1 Distributions for Uncertinty Anlysis 1 Howrd Cstrup, Ph.D. President, Integrted Sciences Group Bkersfield, CA Abstrct In performing mesurement, we encounter errors or bises from number of sources. Such sources include rndom error, mesuring prmeter bis, mesuring prmeter resolution, opertor bis, environmentl fctors, etc. We estimte the uncertinties due to these errors either by computing stndrd devition from smple of mesurements or by forming n estimte bsed on eperience. Estimtes obtined by the former method re lbeled Type A estimtes nd those obtined by the ltter method re clled Type B estimtes. This pper describes sttisticl distributions tht cn be pplied to both Type A nd Type B mesurement errors nd to equipment prmeter bises. Once the sttisticl distribution for mesurement error or bis is chrcterized, the uncertinty in this error or bis is computed s the stndrd devition of the distribution. For Type A estimtes, the distribution or popultion stndrd devition is estimted by the smple stndrd devition. For Type B estimtes, the stndrd devition is computed from limits, referred to s error continment limits nd from probbilities, referred to s continment probbilities. The degrees of freedom for ech uncertinty estimte cn often be determined, regrdless of whether the estimte is Type A or Type B. Bckground Until the publiction of the Guide to the Epression of Uncertinty in Mesurement (GUM) , ccrediting bodies or uditing gencies for test nd clibrtion orgniztions did not tend to focus on uncertinty nlysis requirements. There were two min resons for this: (1) universlly ccepted methodology ws not vilble, nd () ssessors nd uditors did not possess the required epertise. Since the introduction of the GUM, however, ccrediting bodies hve been incresingly insistent tht lbortories implement procedures for uncertinty nlysis nd be ble to demonstrte tht these procedures re being competently followed. Since the publiction of ISO/IEC 1705 , this insistence hs intensified. This hs plced ccrediting bodies nd lbortories like in ctchup mode tht hs led to some hstily contrived mesures, s will be discussed presently. To induce orgniztions to estimte uncertinties, it ws felt necessry by some to dvocte the use of simple lgorithms tht, while they were not pproprite in most cses, would t lest get people on the uncertinty nlysis pth. One such lgorithm involves the indiscriminte use of the uniform distribution to compute Type B uncertinty estimtes. Unfortuntely, orgniztions tht not only wnt to nlyze uncertinties but lso do the job correctly re sometimes penlized by this ill-dvised simplifiction. On one occsion, lbortory ssessor dmitted tht the uniform distribution ws lrgely inpproprite but insisted tht it still be employed. His resoning ws tht it did not mtter if uncertinty estimtes were invlid s long s everyone produced them in the sme wy! This philosophy precludes the development of uncertinty estimtes tht cn be used to perform sttisticl tests, evlute mesurement decision risks, mnge clibrtion intervls, develop meningful tolernces nd compute vible confidence limits. In other words, prt from providing number, the uncertinty estimte becomes useless nd potentilly epensive commodity. Obviously, if vible uncertinty estimtes re to be produced, the blind cceptnce of inpproprite distributions is to be discourged. Accordingly, we need to elborte on lterntive distributions nd discuss the pplicbility of ech Introduction Error nd Uncertinty It is iomtic tht the uncertinty in vlue obtined by mesurement is identicl to the uncertinty in the mesurement error. Additionlly, the uncertinty in the vlue of tolernced prmeter or chrcterized 1 Presented t the 001 IDW Conference, Knoville, TN. Revised 7 My 004, to correct typogrphicl error in the cubic eqution for the qudrtic distribution. Revised 11 April 007 to provide more trctble form of the lognorml distribution.
2 reference stndrd is equl to the uncertinty in the prmeter s devition from its nominl or stted vlue. This iom cn be stted mthemticlly. The nottion is the following X - the true vlue of n ttribute - vlue obtined for the ttribute by mesurement or the ttribute s chrcterized or nominl vlue ε - the error in mesurement or devition from nominl or chrcterized vlue U - mthemticl opertor tht returns the uncertinty in vlue u - the uncertinty in u ε - the uncertinty in ε. We begin by sying tht Mesured Vlue = True Vlue + Mesurement Error, for mesured quntities, nd True Vlue = Nominl Vlue + Devition, for tolernced prmeters or chrcterized reference stndrds. We now rewrite these epressions using the nottion defined bove for mesured ttribute, nd = X + ε, (1) X = + ε, () for tolernced prmeter or chrcterized stndrd. Using the uncertinty opertor U, we obtin u = U( ) = U( X + ε) = U( ε) = u ε, (3) for mesured ttribute, nd ux = U( X) = U( + ε) = U( ε) = u ε, (4) for tolernced prmeter or chrcterized reference. In either cse, the uncertinty in the vlue of interest is equl to the uncertinty in the error or devition in the vlue. Uncertinty Definition We will now define the opertor U. First, however, we need to discuss the nture of mesurement errors nd devitions. We begin by stting tht mesurement errors nd devitions re rndom vribles tht follow sttisticl distributions. For certin kinds of error, such s rndom error, this is esily seen. For other kinds of error, such s prmeter bis nd opertor bis, however, their rndom nture is not so redily perceived. Wht we need to ber in mind is tht, while prticulr error my hve systemtic vlue tht persists from mesurement to mesurement, it nevertheless comes from some distribution of like errors tht cn be described sttisticlly. For instnce, the dimeters of bll berings emerging from mnufcturing process will vry to some finite mount from bering to bering. If one such bering comes into our possession, it will hve systemtic devition from nominl tht is essentilly fied. However, our prticulr devition ws drwn t rndom from popultion of devitions rising from the mnufcturing process. Since this devition is unknown, we cn tret it s rndom vrible whose uncertinty is mesure of the spred of devitions tht chrcterize the process. The wider this spred, the greter the uncertinty. A similr chin of resoning pplies to prmeters emerging from test or clibrtion process nd to errors in mesurement. The upshot is tht, whether prticulr error is rndom or systemtic, it cn still be regrded s coming from distribution of errors tht cn be described sttisticlly. Moreover, the spred in this distribution is synonymous with the uncertinty in the error. It turns out tht there is n idel sttistic for quntifying this spred. This sttistic is the stndrd devition of the distribution. Therefore, to define the opertor U, we need to define the stndrd devition. First, however, we will define the concept of sttisticl vrince. Simply put, the vrince of distribution of errors is the distribution s men squre error. If f() represents the probbility density for popultion of ttribute vlues or mesurement results, nd µ represents the nominl or men or vlue for the popultion, then the popultion vrince or men squre error vr(ε ) is given by u ε = vr( ε ) (5) = f( ε ) ε dε = f( )( µ ) d = vr( ) = u. Notice tht the popultion vrince is sttistic tht quntifies the spred of the distribution. Tht is, the lrger the spred, the lrger the vrince. At first glnce, the vrince or men squre error would seem to be good quntity by which to epress popul-
3 tion s uncertinty. However, the vrince is in the wrong units, nmely, the desired units squred. This is rectified by tking the squre root of the vrince, which yields the stndrd devition. Then, by Eq. (3) or (4) u = U( ) = U( ε ) = vr( ε ). (6) So, we see tht estimting the uncertinty in mesurement is n eercise in which we estimte the stndrd devition of the mesurement error. If we hve smple of mesurements, we cn estimte the stndrd devition due to rndom error in the smple using strightforwrd epression found in sttistics tetbooks 1 u = n ( i ), (7) n 1 i= 1 where n is the smple size nd is the smple men. This is n emple of Type A estimte. For Type B estimtes, we work from error continment limits nd continment probbilities. The process is described in detil in the literture . Stndrd nd Epnded Uncertinty To this point, the uncertinty in mesurement hs been equted with the stndrd devition of the popultion of the mesurement error. In the GUM, this uncertinty is clled the stndrd uncertinty. If the distribution is known, nd the degrees of freedom cn be determined , the stndrd uncertinty cn be used to develop confidence limits for n uncertinty estimte. The GUM refers to confidence limit s n epnded uncertinty. 3 The fctor by which stndrd uncertinty is multiplied to yield n epnded uncertinty is clled the coverge fctor. Unfortuntely, in converstion, it is not lwys cler whether the term uncertinty refers to the epnded uncertinty or to the stndrd uncertinty. In this pper, unless otherwise indicted, it will refer to the stndrd uncertinty. Note the forml similrity between Eq. (7) nd Eq. (5). 3 Actully, the terms stndrd uncertinty nd epnded uncertinty were introduced to supersede the terms stndrd devition nd confidence limit, respectively, in cses where the degrees of freedom for n uncertinty estimte could not be determined. Before the refinement of methods for estimting degrees of freedom , this limittion pplied lmost universlly to Type B estimtes, nd, by etension to mied Type A-B estimtes. In this pper, uncertinty = stndrd uncertinty Sttisticl Distributions In obtining Type A uncertinty estimte, we compute stndrd devition using Eq. (7). In obtining Type B estimte, we work from set of bounding limits, referred to s error continment limits nd continment probbility, which is the probbility tht errors or ttribute vlues lie within these limits. Any one of vriety of distributions my be ssumed to represent the underlying distribution of errors or devitions. In this pper, we consider the uniform, norml, lognorml, qudrtic, cosine, hlf-cosine, U- shped, nd the Student s t distribution. The Uniform Distribution The uniform distribution is defined by the probbility density function (pdf) 1, f( ) = 0, otherwise, where ± re the limits of the distribution. f() 0 The Uniform Distribution. The probbility of lying between - nd is constnt. The probbility of lying outside ± is zero. Acceptnce of the Uniform Distribution Applying the uniform distribution to obtining Type B uncertinty estimtes is prctice tht hs been gining ground over the pst few yers. There re two min resons for this: 1. First, pplying the uniform distribution mkes it esy to obtin n uncertinty estimte. If the limits ± of the distribution re known, the uncertinty estimte is just u =. (8) 3
4 It should be sid tht the "ese of use" dvntge hs been promoted by individuls who re ignornt of methods of obtining uncertinty estimtes for more pproprite distributions nd by others who re simply looking for quick solution. In firness to the ltter group, they sometimes ssert tht the lck of specificity of informtion required to use other distributions mkes for crude uncertinty estimtes nywy, so why not get your crude estimte by intentionlly using n inpproprite distribution? At our present level of nlyticl development [3, 4], this rgument does not hold wter. Since the introduction of the GUM, methods hve been developed tht systemtize nd rigorize the use of distributions tht re physiclly relistic. These will be discussed presently.. Second, it hs been sserted by some tht the use of the uniform distribution is (uniformly?) recommended in the GUM. This is not true. In fct, most of the methodology of the GUM is bsed on the ssumption tht the underlying error distribution is norml. Some of the belief tht the uniform distribution is clled for in the GUM stems from the fct tht severl individuls, who hve come to be regrded s GUM uthorities, hve been dvocting its use. For clrifiction on this issue, the reder is referred to Section 4.3 of the GUM. Another source of confusion is tht some of the emples in the GUM pply the uniform distribution in situtions tht pper to be incomptible with its use. It is resonble to suppose tht much of this is due to the fct tht rigorous Type B estimtion methods nd tools were not vilble t the time the GUM ws published, nd the uniform distribution ws n "esy out." As stted in item 1 bove, the lck of such methods nd tools hs since been rectified. The cceptnce of the uniform distribution on the bsis of its use in GUM emples reminds us of similr prctice tht emerged from the ppliction of Hndbook 5 to the interprettion of MIL-STD- 4566A. In one emple in the Hndbook, hypotheticl lb ws being udited whose nominl operting temperture ws 68 F. Some of the 4566A uditors rected to the emple by citing lbs tht did not mintin this temperture, regrdless of whether it ws pproprite for the lb's opertion. Inevitbly, the 68 F requirement ctully becme institutionlized within certin uditing gencies. Applicbility of the Uniform Distribution The use of the uniform distribution is pproprite under limited set of conditions. These conditions re summrized by the following criteri. The first criterion is tht we must know set of minimum bounding limits for the distribution. This is the minimum limits criterion. Second, we must be ble to ssert tht the probbility of finding vlues between these limits is unity. This is the 100% continment criterion. Third, we must be ble to demonstrte tht the probbility of obtining vlues between the minimum bounding limits is uniform. This is the uniform probbility criterion. Minimum Limits Criterion. It is vitl tht the limits we estblish for the uniform distribution re the minimum bounding limits. For instnce, if the limits ±L bound the vrible of interest, then so do the limits ±L, ±3L, nd so on. Since the uncertinty estimte for the uniform distribution is obtined by dividing the bounding limit by the squre root of three, using vlue for the limit tht is not the minimum bounding vlue will obviously result in n invlid uncertinty estimte. This lone mkes the ppliction of the uniform distribution questionble in estimting bis uncertinty from such quntities s tolernce limits, for instnce. It my be tht out-of-tolernces hve never been observed for prticulr prmeter (100% continment), but it is unknown whether the tolernces re minimum bounding limits. Some yers go, study ws conducted involving voltge reference tht showed tht vlues for one prmeter were normlly distributed with stndrd devition tht ws pproimtely 1/10 of the tolernce limit. With 10-sigm limits, it is unlikely tht ny out-of-tolernces would be observed. However, if the uniform distribution were used to estimte the bis uncertinty for this item, bsed on tolernce limits, the uncertinty estimte would be nerly si times lrger thn would be pproprite. Some might clim tht this is cceptble, since the estimte cn be considered conservtive one. Tht my be. However, it is lso useless estimte. This point will be elborted lter. A second difficulty we fce when ttempting to pply minimum bounding limits is tht such limits cn rrely be estblished on physicl grounds. This is especilly true when using prmeter tolernce limits. It is virtully impossible to imgine sitution where design engineers hve somehow been ble to precisely identify the minimum limits tht bound vlues tht re physiclly ttinble. If we dd to this the fct tht tolernce limits re often influenced by mrketing
5 rther thn engineering considertions, equting tolernce limits with minimum bounding limits becomes very unfruitful nd misleding prctice. 100% Continment Criterion. By definition, the estblishment of minimum bounding limits implies the estblishment of 100% continment. It should be sid however, tht n uncertinty estimte my still be obtined for the uniform distribution if continment probbility less tht 100% is pplied. For instnce, suppose the continment limits re given s ±L nd the continment probbility is stted s being equl to some vlue p between zero nd one. Then, if the uniform probbility criterion is met, the limits of the distribution re given by L =, L. (9) p If the uniform probbility criterion is not met, however, the uniform distribution would not be pplicble, nd we should turn to other distributions. Uniform Probbility Criterion. As discussed bove, estblishing minimum continment limits cn be chllenging prospect. Hrder still is finding relworld mesurement error distributions tht demonstrte uniform probbility of occurrence between two limits nd zero probbility of occurrence outside these limits. Ecept in very limited instnces, such s re discussed in the net section, ssuming uniform probbility is just not physiclly relistic. This is true even in some cses where the distribution would pper to be pplicble. For emple, conjecture hs recently been dvnced tht the distribution of prmeters immeditely following test or clibrtion cn be sid to be uniform. While this seems resonble t fce vlue, it turns out not to be the cse. Becuse of flse ccept risk (consumer s risk), such distributions rnge from pproimtely tringulr to hving "humped" ppernce with rolled-off shoulders. As to whether we cn tret prmeter tolernce limits s bounds tht contin vlues with uniform probbility, we must imgine tht, not only hs the instrument mnufcturer mnged to mirculously scertin minimum bounding limits, but hs lso juggled physics to such n etent s to mke the prmeter vlue's probbility distribution uniform between these limits nd zero outside them. This would be truly mzing fet of engineering for most tolernced quntities especilly considering the mrketing influence mentioned erlier. Cses tht Stisfy the Criteri Digitl Resolution Uncertinty. We sometimes need to estimte the uncertinty due to the resolution of digitl redout. For instnce, three-digit redout might indicte V. If the device employs the stndrd round-off prctice, we know tht the displyed number is derived from sensed vlue tht lies between V nd V. We lso cn ssert to very high degree of vlidity tht the vlue hs n equl probbility of lying nywhere between these two numbers. In this cse, the use of the uniform distribution is pproprite, nd the resolution uncertinty is V u V = = V. 3 RF Phse Angle. RF power incident on lod my be delivered to the lod with phse ngle θ between -π nd π. In ddition, unless there is compelling reson to believe otherwise, the probbility of occurrence between these limits is uniform. Accordingly, the use of the uniform distribution is pproprite. This yields phse ngle uncertinty estimte of π u θ = It is interesting to note tht, given the bove, if we ssume tht the mplitude of the signl is sinusoidl, the distribution for incident voltge is the U-shped distribution. Quntiztion Error. The potentil drop (or lck of potentil drop) sensed cross ech element of n A/D Converter sensing network produces either "1" or "0" to the converter. This response constitutes "bit" in the binry code tht represents the smpled vlue. For ldder-type networks, the position of the bit in the code is determined by the loction of its originting network element. Even if no errors were present in smpling nd sensing the input signl, errors would still be introduced by the discrete nture of the encoding process. Suppose, for emple, tht the full scle signl level (dynmic rnge) of the A/D Converter is volts. If n bits re used in the encoding process, then voltge V cn be resolved into n discrete steps, ech of size / n. The error in the voltge V is thus ε ( V) = V m, where m is some integer determined by the sensing function of the D/A Converter. n
6 The continment limit ssocited with ech step is one-hlf the vlue of the mgnitude of the step. Consequently, the continment limit inherent in quntizing voltge V is (1/)(/ n ), or / n+1. This is embodied in the epression V V quntized = sensed ±. n+ 1 The uncertinty due to quntiztion error is obtined from the continment limits nd from the ssumption tht the sensed nlog vlue hs equl probbility of occurrence between these limits: u V / = 3 n+ 1. One wy of reconciling the prctice is to stte tht the underlying distribution is ctully norml, or pproimtely norml, nd the uniform distribution is used merely s n rtifice to obtin n estimte of the distribution's stndrd devition. This is somewht mzing sttement. If the underlying distribution is norml, why not obtin the uncertinty estimte using tht distribution in the first plce? 4 It cn be shown tht using the uniform distribution s tool for estimting the uncertinty in normlly distributed quntity corresponds to ssuming norml distribution with 91.67% continment probbility. For orgniztions tht mintin high in-tolernce probbility t the unit level, we often see or cn surmise 98% or better in-tolernce probbilities t the prmeter level. Consequently, for these cses, use of the uniform distribution produces uncertinty estimtes tht re t lest 35% lrger thn wht is pproprite. As for those who find this cceptble on the bsis of conservtism, consider the U.S. Nvy's end-of-period relibility trget of 7% for generl purpose items. For single-prmeter items, if the true underlying distribution is norml, use of the uniform distribution cn produce uncertinty estimtes tht re only bout 6% of wht they should be. So much for conservtism. Signl Quntiztion. The smpled signl points re quntized in multiples of discrete step size. Development of Epnded Uncertinty Limits NIST Technicl Note 197  documents the uncertinty nlysis policy to be followed by NIST. In this policy, epnded uncertinty limits for Type B nd mied estimtes re obtined by multiplying the uncertinty estimte by fied k-fctor equl to two. Assuming n underlying norml distribution, this produces limits tht re roughly nlogous to 95% confidence limits. The dvisbility of this prctice is debtble, but this is the subject of seprte discussion. For the present, we consider wht results from the prctice when estimting n uncertinty for cse where the underlying distribution is ssumed to be uniform. Since the uncertinty is estimted by dividing the distribution minimum bounding limit by the squre root of three, multiplying this estimte by two yields epnded uncertinty limits tht re outside the distribution s minimum bounding limits. To be specific, these limits equte to pproimtely 115% continment probbility, which is nonsense. The Norml Distribution When obtining Type A estimte, we compute stndrd devition from smple of vlues. For emple, we estimte rndom uncertinty by computing 4 One recommendtion tht the reder my encounter is tht, if ll tht is vilble for n error source or prmeter devition is set of bounding limits, without ny knowledge of the nture of the error distribution nd with no informtion regrding continment probbility, then the uniform distribution should be ssumed. There re two points tht should be mde concerning this recommendtion. First, fter little reflection on the difficulty of obtining minimum continment limits without knowledge of continment probbility, we cn see tht the recommendtion not dvisble. The prudent pth to follow is to simply put some effort into obtining continment probbility estimte nd scertining most likely underlying distribution. There is relly no wy round this. Moreover, the uthor hs yet to observe n uncertinty nlysis problem where this could not be done. The second point is tht, eperienced technicl personnel nerly lwys know something bout wht they re mesuring nd wht they re mesuring it with. Ecept for the cses described bove, it is difficult to imgine scenrio where n eperienced engineer or technicin would know set of bounding limits nd nothing else.
7 the stndrd devition for smple of repeted mesurements of given vlue. We lso obtin smple size. The smple stndrd devition, equted with the rndom uncertinty of the smple, is n estimte of the stndrd devition for the popultion from which the smple ws drwn. Ecept in rre cses, we ssume tht this popultion follows the norml distribution. This ssumption, llows us to esily obtin the degrees of freedom nd the smple stndrd devition nd to construct confidence limits, perform sttisticl tests, estimte mesurement decision risk nd to rigorously combine the rndom uncertinty estimte with other Type A uncertinty estimtes. Why do we ssume norml distribution? The primry reson is becuse this is the distribution tht either represents or pproimtes wht we frequently see in the physicl universe. It cn be derived from the lws of physics for such phenomen s the diffusion of gses nd is pplicble to instrument prmeters subject to rndom stresses of usge nd hndling. It is lso often pplicble to equipment prmeters emerging from mnufcturing processes. f() In cses where this is not so, other distributions, such s the lognorml distribution cn be pplied. Uncertinty Estimtes In pplying the norml distribution, n uncertinty estimte is obtined from continment limits nd continment probbility. The use of the distribution is pproprite in cses where the bove considertions pply nd the limits nd probbility re t lest pproimtely known. The etent to which this knowledge is pproimte determines the degrees of freedom of the uncertinty estimte [4, 7]. The degrees of freedom nd the uncertinty estimte cn be used in conjunction with the Student's t distribution (see below) to compute confidence limits. Let ± represent the known continment limits nd let p represent the continment probbility. Then n estimte of the stndrd devition of the popultion of errors or devitions is obtined from u =, (10) 1 1+ p Φ where Φ -1 (. ) is the inverse norml distribution function. This function cn be found in sttistics tets nd in populr spredsheet progrms. µ The Norml Distribution. Shown is cse where the popultion men µ is locted fr from physicl limit 0. In such cses, the norml distribution cn be used without compromising rigor. An dditionl considertion pplies to the distribution we should ssume for totl error or devition tht is composed of constituent errors or devitions. There is theorem clled the centrl limit theorem tht demonstrtes tht, even though the individul constituent errors or devitions my not be normlly distributed, the combined error or devition is pproimtely so. An rgument hs been presented ginst the use of the norml distribution in cses where the vrible of interest is restricted, i.e., where vlues of the vrible re sid to be bound by some physicl limit. This condition notwithstnding, the norml distribution is still widely pplicble in tht, for mny such cses, the physicl limit is locted fr from the popultion men. If only single continment limit is pplicble, such s with single-sided tolernces, the pproprite epression is u =. (11) Φ p 1 ( ) The Lognorml Distribution The lognorml distribution cn often be used to estimte the uncertinty in equipment prmeter bis in cses where the tolernce limits re symmetric. It is lso used in cses where physicl limit is present tht lies close enough to the nominl or mode vlue to skew the prmeter bis pdf in such wy tht the norml distribution. is not pplicble. The pdf is given by 1 q f( ) = ep ln σ πσ q m q, where q is physicl limit for, m is the popultion medin nd µ is the popultion mode. The vrible σ is not the popultion stndrd devition. It is referred
8 to s the "shpe prmeter." The ccompnying grphic shows cse where µ = 10, q = 9.607, σ = , nd m = The computed stndrd devition for this emple is u = f() q µ The Lognorml Distribution. Useful for describing distributions for prmeters constrined by physicl limit or possessing symmetric tolernces. Uncertinty estimtes (stndrd devitions) for the lognorml distribution re obtined by numericl itertion. To dte, the only known pplictions tht perform this process re UncertintyAnlyzer  nd AccurcyRtio . The Tringulr Distribution The tringulr distribution hs been proposed for use in cses where the continment probbility is 100%, but there is centrl tendency for vlues of the vrible of interest . The tringulr distribution is the simplest distribution possible with these chrcteristics. f() - 0 The Tringulr Distribution. A distribution tht sometimes pplies to prmeter vlues immeditely following test or clibrtion. The pdf for the distribution is + ( )/, 0 f( ) = ( )/, 0 0, otherwise. The stndrd devition for the distribution is obtined from u =. (1) 6 Like the uniform distribution, using the tringulr distribution requires the estblishment of minimum continment limits ±. The sme reservtions pply in this regrd to the tringulr distribution s to the uniform distribution. In cses where continment probbility p < 1 cn be determined for limits ±L, where L <, the limits of the distribution re given by L =, L. 1 1 p Aprt from representing post-test distributions under certin restricted conditions, the tringulr distribution hs limited pplicbility to physicl errors or devitions. While it does not suffer from the uniform probbility criterion, s does the uniform distribution, it nevertheless displys brupt trnsitions t the bounding limits nd t the zero point, which re physiclly unrelistic in most instnces. In ddition, the liner increse nd decrese in behvior is somewht fnciful for pdf. The Qudrtic Distribution A distribution tht elimintes the brupt chnge t the zero point, does not ehibit unrelistic liner behvior nd stisfies the need for centrl tendency is the qudrtic distribution. This distribution is defined by the pdf 3 1 ( / ) f( ) = 4, 0, otherwise where ± re minimum bounding limits. The stndrd devition for this distribution is determined from u =, (13) 5 i.e., bout 77% of the stndrd devition estimte for the uniform distribution.
9 f() f() - 0 The Qudrtic Distribution. Ehibits centrl tendency without discontinuities nd does not ssume liner pdf behvior. For continment probbility p nd continment limits ±L, the minimum bounding limits ± re obtined from L 1 = p p p + < < 3 1 cos rccos(1 ) 1 1 The Cosine Distribution While the qudrtic distribution elimintes discontinuities within the bounding limits, it rises bruptly t the limits. Although the qudrtic distribution hs wider pplicbility thn either the tringulr or uniform distribution, this feture nevertheless diminishes its physicl vlidity. A distribution tht overcomes this shortcoming, ehibits centrl tendency nd cn be determined from minimum continment limits is the cosine distribution. The pdf for this distribution is given by 1 π 1 + cos f( ) =,. 0, otherwise The uncertinty is obtined from the epression 6 u = 1, (14) 3 π which trnsltes to roughly 63% of the vlue obtined using the uniform distribution The Cosine Distribution. A 100% continment distribution with centrl tendency nd lcking discontinuities. Solving for when continment probbility nd continment limits ±L re given requires pplying numericl itertive method to the epression 1 sin( π ) p + = 0, L/ ; L. π The solution lgorithm hs been implemented in the sme softwre lluded to in the discussion on the qudrtic distribution. It yields, for the ith itertion, where nd = F F, i i 1 / 1 F = sin( π ) p + π F = 1+ cos( π ). The Hlf-Cosine Distribution The hlf-cosine distribution is used in cses where the centrl tendency is not s pronounced s when norml or the cosine distribution would be pproprite. In this regrd, it resembles the qudrtic distribution without the discontinuities t the distribution limits. The pdf is π π cos f( ) = 4,. 0, otherwise If the minimum limiting vlues ± re known, the uncertinty is obtined from the epression u = 1 8/ π. (15) If continment limits ±L nd continment probbility p re known, the limiting vlues my be obtined from the reltion π L =, L. 1 sin ( p)
10 f() The Student's t Distribution If the underlying distribution is norml, nd Type A estimte nd degrees of freedom re vilble, confidence limits for mesurement errors or prmeter devitions my be obtined using the Student's t distribution. This distribution is vilble in sttistics tetbooks nd populr spredsheet pplictions. Its pdf is ν + 1 Γ f( ) = (1 + / ν ) ν πν Γ ( ν + 1)/, - 0 The Hlf-Cosine Distribution. Possesses centrl tendency but ehibits higher probbility of occurrence ner the minimum limiting vlues thn either the cosine or the norml distribution. The U Distribution The U distribution pplies to sinusoidl RF signls incident on lod. It hs the pdf 1, < < f( ) = π 0, otherwise, where represents the mimum signl mplitude. The uncertinty in the incident signl mplitude is estimted ccording to u =. (16) where ν is the degrees of freedom nd Γ(. ) is the gmm function. The degrees of freedom quntifies the mount of knowledge used in estimting uncertinty. This knowledge is incomplete if the limits ± re pproimte nd the continment probbility p is estimted from recollected eperience. Since the knowledge is incomplete, the degrees of freedom ssocited with Type B estimte is not infinite. If the degrees of freedom vrible is finite but unknown, the uncertinty estimte cnnot be rigorously used to develop confidence limits, perform sttisticl tests or mke decisions. This limittion hs often precluded the use of Type B estimtes s sttisticl quntities nd hs led to such discomforting rtifices s fied coverge fctors. f() f() 0-0 The U Distribution. The distribution is the pdf for sine wves of rndom phse incident on plne. If continment limits ±L nd continment probbility p re known, the prmeter cn be computed ccording to L = L. sin π p / ( ), Student's t Distribution. Shown is the pdf for 10 degrees of freedom. Fortuntely, the GUM provides n epression for obtining the pproimte degrees of freedom for Type B estimtes. However, the epression involves the use of the vrince in the uncertinty estimte, nd method for obtining this vrince hs been lcking until recently . A rigorous method for obtining this quntity hs been implemented in commercilly vilble softwre  nd in freewre ppliction . Once the degrees of freedom hs been obtined, the Type B estimte my then be combined with other estimtes nd the degrees of freedom for the combined
11 uncertinty cn be determined using the Welch- Stterthwite reltion . If the underlying distribution for the combined estimte is norml, the t distribution cn be used to develop confidence limits nd perform sttisticl tests. The procedure is to first estimte the uncertinty using Eq. (10) nd then estimte the degrees of freedom from the epression where ( u) 1 1 σ ν B u 3ϕ ϕ ( ) + π e ( p) ϕ 1 1+ p ϕ =Φ., (17) The vribles nd p represent "give or tke" vlues for the continment limits nd continment probbility, respectively. At first glnce, Eq. (17) my seem to be nything but rigorous. However, severl dt input formts hve been developed tht rigorize the process of estimting nd p . They re vilble in the referenced softwre pplictions cited bove [3, 7]. Striving for Conservtive Estimtes If n uncertinty estimte is viewed s n end product tht will be filed wy without ppliction of ny kind, then employing unrelistic distributions nd fied coverge fctors my be considered cceptble by some. Such distributions cn yield sttisticlly vlid estimtes, regrdless of whether or not these estimtes re physiclly vlid. However, if n uncertinty estimte is to be employed in mking decisions, such s my result from hypothesis testing or decision risk nlyses, employing physiclly unrelistic distribution is to be discourged. In these cses, dvocting the use of such distribution on the grounds tht it yields conservtive uncertinty estimtes is s irresponsible s employing intentionlly bised instruments to obtin mesurements tht re fvorbly skewed in one direction or nother. In ddition, the use of unrelistic distributions my yield estimtes tht re considerbly smller thn wht is pproprite under certin conditions. The emple of estimting bis uncertinty for singleprmeter Nvy generl purpose items, mentioned erlier, is cse in point. Another considertion tht rgues ginst employing conservtive uncertinty estimtes is tht this prctice sometimes leds to "reckless" conclusions. This is the cse when mesurement from one lbortory is tested ginst mesurement from nother to ssess equivlence between lbortories. If conservtive estimtes re used, the test ctully becomes less stringent thn otherwise. The bottom line is tht conservtive uncertinty estimtes re essentilly zero-informtion quntities tht hve no legitimte use. If conservtism is desired, it cn be implemented by insisting on high confidence levels in estimting confidence limits fter vlid uncertinty estimte is obtined. The higher the confidence level, the wider (more conservtive) the confidence limits. Recommendtions for Selecting Distributions Unless informtion to the contrry is vilble, the norml distribution should be pplied s the defult distribution. For Type B estimtes, the dt input formts lluded under the discussion of the Student's t distribution should lso be employed to estimte the degrees of freedom. If it is suspected tht the distribution of the vlue of interest is skewed, pply the lognorml distribution. In using the norml or lognorml distribution, some effort must be mde to estimte continment probbility. If set of continment limits is vilble, but 100% continment hs been observed, then the following is recommended: 1. If the vlue of interest hs been subjected to rndom usge or hndling stress, nd is ssumed to possess centrl tendency, pply the cosine distribution. If it is suspected tht vlues re more evenly distributed, pply either the qudrtic or hlf-cosine distribution, s pproprite. The tringulr distribution my be pplicble, under certin circumstnces, when deling with prmeters following testing or clibrtion.. If the vlue of interest is the mplitude of sine wve incident on plne with rndom phse, pply the U distribution. 3. If the vlue of interest is the resolution uncertinty of digitl redout, pply the uniform distribution. This distribution is lso pplicble to estimting the uncertinty due to quntiztion error nd the uncertinty in RF phse ngle.
12 Generl Procedure for Obtining Uncertinty Estimtes Type A Estimtes In mking Type A estimte nd using it to construct confidence limits, we pply the following procedure tken from the GUM nd elsewhere: 1. Tke rndom smple of size n representtive of the popultion of interest. The lrger the smple size, the better. In mny cses, smple size less thn si is not sufficient.. Compute smple stndrd devition, u using Eq. (7). 3. Assume n underlying distribution, e.g., norml. 4. Develop coverge fctor bsed on the degrees of freedom (n 1) ssocited with the smple stndrd devition nd desired level of confidence. If the underlying distribution is ssumed to be norml, use either t-tbles or Student s t spredsheet functions. In Microsoft Ecel, for emple, twosided coverge fctor cn be determined using the TINV function: t = TINV((1 p), ν ), where p is the confidence level nd ν is the degrees of freedom. 5. Multiply the smple stndrd devition by the coverge fctor to obtin L = tu nd use ±L s p 100% confidence limits. References        ISO/TAG4/WG3, Guide to the Epression of Uncertinty in Mesurement, Interntionl Orgniztion for Stndrdiztion (ISO), Genev, ISO/IEC (E), Generl Requirements for the Competence of Testing nd Clibrtion Lbortories, ISO/IEC, December 15, UncertintyAnlyzer, , Integrted Sciences Group, All Rights Reserved. Cstrup, H., "Estimting Ctegory B Degrees of Freedom," Proc. Mesurement Science Conference, Jnury 000, Anheim. AccurcyRtio, , Integrted Sciences Group, All Rights Reserved. Tylor, B. nd Kuytt, C., NIST Technicl Note 197, "Guidelines for Evluting nd Epressing the Uncertinty of NIST Mesurement Results," U.S. Dept. of Commerce, ISG Ctegory B Uncertinty Clcultor, 000, Integrted Sciences Group, All Rights Reserved. Avilble from Type B Estimtes In mking Type B estimte, we reverse the process. The procedure is 1. Tke set of confidence limits, e.g., prmeter tolernce limits ±L (continment limits).. Estimte the confidence level, e.g., the in-tolernce probbility (continment probbility). 3. Estimte the degrees of freedom using Eq. (17). 4. Assume n underlying distribution, e.g., norml Compute coverge fctor, t, bsed on the continment probbility nd degrees of freedom. 6. Compute the stndrd uncertinty for the quntity of interest (e.g., prmeter bis) by dividing the confidence limit by the coverge fctor: u = L/ t. 5 The Type B estimtion procedure hs been refined so tht stndrd devitions cn be estimted for non-norml popultions nd in cses where the confidence limits re symmetric or even single-sided [3, 5].