Propagation of conformity statements in compliance with the GUM and ISO 17025

According to ISO/IEC 17025:2017 statements of conformity must identify the decision rule applied, they must be risk-based and account for uncertainty. In legal metrology and often among testing and calibration laboratories, there is the need to reuse measurement-based conformity statements to disseminate acceptability of measurement results. In particular, decision rules are required that allow the statement of conformity for a linear combination of quantities for which, in turn, conformity statements are available. These decision rules should be simple and use information that is typically available, and they should comply with ISO/IEC 17025:2017, again by accounting for the level of risk and for uncertainty following the suite of documents of the GUM (the Guide to the Expression of Uncertainty in Measurement). Existing guidance requires the input and evaluation of standard uncertainties, or even of distributions, to derive statements of conformity, and as such may be inapplicable, or the required effort may deter practitioners. After reviewing the existing guidance, this research will identify typical settings which lead to particularly simple decision rules for stating conformity for linear combinations of quantities. These new decision rules are based on the specification limits and on information implicitly available in the decision rules of each input quantity. The rules will be proven, they are generalizable, intended to comply with ISO/IEC 17025:2017 and the GUM documents, and suitable to easily state the risk of not conforming to the weighted sum of the input specifications. For practitioners, a quick reference on feasible conformity statements for linear combinations of quantities is provided. The applicability of and need for the new decision rules are illustrated by two examples involving the legally regulated weighing of long vehicles and of net loads.


Introduction to statements of conformity and requirements for their propagation
Conformity assessment as understood in [1,2] and here uses measurement results to decide whether measurable properties of an item conform to specified requirements.These requirements are typically expressed as tolerance limits or intervals.Statements of measurement-based conformity are made according to ISO/IEC 17 025:2017 [2] by competent laboratories, according to OIML or WELMEC guidelines (such as OIML G 19:2017 [3] or WELMEC 4.2:2006 [4]) in the legal metrology field, according to ILAC-G8:09/2019 [5] for accreditation purposes, or according to national regulations or related standards and recommendations.Such statements of measurement-based conformity are the basis for and the aim of this research.More broadly defined (see [6]), conformity assessment may include preparatory and downstream activities, and may be applied to entities such as products, processes, services, persons or organizations.Different concepts of conformance may be prevalent elsewhere, such as in statistical process control.
Statements of measurement-based conformity may contain very little information, especially in legal metrology (see clauses 4, 6 in [3]), at the end of the metrological traceability chain or outside of a laboratory environment.Apart from general information, [2] prescribes the inclusion of the following minimum information for statements of conformity: a) to which results the statement of conformity applies; b) which specifications, standards or parts thereof are met or not met; c) the decision rule applied (unless it is inherent in the requested specification or standard).
The international standard makes it clear that conformity statements are required to be risk-based and need to account for measurement uncertainty ([2] see foreword, clauses 3.7, 7.8.6.1 and A.2.3), while for details it references [7] (the version of JCGM 106:2012 [1] adopted by ISO and IEC as a guide) 4 .However, explicit information on the measured value, associated uncertainties or even distributions is often not available in conformity statements.In contrast, calibration certificates must include measurement uncertainty in addition to many other pieces of information (see [2]).
This research considers how conformity statements that may contain the little information described above can be reused to disseminate acceptability and equivalence of measurement results.In particular, it aims to enable the statement of conformity for linear combinations of quantities for which in turn conformity statements are available.The resulting (i.e.propagated) conformity statement is intended to fulfill the requirements of [2], meaning that it will account both for the level of risk and for uncertainty.
Section 2 demonstrates that it is insufficient to simply add together the weighted tolerance limits.Even if the specific risk of non-conformance is known to be small for each input quantity, these risks may accumulate and thus invalidate the conformity of the linear combination of these quantities.Similarly, [8, E.3.3] makes it clear that a quadratic sum of coverage intervals, each with known coverage probability, also does not allow a statement on the risk of non-conformance.
Nevertheless, [2, annex A.2.3] considers the use of conformity statements to disseminate metrological traceability, even if these statements omit the measurement result and associated uncertainty.While [2] describes the case of importing tolerance limits into uncertainty budgets, it provides no details on how to do so and it does not consider the subsequent derivation of a statement of conformity.On the other hand, [1] gives guidance on conformity assessment for (scalar) quantities and binary decisions.This guidance can be applied when the information on the output quantity is expressed by a probability distribution or by a best estimate with coverage interval (see [1, scope]).
Conformity assessment in [1] has been placed in a wider context (such as quality assurance for products or nominal and ordinal properties in [9,10]), and its potential has been illustrated in applications (for instance in legal metrology [11]).Conformity assessment in [1] was extended to multivariate measurements (e.g.[11][12][13]), to account for the impact of false decisions (e.g.[11,14]), for sampling and for the presence of correlated systematic effects [15].Various guidance has been derived or updated (e.g.[5,16,17]).
The authors are not aware of research or guidelines on the conformity of linear combinations of quantities for which only information items a) to c) are available.The decision rules that will be proposed are confined to univariate, measurable quantities (as defined in [1]).For each of these different input quantities, we assume that it is measured by an individual item (a measuring instrument) whose measurement errors conform to a specified tolerance interval.This conformance was established according to [2] based on a calibration or test.If the linear combination of these input quantities describes a measurand (as defined in [8]), we are interested in the specific consumer's risk that the linear combination of values measured by the set of instruments will lie outside a specified tolerance interval.We will call this the risk of non-conformance for the linear combination of quantities.
Section 3 focuses on which and how information in conformity statements of input quantities may yield knowledge about the distribution or coverage interval of a linear combination of these quantities so that [1] can subsequently be applied.The section starts by briefly reviewing the guidance available in [8,18].Distribution-free intervals are listed as useful extensions of [8] that require fewer assumptions.These approaches rely, among other things, on knowledge of standard uncertainties or even whole distributions for each input quantity, information which may be not, or not directly, available in statements of conformity.In addition, these approaches require the evaluation of the combined standard uncertainty and/or the approximation of the distribution for the output quantity, both of which may be considered too complex in legal metrology and outside the laboratory environment.Propagating uncertainties or distributions may be disproportionate to the simplicity of many statements of conformity.
Section 3 therefore also develops tailored solutions.These solutions are based on information that is typically available in conformity statements, i.e. specifications and the decision rules or non-conformance risks used to establish conformance.Given this information, settings are identified which lead to particularly simple rules for making a risk-based statement on the conformity of linear combinations of quantities.The derivation of these decision rules is developed in the appendix and may be extended to related settings.As a result, linear combinations of quantities which conform to specifications with known decision rule or non-conformance risk in turn conform to the weighted sum of the input specifications with a specific risk of non-conformance that is the sum of the input risks or can be given in simple lookup tables.The new decision rules do not rely on the Monte Carlo method or on explicitly evaluating combined standard uncertainties.
The general need for risk-based decision rules that account for uncertainty arises in [2], in particular in its definition 3.7.The need for decision rules that enable a simple propagation of conformity statements is evident in Germany's legal metrology sector.A recent regulation [19, change 13 d) in no.7 of paragraph 25] allows the addition, subtraction, multiplication and division of measurement results that comply with the law.To implement this regulation, a rule has been published [20] which is not in accordance with [2] as it is not riskbased and does not account for uncertainties.In practice, valid and simple decision rules are needed at the international level for actions such as weighing vehicles that are too long for a single scale and determining net weights outside verified tare devices, for example, in zero waste shops.In section 4, these two examples will be used to illustrate the newly developed decision rules.Beyond legal metrology regulations, this research will ease the dissemination of the acceptability of measurement results via conformity statements and at the same time assure compliance with international standards and guides.It will support and extend the application of [2, annex A.2.3].

Insufficient knowledge to propagate conformance
One may be tempted to claim that if the risk of nonconformance is small for each measurement result of a conforming instrument, then any, possibly weighted, sum or difference of these results also bears a small risk of violating their added and possibly weighted tolerance limits.This claim may be due historically to the 'error analysis' of systematic errors (see [3]) or to convolving distributions with limited support (such as the rectangular distribution, see [8, clause F.2.3.3, note in E. 3.3]).In fact, for commercial and official transactions as well as in medical and pharmaceutical laboratories in Germany, the current regulation [20, section 14 in part II] allows reporting and using results for quantities derived as sums of measurement results and states that they conform to the sum of the maximum permissible errors (MPEs) in service.(See appendix A for a translation of relevant parts of [20].)Similarly, for weighing long vehicles, the traffic surveillance documents of several countries [21, clause 3], [22, clause 6.2] and [23, part 1 section 2] specify the summation of conforming measurement results.However, the underlying assumption that the risk of non-conformance is zero for each input quantity is usually invalid.For example, [4] states 'there is no doubt that there will always be some measurements that will exceed the maximum permissible error', and this is confirmed by statistics of such exceedances, for example in Germany [24][25][26].
This section will demonstrate the invalidity of the claim that the non-conformance risk always remains small when combining results of conforming instruments.In fact, the specific risks of non-conformance from each input quantity may accumulate, possibly making the resulting risk of non-conformance for their linear combination unacceptably large.Example 1 illustrates this before proposition 1 gives a precise and provable formulation.
Example 1.Let us consider three quantities, each measured by a conforming instrument.The probability for the values of these quantities is illustrated on the left hand side of figure 1.Each instrument is assumed to conform to a tolerance interval of 10 ± 1 with a small risk of non-conformance of, say, 6%.For values within the interval (9,11), the probability is assumed to be equal.In contrast, the residual probability of violating this tolerance interval may for each quantity be located at quite extreme values, for example 15.
For the sum of the three quantities, the probability can be calculated that it conforms to a tolerance interval of 30 ± 3, i.e. to the sum of the three intervals.If the input quantities are negatively correlated such that extreme values of one quantity always coincide with conforming values of the other two quantities, then the risk of not conforming to the interval 30 ± 3 is 18%, as shown in figure 1 on the right hand side.
While a risk of non-conformance of 6% may be considered small, a risk of 18% will usually not be.The sum of three quantities will thus not always conform to the sum of the tolerance intervals to which these three quantities individually conform.
The above example easily generalizes to other risks of nonconformance, to any linear combination of an arbitrary number of results from instruments, and to any distribution of values in the tolerance interval: Proposition 1.For quantities Y i , i = 1, 2, . .., let p i be the risk that values of the quantity lie outside the tolerance interval y i ± MPE i .Then the risk may reach but not exceed min ( ∑ p i , 1) that values for the linear combination of quantities ∑ c i Y i (with c i ̸ = 0 for all i) do not lie in any tolerance interval including That is, the linear combination of results from conforming instruments may not conform.
In proposition 1 as well as in sections 3.2 and 3.3, the notation y i ± MPE i is used for simplicity.All statements generalize to arbitrary tolerance intervals, i.e. to in-service or other specifications, that were established for the instruments measuring the input quantities.
From this proposition and its proof in appendix B, we conclude that knowing each input quantity conforms to specifications is alone insufficient to judge the conformity for any combination of these quantities.Non-conformity will occur, especially when the risk of non-conformance for each quantity is unknown or insufficiently small, or when the number of quantities being combined is large.Therefore, rules such as [20, section 14 in part II] are not risk-based, do not comply with [2], and counteract the aim of legal metrological control to establish confidence in conformity to legal requirements [27, section 2].At the same time, proposition 1 cautions against approximating the distribution of input quantities in uncertainty evaluations by a finite support distribution, as indicated in [8, F.2.3.3] and as frequently occurs in practice (see guidance in calibration [28, supplements], in legal metrology [3, annex F] and [29] or in the automotive industry [30], etc).
In general, one cannot infer conformance of a linear combination of conforming results based on specifications alone.Additional knowledge is needed, for example whether the sum of the risks of non-conformance is small, on how the risk of non-conformance is spread outside each of the tolerance intervals, or about the relation between the input quantities.
The following section reviews and develops various settings for which information in conformity statements of input quantities is sufficient to yield risk-based conformity statements for linear combinations of these quantities.

Sufficient knowledge to propagate conformance
Conformity assessment according to [1, scope] can be applied when the measurement result for a quantity is expressed by a probability distribution (usually a probability density function [PDF], a distribution function or a numerical approximation thereof) or by a best estimate with coverage interval and associated coverage probability.In order to apply [1], we will investigate how one can arrive at this information for linear combinations of quantities for which conformity statements are available.In particular, section 3.1 will provide an overview of information on each input quantity that is required to apply the GUM documents [8,18] and to subsequently derive conformity statements for linear combinations of these quantities.A portion of this information is typically available in calibration certificates.Section 3.2 will then identify settings that lead to simple, risk-based conformity statements in cases where all that is available are the specifications for each input quantity together with the uncertainty associated with the measurements performed to establish these specifications.The resulting conformity statements comply with the GUM [1,8] but do not require the evaluation of the combined standard uncertainty, applicability of the Central Limit Theorem [8], or implementation of the Monte Carlo method [18].The statements will be proven in appendix C and are based on predominantly analytical summations of Normally and uniformly distributed random variables.Finally, section 3.3 will use proposition 1 to show that knowledge of only the specifications and non-conformance risks for each input quantity may lead to a risk-based conformity statement for their linear combination.Conformity established according to sections 3.2 or 3.3 is easy to state beyond the laboratory environment, and section 4 showcases typical application examples.All the statements of conformity in this section as well as those for the examples in section 4 are intended to comply with the requirements in [1] and [2] and are summarized in look-up table 1. Table 1   • the risk of non-conformance is p applies also to non-linear models G2

Overview of common information on
• large number of quantities • conformity established by a device with The PDF expresses the state of knowledge of a quantity, and this section provides an overview of existing guidance on how the PDF of an output quantity can be derived.The methods in these guides are either themselves based on PDFs or on estimates, uncertainties and other assumptions about all input quantities.Conservative variants will be added and allow simpler conformity statements with fewer assumptions.The PDF of an output quantity is uniquely characterized when the PDF of those input quantities is given that relate to the output quantity, for instance, by a linear measurement model as considered here (see [18,

introduction and scope]). That is, if
• for independent quantities Y i , each PDF is known, or • for dependent quantities Y i , Y j , the joint PDF is known, the PDF for any combination of these quantities can be derived.This derivation can rarely be done analytically, but the Monte Carlo method as described in [18, clause 7] provides a numerical approximation.From the resulting PDF of the output, the risk of non-conformance can be calculated for a given tolerance interval (see coverage interval and probability in [18, clause 7.7] and [1, clause 7.1], and software such as [31]).Both of the above settings are listed in table 1 as case G1 and G2, respectively.
While the Monte Carlo method is very flexible, its implementation is not widespread among calibration laboratories, and even less so towards the end of the metrological traceability chain, for example among testing laboratories or verification authorities (see, e.g.[33,34]).In addition, the Monte Carlo method requires a case by case evaluation of the output PDF for each linear combination of quantities.There are usually no simple rules defining how different input will affect the conformity statement for the output.
Simplifications to evaluate the PDF of a measurand are available in [8, G.6.5].These apply when the PDF for the output quantity is a Normal or t-distribution, which is known to be the case when the Central Limit Theorem is applicable, or when the PDF for each input quantity is Normal.Further simplifications are described in case C1 and C2.Conservative, distribution-free coverage intervals can be calculated based on estimates, uncertainties, and few additional assumptions.No assumptions for the Central Limit Theorem or the particular distribution of each input quantity are required.(This approach was utilized in the proposed draft revision JCGM_100_201X_CD of [8].)The expanded uncertainty is the product of the combined standard uncertainty and a particularly simple coverage factor that depends only on the desired coverage probability [35].The risk of nonconformance is smaller than one minus this conservative coverage probability, or it can be calculated by applying guarded acceptance, simple acceptance, or other decision rules.
The above cases G3 to G5, C1 and C2 are based on the knowledge of standard uncertainties for each measurement result and can be easily generalized to the case where expanded uncertainties with coverage factors or bounded uncertainties are available instead, as noted in case C3 and C4.
Estimates, uncertainties and coverage factors (or other representations if needed) are typically available in calibration certificates.See, for instance, the policy documents [36, clause 5.2] and [28, clause 6.1], which are mandatory for participating accreditation bodies and conformity assessment bodies.For conformity statements that are based on calibration certificates, the estimates and standard uncertainties required to apply the law of propagation of uncertainty [8] may thus be available.However, additional information is required to calculate risks, such as a justification of the independence assumption or quantification of correlation in cases C1 and C2, and additionally justification of the Normality assumption or quantification of the degrees of freedom for G3 to G5, C3 and C4.Furthermore, all of these cases require the evaluation of the combined standard uncertainty in order to state conformity for the linear combination of quantities.

Information required to apply simplified rules: specifications and decision rules
This section focuses on information that is typically available in statements of conformity.Settings will be identified which lead to particularly simple rules for making a risk-based statement on the conformity of linear combinations of quantities.Derivations of these rules are either based on evaluating the distribution of the output by analytically convolving the input distributions (as acknowledged in [8, G.1.4])and by numerical integration, or they are based on distribution-free approximations.
In particular, it is assumed that for each input quantity Y i the following information is available: (i) instrument measurements conform to the specification ±MPE i , and (ii) conformity was established by an independent instrument whose standard uncertainty is known to be, or to be limited by, MPE i /m for some divisor m > 1, and (iii) all quantities involved are independent.
Assumptions i) and ii) suggest that each input quantity may be described as the sum of two independent quantities Y i = X i + U i , where U i describes the knowledge as if conformity had been established by a perfect instrument (i.e. one that measures with zero uncertainty), and X i characterizes the remaining doubt arising from the actual instrument used to establish conformity.The first quantity is best described by a uniform distribution because nothing is known but the specifications (see [8, clause 4.3.7]and [18, clause 6.4.2]).That is, U i ∼ Unif(y i − MPE i , y i + MPE i ), with y i being the midpoint, usually the indication value, and MPE being the maximum permissible error (adapted from [1, clause 3.3.18]).
Simple conformity statements can be provided for the common setting where each calibrating instrument X i follows a Normal distribution with expanded uncertainty that is limited by kMPE i /c for some coverage factor k (typically k = 2).Theorem 1 in appendix C shows that the risk that sums and differences of such quantities Y i do not conform to the specification ∑ MPE i , can be calculated by an essentially onedimensional numerical integration.This decision rule is listed as case S0 in table 1 and supported by the web app [32], which calculates non-conformance risks following theorem 1. Case S1 in table 1 further simplifies these conformity statements under the additional assumption that specifications for all input quantities are identical: MPE i = MPE.The risk that sums and differences of Y i do not conform to the specification nMPE can then be tabulated (see the top of table 2 and proof of corollary 1 in appendix C), and this risk quickly decreases with an increasing number of input quantities.For intervals [T L , T U ] that include the specification ±nMPE (i.e. when the estimate y lies in the acceptance interval [T L + nMPE, T U − nMPE]), the decision rule of guarded acceptance gives risks of non-conformance smaller than those displayed in table 2. The decision rule of simple acceptance can be applied when nMPE is smaller than some U max .Likewise, case S2 in table 1 provides tabulated conformity statements for sums and differences of quantities with partially different specifications MPE 1 , MPE 2 , MPE 2 , . . .(see bottom of table 2 as well as figure C1 for selected cases, and proof of corollary 2 in appendix C).
Similarly simple conformity statements can also be derived and tabulated for linear combinations with arbitrary combinations of specifications MPE i and arbitrary coefficients c i , for calibrating instruments with an expanded uncertainty having other limits kMPE/m, as well as for quantiles other than ∑ |c i |MPE i and for fixed risk levels.
Simple conformity statements can also be provided when each calibrating instrument X i follows a single-peaked and symmetric or even an arbitrary distribution.For sums and differences of quantities Y i with identical specifications, MPE i = MPE, the standard uncertainty and conservative expanded uncertainty then take a particularly simple form, as given in cases S3 and S4 in table 1.For example, the risk of not conforming to the specification nMPE is smaller than 0.05 if no less than n = 4 (see case S3) or n = 8 (see case S4) quantities Y i are combined for c ⩽ 6. Similarly simple and conservative conformity statements can be tabulated for linear combinations of quantities with varying specifications MPE i and for calibrating instruments with a standard uncertainty having other limits MPE i /m.Conformity statements based on conservative, distribution free intervals (as in cases S3 and S4) will generally lead to larger limits for the risk of non-conformance or to larger specifications than exact evaluations of the output distribution (as in cases S0 to S2).
While assumption i) is part of all conformity statements, assumption ii) may be available in the calibration certificate of the instrument used to establish conformity or from the decision rule applied.For example, decision rules based on simple acceptance define requirements (often limits) on the measurement uncertainty when establishing conformity.Decision rules based on guarded acceptance allow us to derive assumption ii) from assumption i), the acceptance interval and the limit for the risk of non-conformance, as illustrated in [37, E2.5.3.3 and E2.5.3.6].In legal metrology, assumptions i) and ii) are often justified, see [1, section 8.2.3] or [3, clauses 5.3.5, 7], for example.Assumption iii) may be tested statistically (e.g.[38]) and holds when no common influences exist for pairs of conforming measurements, for instance, when the calibrating instruments and environmental conditions differ.
The advantage of decision rules S0 to S2 and even S3 and S4 is that they are applicable when combining few or dominating quantities-as opposed to cases G3 and G4.In addition, cases S1 to S4 simplify statements of conformitycompared to cases G1 to G5, C1 and C2-because they do not require the evaluation of combined standard uncertainties or even the application of the Monte Carlo method.The suggested decision rules are particularly simple for sums or differences of quantities whose specifications are almost all identical.Furthermore, information on i) and ii) is commonly available under [2].In any case, the statements of conformity resulting from S0 to S4 are risk-based as required in [2] and were derived applying the rules of the GUM suite of documents.

Minimum information required: specifications and non-conformance risks
Compared to section 3.2, let us now consider the case where even less information is available on each quantity Y i .Specifically, let (i) instrument measurements conform to the specification ±MPE i , and (ii) the risk of non-conformance or its maximum p i be known.
Proposition 1 can be applied to state that the risk of nonconformance outside the interval y ± ∑ |c i |MPE i is limited by min ( ∑ p i , 1), where y is the linear combination of the indications y i .If ∑ p i does not exceed some pre-specified threshold, such a conformity statement is risk-based and can be considered compliant with [2, clauses 3.7, 7.8.6.1] and [1, clause 7.5].Measurement uncertainty is accounted for in this decision rule by viewing y ± ∑ |c i |MPE i as a coverage interval with known (or limited) coverage probability.
Assumption i) is part of all conformity statements.Assumption ii) may be available directly in the conformity statement or it may be inferable from the decision rule, e.g.under guarded acceptance from the width of the guard band [1, clause 8.3.2.3], or under simple acceptance from the requirements on the uncertainty when establishing conformity (see section 3.2 with n = 1).If information is available beyond the mere assumptions i) and ii), risks of non-conformance lower than ∑ p i may be deducible, as proposition 1 states an upper limit.
The advantage of this decision rule (case M1 in table 1) is its utter simplicity.Moreover, it is applicable even when the GUM suite of documents is not.That is, if knowledge about correlation or distributional assumptions outside the specifications (such as a limited standard uncertainty) are unavailable, the standard uncertainty, coverage interval, distribution-free interval or PDF for the linear combination of input quantities cannot be evaluated [8,18] and thus [1] cannot be applied.Decision rule M1, however, can be applied.

Examples
When using weighing instruments that are known to conform, several scenarios exist that require combining measurements linearly.Two scenarios from the field of legal metrology will illustrate the decision rules introduced in section 3: the weighing of trucks (section 4.1) and tare calculations (section 4.2).The conformity of the non-automatic weighing instruments involved in all examples is assessed according to [39], which in clause 3.7.1 requires using verified weights with an 'error' or uncertainty no greater than one third the maximum permissible error that is to be established.

Weighing of trucks
In many countries, overloaded vehicles are prohibited on public roads.For purposes of traffic surveillance, weighing the vehicle directly on a weighbridge of appropriate size usually determines the weight most precisely.However, if the vehicle is too long to fit onto the platform(s) of a weighbridge, determining the overall vehicle weight will almost always involve summing several partial weights.Procedures can be classified into sequential (or repeated) and simultaneous weighing, and the decision rules M1 and S1 or S2, respectively, are applied below to derive the conformity of the vehicle weight measurement procedure in compliance with [2] and [1].Currently implemented procedures vary from country to country, and at least [21, clause 3], [22, clause 6.2] and [23, part 1 section 2] do not consider the risk of non-conformance for the weighing of vehicles that are too long for a single weighbridge.

Case M1-sequential weighing on the same instrument.
Following [22, clause 6.2], [21, clauses 3.4.2, 3.5] and [23, part 1 sections 2.3, 2.4], a long vehicle can be weighed in two or more steps on the same weighing instrument.In each step, the wheel, axle or axle group load Y i is measured and the vehicle is then repositioned.The overall weight is the sum of all loads Y i .
Measurements y i are made by a weighing instrument that conforms to the in-service specification ±2MPE, and the weights used to establish the specification have an expanded uncertainty U ⩽ MPE/3 with coverage factor k = 2 (see [39, clause 3.7.1]and [40, clause 5.2]).The quantities Y i are correlated due to the common weighing instrument.The degree of correlation is, however, rarely known because contributions from common influencing factors, such as long term stability and common testing equipment or measurement conditions, may vary greatly between measurements.
The conformity of the sum of all weight measurements ∑ y i to the sum of the input specifications ±2nMPE can be established according to decision rule M1 (see table 1).If the risk of non-conformance p i for each input measurement is smaller than 0.033, the risk that the sum ∑ y i does not conform to ±2nMPE is limited by 0.033 multiplied by the number of weighing steps n, e.g. by 0.133 for n = 4.The assumption that p i ⩽ 0.033 is based on the assumptions of case S1 and applying n = 1, m ⩾ 12 (see table 2 top).That is, a linear relation Y i = X i + U i has not been disproven and Normally distributed values for the weights X i used to establish conformity may be a reasonable assumption given their coverage factor k = 2.In addition, results from national verifications in Germany [24-26, p. 5 in each] show rates of 6.3, 6.8 and 6.7% for the nonconformance of non-automatic weighing instruments to MPE in 2014, 2013 and 2012, respectively, and thus indicate the conservativeness of the assumption p i ⩽ 0.033 for 2MPE.
Based on decision rule M1, regulatory bodies are able to decide on acceptable (and unacceptable) levels of risk of nonconformance for the sequential weighing of long vehicles.For each measurement y i , the assumptions from section 4.1.1hold, and in addition all quantities involved may be assumed to be independent, e.g. if common environmental factors are negligible and if different weights were used to establish conformity.The conformity of the sum of all weight measurements ∑ y i to the sum of the input specifications ±2 ∑ MPE i can then be established according to case S0 (based on theorem 1).
When the input specifications are identical, MPE 1 = . . .= MPE n , decision rule S1 (see table 1) applies.This is the case, for example, when the weight on the vehicle is evenly distributed and only one type of weighing instrument is used.The risk that the sum ∑ y i does not conform to ±2nMPE is then limited by the values given in the top of table 2 for m ⩾ 12, e.g. by 6.0 • 10 −6 for n = 4.When all input specifications but one are identical, MPE 2 = . . .= MPE n , decision rule S2 (see table 1) applies.This is the case, for example, when a weighbridge is used in combination with wheel or axle load weighing instruments.The in-service specification ±2MPE 2 of the latter instruments is typically larger than the in-service specification ±2MPE 1 of the weighbridge, with ratios of around 0.5 being typical for mobile weighing instruments.The risk that the sum ∑ y i does not conform to ±2 ∑ MPE i is then limited by the values on the right hand side of figure C1, e.g. by 0.000 15 for 2MPE 1 /2MPE 2 = 40/100 = 0.4 and n = 3.The application of decision rule M1 is possible as well but leads to much larger limits for the risk of non-conformance.
If the assumptions of theorem 1 hold, the simultaneous weighing of long vehicles will have a level of risk of not conforming to the sum of the input specifications that is smaller than the risk of each input measurement not conforming to its specification-at least for the scenarios depicted in figure C1.

Tare calculations
The conformity of net loads may be established directly by non-automatic weighing instruments with tare device verified according to [39].In contrast, if a tare weight has been determined on a different weighing instrument or without the tare device, the net weight is calculated as the difference between gross and tare weight, say Y 1 − Y 2 , and often the risk of nonconformance is not accounted for.
Nowadays, the need for tare calculations becomes evident, among other places, in stores where customers bring their own reusable packaging (see [41]).Often the customer weighs the packaging and the measurement result is recorded (e.g.printed as a bar code, saved to a database, written onto the packaging), but this result may not comply with the tested tare function of the weighing instrument at the checkout.
Depending on whether the same or a different weighing instrument is used for gross and tare weighing, the conformity of the difference y 1 − y 2 to the specification ±2(MPE 1 + MPE 2 ) can be established and the risk of nonconformance calculated along the lines of section 4.1.1(case M1) and 4.1.2(cases S0 to S2), respectively.Conformity statements of the net weight then comply with [2] and [1].

Conclusions and discussion
Making risk-based statements on the conformity of a set of instruments that measure a linear combination of quantities for which, in turn, conformity statements are available, is often difficult due to a lack of sufficient or direct information.
This research has demonstrated that information on the specifications of the instruments measuring the input quantities is, alone, insufficient to infer the conformity of their linear combinations.As a consequence, regulations and current practice, which ignore risks when propagating conformity statements, will need to be adapted.
With a view to supporting statements of conformity for linear combinations of quantities, this research summarized existing guidance.Guidance on the elicitation and propagation of input distributions requires a case by case evaluation via the Monte Carlo method and is thus ill-suited for regulations and in industrial practice.It may also be unclear how to arrive at full distributions for all input quantities from the information available in conformity statements.Alternative guidance relies on inputting and propagating estimates and uncertainties, and on making additional assumptions.Such guidance is ill-suited when this information is not available, if the case involves a small number of non-Normal quantities, dominating non-Normal quantities, or dependent quantities, and when the law of propagation of uncertainty or the quantification of degrees of freedom is considered too difficult.
Therefore, this research developed decision rules that fill gaps in the existing guidance and that are tailored to input quantities for which conformity statements are available.These decision rules rely on knowledge about the input specifications that is given in all conformity statements, as well as about the input risk of non-conformance or about the uncertainties assigned at the previous level of the metrological traceability chain.The latter two pieces of information may be available directly or they can be derived from the decision rules.The newly developed decision rules comply with ISO/IEC 17 025:2017 and the GUM documents, and they also apply when combining few or dominating quantities.One decision rule even applies for dependent quantities with an unknown level of correlation, and this rule is particularly simple.For other decision rules, the risk of non-conformance involves a one-dimensional numerical integration, which can be, and was, tabulated for common settings.The new decision rules do not rely on the Monte Carlo method or on explicitly evaluating combined standard uncertainties.
Further generalizations on stating conformity for linear combinations of quantities were pointed out, such as statements for fixed levels of risk.In addition, information from even higher levels of the metrological traceability chain or on other closed families of distributions may be utilized in analogy to the theorem developed here.The developed decision rules may generalize to instruments whose conformity assessment is based on the conformity of its measuring components, if a measurement model suitably describes the combination of underlying quantities.Not considered here, but something industry may be interested in investigating next, are simple decision rules for non-linear combinations of quantities.
Altogether, this research supports the dissemination of acceptability of measurement results.It provides practitioners at the end of the calibration chain and metrologists developing regulations and guidance documents for them with new and simple decision rules and exemplification of their use, as well as with a quick reference summarizing how to propagate conformity statements-with certainty.∑ MPE i and setting A n = 2 n ∏ MPE i , J k = {(j 1 , . . ., j k ) : 1 ⩽ j 1 < j 2 < . . .< j k ⩽ n}.

Proof. Let
Ūi = U i + MPE i ∼ Unif(0, 2MPE i ) for i = 1, . . ., n.Then Ū = ∑ n i =1 Ūi follows a generalized Irwin-Hall distribution with parameters (2MPE 1 , . . ., 2MPE n ) (see [42]) and X = ) .The quantile −MPE of ∑ n i =1 (X i + U i ) has the same probability as the quantile 0 of Ū + X, which can be evaluated by analytically integrating over z up to 0 and numerically solving the remaining integration, that is Corollary 1.In addition to the assumptions of theorem 1, let MPE 1 = . . .= MPE n and m ⩾ 6 or m ⩾ 12, then the risk that the sum ∑ n i =1 (U i + X i ) does not conform to ±nMPE 1 is smaller than the risk indicated in table 2. ) dv with f IH being the density of the Irwin-Hall distribution [43].
That is, the probability of the quantile can be calculated independently of MPE 1 and by integrating numerically in essentially one dimension.
In addition, the distribution of X + Ū is symmetric and the probability of its quantile 0 is for any m > 6 smaller than for m = 6, as Φ is strictly increasing.We argue analogously for m > 12.
Table 2 lists twice the probability for the quantile −nMPE 1 of ∑ (X i + U i ) for up to n = 11 instruments and for m = 6 and m = 12. with fŪ being the density of the generalized Irwin-Hall distribution with parameters (2a, 1, . . ., 1) here.That is, the probability of the quantile does not depend on MPE 1 and MPE 2 but only on their ratio, and it can be calculated by integrating numerically in essentially one dimension.
In addition, the distribution of X + Ū is symmetric and the probability of its quantile 0 is for any m > 6 smaller than for m = 6, as Φ is strictly increasing.We argue analogusly for m > 12.
Figure C1 shows twice the probability for the quantile −(MPE 1 + (n − 1)MPE 2 ) of ∑ n i =1 (X i + U i ) for different ratios MPE 1 /MPE 2 , for n = 1, . . ., 7, m = 6 and m = 12.Some of these settings are listed in table 2. The R [44] code used to produce this figure was validated with the results of corollary 1 for MPE 1 = MPE 2 and with the Monte Carlo method [18] for other selected settings.

Figure 1 .
Figure 1.Illustration of an example, where the risk of non-conformance accumulates to 18% (right panel) for the sum of three quantities.It is assumed that each input quantity is measured by an instrument that is known to conform with a risk of 6% (left panel).
the measurement result from conforming instruments and how it leads to conformity statements of their linear combinations.Let Y i denote the quantities that are each measured with a conforming instrument and let Y = ∑ c i Y i be their linear combination.Let [T L , T U ] denote the tolerance interval that the output quantity Y shall conform to and p the risk of not conforming to this interval.

4. 1 . 2 .
Case S1 or S2-simultaneous weighing on different instruments.Following [23, part 1 sections 2.2-2.5] a long vehicle can be weighed statically and simultaneously on several weighing instruments-each measuring load Y i .The overall weight is the sum of all loads Y i .

Figure C1 .
Figure C1.For case S2 in table 1 the risks of non-conformance to the specifications MPE 1 + (n − 1)MPE 2 are displayed for several ratios of specification MPE 1 /MPE 2 , for m = 6, m = 12 and for n input quantities known to conform to MPE 1 , MPE 2 , . . ., MPE 2 .(See corollary 2 and theorem 1 for details.)The dots indicate the values listed in table 2 and in the example in section 4.1.2.

Table 1 .
is

Table 1 .
a For non-empty acceptance regions [TL + U, TU − U] ∋ y one has [y ± U] ⊂ [TL, TU].The risk of non-conformance is thus P(Y / a complete review of possible ways to propagate conformity statements, but rather gives an overview of existing guidance (where 'G'-cases summarize GUM methods and 'C'cases are based on conservative assumptions) together with new decision rules (with 'S' referring to simple statements and 'M' to minimum information).The table is sorted by increasing simplicity and serves as a quick reference for practitioners.
[32]onforming to y ± U with risk of non-conformance p following theorem 1 (use e.g.web app[32]), or not That is, firstly, if for each input quantity (i) the estimate, uncertainty and degrees of freedom are given, and (ii) the assumptions of the Central Limit Theorem (see case G3 in table 1) apply, then the PDF for the linear combination of these quantities is a t-distribution, which is scaled by the combined standard uncertainty [8, clause 5.1.2],shifted by the estimate [8, equation (2)], and has degrees of freedom that can be approximated [8, equation (G2.b)].Using appropriate tables or software, the risk of non-conformance can be calculated directly from the PDF or by calculating a coverage interval and applying guarded acceptance, simple acceptance, or other decision rules.Additionally assuming large degrees of freedom (see case G4) simplifies the statement of conformity due to an approximately Normal distribution for the output PDF, thus avoiding calculations to approximate the degrees of freedom.Alternatively, if for each input quantity (i) the estimate and uncertainty are given, and (ii) the PDF is a Normal distribution (or can be approximated by it), and (iii) all quantities are independent, then the PDF for the output is known to be a Normal distribution without additional assumptions for the Central Limit Theorem.This is a common setting ([8, clause 6.3.3],[1, clauses 4.3, 7.2]) and is described in case G5.The risk of non-conformance can be calculated directly from the PDF [1, clause 7.4], or from decision rules, such as guarded or simple acceptance [1, clauses 8.2, 8.3].

Table 2 .
The maximum risk of non-conformance to the specification nMPE and MPE 1 + (n − 1)MPE 2 , respectively, is listed for sums or differences of n input quantities-at the top in the case of identical specifications, MPE i = MPE (case S1 from table1), and at the bottom in the case of mostly identical specifications MPE 1 , MPE 2 , MPE 2 , . . .(case S2).The calibrating instrument measured with an uncertainty ⩽ MPE i /m.