A New Conception of Measurement Uncertainty Calculation

Transcription

1 Vol. 14 (013) ACTA PHYSICA POLONICA A No. 3 Optical ad Acoustical Methods i Sciece ad Techology 013 A New Coceptio of Measuremet Ucertaity Calculatio J. Jakubiec Silesia Uiversity of Techology, Faculty of Electrical Egieerig Istitute of Metrology, Electroics ad Automatio, Akademicka 10, Gliwice, Polad The paper presets a coceptio of ucertaity calculatio of a result obtaied i a direct measuremet realized i coditios described by radom errors. The coceptio basis o the error deitio beig a eect of aalysis of a quatizatio process ad, rst of all, it permits to determie ucertaity of a sigle measuremet result i measurig ad cotrol systems processig sigals varyig i time. Divisio of the errors ito two types A ad B permits elaboratio of such a procedure which eables ucertaity calculatio for a average value of a series of measuremets i the way close to this oe proposed by GUM ad widely discussed i last years. Theoretical cosideratios are illustrated by examples showig practical properties of the preseted ucertaity calculatio procedures. DOI: /APhysPolA PACS: 06.0.F, Ta, f 1. Itroductio The formula commoly used for ucertaity evaluatio of measuremet results is based o the followig equatio: N ( ) f u (y) = u (x i ) (1) x i take from GUM [1] which is treated i metrology as a cotaier of fudametal rules dealig with iaccuracy of measuremet data. Equatio (1) is called i it as the law of ucertaity propagatio ad determies relatios betwee variatios of quatities described i probabilistic categories. Quatities x i, i = 1,,..., N, are measured directly ad the, o the base of its estimates, a value of the quatity y is calculated with assumptio that a relatio betwee these quatities is kow as the fuctio which geerally ca be writte as y = f (x 1, x,..., x N ). () Variaces u (x i ), i = 1,,..., N, i Eq. (1) are deed as squares of suitable so-called stadard ucertaities which are i fact stadard deviatios (or their estimates) of quatities measured directly. Variace u (y) is treated as a square of stadard ucertaity u(y) of the quatity y measured idirectly. Equatio (1) ca be also used to determie stadard deviatio (stadard ucertaity) of a quatity measured directly [1]. I this case, all dieretials i (1) are equal to 1, therefore, it may be writte i the form: u = u A + u B1 + u B u BJ, (3) where the ucertaities have bee divided ito two categories. Ucertaity u A, amed as type A, is calculated statistically as a parameter of a series of measuremet results. The ucertaities of type B: u B1, u B,..., u BJ are determied o the base of kowledge about measuremet coditios. jerzy.jakubiec@polsl.pl The fudametal problem, dealig with the formulae applied for ucertaity calculatio obtaied o the basis of (3), cosists i lack of formal agreemet betwee its elemets. Oe should otice that the cosidered ucertaities have to be see as stadard deviatios of the same kid radom quatities iueced o iaccuracy of a direct measuremet. Takig ito accout the fact that type A ucertaity is calculated o the basis of a series of measuremet results, oe caot poit ay other series which is obtaied from the same measuremet experimet. Therefore, the ucertaity u A caot be combied with ay other ucertaity. Oe may poit some more weakess of the ucertaity calculatio procedure based o GUM [1]. The most essetial oe results from the fact that this procedure eeds a series of measuremet results to determie ucertaity of type A, thus, it caot be used for ucertaity calculatio of a sigle measuremet result. I may situatios, especially happeig i idustrial measurig ad cotrol systems [5], measured quatities are varyig i time ad there are o possibilities to obtai more measuremet results tha oe. Let us aalyze (3) oce more. The basic questio is: which assumptios should be take i order to obtai formal acceptace of this equatio? It is obvious that variaces (squares of stadard deviatios or stadard ucertaities) ca be added if they are parameters describig some ucorrelated radom quatities. I the discussed situatio, these quatities have to be treated as other quatities weighted accordig to how the measuremet result varies with chages i these quatities [1], which meas that they are agets iuecig o accuracy of the measuremet result. From the mathematical poit of view, such agets are cosidered as radom errors [6]. Therefore, oe ca say i coclusio that the variaces, added accordigly with (3), are parameters of suitable radom errors which describe iueces of factors essetial for iaccuracy of the result measured directly [7]. Takig ito accout above, the error equatio i the form (436)

2 A New Coceptio of Measuremet Ucertaity Calculatio 437 e = e A + e B1 + e B e BJ (4) ca be treated as a formal basis of (3). Accordig to (4), the realizatio of the total (combied) radom error e is the sum of realizatios of the ucorrelated radom partial errors deoted as e A, e B1,..., e BJ. I this case, the error variaces ca be summed as it is described by (3). Oe should otice that the partial errors i (4) have the same idexes as the variaces i (3) because divisio of the errors ito two categories A ad B, appropriate to the ucertaities A ad B [1], is useful i some situatios occurrig i practice. Let us take the that the error of type A is such a kid of a error, probability distributio of which is determied by usig statistical meas while descriptios of the errors of type B are obtaied i the other ways based o the degree of belief that a evet (i.e. realizatio of the error) will occur [1]. Havig give variatio of the error, oe ca add it to the variatios of the other errors accordigly with (3) ad extract the square root which ally results i obtaiig the stadard deviatio of the total error. As a rule, the ext operatio cosists i calculatio of the expaded ucertaity, which is performed as multiplicatio of the stadard deviatio by the coverage factor [1]. This factor depeds o distributio of the total error which meas that (3) i its clear form may be used for the expaded ucertaity evaluatio oly if the total error ca be described by the ormal distributio, i.e. the cetral limit theorem [8] is compulsory durig combiig the partial errors. I other cases Eq. (3) is useless. From these begiig cosideratios, oe ca draw the coclusio that moder metrology eeds a ew look to the problem of ucertaity calculatio of a measuremet result. Mai part of the paper is devoted to presetatio of the coceptio which permits to solve this problem for measuremets performed both i laboratories ad i idustrial measurig systems. The base of this coceptio is a measuremet error deitio.. Measuremet error i descriptio of a measured result To obtai the measuremet error deitio, oe ca aalyse the case whe a sigle measuremet is realized. This case is typical for measurig systems [4, 5], iput sigals of which usually vary i time. Therefore, every measuremet has to be performed i two stages: at rst, the sigal is sampled at the precisely determied momet ad, after that, the sample is measured by a aalog-to- -digital (AD) coverter. AD coversio is realized oly oe time, so oe obtais oly oe measuremet result for every sample ad there is o possibility to repeat this actio for the same istataeous value of the sigal. From measuremet poit of view, AD coversio ca be described as a direct compariso of the measured quatity with a sum of the same kid elemetary measuremet stadards called quata [4, 5]. Every quatum has the same value which is much less tha a workig rage of a AD coverter. I voltage AD coverters, the quata sum is usually obtaied by usig a voltage divider built from resistors havig the same values R or coectios R with R [5]. Geerally, every measurig istrumet workig o the described above rule is called a quatizer. Its scheme, i the simplest form, is show i Fig. 1a. Fig. 1. (a) Basic scheme of a quatizer, (b) iterpretatio of a quatizatio result. A quatizatio process ca be performed i may differet ways [5], but i every case the essece of this process cosists i compariso of a value of the quatized quatity x with the sum of quata (every has the value q) delivered from the quata cotaier. Let us take that if x q > 0 state of comparator COMP s = 1, i the other case s = 0. The cotrol circuit icreases umber of quata as log as s = 0. Deotig the al umber of quata as q +1, where q is the dimesioless idicatio of the quatizer beig the umber of quata assiged to the value of measured quatity x, oe ca describe the quatizatio result as a iterval i the way show i Fig. 1b. Accordigly with Fig. 1b, the quatizatio process cosists i assigmet of the selected iterval, determied i real umber set R, to the measured quatity x. Such a assigmet is caused by the quatum character of the stadard used i this process ad let us to write that the true value of the measured quatity fulls the iequality q q < x ( q + 1) q, (5) which describes the relatio betwee value of measured quatity x ad umber q obtaied as a dimesioless result of the quatizatio process. For further cosideratios, oe deotes x = q q, (6) where x is the dimesioed idicatio expressed i uits of the measured quatity x ( x ca be called also a raw measuremet result). After itroducig expressio (6) to (5), oe obtais x < x x + q (7) ad the 0 < x x q. (8) Such a kid of dierece as i (8), i.e. betwee a value of a measured quatity ad a umber obtaied as a result of a measuremet is commoly called a measuremet error. I the described situatio, measuremet is performed by a quatizer ad the error is deed as:

3 438 J. Jakubiec e x = x x. (9) Equatio (9) describes how much the measured value x diers from the suitable idicatio x of a quatizer. Geerally, this deitio ca be applied for every error of a measuremet result. After itroducig the deitio (9) to iequality (8), we have 0 < e x q, (10) which meas that, i the cosidered situatio, values of the quatizatio error chage i limits from 0 to q. Iterval (10) is ot symmetrical. I this case, values of the error ca be reduced (corrected) by subtractig its average value, equal to 0.5q (see Fig. ), i the followig way: eˆx = e x 0.5q. (11) Couplig Eq. (11) ad iequality (10) oe obtais 0 < e x + 0.5q q, (1) which meas that the corrected quatizatio error chages i symmetrical iterval 0.5q < eˆx 0.5q. (13) The same result, as described by expressio (13), ca be obtaied by replacig the idicatio x with its corrected value [5]: ˆx = x + 0.5q. (14) Accordigly with iequality (13), error eˆx takes the least possible values. Takig it ito accout, corrected idicatio ˆx ca be treated as the best represetatio of the true value obtaied from the measuremet process ad called a evaluatio of a measured quatity. Fig.. Distributios of the quatizatio error: (a) with systematic compoet, (b) after correctio of the systematic compoet. It should be oticed that the correctio may be doe at the level of the dimesioless result q. After addig 0.5 to q i the expressio (6), oe obtais the same eect as described by iequality (13) ad Eq. (14). Iequalities (10) ad (13) determie oly the limits of the quatizatio error. If oe assumes that every measured value is as probable as the other values from the iput rage of the AD coverter, the quatizatio error ca be described i probabilistic categories. I this case the probability desity fuctios of the error have the rectagular shapes show i Fig. a ad b, for Eqs. (10) ad (13), respectively [5]. The distributios from Fig. a ad b dier oly with the expected values: the distributio from Fig. a has this value equal to q/ while this oe from Fig. b has the expected value equal to 0. This property lets us to put the hypothesis that, from the ucertaity calculatio procedure poit of view, the measuremet errors ca be divided ito two categories: systematic ad radom. Values of the systematic errors are kow ad may be elimiated by addig correctios to a measuremet result, as it has bee doe i (14), or to the error (11). Such a elimiatio results i reducig the expected value of the radom error to 0. Therefore, the radom errors are described i probabilistic categories, usually by usig a probability desity fuctio (geerally: by a distributio) with the zeroed expected value. A source of a measuremet error is geerally characterized by a set of possible values which ca be take by this error i the determied measuremet coditios. A error value set ca be described i two ways: i probabilistic categories [6] or by a determiistic fuctio [7]. A descriptio of radom properties of the error takes usually the form of a distributio which is a fuctio of the error value. A determiistic descriptio is used whe it is possible to determie a quatity, which a error depeds o, ad a fuctio that describes this depedece. Such a quatity is called a iuece quatity. There are two ways of usig the determiistic fuctio describig properties of a error. The rst oe cosists i calculatig the error value for the kow value of the iuece quatity. After that it is possible to use the calculated value as the correctio i the way described above. The secod way is applied if there is ot possibility to kow the value of the iuece quatity but there are kow limits of its chageability i measuremet coditios. I this case, oe ca assume that the iuece quatity chages uiformly withi the give limits. It permits to determie frequecy of the error value occurrece which ca be iterpreted as a descriptio of the probability desity fuctio of the error [5]. The same error ca be described i a determiistic way or i probabilistic categories depedetly o measuremet coditios. This property is illustrated by Example 1. Example 1. Let assume that temperature t varies from t = 10 C to t + = 30 C ad iueces liearly o error e as it is show i Fig. 3a. Therefore, the iuece fuctio ca be writte as e(t) = at + b, where a = e (t +) e (t ) = t + t = /K ad b = e (t ) a t = = 0. Havig temperature t kow, oe ca calculate e(t) ad use it as the correctio. It is impossible if oly limits t ad t + are kow. I this case oe ca assume that all values of the temperature are of the same probability i iterval [t, t + ], which permits to determie probability desity fuctio g(e) of error e show i Fig. 3b [5]. Distributio from Fig. 3b is rectagular withi limits: e(t ) = ad e(t + ) = I such a case,

4 A New Coceptio of Measuremet Ucertaity Calculatio 439 Fig. 3. (a) Determiistic descriptio of exemplary error, (b) its probabilistic descriptio with systematic compoet, (c) probabilistic descriptio of the error after correctio of the systematic compoet. the expected value of error e is equal to E(e) = ( )/ = 10 3, which meas that oe ca decrease every value of the error by subtractig correctio c = E(e) = The same result ca be obtaied addig correctio c to the measuremet result (see (14)). After correctio (see Fig. 3c), the error distributio becomes symmetrical i relatio to the vertical axis ad its chageability rage is described by limits lower : e = e(t ) c = = ad higher : e + = e(t + ) c = = Ucertaity as a parameter of a measuremet error value set From the preseted above cosideratios, oe ca draw the coclusio that iaccuracy expressio of a measuremet result has to be based o mathematical descriptio of error burdeig this result. The error deitio itroduced as (9) is ot take arbitrary but it has the form resultig from aalysis of the quatizatio process. Takig ito accout that properties of this process ca be treated as represetative for may measuremet situatios, the deitio of a measuremet error e ca be geerally writte i the form e = x ˆx, (15) where ˆx is a evaluatio of a measuremet result, the value of which is the most closer to the true value of a measured quatity. It meas that performig a sigle measuremet oe caot obtai a better value of a measured quatity tha a evaluatio. To the ext cosideratios, oe assumes that the evaluatio is obtaied by addig correctio to the raw measurig result i this way that, after correctio, the measuremet error has expected value equal to 0. Therefore, a error source burdeig a evaluatio is radom ad it is described by probability desity fuctios with zeroed expected values. Equatio (15) is very importat from ucertaity defiitio poit of view. I GUM [1] the term ucertaity of measuremet is geerally iterpreted as doubt about the exactess or accuracy of the result of a measuremet. Usig probabilistic categories, this descriptive deitio of the ucertaity ca be writte i the mathematical form as the expressio: Pr [ x ˆx U] = p, (16) where Pr meas probability of such a evet that absolute value of the dierece betwee ukow true value of a measured quatity x ad its evaluatio ˆx is equal or less tha ucertaity U. Accordigly with (16), the probability is equal to codece level p, the value of which is typically take as p = Havig give error deitio (15), oe ca itroduce it to expressio (16) which after that takes the form P r [ e U] = p. (17) If the error is described by symmetrical probability desity fuctio g(e) with the expected value equal to 0, oe ca write relatio (17) as the fuctioal U U g(e)de = p. (18) Example. Let us calculate ucertaity caused by the error described by probability desity fuctio show i Fig. 3c. This fuctio ca be writte as g(e) = a for e e e +, (19) g(e) = 0 otherwise, where the value of coeciet a ca be determied by usig the equatio resultig from the fact that fuctio g(e) has to satisfy the ormalizig coditio g(e)de = 1. (0) Takig values e ad e + from Example 1 ad basig o (19) ad (0), oe obtais a = 500. Fig. 4. Graphical iterpretatio of ucertaity U calculated i Example. Havig give the error distributio as (19), oe ca determie value of the ucertaity from expressio U U ade = (1) which has bee obtaied o the basis of (18). Takig ito accout that a = 500, Eq. (1) is satised for U = Graphical iterpretatio of ucertaity U is give by usig Fig. 4. It shows that ucertaity is a parameter of a error value set: it describes limits withi which the area uder the probability desity fuctio g(e) is equal to p = 0.95 (95% of the whole area uder this fuctio). 4. Expressio of a sigle measuremet result as a iterval The commo approach to descriptio of a measuremet result cosists i writig it as a umerical iterval,

5 440 J. Jakubiec the limits of which are determied by a ucertaity [1]. Such a kid of descriptio of measuremet data is frequetly used i the case whe dieret mathematical meas are applied for presetatio these data i metrological categories [915]. Iterpretatio of the ucertaity give by Fig. 4 directly leads to the presetatio of a sigle measuremet result as a umerical iterval. From (16) it results that probability of dig the true value of the measured quatity iside the iterval described as x ˆx U () is equal to p. Trasformig iequality (), oe obtais expressio: ˆx U x ˆx + U (3) describig limits of the iterval which represets measurig quatity x after its sigle measuremet. The lower limit of the iterval is x = ˆx U, (4) while the upper oe x = ˆx + U. (5) Therefore, the middle of the iterval has the form [15]: mid(x) = x + x ˆx + U + (ˆx U) = = ˆx (6) ad the radius rad(x) = x x ˆx + U (ˆx U) = = U. (7) The expressios preseted above show that every sigle measuremet result should be determied as the iterval cotaiig real umbers which meas that measured quatity x is trasformed to iterval x i eect of a measuremet process. The iterval is built aroud the evaluatio ˆx beig its middle (6) while iterval radius is equal to ucertaity U. Therefore, the iterval represetig a measuremet result with probability p (it is called 95% coverage iterval [16] for p = 0.95) ca be writte as x = [x, x] = [ˆx U, ˆx + U]. (8) Processig measuremet data by usig algorithms [5] eeds descriptio of the iterval i form (8) as two additive parts x = ˆx + [ U, U]. (9) I form (9), the iterval is composed of evaluatio ˆx ad symmetrical iterval [ U, U] (its radius is equal to 0) which ca be called a ucertaity iterval. If the probability desity fuctio is o-symmetrical, the ucertaity iterval is ot symmetrical, too. Mathematical meas that ca be used i this case for determiatio of a measuremet result as the iterval have bee described i [5]. 5. Combied ucertaity I practice, every measuremet result is burdeed by may errors which meas that the total ucertaity is a combiatio of partial ucertaities describig iueces of the partial errors o iaccuracy of a measuremet result. Oe ca poit four mai ways of calculatig the combied ucertaity. The rst oe cosists i determiatio of stadard deviatio (stadard ucertaity) accordigly with (3) ad multiplicatio it by the coverage factor [16], value of which depeds o both p ad shape of distributio of the combied error beig sum (4) of the partial errors. I the secod way, all calculatios are performed o ucertaity itervals determied for each partial error [17]. Possibility of takig ito accout correlatios betwee partial errors is importat property of this method [18]. The third way ca be realized if combied error e c is described as the sum of N ucorrelated errors N e c = x i. (30) Kowig distributios of partial errors e i, i = 1,..., N, oe ca determie the probability desity fuctio of the combied error makig multi-step covolutio accordigly with [4]: g c (e c ) = g 1 (e 1 ) g (e )... g N (e N ), (31) where g i (e i ) is probability desity fuctio of i-th error ad is symbol of covolutio. Hawig determied g c (e c ), oe ca use (18) to calculate the combied ucertaity. The last procedure, most useful i practice ad recommeded by [16], is based o error model (30) ad applied probabilistic simulatio called Mote Carlo method. Except of makig covolutio (31), it applies (30) for summig up realizatios of the partial errors, as it is show i Example 3. The procedure is performed i may steps (i practice oe uses about steps). Every step cosists i summig values of partial errors take from populatios described by kow probability desity fuctios. The sum of realizatios is located i the set cotaiig values of the combied error. At the ed of this procedure, the ucertaity is calculated o the base of this set by usig fuctioal (18). Fig. 5. Histogram of the combied error obtaied i Example 3. Ucertaity calculated from this histogram for codece level p = 0.95 is U = Example 3. Let assume that the combied error is the sum of two ucorrelated partial errors, i.e.: e c = e 1 + e. Error e 1 is described by rectagular probability desity fuctio show i Fig. 3b while the distributio of error e is ormal with stadard deviatio σ = 0.33

6 A New Coceptio of Measuremet Ucertaity Calculatio The combied error calculated by usig Mote Carlo method is show i Fig. 5. More details about applicatios of this method to calculate ucertaities as parameters of errors ca be foud i [5]. 6. Errors of type A ad B To calculate ucertaity of a measuremet results accordigly with the described procedure, it is ecessary to have give the probability desity fuctio of the total error. It meas that if the total error is sum (30), distributios of all partial errors must be kow. Therefore, descriptios of the errors have to be determied a priori, i.e. before a measuremet realizatio. However, this requiremet ot always has to be fullled. Oe ca poit the situatio whe the error distributio may be determied after a measuremet process o the basis of a series of measuremet results. I this case, it is ecessary to divide partial errors ito two categories: type A ad type B, accordig to (4). Takig ito accout cosideratios preseted above ad these oes from Sect. 1, oe ca qualify a error as type B if its distributio is kow a priori while the distributio of type A error should be determied o the basis of a measuremet series. Acceptace of this deitio imposes specic requiremets o properties of the errors of type A ad B if oe takes ito accout that both types of the error are radom. At rst, it meas that all realizatios of errors burdeig measuremet results i a series should be described as arisig from oe source, i.e. as the error of type A. At secod, distributio of this kid of error should be described as ormal because i practice oly oe parameter, i.e. a variace, is estimated o the basis of a series of measuremets, which is caused by a limited umber of results formig such a series. Oe has to poit a essetial feature of errors if they are divided ito these two categories. I the coditios, i which a measuremet series is obtaied, all errors of type B take costat values because if ot every chage of some B type error is qualied to the set of realizatios of A type error. Therefore, accordigly with (4), sum of B type errors ca be writte as e B1 + e B e BJ = c, (3) where c is costat. The described above feature is importat because it eables obtaiig a estimate of the variatio of the A type error o the base of a series of measuremet results. Let us otice that after itroducig (4) ad (3) to deitio of a measuremet error (15) it takes the form e A + c = x ˆx. (33) I Eq. (33), measured quatity x has costat value because its ivariability is the basic assumptio take for realizatio of a measuremet series. Value of c is also costat, therefore, from (33) it results that realizatios of two radom variables: error e A ad evaluatio ˆx dier i a costat value ad have opposite sigs. It meas that both variables have the same variaces. Takig this ito accout, oe ca calculate a estimate of the variace of error e A o the base of a series of measuremet results {ˆx i, i = 1,..., }, is the total umber of measuremets. The maximum likelihood estimator of the variace is determied as [8]: s = 1 (ˆx i x), (34) 1 where x is the average value of the series described by expressio x = 1 ˆx i. (35) The followig example illustrates usig a measuremet series to determiatio of A type error distributio. Example 4. Measuremet process has delivered the series of 10 results: ˆx i = {1.0003, , , , , , , , , }. O the basis of these results the average value, calculated accordigly with (35), is equal to x = ˆx i = = I this case, the variace estimator, described by (34), has the value s = 1 10 (ˆx i x) = = Error of type A has the same variace as the series of measuremets. Oe assumes that all systematic errors have bee corrected, thus, the expected value of this error is equal to 0. I this situatio, takig ito accout that the ideticatio procedure may be used oly for radom errors with ormal distributio, oe ca describe the distributio of A type error as ormal N (0, σ A ), where σ A is stadard deviatio estimated as σ A = s = = Ucertaity of a average value of a measuremet series Havig determied distributios of A type error ad errors of type B, oe ca calculate ucertaity of a sigle measuremet result i the described way. However, i practice the same value of quatity is ofte measured may times which eables reducig iaccuracy if radom error of type A domiates over the other errors (of type B). I this case, it is ecessary to perform a series of measuremets, o the basis of which average value (35) is calculated. Its ucertaity ca be obtaied i the way described i [5] with assumptio that calculatio of the average value is a kid of processig algorithm realized o a series of measuremet results. This method is characterized below. A processig algorithm operates o measuremet data described by usig a sigle measuremet result model. This model, built o the base of error deitio (9), has the form x = ˆx + e (36) obtaied with assumptios that all systematic errors have

7 44 J. Jakubiec bee corrected. Equatio (36) meas that a ukow true value of measured quatity x is the sum of its evaluatio ˆx (obtaied after correctio of a sigle measuremet result) ad a realizatio of total (combied) error e. Geerally, the combied error is give by sum (30) of partial errors which are described i probabilistic categories ad ucorrelated as well as its probability desity fuctios have expected values equal to zero. Let us use model (36) i situatio whe a result of a measuremet process is give as a series {ˆx i, i = 1,..., } composed o evaluatios of quatity x, which has a costat value durig the process performed i the same measuremet coditios. If every measuremet result from this series is described by model (36), the average value (35) takes the form 1 x i = 1 (ˆx i + e i ). (37) Takig ito accout that durig the measuremet experimet the true value of the measured quatity does ot chage, i.e. x i = x = cost, i = 1,...,, expressio (37) ca be writte as x = 1 (ˆx i + e i ) = 1 ˆx i + 1 e i = x + 1 e i, (38) which meas that if the evaluatio of the measured quatity is give as average x (35), the total error of this average value is described by equatio e x = 1 e i. (39) The measuremet results formig the series are obtaied i the same coditios which meas that every result is burdeed by errors of the same kid. Takig this ito accout ad usig descriptio of errors type A ad B i the form (4), oe ca write (39) as e x = 1 [e A (i) + e B1 (i) + e B (i) e BJ (i)]. (40) As it has bee discussed above, the series is measured with assumptio that errors of type B have costat values. Therefore, for every error of type B, it takes place 1 [e Bj (i)] = 1 [e Bj ] = e Bj, j = 1,..., J. (41) I this case, Eq. (40) takes the form e x = 1 [e A (i)] + e B1 + e B e BJ. (4) Error of type A is described i Eq. (4) as the sum of errors which burde measured results beig elemets of the series. Realizatios of every partial error are take from the same populatio of the radom error e A, thus variatio of this sum ca be writte as σ A x = [ ] σ A = σ A = σ A, (43) where σa is variatio of A type error. Takig ito accout the fact that this variatio may be estimated from the measuremet series by usig (34), oe ca determie estimate of the stadard deviatio of the A type error for the average value (35) of the series o the base of equatio σ σ A x = A s = = 1 (ˆx i x). (44) ( 1) Equatio (4) shows that errors of type B should be itroduced if there is a eed to expad measuremet coditios i relatio to these oes i which the series has bee obtaied. Errors of type B describe all additioal factors, radom properties of which appeared if iuece quatities chage their values. I such a case, basig o (4) ad (44), oe ca calculate stadard deviatio of the total error as σ x = σa x + σ B1 + σ B σ BJ, (45) where σ B1, σ B,..., σ BJ are stadard deviatios of type B errors. Equatio (45) is usable for calculatio of the expaded ucertaity if oe disposes iformatio that the combied error has the distributio close eough to the ormal oe. I this case, the ucertaity is obtaied by multiplyig stadard deviatio (45) (stadard ucertaity) by the proper coverage factor suitable for take value of the codece level p [16]. Otherwise, oe ca use two ways. The rst way cosist i determiatio of the total error distributio by usig covolutio (31) or Mote Carlo method ad the, usig fuctioal (18) to calculate the ucertaity. Applicatio of the secod way eeds kowledge about values of expaded ucertaities of the partial errors. Havig it, oe ca calculate the composed ucertaity usig reductive iterval arithmetic described i [17, 18]. 8. Basic procedures of ucertaity calculatio Takig the preseted cosideratios ito accout, oe ca poit two basic procedures which ca be used i practice to calculate ucertaity of a measuremet result: Procedure 1 applied if oly a sigle measuremet has bee performed. Procedure the al measuremet result is calculated as average value (35) o the basis of the measuremet series. Procedure 1. This procedure is realized i the followig steps: 1. I the begiig, oe should determie all partial error sources ad describe them by usig symmetrical probability desity fuctios with zeroed expected values. I the case whe expected value of

8 A New Coceptio of Measuremet Ucertaity Calculatio 443 the partial error diers from zero, it is eeded itroducig the proper correctio i the way described i Sect... The preseted procedure ca be used oly for ucorrelated errors (whe they are correlated oe ca use procedure described i [16]), therefore, the ext step cosists i determiatio of the total error as the set of values obtaied as sum (30) of realizatios of the partial errors. The probability desity fuctio of the total error ca be determied usig covolutio (31) but the simpler way is to determie the histogram of the error occurrece by usig Mote Carlo method as it is show i Example Havig described the total error oe ca calculate ucertaity of a sigle measuremet result solvig fuctioal (18) for take codece level p. The ucertaity value describes a priori iaccuracy of every sigle measuremet result obtaied i the coditios characterized by the selected errors. 4. The last step cosists i writig the measured result as iterval (8), limits of which are determied by the evaluatio of the measured quatity ad the calculated ucertaity. Example 5. Let us take that coditios, i which a sigle measuremet of voltage x is performed, are characterized by two errors: e 1 with ormal distributio N (0, ) V ad e with rectagular distributio i limits [ 0.001, 0.001] V. These errors have bee determied before realizatio of measuremets, therefore oe calculate ucertaity a priori as the parameter describig iaccuracy of every sigle measuremet. I this case, the total error e c is the sum of two errors e 1 ad e. Usig Mote Carlo method, oe obtais the histogram of the total error show i Fig. 6. Fig. 6. Histogram of the combied error obtaied i Example 5. Ucertaity calculated from this histogram for codece level p = 0.95 is U = O the base of this histogram, the ucertaity calculated accordigly with (18) for codece level p = 0.95 has value U = V. Let us assume ow that a sigle measuremet has bee performed ad the obtaied value of the evaluatio is for example: ˆx = V. The measuremet result ca be writte the as iterval: x = [ , ] = [0.8559, ] V. Procedure 1a. This procedure is performed i the same steps as procedure 1 with this dierece that the error sources are divided ito types A ad B. Error of type A is determied o the basis of the series obtaied from a measuremet experimet performed especially to this aim. The variace of this error is estimated by usig (34). Determiatio of distributios of the B type errors is realized i the same way as for procedure 1, i.e. by usig kowledge about coditios of the measuremet process which are characterized by iuece quatities. Example 6. Let us assume that the error model of a sigle measuremet result of a voltage cotais two errors: e A of type A ad e B of type B. Error of type A is idetied o the basis of a series of measuremets. Let us take that realizatio of the ideticatio experimet has delivered the series of results take i Example 4. The estimate of the stadard deviatio, calculated i Example 4 o the basis of this series, has the value: σ A = V. Therefore, type A error is described by the ormal distributio N (0, ) V. The error of type B describes such a compoet of iaccuracy which results from the fact that measuremets are performed i temperature that iueces o measured results but its value is ot kow. Takig that error of type B has properties described i Example, its distributio is rectagular i limits [ 0.001, 0.001] V. The histogram of the total error has bee determied by usig Mote Carlo method i the same way as i Example 3. O the base of this histogram, the ucertaity calculated accordigly with (18) for codece level p = 0.95 has value U = V. If oe assumes that the evaluatio value of a sigle measuremet is ˆx = V, it ca be writte as iterval x = [ , ] V = [0.8556, ] V. Procedure. It is realized as follows: 1. Havig obtaied a series of measuremet results, at rst oe should calculate the average value of the measured quatity accordigly with (35).. The ext step cosists i estimatio of the stadard deviatio of the type A error for the average value of the series o the base of (44) with assumptio that it is of the ormal distributio. 3. Distributios of the type B errors are determied o the base of kowledge about coditios of the measuremet process. 4. I the ext step, the probability desity fuctio of the total error is determied usig covolutio (31) or Mote Carlo method.

9 444 J. Jakubiec 5. Havig described the total error, oe performs ucertaity calculatio of the average value of the series by usig (18). 6. The last step of the procedure cosists i writig the measured result as the iterval (8), the boudaries of which are determied by the evaluatio of the measured quatity ad the calculated ucertaity. Example 7. The series of measuremet results, take i Example 4, has bee obtaied. The al result is determied i this case as average (35) of the series. The average is calculated i Example 4 ad its value is x = V. Error of type A ca be idetied i this case o the basis of the same series of measuremets as the average value, therefore, its stadard deviatio is estimated accordigly with (44) ad has value σ A x = V = V The distributio of error of type A has to be assumed as ormal, thus it is described as N (0, ) V. Error of type B has the same properties as this oe i Example 6, i.e. its distributio is rectagular i limits [ 0.001, 0.001] V. The histogram of the total error, determied by usig Mote Carlo method, is the same as show i Fig. 5 (the errors i Example 3 have the same distributios as i the preseted example). O the basis of this histogram, the ucertaity calculated accordigly with (18) for codece level p = 0.95 has value U = V. Therefore, the al result beig the average value of the series ca be writte as iterval x = [ , ] V = [0.9989, ] V. 9. Coclusios The preseted coceptio of ucertaity calculatio of a result measured directly bases o the deitio of the measuremet error. A ucertaity is determied as a parameter describig a set of the error values treated as realizatios of a radom quatity. Although the form of the error deitio is the eect of the quatizatio process aalysis, oe ca use this deitio i every measuremet situatio. The reaso of such a poit of view results from the fact that i moder metrology oe ca otice the tedecy to build stadards basig o quatum pheomea. Moreover, every measuremet istrumet is characterized by its resolutio which causes that umber of possible measuremet results is limited i the way close to this oe occurig i the quatizatio process. Usig the error deitio as a startig poit of this coceptio eables obtaiig the procedure of ucertaity calculatio which is relatively simple ad formally well grouded. The procedure is maily dedicated to calculate ucertaity of a sigle measuremet result which eables applicatio of this procedure i measurig ad cotrol systems to obtai istataeous values of sigals varyig i time. As it is show i the paper, the procedure may be used for determiig ucertaity of the average value of a series of measuremets ad, i geeral, for calculatig ucertaity of algorithms [5] commoly used for measuremet data processig. Divisio of the errors ito separated types A ad B permits to obtai the ucertaity calculatio procedure close to this oe proposed i GUM [1] ad i its supplemet [16]. Refereces [1] Guide to the Expressio of Ucertaity i Measuremet, ISO 199, [] R. Kacker, K. Sommer, R. Kessel, Metrologia 44, 513 (007). [3] [4] W. Bich, Metrologia 49, 70 (01). J. Jakubiec, Metrol. Measur. Syst. 1, 406 (004). [5] J. Jakubiec, Errors ad Ucertaities i Measurig ad Cotrol Systems, o. 176, SUT, Gliwice 010, (i Polish). [6] W.F. Fuller, Measuremet Error Models, Wiley, New York [7] A. Va der Vee, M.G. Cox, Metrologia 40, 45 (003). [8] A. Papoulis, Probability, Radom Variables, ad Stochastic Processes, Wiley, New York [9] W. Jakubik, M. Urbaczyk, E. Maciak, T. Pustely, Bull. Pol. Acad. Sci. Tech. Sci. 56, 133 (008). [10] W. Batko, P. Pawlik, Acta Phys. Pol. A 11, 15 (01). [11] T.R. Birdie, K.S. Suraa, Reliable Comput. 4, 69 (1998). [1] D.I. Doser, K.D. Crai, M. Baker, Reliable Comput. 4, 41 (1998). [13] C. Tyszkiewicz, T. Pustely, Opt. Appl. 34, 507 (004). [14] O. Kosheleva, V. Kreiovitch, Reliable Comput. 5, 81 (1999). [15] A. Neumaier, Iterval Methods for Systems of Equatios, Cambridge Uiversity Press, Cambridge [16] Guide to the Expressio of Ucertaity i Measuremet. Supplemet 1. Numerical Methods for the Propagatio of Distributios, ISO, 004. [17] J. Jakubiec, Applicatio of Reductive Iterval Arithmetic to Ucertaity Evaluatio of Measuremet Data Processig Algorithms, Moograph o. 6, SUT, Gliwice 00. [18] J. Jakubiec, Metrol. Measur. Syst. X, 137 (003).