264,00 1384,0 1272,0 15 1316,67 35,61 0,000 1316,67 0,30710 1236,00 1270,0 1336,0 15 1303,07 38,35 0,000 1303,07 0,30111 1274,00 1324,0 1338,0 15 1306,40 37,64 0,000 1306,40 0,30312 1352,00 1392,0 1370,0 15 1319,20 46,90 0,000 1319,20 0,30813 1336,00 1342,0 1278,0 15 1317,87 48,51 0,000 1317,87 0,30714 1256,00 1276,0 1304,0 15 1312,27 45,41 0,000 1312,27 0,30515 1326,00 1344,0 1344,0 15 1323,73 38,67 0,000 1323,73 0,31016 1256,00 1360,0 1314,0 15 1323,33 41,62 0,000 1323,33 0,31017 1344,00 1324,0 1332,0 15 1315,73 33,99 0,000 1315,73 0,30618 1336,00 1338,0 1384,0 15 1322,53 36,37 0,000 1322,53 0,30919 1366,00 1310,0 1382,0 15 1337,33 31,49 0,000 1337,33 0,31520 1282,00 1312,0 1364,0 15 1333,60 35,73 0,000 0,000 40,433 1333,60 1333,60 OK 0,31421 1396,00 1316,0 1364,0 15 1343,33 32,16 0,000 1343,33 OK 0,31822 1330,00 1384,0 1376,0 15 1349,33 34,16 0,000 1349,33 OK 0,32023 1342,00 1270,0 1236,0 15 1335,33 47,11 0,000 1335,33 OK 0,31424 1324,00 1324,0 1238,0 15 1323,87 50,05 0,000 1323,87 OK 0,31025 1276,00 1392,0 1336,0 15 1326,93 52,45 0,000 1326,93 OK 0,31126 1346,00 1342,0 1334,0 15 1323,33 48,32 0,000 1323,33 OK 0,31027 1310,00 1364,0 1324,0 15 1317,20 44,29 0,000 1317,20 OK 0,30728 1344,00 1324,0 1344,0 15 1328,13 35,57 0,000 1328,13 OK 0,31229 1344,00 1304,0 1344,0 15 1335,20 26,58 0,000 1335,20 OK 0,31430 1324,00 1324,0 1338,0 15 1334,00 15,58 0,000 1334,00 OK 0,31431 1256,00 1032,0 1352,0 15 1308,53 80,66 0,000 1308,53 OK 0,30432 1334,00 1336,0 1396,0 15 1313,07 83,02 0,000 1313,07 OK 0,30533 1238,00 1338,0 1252,0 15 1300,80 85,44 0,000 1300,80 OK 0,30034 1304,00 1370,0 1304,0 15 1299,87 85,88 0,000 1299,87 OK 0,300

NB-CPD/SG10/03/006r2 of 74

Example 13


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 declared 90 % characteristic net dry density is 1400 kg/m3







0,63 σ ≤ ss ≤ 1,37 σ


NB-CPD/SG10/03/006r2 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 non-conforming inspection lot has to be treated separately as described in the text.





OK?

λ 10,dry (90/90)‐material

1 1206,00 1264,0 1362,0 3 1277,33 78,85 4,259 1613,16 0,4272 1326,00 1344,0 1314,0 6 1302,67 57,86 2,494 1446,98 0,3603 1344,00 1360,0 1386,0 9 1322,89 55,90 2,133 1442,13 0,3584 1318,00 1324,0 1276,0 12 1318,67 49,55 1,967 1416,14 0,3475 1344,00 1338,0 1364,0 15 1324,67 45,94 1,867 1410,43 0,3456 1252,00 1310,0 1324,0 15 1328,27 33,47 1,867 1390,76 0,3377 1306,00 1312,0 1304,0 15 1324,13 34,15 1,867 1,867 1,613 41,857 1387,88 1402,28 0,3368 1336,00 1316,0 1324,0 15 1316,53 26,93 1,847 1393,86 0,3389 1264,00 1384,0 1272,0 15 1316,67 35,61 1,828 1393,18 0,33810 1236,00 1270,0 1336,0 15 1303,07 38,35 1,808 1378,76 0,33211 1274,00 1324,0 1338,0 15 1306,40 37,64 1,789 1381,28 0,33312 1352,00 1392,0 1370,0 15 1319,20 46,90 1,769 1393,26 0,33813 1336,00 1342,0 1278,0 15 1317,87 48,51 1,750 1391,11 0,33714 1256,00 1276,0 1304,0 15 1312,27 45,41 1,730 1384,69 0,33515 1326,00 1344,0 1344,0 15 1323,73 38,67 1,711 1395,34 0,33916 1256,00 1360,0 1314,0 15 1323,33 41,62 1,691 1394,12 0,33817 1344,00 1324,0 1332,0 15 1315,73 33,99 1,672 1385,70 0,33518 1336,00 1338,0 1384,0 15 1322,53 36,37 1,652 1391,68 0,33719 1366,00 1310,0 1382,0 15 1337,33 31,49 1,633 1405,67 0,34320 1282,00 1312,0 1364,0 15 1333,60 35,73 1,613 1,613 40,433 1398,82 1398,82 OK 0,34021 1396,00 1316,0 1364,0 15 1343,33 32,16 1,613 1408,55 OK 0,34422 1330,00 1384,0 1376,0 15 1349,33 34,16 1,613 1414,55 OK 0,34723 1342,00 1270,0 1236,0 15 1335,33 47,11 1,613 1400,55 OK 0,34124 1324,00 1324,0 1238,0 15 1323,87 50,05 1,613 1389,09 OK 0,33625 1276,00 1392,0 1336,0 15 1326,93 52,45 1,613 1392,15 OK 0,33826 1346,00 1342,0 1334,0 15 1323,33 48,32 1,613 1388,55 OK 0,33627 1310,00 1364,0 1324,0 15 1317,20 44,29 1,613 1382,42 OK 0,33428 1344,00 1324,0 1344,0 15 1328,13 35,57 1,613 1393,35 OK 0,33829 1344,00 1304,0 1344,0 15 1335,20 26,58 1,613 1400,42 OK 0,34130 1324,00 1324,0 1338,0 15 1334,00 15,58 1,613 1399,22 OK 0,34031 1256,00 1032,0 1352,0 15 1308,53 80,66 1,613 1373,75 OK 0,33032 1334,00 1336,0 1396,0 15 1313,07 83,02 1,613 1378,29 OK 0,33233 1238,00 1338,0 1252,0 15 1300,80 85,44 1,613 1366,02 OK 0,327

NB-CPD/SG10/03/006r2 of 74

Example 14

Example of TWO-SIDED statistical analysis of dimension using “Rolling” inspection.




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







0,63 σ ≤ ss ≤ 1,37 σ


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 non-conforming inspection lots give a warning to the manufacturer to take some corrective actions.

NB-CPD/SG10/03/006r2 of 74

EXAMPLE 14 2,166

TWO SIDED TOLERANCE INTERVAL-lower limitfractile p 50 Start correction 7 Series of inspection lots 4 Declared Value lower limit 242confidence level 75 End correction 20 Declared Value upper limit 247 1,677

Inspection lot test 1 test 2 test 3 n Xm Ss k1,u kc k1,k σXest lower

limit

Xest upper limit

Xest lower limit

Xest upper limit

Xest lower limit

Xest upper limit

Equation OK?

1 244 245 246 3 245,00 1,00 1,492 243,508 246,4922 245 247 246 6 245,50 1,05 1,009 244,442 246,55823 246 247 247 9 245,89 1,05 0,9 244,940 246,83764 247 246 246 12 246,00 0,95 0,85 245,190 246,81045 242 242 241 12 245,17 2,21 0,85 243,289 247,04416 241 242 242 12 244,08 2,57 0,85 241,895 246,27187 245 246 247 12 243,92 2,43 0,85 0,850 0,713 2,166 241,852 245,9816 242,076 245,75768 247 246 246 12 243,92 2,43 0,839 242,099 245,73479 243 244 243 12 244,33 2,10 0,829 242,538 246,128610 244 244 243 12 244,83 1,53 0,818 243,061 246,605811 246 246 246 12 244,83 1,47 0,808 243,084 246,582912 245 245 246 12 244,58 1,24 0,797 242,857 246,310113 245 245 244 12 244,92 1,00 0,787 243,213 246,620614 246 245 245 12 245,33 0,65 0,776 243,652 247,014515 245 244 243 12 244,83 0,83 0,766 243,175 246,491616 244 245 244 12 244,58 0,79 0,755 242,948 246,218817 248 248 247 12 245,33 1,61 0,745 243,721 246,94618 247 246 248 12 245,75 1,82 0,734 244,160 247,339819 245 245 245 12 246,00 1,54 0,724 244,433 247,56720 244 245 245 12 246,08 1,44 0,713 0,713 1,677 244,888 247,2791 244,888 247,279 OK21 246 245 248 12 245,75 1,29 0,713 244,554 246,946 OK22 245 246 248 12 245,58 1,24 0,713 244,388 246,779 OK23 245 246 247 12 245,83 1,27 0,713 244,638 247,029 OK24 247 246 247 12 246,33 1,07 0,713 245,138 247,529 OK25 245 244 244 12 245,83 1,27 0,713 244,638 247,029 OK26 243 244 245 12 245,25 1,36 0,713 244,054 246,446 OK

NB-CPD/SG10/03/006r2 of 74

UE-Nr_8

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