CEN-TC-125-N 1037-2012

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CEN/TC 125 N 1037

CEN/TC 125CEN/TC 125 - Masonry

Email of secretary: [email protected] Secretariat: BSI (United Kingdom)

N1037 CEN TR Evaluation of conformity for masonry units

Document type: Other meeting document

Date of document: 2012-03-01

Expected action: MEET

Background: Document N 1037 - to consider the proposal to activate the preliminary work item and to decide onthe progress of the document.

Committee URL: http://cen.iso.org/livelink/livelink/open/centc125

http://cen.iso.org/livelink/livelink/open/centc125mailto:[email protected]

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Document type: Technical Report

Document subtype:Document stage: Working DocumentDocument language: E

D:\ISO\isomacroserver-prod\temp\DOCX2PDFISOTC\DOCX2PDFISOTC.lliadmin@SRVWEB32_213\3589023_1.doc STD Version 2.4a

CEN/TC 125 N 1037 Date: 2012-02

WI 00125157 – Pr CEN TR XXX: 2012

CEN/TC 125

Secretariat: BSI

Specifications for masonry units — Evaluation of conformity for masonry

units according to EN 771 series

Einführendes Element — Haupt-Element — Ergänzendes Element

Élément introductif — Élément central — Élément complémentaire

ICS:

Descriptors:

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prCEN/TR XXX:2012 (E)

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Contents Page

Foreword ..............................................................................................................................................................3

1 Scope ......................................................................................................................................................4

2 Symbols ..................................................................................................................................................4

3 Reference list .........................................................................................................................................5

4 General ....................................................................................................................................................5

5 Factory production control ...................................................................................................................6 5.1 General ....................................................................................................................................................6 5.2 Testing and measuring equipment ......................................................................................................7

5.3 Production equipment ...........................................................................................................................7 5.4 Raw materials .........................................................................................................................................7 5.5 Production process ...............................................................................................................................8 5.6 Finished product testing .......................................................................................................................9 5.6.1 Inspection lot ....................................................................................................................................... 10 5.6.2 Spot sampling and sample sizes ...................................................................................................... 10 5.6.3 Production types................................................................................................................................. 11 5.6.4 Method A: Batch control .................................................................................................................... 12 5.6.5 Method B:‖Rolling‖ inspection ......................................................................................................... 13 5.6.6 Evaluation of test results ................................................................................................................... 15 5.6.7 How to come from unknown to known standard deviation? ......................................................... 19 5.6.8 Conformity ........................................................................................................................................... 19 5.6.9 A simple and conservative approach ............................................................................................... 25

5.6.10 Non-conforming products ................................................................................................................. 25 5.6.11 Guidance .............................................................................................................................................. 26 5.6.12 Records ................................................................................................................................................ 28

6 Initial type tests ................................................................................................................................... 28

Annex A (informative) Tables for acceptance coefficient kn depending on the used fractile p andconfidence level γ (taken from ISO 16269-6 (2005)) ............................................................................ 30

Annex B (informative) Examples of statistical evaluation ................................................................................ 47

Bibliography ..................................................................................................................................................... 75

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Foreword

This document (prCEN/TR XXX:2012) has been prepared by Technical Committee CEN/TC 125 “Masonry”,the secretariat of which is held by BSI.

This document is a working document.

The initial draft of this document was prepared by the joint working group CEN/TC 125/TG 5 and the SectorGroup 10 of Notified Bodies for the Construction Products Directive. The CEN/TR gives a tool available formanufacturers and Notified Bodies.

It is laid down in the hENs that the manufacturer shall demonstrate compliance for his product with therequirements 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 theevaluation of different properties at the different stages of the FPC with the aim to minimise testing costs forthe manufacturer and to ensure that the requirements are fulfilled. Detailed examples are given in the

Annexes.

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1 Scope

This document contains guidance for manufacturers and Notified Bodies (NBs) involved in the evaluation ofconformity of FPC of masonry units according to EN 771-1 to EN 771-6.In the masonry unit standards and in national legislation some properties are requested to be declared basedon a certain fractile and confidence level. To demonstrate compliance with that a statistical tool may be used.The purpose of this guidance document is to exemplify how a statistical tool can be used in practice. Thisdocument should not contradict nor extend the scope of the work and role of a NB, nor impose additionalburdens on the manufacturer, beyond those laid down in the CPD and EN 771-1 to EN 771-6.

2 Symbols

k n is the acceptance coefficient

k 1 is the acceptance coefficient one-sided tolerance interval

k 2 is the acceptance coefficient two-sided tolerance interval

k c is the corrected acceptance coefficient

k k is the acceptance coefficient for known standard deviation

k u is the acceptance coefficient for unknown standard deviation

n is the number of test samples within the spot sample

x m is the mean test result

x i is the test result for test sample i

i is the number of the individual test sample

x est 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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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 (Denseand lightweight aggregates)

EN 771-4:2011 Specification for masonry units – Part 4: Autoclaved aerated concrete masonryunits

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 reinforcedand unreinforced masonry structures

EN 1996-1-2:2005 Eurocode 6: Design of masonry structures - Part 1-2: General rules - Structural firedesign

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 methodsfor unreinforced masonry structures

4 General

It is specified in the EN 771 series that the manufacturer shall demonstrate compliance for his product with therequirements of the relevant European Standard and with the declared values for the product properties bycarrying 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 thedefinition of Category I units which is”Units with a declared compressive strength with a probability of failure toreach 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 itis 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 fulfilledby 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, andfor properties tested by a non-destructive test it will be too expensive to operate with a 100 % confidencelevel. 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 thatdeclared 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 acontract to be fulfilled, the manufacturer is free to fix the confidence level he will operate with, and the higherthe chosen level is, the lower is the risk that the manufacturer is running that the delivered products do notfulfil the declared values. The risk the manufacturer is running is fixed by a combination of the actual variationin test results over time, the frequencies of checking and testing, the way the FPC system is developed andhow 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 buildingsite. 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 itis impractical to test each consignment the manufacturer has to plan the FPC system in such a way that theeffect of the variations of product characteristics during the production is taken into account when declaringthe characteristics for the consignment. In some production processes products are naturally separated intobatches and a consignment is quite often only a part of a batch. If a production is based on a continuous flowa 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 proceduresare:

mainly related to the process only (full process control and consequently only a small amount offinished 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 theproperty to be assessed. If the test for the property is low cost, e.g. test of dimensions, and if the property isless important in relation to the end use then it may be the right solution to use finished product testing. But ifthe testing of the property is expensive, e.g. frost resistance tests, then the solution may be to base theassessment on process control using proxy tests.

In some companies responsibility for the production is placed only on one person, and if this person is notavailable, the responsibility for taking decisions is unclear. This can result in unnecessary and costly stops of

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the production or the manufacture of non-conforming products. It should be in the interest of the manufacturerto avoid this by establishing the responsibility, authority and interrelation 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 bestquality of the products and therefore to have an interest in clear procedures. The best way of achieving this isto have them in a written form. Procedures for what to do, when control and check parameters during theproduction are not obtained or fulfilled, are of the same or may be of greater importance. Therefore the needfor having them in a written form is crucial.

The manufacturer may define product groups. A product group consists of products from one manufacturerhaving common values for one or more characteristics. That means that the products belonging to a productgroup may differ according to the characteristics in question. If a product group is defined, then the FPCsystem shall ensure that all types of units within a group are controlled and over time also in the finishedproduct 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 accordancewith the test standard. If it is not defined there, then a „rule of thumb‟ can be 1/5 – 1/10 times of the accuracyof the value to be declared. Testing or measuring data are not helpful in itself, unless you know that the dataare accurate. It should be in the interest of the manufacturer to know that testing and measuring data arereliable. To obtain that, all relevant weighing, measuring and testing equipment that have an influence on thedeclared 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 thelength of a unit is 300 mm, then the measuring area for the length is approximately 290 – 310 mm and can beverified 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. Duringproduction wear and tear can also happen. For that reason, it is recommended that all parts of productionequipment 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 thisinfluence as much as possible the manufacturer has to define his own acceptance criteria of raw materialsand the procedures with which to operate to ensure that these are met. This is independent of the way theconstituents are received in the factory – bought from a supplier or delivered from the manufacturer‟s ownsources. If the constituents or some of them are bought from a supplier, the manufacturer is advised to besure that the control system for the constituents carried out by the supplier is sufficient. Normally it isacceptable if the control system of the supplier is supervised by a third party, and then the manufacturer hasonly to check the delivery notes and make a visual inspection to ensure that the delivery is in line with theorder. If the raw materials are delivered from the manufacturer‟s own sources, for example the manufacturer‟sown clay pit, then a procedure to check, if the grain size distribution of the clay is kept constant, could be tomeasure regularly the amount of clay in a test sample passing a 90 μm test sieve. An example of control data

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is given in Figure 1 along with the acceptance criteria fixed by the manufacturer, the upper limit (UL) and lowerlimit (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 theproducts and variation in the properties. It should be in the interest of the manufacturer to obtain the bestquality 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 eachparameter requirements to be fulfilled or limits (upper and lower limits, UL and LL) between which theparameter is allowed to vary. These limits and the frequency of measuring or checking the parameter have tobe based on the manufacturer‟s experience and on the importance and the variation of the parameter. Themanufacturer should also specify what should be done, when control and check parameters during theproduction 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 wearand tear of the mould in which the units are produced. In the following part of the production process the unitswill shrink 0,1 mm, and the intension is to declare a length of 228,5 mm and a tolerance of ± 0,5 mm. Bothaspects need to be taken into account when fixing the control limits. The reason for the dramatic drop is arenewal of the mould. The renewal of the mould should have taken place at spot sample 11 as it was leadingto a situation where all units in the inspection lot produced between spot sample 11 and 12 did not conform to

the fixed upper limit.

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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 passingthe control limit of the narrow range, it can be looked upon as a warning, and a small correction of the processmay be made, but when the parameter passes the control limit for the wider range, a more radical correctionof 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 beestablished 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 inspectionlot (see clause 5.6.1). If an evaluated test result is not conforming, the whole production since the last test hasto be looked upon as non-conforming. For that reason it can be recommended, that for properties where thereference test is time consuming and may be costly, alternative tests or proxy tests that are less timeconsuming and costly are used. By doing so the time span between the tests can be shortened and theamount of products covered by a non-conforming test result will be less and thereby reduce themanufacturer‟s risk.

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.

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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 canbelong to the same product family.

That means that an inspection lot for the characteristic in question can only consist of products belonging tothe 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 arepresentative 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 away that the spot sample is representative for the inspection lot.

Figure 3 — An example of representative sampling

In the European Product Standard sampling procedures for stacks and banded packs are given. It is alsopossible 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

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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 isaccepted or not, see clause 5.6.8. In this respect the test results can be dealt with separately or treatedtogether with the previous results. It depends on the type of production (batch production or seriesproduction).

5.6.3 Production types

A production, which is naturally separated into batches, is named a batch production. In the case of thebatch production the properties of the units may change batch by batch. A batch is normally looked upon as aseparate inspection lot. If the process control minimises the changes from one batch to another, an inspectionlot can cover more than one batch.

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A production, which is based on a continuous flow, is named a series production. In the case of seriesproduction the properties of the units are the same within a series. A series production contains normally morethan 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, whichmeans, that each batch is controlled separately.

In clause 5.6.6 when dealing with the evaluation of test results the acceptance coefficient k n is given in Tables

1 and 2. These tables show that there is a great difference in using k n for 3 or for 6 test results and for thatreason 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 thefollowing 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 2

nd 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 togetherwith the ones from the 1

st and the 2

nd inspection lot and therefore the spot sample size will be 9. For the 4

th

inspection lot 3 new samples are taken and tested and evaluated together with the ones from the 1 st, 2nd and3

rd inspection lot and therefore the spot sample size will be 12. For the 5

th inspection lot 3 new samples are

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taken and tested and evaluated 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. Therolling system is illustrated in the following Figure. In clause 5.6.6 when dealing with the evaluation of testresults the acceptance coefficient k n is given in Tables 1 and 2. These tables show that there is a greatdifference 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 inspectionlot 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 5

th 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 thelast 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 15inspection 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 ofstatistical techniques, by attributes or by variables to verify the product characteristics and to determine if theproduction conforms to the compliance criteria and the products conform to the declared values. One methodof satisfying this conformity criterion is to use the approach given in ISO 12491. This approach is shown indetail in this section.

When using the test results of a spot sample with a limited number of samples to estimate the characteristicsof 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 dealtwith in the evaluation by taking into account the standard deviation s of the test results of the spot sample.The second uncertainty is dealt with by using an acceptance coefficient k n . The acceptance coefficient k n canbe regarded as a factor minimising the statistic uncertainties from spot sampling. k n is dependent on several

factors:

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The number of samples in the inspection lot n

The confidence level γ

The fractile p *)

The standard deviation is unknown. The symbol used is k u

The standard deviation is known. The symbol used is k k

One-sided limit evaluation. The symbol used is k 1 Two-sided limit evaluation. The symbol used is k 2

*) Be aware that a 5 % characteristic value corresponds with a fractile p = 95 and a 95 % characteristic valuecorresponds also with a fractile p = 95. 50 % characteristic value corresponds with a fractile p = 50.

When evaluating the test results from a spot sample, then use the following procedure:

Calculate the mean value of the test results using the following equation:

xm

1

n x

i

i1

n

(1)

where

x m is the mean test result x i 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 s s for the test results of the spot sample using the following equation:

s

xi x

m 2

i1

n

n 1 (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 x i is the test result for test sample i x m is the mean test result

If the standard deviation is unknown and if the test results have to be compared with a lower limit value thencalculate the estimated test result x es t using the following equation:

x est = x m – k 1,u × ss (3)

If the standard deviation is unknown and if the test results have to be compared with an upper limit value thencalculate the estimated test result x es t using the following equation:

x est = x m + k 1,u × ss (4)

If the standard deviation is unknown and if the test results have to be compared with a two-sided limit valuethen calculate the estimated test result x es t using the following equation:

x est = x m ± k 2,u × ss (5)

If the standard deviation σ is known and if the test results have to be compared with a lower limit value thencalculate the estimated test result x es t using the following equation:

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x est = x m – k 1,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 x es t using the following equation:

x est = x m + k 1,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 x es t using the following equation:

x est = x m ± k 2,k × σ (8)

where

x est is the estimated test result of the spot sample x m is the mean test result k 1,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 k 2,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 k 1,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 k 2,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

Table 1 — kn for 50 % characteristic value (50 % fractile) and 95 % confidence level

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 2 — kn for 5 % characteristic value (95 % fractile) and 95 % confidence level

Standarddeviation

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

More tables are given in Annex A.

The method of using the acceptance coefficient for known standard deviation k k is only valid when the

standard deviation ss of the spot sample corresponds to the following equation:

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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 k u. This means that the inspection lots have to betreated separately.

The effect of the size of the spot sample and the standard deviation of the test results of the sample on theacceptance coefficient k n 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 theresults of the compressive strength are given on each unit. The mean value and the standard deviation arecalculated. From the table for “kn for 50 % fractile and 95 % confidence level” the acceptance coefficient k n forunknown standard deviation and n = 6 are taken and the estimated compressive strength for the inspection lotis 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 estimatedcompressive strength. A higher standard deviation is demonstrating less control compared to the firstexample. When keeping the confidence level the estimated compressive strength for the inspection lot needsto be lower.

In the third example the two previous spot samples are looked upon as one spot sample consisting of 12units. The mean value and the standard deviation are calculated. From Table 1 the acceptance coefficient k n for unknown standard deviation and n = 12 are taken and the estimated compressive strength for the lot iscalculated. By enlarging the number of units to be tested of the spot sample the estimated value is morecertain leading to a higher estimated compressive strength of the inspection lot compared to the secondexample, where the mean value and the standard deviation are about the same.

Table 3 — Example showing the effect of spot sample size and deviation

Spotsample

size

Meanvalue

In MPa

Standard.deviation

in MPa

Coef-ficient95 %,

unknown k n

Estimatedcomp.

strengthin MPa

6 20 1,3 0,82 19

6 20 3,2 0,82 17

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12 20 3,0 0,52 18

As you see, when reducing the variation in the test results by operating a better process control the estimatedvalue for the tested property will be higher. The same will be achieved by increasing the number of units of thespot sample.

5.6.7 How to come from unknown to known standard deviation?

Looking at the tables for k n , Tables 1 and 2, it is clear, that there is a considerable effect in going from anunknown to known standard deviation. In control method A (clause 5.6.4) the standard deviation of thepopulation is considered to be unknown at least for the first 40 test samples and the acceptance coefficient k u has to be taken from tables for unknown standard deviation. For the next 80 test samples the standarddeviation can be considered to be known, but the used acceptance coefficient is corrected ( k c ). Theacceptance coefficient for the known standard deviation k k is taken from tables for known standard deviation.The corrected acceptance coefficient k c is calculated by a linear interpolation between the acceptancecoefficient k u and k k . 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 atleast for the first 20 test samples and the acceptance coefficient k u 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, butthe used acceptance coefficient is corrected (k c ) as above. The acceptance coefficient for the known standarddeviation k k is taken from tables for known standard deviation. The known standard deviation σ is calculatedbased on the first at least 20 test results.

If “progressive sampling” is used the standard deviation of the population is considered to be unknown at leastfor the first 30 test samples and the acceptance coefficient k u has to be taken from tables for unknownstandard deviation. For the next 30 test samples the standard deviation can be considered to be known, butthe used acceptance coefficient is corrected (k c ) as above. The acceptance coefficient for the known standarddeviation k k is taken from tables for known standard deviation. The known standard deviation σ is calculatedbased on the first at least 30 test results.

5.6.8 Conformity

After calculating x es t by testing the inspection lots the result has to be compared with either the declared valueor a lower or upper limit depending on the property. For compressive strength it is the declared value or thelower limit and for dimension it is the upper and lower declared value or the upper and lower limit. In Figure 8the estimated mean compressive strength is based on 95 % confidence level for the different spot samplesusing the calculations of the test data shown in Figure 4. In Figure 9 the estimated 5 % characteristiccompressive strength based on 95 % confidence level is shown using the same test data.

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Figure 9 — Example of variation in the estimated mean compressive strength over time

LL DV

UL

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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 andtherefore conforming to the fixed limit values. The declared value needs to be equal to or lower than the lowerlimit value.

In Figure 11 the variation in the length of the units over time is given. The units are from the same productionas the ones checked as green units, see Figure 2.

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Figure 12 — Example of a shape of a masonry unit

By carrying out tests for masonry made of specific units it is possible for these units to establish a relationshipbetween 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

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By testing and controlling the gross dry density it is possible to declare the thermal conductivity, λ10,dry,unit , ofthe 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 adramatic drop occurs periodically the probable reason for the variation in the gross dry density is a renewal ofthe mould and therefore the variation in the shape and not a variation in the net dry density.

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 testresults of the spot samples have to be evaluated, e.g. by the calculation procedures described in clause 5.6.6using 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 givenproperty and calculate the mean value and the standard deviation and fix then a band in which new testresults have to fit in. The upper band limit and lower band limit then can be 2 times of the standard deviationaway 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 singletest results including the non-conforming values shall be repeated and the band limit values adjustedaccordingly. The same shall happen for the declared value. The non-conforming inspection lot can be treatedas 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 inFigure 11, it is important to avoid that the whole inspection lot is mixed up with the other inspection lots. Thenon-conforming inspection lot has to be treated separately. It may be reclassified by the manufacturer andgiven 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.

It should be in the interest of the manufacturer to avoid that the same non-conformity occurs again. When

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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 valuesare 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 toreduce 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 theproduct 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 controlmethods A and B can be used. For product 2 it is even possible to use control method A for someproperties and for some properties control method B. If using method B a re-declaration in connectionwith a non-conformity is possible based on test results obtained by testing a new spot sample taken atrandom from the inspection lot following control method A, but it is necessary to keep the test resultsleading 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 drydensity and dimension 75 % may be chosen. For gross dry density or net dry density used as a proxyproperty to thermal conductivity a confidence level of 50 % or 90 % may be chosen.

If the spot sample size is 6 units, the acceptance constant k n for mean compressive strength at a 95% confidence level is k 1,u = 0,82 for unknown standard deviation and k 1,k = 0,67 for known standarddeviation.

If the spot sample size is 6 units, the acceptance constant k n for 5 % characteristic compressive

strength at a 95 % confidence level is k 1,u = 3,71 for unknown standard deviation and k 1,k = 2,32 forknown standard deviation. If the spot sample size is 6 units, the acceptance constant k n for mean compressive strength at a 75

% confidence level (Category II units) is k 1,u = 0,30 for unknown standard deviation and k 1,k = 0,28 forknown standard deviation.

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 drydensity and dimension 75 % may be chosen. For gross dry density used as a proxy property to

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thermal conductivity a confidence level of 50 % or 90 % may be chosen.

If the spot sample size is 3 units, the acceptance constant k n for mean compressive strength at a 95% confidence level is k 1,u = 0,52 for unknown standard deviation and k 1,k = 0,47 for known standarddeviation. If a sample size of a spot sample is raised to 6 units instead of 3 then the acceptanceconstant k n for mean compressive strength is k 1,u = 0,35 for unknown standard deviation and k 1,k =

0,34 for known standard deviation. If the spot sample size is 3 units, the acceptance constant k n for 5 % characteristic compressive

strength at a 95 % confidence level is k 1,u = 2,74 for unknown standard deviation and k 1,k = 2,12 forknown standard deviation. If a sample size of a spot sample is raised to 6 units instead of 3 then theacceptance constant k n for mean compressive strength is k 1,u = 2,31 for unknown standard deviationand k 1,k = 1,98 for known standard deviation.

If the spot sample size is 3 units, the acceptance constant k n for mean compressive strength at a 75% confidence level (Category II units) is k 1,u = 0,20 for unknown standard deviation and k 1,k = 0,19 forknown standard deviation. If a sample size of a spot sample is raised to 6 units instead of 3 then theacceptance constant k n for mean compressive strength is k 1,u = 0,14 for unknown standard deviationand k 1,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 leadingto a non-conformity and evaluate the test results using a reduced acceptance constant (e.g. 0,52instead 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 thetest results using a reduced acceptance constant, according to the number of the units in the testsample and change eventually the declaration accordingly. *)

*) Always keep the results of the inspection lot within the system when evaluating the next inspection lot orstart from the very beginning.

When a non-conformity is identified in the finished product testing it is not possible to take any correctiveactions for the tested inspection lot. It can only be discarded or re-declared. The longer the production processof 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 aboutmeasurement of the length of the green units, see Figure 2, demonstrates that it is possible to detect aproblem (wear and tear of the mould) early in the process, which leads to a non-conformity of the finishedproduct in the process. Checking dimensions, weights and temperatures are quite simple, but done at the rightplaces in the process they will give a lot of information valid for the control of the process and the properties ofthe 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 mixof process control and finished product testing, and to consider also the possibility of using internal proxy testsin the process control.

The manufacturer may define product groups. A product group consists of products from one manufacturerhaving common values for one or more characteristics. That means that the products belonging to a productgroup may differ according to the characteristics in question. If a product group is defined, then the FPCsystem shall ensure that all types of units within a group are controlled and over time also by the finishedproduct 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.

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Traceability in the process

The clause deals with the traceability in the process from raw materials to finished products. It is not dealingwith the traceability on the market.

As mentioned earlier it should be in the interest of the manufacturer to avoid that the same non-conformityoccurring again. It is therefore important to try to identify the reason why, when it occurred, otherwise it isdifficult 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 differentparts of the process and their influence on the variation in the properties of the finished product the better hewill be able to identify the reason for non-conformity. To be able to obtain that knowledge it can berecommended that the manufacturer follows the same units all through the process from time to time if not onevery occasion and to evaluate all the checks and measurements together and to compare the results withother similar evaluations done. Based on such an exercise it may be possible to establish traceability in theprocess.

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 isthe need for instructions dealing with the marking procedure and how to handle and to control the stock. It isimportant that 2 types of units with the same shape but not the same properties are marked in such a way thatthey will not be mixed up. Inspection lots of products should be identifiable and traceable.

5.6.12 Records

Many years of experience have shown that it can be dangerous to have only one person who knows all theinformation 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 inwriting. It can be recommended to describe step by step what needs to be done in the whole process from theraw material to the finished product leaving the gate of the factory in order to be able to produce a high qualityproduct. This can include specifying the position of each check and observation points and controlprocedures. It is really valuable for the machine operator to have information of corrective actions availablewhen control parameters are passing the control limits.

Experiences show also that on a busy day it is easy to forget important observations made during theproduction control if these observations are not recorded. To make it easier to record observations it can berecommended to use tables.

Samples are taken during the process and from finished products and these samples need to berepresentative 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‟srisk the frequency should be carefully considered, decided and recorded. If test results and FPC system giveevidence 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 toproduce. A manufacturer would like to fulfil the market needs and therefore intends to develop and to produceunits with specific properties. To ensure that these properties are available it is necessary after completion ofthe 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

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test done as part of the control method A can act as an initial type test if the reference test methods are usedand the sampling procedure for ITT. In that respect the declared values, which may vary from batch to batchhave to be determined batch by batch and have to be based on an evaluation of the same test results (seeclause 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 Ausing the reference test methods and the sampling procedure for ITT, then the test can be regarded as aninitial type test.

Whenever a major change in the source, blend, or nature of raw materials occurs, or when there is a changein processing conditions, leading to what the manufacturer considers will constitute a new product type beingproduced, the appropriate initial type test shall be repeated. If the manufacturer has doubts it can berecommended to check whether some of the characteristics have changed or not by using the FPC testprocedures.

The manufacturer may define product groups. The products belonging to a product group may differ accordingto the characteristics in question.

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 anassociation) to justify his own declaration of conformity regarding a product that is manufactured according tothe same design and with raw materials, constituents and manufacturing methods of the same kind, providedthat permission is given, and the test is valid for both products.

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Annex A (informative)

Tables for acceptance coefficient kn depending on the used fractile p and

confidence level γ (taken from ISO 16269-6 (2005))

Table A.1 — k1 for one-sided statistical tolerance, standard deviation: knownand confidence level γ = 50 %

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 1,6455 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 1,6458 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 1,64513 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 1,64519 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 1,64523 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 1,64527 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 1,64532 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 1,64537 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 1,64542 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 1,64546 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

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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 1,64550 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

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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 1,7403 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 1,7377 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 1,73212 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 1,73016 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 1,72622 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 1,72427 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 1,72230 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 1,71741 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 1,71544 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 1,8225 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 1,8169 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 1,80616 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 1,79724 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 1,78734 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 1,78339 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 1,77943 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 1,77649 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

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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 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 1,8725 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 1,8649 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 1,85216 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 1,83924 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 1,82734 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 1,82239 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 1,81843 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 1,81349 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

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Table A.5 — k1 for one-sided statistical tolerance, standard deviation: unknownand confidence level γ = 50 %

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 0,888 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 1,6555 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 1,6549 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 1,65316 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 1,65224 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 1,65234 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 1,65139 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 1,65143 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 1,65049 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

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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 0,708 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 1,8065 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 1,8009 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 1,79016 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 1,78124 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 1,77134 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 1,76739 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 1,76443 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 1,76049 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

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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 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 1,9565 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 1,9449 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 1,92416 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 1,90624 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 1,88734 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 1,88039 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 1,87443 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 1,86649 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

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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 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 2,0525 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 2,0369 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 2,01016 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 1,98524 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 1,96134 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 1,95039 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 1,94243 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 1,93249 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

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39

Table A.9 — k2 for two-sided statistical tolerance, standard deviation: knownand confidence level γ = 50 %

n fractile : p

n fractile : p

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 1,9695 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 1,9689 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 1,96816 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 1,96724 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 1,96634 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 1,96539 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 1,96543 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 1,96549 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

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