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E L S E VI E R Measurement 16 (1995) 51 57Measurement On the application of the Guide to the Expression of Uncertainty in Measurement to measuring instruments E r n e s t o A r r i a , , , F r a n c o C a b i a t i b, S a v e r i o D ' E m i l i o b, L u i g i G o n e l l a a a Politecnico di Torino, Corso Duca Degli Abruzzi, 24, 1-10129 Torino, Italy b I EN Galileo Ferraris, Torino, Italy Abstract The problem situation related to the calibration and use of measuring instruments is addressed in general terms with reference to the ISO Guide to the Expression of Uncertainty in Measurement. Some relevant concepts and definitions, not explicitly addressed in the Guide, are considered, such as the role of the compatibility of different measurements of the same measurand and the calibration diagram of the measuring instruments. These additional concepts are expressed in terms coherent with the philosophy of the Guide. Situations commonly met in measurement practice, such as calibration at different levels of accuracy, evaluation of traceability levels throughout the measurement hierarchy (from the primary standards to the calibration laboratories and shop floor tests) and interlaboratory comparisons, are analyzed and discussed. Keywords: Measurement uncertainty; Measurement compatibility; Measuring instrument; Calibration diagram; Traceability level; Interlaboratory comparison 1. Introduction The Gui de t o the Expr essi on of Uncer t ai nt y in Meas ur ement [ 1] represent s the mos t compr e- hensi ve document agreed on at i nt er nat i onal level and consi st ent wi t h t he r ecommendat i ons of the Wor ki ng Gr o u p convened by BI PM in 1980 and of CI P M in 1981 and 1986. The Gui de, as st at ed in its scope, is meant t o be appl i cabl e t o a l arge spect r um of scientific and t echni cal appl i cat i ons, "f r om the shop fl oor t o f undament al research", i ncl udi ng bot h the cal i br at i on of meas ur i ng i nst rument s and tests t hr oughout a nat i onal measur e- ment system, and t he devel opment , mai nt enance and compar i s on of reference st andar ds. However , t he act ual di scussi on and t he exampl es are limited to the eval uat i on of t he uncer t ai nt y in i ndi rect *Corresponding author. 0263-2241/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0263-2241(95)00017-8 meas ur ement s where the meas ur and is funct i onal l y tied to di rect l y measur ed quant i t i es. I n the det ermi - nat i on of t he uncer t ai nt y component s of these var i ous di rect meas ur ement s a f undament al rol e is pl ayed by the uncer t ai nt y due to t he di r ect - r eadi ng i nst r ument itself, a rol e t hat overri des all ot hers in the case, t he mos t usual in i ndust ri al pract i ce, where t he meas ur ement is carri ed out by a single meas ur ement action. The i nst r ument al uncer t ai nt y is referred to by t he Gui de as a dat um suppl i ed by t he manuf ac- t urer, mat hemat i cal l y t reat ed as a squar e di st ri bu- tion. Of course the Gui de' s aut hor s assume t hat the eval uat i on of such an i mpor t ant i nst r ument al charact eri st i c is t o be per f or med t hr ough a cal i bra- t i on process. However , in deal i ng wi t h measur i ng i nst rument s, mor e at t ent i on shoul d be pai d to this cal i br at i on process and t o the rol e of t raceabi l i t y. Obvi ousl y, some concept ual and t er mi nol ogi cal 52 E. Arri et al./Measurement 16 (1995) 51-57 integration is required in additional documents to be issued to address these topics. We are here endeavouring to suggest a path for these documents, which are needed as a frame of reference for the standards related to specific types of instruments. To this purpose, we must add to the terms and definitions given in the Guide and the last edition of VIM [2] a few terms and concepts concerning the calibration and use of measuring instruments, the measurement compatibility and its role in evaluating measurements of the same measurand carried out by different instruments, the traceability through the successive steps of any hierarchical chain of measurement. For such an integration we shall follow the operative approach adopted in the Italian frame standard on terms and definitions in measurement [3] and the proposal presented to IEC TC 66 [4]. For some terms and concepts concerning the national standards and the dissemi- nation activity more specific references are considered [5-7]. To some extent, these documents are in substantial agreement with the lines of the Guide, even if some mismatch is present and needs to be overcome in future specific standards. As stated in the Guide, the measurement uncertainty is a parameter characterizing "the quality of a result of a measurement" (clause 0.1) or "the dispersion of the values that could be reasonably attributed to the measurand" (clause 2.2.3 and VIM entry 3.9). This parameter is expressed, according to the recommendation of the CIPM Working Group, by a value related to the estimated standard deviation. In the standard [3] the uncertainty is considered an integral part of the information given by the measurement result, rather than an additional or supplementary part of this information, to the point that the measurement result is not expressed just by a single value, but by a whole interval of values. This interval is identified by a measurement value (usually its mid element) and its measurement uncertainty (usually the half width of the interval). Both approaches renounce a more detailed description of the uncertainty (e.g. statistical distribution, fuzzy set), which would depend on the specific features of the measurement conditions, and limit the information on the uncertainty to its amplitude only, for the sake of simplic- ity and uniformity. 2. Additional concepts and definitions The statement in clause 1.2 "This Guide is primarily concerned with the expression of uncertainty in the measurement of a well defined physical quantity, the measurand, that can be characterized by an essentially unique value" is in line with the concept of true value, that, however, the Guide overcomes for the reasons expounded in Annex D. In fact, every measurand has an intrinsic uncertainty, given by the minimum uncertainty that can be assigned in its description (clause 2.3.1 in [3]). If one has to measure a given quantity with uncertainty lower than its own intrinsic uncertainty, one is compelled to redefine the quantity using a more sophisticated model. In any case, the intrinsic uncertainty has to be included as a component of the total uncertainty, as acknowledged in clause D.I.1 of the Guide. From this point of view, the measurement uncertainty reflects both the level of definition and the level of knowledge of the measur- and rather than just "the lack of exact knowledge of the value of the measurand", as stated in the Guide (clause 3.3.1). The result of a measurement carried out with an uncertainty equal to the intrinsic uncertainty of the measurand may be called the best measurement of the measurand (clause 2.3.1 in [3]). The fact that a quantity can be defined only at a finite level of detail makes the true value concept questionable even before the measurement uncertainty is accounted for. As well known, even in dealing with primary or national standards, the metrologists evaluate and declare the uncertainty associated with the value(s) of each standard [5]. Another basic concept strictly linked to the uncertainty, mentioned but not expounded in the Guide (clause 1.4), is the measurement compatibility. This is an important concept because from the operative point of view it may be taken as the basis of the definition of uncertainty. The concept of measurement compatibility is in complete agreement with the CIPM Recommendation, which eliminates the traditional concept of true value of the measurand, substituted in the operative E. Ar r i et a l . / Me a s u r e me n t 16 ( 1 9 9 5 ) 5 1 - 5 7 53 appr oach by t hat of best measur ement of the measurand. Specifically, t he measur ement compat i - bility plays a cent ral role in evaluating: - measur ement s of the same measur and in the same measur ement condi t i ons obt ai ned using different i nst rument s or by different l aborat ori es (e.g. i nt er compar i sons of nat i onal st andards, i nt er l abor at or y compar i sons in t he f r amewor k of cal i brat i on services); - t r a c e a b i l i t y checks in t he accredi t at i on of cal i brat i on and testing l aborat ori es, whose final aim is t o assure t he i nt erchangeabi l i t y and integ- rat i on of product s. In the operat i ve appr oach pr omot ed in [ 3] and confi rmed in [ 4] , a measuri ng i nst r ument can be represent ed as a funct i onal block, which ei t her (a) supplies an out put readi ng dependent on the quant i t y appl i ed to its input, or (b) makes available an out put quant i t y identified by a nomi nal value. In the VI M t he i nst rument s of t ype (a) are called "measuri ng t ransducers or i ndi cat i ng and record- ing i nst r ument s" and the i nst rument s of t ype (b) are called "mat eri al measures". The l at t er ones include an increasing number of i nst rument s, e.g. the mul t i funct i on and mul t i range calibrators, where the nomi nal value may be single or selected within a set of discrete or cont i nuous values. The t r eat ment of uncer t ai nt y and t he cal i brat i on process are essentially the same for bot h types of i nst rument s, because the readi ng of type-(a) instrument s plays the same role as the nomi nal val ue of t ype-(b) instruments. A clear di st i nct i on has t o be made bet ween the r e adi ng val ue r of an i nst rument (or the nomi nal v al ue of a mat eri al measure) and the me a s u r e me n t r e s ul t M = I r a - u, m + u] assigned t o represent t he measurand, where m is the me a s u r e me n t val ue and u t he me a s u r e me n t u n c e r t a i n t y . The rel at i on between the readi ng val ue r and t he measur ement result M is det ermi ned t hr ough a cal i brat i on pr ocedur e that involves a cal i brat ed i nst rument , havi ng in general an uncer t ai nt y l ower or negligible in compar i son with t hat of the i nst r ument under cal i brat i on. Thi s is a general rule, except in t he br anch of met r ol ogy t hat concerns the devel opment of t he pr i mar y standards. In fact, these i ncor por at e a real i zat i on of t he relevant SI units, for which t he knowl edge of t he realized physical quant i t y cannot rely directly on the cal i brat i on of anot her reference st andard, but can be val i dat ed t hr ough inter- nat i onal compari sons. The result of a cal i brat i on pr ocedur e is generally issued in t he form of tables. However, t hi nki ng in t erms of a di agram of M versus r can be useful from t he concept ual poi nt of view: accordi ng t o [ 3, 4] , this di agr am is named c al i br at i on di agr am. Due t o t he looseness i nt r oduced by uncer t ai nt y u, the di agram must be conveni ent l y t hought of as a strip di agram (see Fig. 1 ). For the maj ori t y of moder n i nst rument s, the mid line of the cal i brat i on di agram ( cal i br at i on curve) is close to a straight line and can be expressed by a par amet er dependi ng on the readi ng value r. This par amet er may be either the c al i br at i on f a c t o r k(r) = m/ r or the fractional devi at i on d(r) from the straight-line value knr, where kn is the nomi nal val ue of k. In the l at t er case it is d(r) = ( m - k , r ) / k , r , m = k , r [ 1 + d(r)]. ( 1 t In the cal i brat i on process, the same measur and is put in common to the i nst rument under test and to t he measuri ng system assumed as reference, whose cal i brat i on di agr am is known. As a result, the cal i brat i on di agram of the i nst r ument under test can be generat ed from the readings of the t wo Measurement values / ~ calibration curve m + m - u i I D - r Reading wdues Fig. 1. Conceptual representation of the calibration diagram, obt ai ned by combi ni ng t he reading values (r) with the correspondi ng measurement values (m) and the associated uncertainty (u). The calibration di agram is intended as a strip diagram whose mid line is the calibration curve. 54 E. Arri et al./Measurement 16 (1995) 51- 57 i nst rument s and using t he known cal i brat i on diagram. The i nst rument s of t ype (a) and (b) can be combi ned in different ways, as bot h t he reference and t he i nst r ument under test can bel ong to ei t her type. It is i nt erest i ng t o not e t hat onl y readi ng values r and measur ement values m are i nvol ved in the cal i brat i on process, while no assumpt i on is needed beside the val i di t y of t he cal i brat i on di agram of the i nst r ument used as reference and t he measur- and stability duri ng t he calibration. Measur ement i nt eract i ons are our onl y source of knowl edge on t he physical quantities, and t he cal i brat i on of an i nst r ument against known measur ands is t he onl y source of i nf or mat i on on the rel at i on bet ween t he readings r and t he measur ement values m of the measurands. Therefore, at the ul t i mat e levels of t he cal i brat i on chai n the stability can onl y be ascer- t ai ned by t he met rol ogi cal experi ence on t he over- all consi st ency of the readi ngs of t he reference systems i nt eract i ng with given measurands. At t he next levels of cal i brat i on, one can assume t hat t he reference systems and t he measur ands are stable, i.e. t hat t hei r vari at i ons are negligible with respect t o the uncer t ai nt y of t he i nst r ument under test, or at least one knows enough about such vari at i ons (with time or ot her condi t i ons) as t o be able t o account for t hem when eval uat i ng t he uncer t ai nt y u in t he cal i brat i on diagram. It is wor t h poi nt i ng out t hat t he knowl edge of t he measur and is t he nor mal ai m of scientific and technical measurement , whereas in the field of appl i ed met r ol ogy one most l y deals with "known" measur ands and "unknown" instruments: t he aim of t he measur ement is not t o assign a val ue t o a physical quant i t y but t o assign a cal i brat i on diagram to an i nst rument . Thi s makes evi dent the i mpor t ance of the cal i brat i on di agram, what ever t he f or m adopt ed for the cert i fi cat i on of cal i brat i on results is. The me a s u r e me n t c o mp a t i b i l i t y is defined in [ 3,4] as t he pr oper t y of measur ement results of the same measur and consisting in t hei r being expressed by non-di sj oi nt intervals. Accordingly, t wo measur ement results M l =[ ml - ux, m l + u l ] and M z = [m E -- u2, m 2 -n t- U2] are compat i bl e if Ira1 - m z l ~ < u ~ --~-u 2 . (2) A different cri t eri on is suggested by WECC in connect i on with t he eval uat i on of an i nt erl abora- t or y compar i son [ 7] : I m l - m 2 1 < ux/~l +Uez. ( 3 ) In bot h cases the difference of t he measur ement values m~ and m2 is compar ed with a combi nat i on of the uncert ai nt i es Ul and u2 of M~ and M2. A mor e general definition of compat i bi l i t y, valid not onl y in the case of compl et el y i ndependent measurement s but also when t hey are part i al l y corre- lated, coul d be 2 2 I ml - m21 < u~2 = x / u~ + U 2- 2 R l z u l u 2 , ( 4 ) where u12 is the uncer t ai nt y of the difference ml - m2 eval uat ed accordi ng t o the Gui de, and R12 is t he cor r el at i on coefficient bet ween M~ and M 2. Fr om this rel at i on one can derive the ones valid in any specific situation, e.g. for measurement s compl et el y i ndependent (R12=0) or compl et el y correl at ed (R12 = + 1). In addi t i on, a compat i bi l i t y i ndex bet ween t wo measur ement results M~ and M a mi ght be defined as follows: I c = Ira1 - - mz [ / U1 2 , (5) and M1 and M 2 are compat i bl e if I~ is not larger t han 1. The definition (4) of compat i bi l i t y is not far from t he classical conf or mi t y test criteria or from t hose discussed in [-8]: in any case the difference bet ween t wo measur ement values ml and m2 of the same measur and is compar ed with the est i mat ed st andar d devi at i on of this difference multiplied by a suitable factor. The above consider- ations show how much the concept of uncert ai nt y is tied in with compat i bi l i t y. In a met rol ogi cal system, t he compat i bi l i t y of measurement s is at t ai ned t hr ough t he t raceabi l i t y of t he measuri ng i nst rument s t o pr i mar y standards. A commonl y used definition of traceability, in subst ant i al agreement with the one given in [ 2] , is: t raceabi l i t y is the pr oper t y t hat a measur ement i nst r ument acquires when it is cal i brat ed using measur ands whose values and uncert ai nt i es have been assigned with reference t o recogni zed nat i onal or i nt ernat i onal st andards (clause 7.1.1 in [ 3] ) . E. Arri et al./Measurement 16 (1995) 51- 57 55 3. Appl i cat i on t o s ome pract i cal s i t uat i ons The concept of t raceabi l i t y implies a ranki ng of the cal i brat i on l aborat ori es accordi ng to t hei r met- rol ogi cal capabilities, from the nat i onal l abor at o- ries, devot ed t o mai nt ai n and i mpr ove the pr i mar y st andards, down t o t he cal i brat i on st at i ons at the shop floor. It is wort hwhi l e to define quant i t at i vel y a t r a c e a b i l i t y l e v e l able to assess the posi t i on of a l abor at or y in the t raceabi l i t y chain, a level t hat is of course different for different quant i t i es and ranges. Thi s level may be defined by t he uncert ai nt y resulting from the combi nat i on of the uncert ai nt i es of the reference i nst rument s used in the cal i brat i on pr ocedur e and the uncert ai nt i es i nt r oduced in the cont r ol of the influence quantities, of t he coupl i ng of the measur ands to the i nst rument under test, etc., i.e. all t hose component s of uncer t ai nt y t hat are i ndependent of the tested i nst rument itself. In nor mal oper at i ons of cal i brat i on checking the uncer t ai nt y expressing the t raceabi l i t y level ought to be negligible with respect to the uncert ai nt y of the i nst r ument under test. If this is not so, the t raceabi l i t y level has to be t aken i nt o account in the uncert ai nt y budget with procedures t hat ought to be st andardi zed. Within a met rol ogi cal system based on traceability, the definition of uncer t ai nt y given in the Gui de in general t erms (clause 2.2.3) assumes a mor e definite meaning. In such a system, indeed, the measur ement values "t hat can be reasonabl y at t ri b- ut ed to the measur and", whose di spersi on is charact eri zed by the uncert ai nt y, are the measur ement values t hat can be obt ai ned with a t raceabl e measuri ng i nst rument . The t raceabi l i t y level of a given l abor at or y for a given ki nd of i nst r ument can be assessed by a pr ocedur e usually called "audi t " or "i nt erl abora- t or y compar i son". The pr ocedur e consists in circu- lating a travelling st andar d of a suitable and known cal i brat i on di agr am among a gr oup of N l abor at o- ries, each of t hem member of an established met r o- logical system. As the lines of t raceabi l i t y for such a system are clearly defined, following explicitly stated cal i brat i on chains, t he t raceabi l i t y levels are det ermi ned. Thus, t he measur ement results Mi, i = l , 2 , . . . , N , obt ai ned by each l abor at or y are expect ed to be compat i bl e, so t hat the exercise coul d be seen as a compat i bi l i t y test and t he results eval uat ed on t he basis of a compat i bi l i t y criterion. Mor e precisely, if M1 and M 2 a r e the results obt ai ned by the l aborat ori es 1 and 2, the compat i - bility index is given by (5) and (4), where now ml, m 2 are t he measur ement values, and u~, u2 their uncert ai nt i es eval uat ed t aki ng i nt o account the t raceabi l i t y levels. Fr om the N measur ement results a global measur ement result Mm can be derived, whose measurement value m and uncert ai nt y Um are given by the weighted mean of t he measur ement values m~ and t hei r uncer t ai nt y ui. An eval uat i on of every single result M~ coul d be based upon a compat i bi l i t y i ndex obt ai ned using (5). If the N measur ement results can be consi dered i ndependent , it is (see Appendix): I ¢i = I m i - m [ / x / u 2 - UZm . (61 Also, a general i ndi cat i on of the efficiency of the met rol ogi cal system comes from the mean of the indexes Ici of all l aborat ori es. Since the result of a cal i brat i on may be expressed in t erms of the cal i brat i on factor, one is t herefore easily t empt ed to t hi nk of the cal i brat i on fact or as embodyi ng the whol e i nf or mat i on on the calibration, and to t reat the uncert ai nt y of the i nst rument as it were an uncert ai nt y of its cal i brat i on factor, a posi t i on from which simple algebraic t r eat ment mi ght be derived. One must keep in mind, however, t hat by itself t he cal i brat i on fact or does not yield any i nf or mat i on on the uncert ai nt y of the instrument, being onl y an algebraic device poi nt i ng out the posi t i on of the mid poi nt s of the cal i brat i on di agram of which the uncert ai nt y is the hal f wi dt h (see Fig. 1). The nor mat i ve t er mi nol ogy ought to make a clear di st i nct i on bet ween the operat i ons of cal i brat i on (or re-calibrationt, meant to det ermi ne the cal i brat i on di agram of an i nst rument , hence its i nst rument al uncert ai nt y, and the operat i ons of cal i brat i on check, meant onl y to verify whet her t he act ual readings of an i nst rument fall within the range prescri bed by its cal i brat i on diagram: the former i mpl y the statistical el abor at i on of readi ngs under different condi t i ons sweeping over the whol e range of quant i t i es of influence and ot her oper at i ng condi t i ons, while t he l at t er may 56 E. Ar r i et al . / Measurement 16 ( 1995) 51 57 be carri ed out with few readings and be simply expressed in t erms of the cal i brat i on factor. When a quant i t y Y depends on N ot her quant i t i es X 1 , . . . , X N t hr ough a funct i onal rel at i on Y = f ( X ~ , . . . , X N ) and the measur ement result of Y is obt ai ned f r om the measur ement results of X I , . . . , X N , t he same funct i onal rel at i on t hat describes the physical system, t o which the measur ed quant i t i es bel ong as paramet ers, is assumed t o hol d also for t he measur ement values and t he usual rules for the eval uat i on of uncer t ai nt y in indirect measurement s are applied. Specifically, t he measur ement value y of Y is det ermi ned using the following rel at i on bet ween the measur ement values: y = f ( k l r l , . . . , k N r N ) , where k i r l is t he measur ement value of X~, k~ the cal i brat i on f act or and r i the readi ng val ue of the i nst r ument used for measuri ng or generat i ng Xi. As suggested in clause A.1 of [ 6] , t he funct i on f shoul d express not simply a physical law but the measur ement process used, and in part i cul ar, it shoul d cont ai n all quant i t i es X~ t hat can cont r i but e a significant uncer t ai nt y component t o t he measur ement result (e.g. cor r ect i on factors, quant i t i es t hat t ake i nt o account ot her sources of vari abi l i t y such as time and measur ement conditions). given by the rel at i on ~{m~ m ~ _ rnN~ rh " = u" / - - ~- r "~- ± "'" + ~-N) \ u~ u2 ( A . 1 ) where u~ is t he uncer t ai nt y associated with m i , and u,, is the uncert ai nt y associated with m, for which it is 1 2 (A.2) U m - - 1 1 1 . ~ + .75 +... + .-i- U 1 U2 UN Usi ng (A.1) and (A.2) the uncert ai nt y of m ~ - m results: U i m = N / N 2 __ Um2. (A.3) Fr om (A.3) the expressi on (6) of t he compat i bi l - ity i ndex bet ween Mi and M is obt ai ned. Mor eover , from (A.3) and (4) it follows t hat the correl at i on coefficient bet ween M~ and M is given by U m Ri,, - (A.4) U i " 4 . C o n c l u s i o n s Fur t her concept s and definitions not explicitly consi dered in t he I SO Gui de (such as cal i brat i on di agr am and factor, measur ement compat i bi l i t y and t raceabi l i t y level) seem necessary to fulfil t he scope of t he Gui de, part i cul arl y in it bei ng suitable to appl i cat i ons at all levels of the measur ement hi erarchy. The utility of i nt r oduci ng these concept s in addi t i onal , mor e specific guidelines or st andar ds is shown by t hei r appl i cat i on t o some significant cases of cur r ent measur ement practice. A p p e n d i x Let mi, with i =l , 2, . . , N, be t he measur ement values obt ai ned by different l abor at or i es considered i ndependent . Then the weighted mean m is It can also be not ed t hat in the part i cul ar case N= 2 from (6) it follows: I~1 = I~2 = I ml - m21/ V / ~ l + uZ2 , (A.5) i.e. t he compat i bi l i t y indexes with the weighted mean M of ei t her M1 or M2 are equal and, in addi t i on, are equal to t he compat i bi l i t y index bet ween M1 and M2. R e f e r e n c e s [1] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML, Guide t o t he Expressi on o f Uncert ai nt y in Measurement . Fi r s t Edition, 1993, ISBN 92-67-10188-9. [2] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML, Int ernat i onal Vocabul ary o f Basi c and General Ter ms in Met rol ogy, Second Edition, 1993, ISBN 92-67-01075-1. 1-3] Norma UNI 4546, Mi sure e misurazioni. Termi ni e def i ni zi oni f ondament al i , Novembre 1984. [4] Document 66(Italy)ll, New Edition of IEC 359 2nd E. Arri et al./Measurement 16 (1995) 51-57 57 Edition 1987 Expression of the Performance of Electrical and Electronic Measuring Instruments, June 1990. I-5] S. D' Emilio, M.L. Rastello, P. Soardo, A. Peut o and S. Sartori, I campioni de| sistema nazionale di t arat ura, AEI 80/ 5) (1993) 36-43. I-6] B.N. Taylor and C.E. Kuyatt, Guidelines for evaluating and expressing the uncertainty of NIST measurement results, NI S T Technical Note 1297, 1994 edition. [ 7] WECC Doc. 15, Gui de for the organization of interlaboratory comparisons, 1992 edition. [ 8] K. Weise and W. W6ger, A Bayesian theory of measurement uncertainty, Meas. Sci. Technol. 411993) 1 11. 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