DEGREES OF EQUIVALENCE IN A KEY COMPARISON 1 Thang H. L., Nguyen D. D. Vietnam Metrology Institute, Address: 8 Hoang Quoc Viet, Hanoi, Vietnam

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1 DEGREES OF EQUIVALECE I A EY COMPARISO Thang H. L., guyen D. D. Vetnam Metrology Insttute, Aress: 8 Hoang Quoc Vet, Hano, Vetnam Abstract: In an nterlaboratory key comparson, a ata analyss proceure for ths comparson was propose an recommene by CIPM [,, 3], theren the egrees of equvalence of measurement stanars of the laboratores partcpate n the comparson an the ones between each two laboratores were ntrouce but a corresponng clear an plausble measurement moel was not gven. Authors n [4] offere possble measurement moels for a gven comparson an a sutable moel was selecte out after rgorous analyzng steps for epectaton values of these egrees of equvalence. The systematc laboratory-effects moel was then selecte as a rght one n ths report. Those moels were all base on the one true value estence assumpton. However n the year 008, a new verson of the Vocabulary for Internatonal Metrology (VIM [7] was ssue where the true value of a gven measurement stanar shoul be now perceve as multple true values whch followng a gven statstcs strbuton. Applyng ths percepton of true values of a measurement stanar wth combnaton of the steps n [4], measurement moels have been evelope an egrees of equvalence have been analyze. The results show that although wth new efnton, the systematc laboratory-effects moel s stll the reasonable one n a gven key comparson. I. Introucton In reference [], concept of egrees of equvalence between laboratores was state as one of mportant crtera n Mutual Recognton Arrangement (MRA between atonal Metrology Insttutes (MIs. Degrees of equvalence are efne n [] as followng: Degree of equvalence of a measurement stanar: the egree to whch the value of a measurement stanar s consstent wth the key comparson reference value. Ths s epresse quanttatvely by the evaton from the key comparson reference value an the uncertanty of ths evaton. The egree of equvalence between two measurement stanars s epresse as the fference between ther respectve evatons from the key comparson reference value an the uncertanty of ths fference. Mathematcally, the egree s epresse as = - an u ( = u ( - u (. The egree of equvalence between two measurement stanars s epresse as = - an u ( = u ( + u ( [3]. To llumnate the statstcs natures of those quanttes, measurement moels for a key comparson have been offere an analyze n [4]. In those moels, a gven measurement stanar s assume havng only one true value. Actually, as scusse n [7], a Emal: thanglh@vm.gov.vn

2 more general vew shoul be of unerstanng that for a gven measurement stanar, there est a set of true values whch we then assume followng a gven statstcs strbuton. II. Mathematcal moelng Let conser a gven key comparson where a measurement quantty havng a set of true values Y, = to ( s the number of partcpants whch s followng a unque stable strbuton urng the comparson tme. The epectaton an varance of Y wll be an V(Y = s (Y. Call X, X X an, are epectaton values an measure values of the measurement quantty measure an prove by the th laboratory. Each measure value wll have a relable measurement uncertanty u(. Call b = (X Y, b = (X Y,, b = (X Y. The set of b, b,, b are not always zero ue to some unrecognzable errors urng the measurement but all of the measurement values of a certan laboratory shoul stll have the same epectaton value. et some measurement moels wth fferent assumptons wll be evelope an ther analyss wll be carre out.. one laboratory effect In ths case the measurement equaton wll be of the form: ( The equaton for epectaton values wll be: E( = X. Here b = 0 mples the partcpatng laboratory makes no errors on the measurement or all the errors were recognzable an correcte. The corresponng varance equaton wll be: V( = V(Y + V(e or V( = s (Y (. Ranom laboratory effect The measurement equaton wll be: (3 The epectaton equaton: E( or E( (4 where b s assume to follow a statstcs strbuton wth zero epectaton. The varance equaton: V( = V(Y + V(b + V(e or V( = s (Y (5 3. Systematc laboratory effect The measurement equaton wll be: (6 where b becomes a constant now. The epectaton an varance equaton:

3 or E( (7 V( = V(Y + V(b + V(e (8 E( (9 III. ey reference values. one laboratory effect The key reference value:. Ranom laboratory effect The key reference value: 3. Systematc laboratory effect The key reference value: V( = V(Y (0 = (Σ, u( = / (Σ ( = (Σ, u( = / (Σ ( = (Σ, u( = / (Σ (3 IV. Degrees of equvalence. one laboratory effect Measurement moels of any two partcpatng laboratores: an (4 Devaton of measure values of two laboratores: = - - Y - e (5 Devaton of a measure value an the key reference value: = - (6 The epectaton values: E( - - E(e = 0, E( = E( - E( E(

4 - = 0 (7. Ranom laboratory effect Measurement moels of any two partcpatng laboratores: an (8 Devaton of measure values of two laboratores: = - - Y - b - e (9 Devaton of a measure value an the key reference value: = - The epectaton values: (0 E( E( - = 0 an = 0 ( 3. Systematc laboratory effect Measurement moels of any two partcpatng laboratores: và ( Devaton of measure values of two laboratores: = - - Y - b - e (3 Devaton of a measure value an the key reference value: = - The epectaton values: / (u (Y (4 E( / (u (Y / (u (Y - - Σ b / (u (Y = E(b - Σ b / (u (Y = b b / (u (Y an E( - - E(b - E(e = b - b (5

5 V. Dscusson The approach n ths report accepte the assumpton of estence of a set of true values nstea of the estence of only one unque true value for a gven measurement stanar of the artfact n a key comparson. Those true values are strbute n a common probablstc ensty functon. The corresponng egrees of equvalence, or n other wors, the evatons an ther measurement uncertantes are then analyze. It s then seen that f a gven partcpatng laboratory not contrbute any error to the measurement or the error contrbute of ths laboratory to the measurement s ranom n nature as seen n equatons (7 an (, then the epectatons are always zero. These mply that the laboratores uner queston are always equvalent whch s not a reasonable acceptance. Ths fact mples that they shoul not be goo moels for a key comparson. In contrast, f a partcpatng laboratory contrbute to the measurement a systematc error then the epectatons of evatons are not possbly zero n all cases as seen n equaton (5. The systematc errors commtte by each one b an b an ther uncertantes wll efntely ece f they are equvalent or not. An then ths moel coul be assgne to be a goo moel to escrbe the measurement process. It s worthy to notce that ths concluson s concent to the one n [4]. VI. Concluson In ths report, the egree of equvalence s consere n three fferent moels. The eplct evatons of each laboratory pars an that of one laboratory wth the key reference value are erve. The epectatons of the evatons an then the egrees of equvalence are analyze for each moel wth the assumpton of multple true values. The result support that the laboratory s systematc error moel s the accepte one. The result s smlar to the one n [4]. Acknowlegement: Dr. guyen Duc Dung an Dr. Tran Bao have contrbute to the scusson an mnutes of the report. References [] Internatonal Commttee for Weghts an Measures (CIPM: Mutual recognton of natonal measurement stanars an of calbraton an measurement certfcates ssue by natonal metrology nsttutes, Techncal Report, 999 ( [] Internatonal Commttee of Weghts an Measures (CIPM: Guelnes for CIPM key comparsons, March 999, ( [3] M. G. Co: The evaluaton of key comparson ata, Metrologa 39, pp , 00. [4] R.. acker, R. U. Datla, an A. C. Parr, atonal Insttute of Stanars an Technology, Gathersburg, MD USA: Statstcal Interpretaton of ey Comparson Reference Value an Degrees of Equvalence, J. Res. atl. Inst. Stan. Technol. 08, (003. [5] BIPM key comparson ata base ( [6] Evaluaton of measurement ata Gue to the epresson of uncertanty n measurement (

6 [7] Internatonal vocabulary of metrology Basc an general concepts an assocate terms (VIM (