Checking Out the Doght Stadard Odors in Polygamy

Transcription

1 Cosstey Test o Mass Calbrato of Set of Weghts Class ad Lowers Lus Oar Beerra, Igao Herádez, Jorge Nava, Fél Pezet Natoal Ceter of Metrology (CNAM) Querétaro, Meo Abstrat: O weghts albrato oe by oe there s the roble of hekg the valdty of the foud values that ould be devated due to systeat errors e.g. a ossble ass devato of the weght stadard used. The reset work rooses a way to hek the ass values resultg fro the albrato that would suort albrato desos, eseally alable for the seodary albrato laboratores. 1. Itroduto O weghts albrato there s frequetly the roble of the truthfuless of the resultg values for the albrato. Weghts are delate struets to hadle ad t s oo that the stadard weghts ould drft the value of ther ass, ad there s ot a way to ote t before ther et albrato. It s roosed here a osstey test that offers the etrologst the ossblty to ereve f the ass value of ay of the stadard weghts s ot osstet the albrato of weght sets oe the etrologst has a dato of oosstey, he or she would have the ossblty to take desos as orretve atos.. Bakgroud The Iteratoal Reoedato OIML R111 [1,] defes seve auray lasses of weghts fro 1 to M 3. The Reoedato OIML R111 defes the oal values of ass stadards as 1 10 kg, 10 kg or 510 where ould be a egatve uber, ostve uber or zero, ad the sequee of the sets ould be fored fro the et otos (1;1;;5) 10 kg; (1;1;1;;5) 10 kg; (1;;;5) 10 kg or (1;1;;;5) 10 kg [1]. O weght albratos of lass ad lower; weght stadards of better lass of auray are used, the atheatal odel for the ovetoal ass value s the followg [3], Where, ( ρ, )( V V ) 1 (1 a Covetoal ass of the ukow weght Covetoal ass of the stadard weght ρ a Ar desty 1, Covetoal ar desty V Volue of the stadard weght V Volue of the ukow weght Mass dfferee betwee stadard weght ad the ukow weght read o the balae The obed stadard uertaty of the ovetoal ass of the ukow weght s obtaed fro equato (), [4] ( u u ( V V ) u (, ) ( u u ) u ρ ρ 1 a a V V Where, u Stadard uertaty of the varable The eaded uertaty of the ovetoal ass s obtaed by ultlyg the obed stadard uertaty by a ofdee fator, usually k that alfy the ofdee tervals of the uertaty to aroately 95%, the eaded uertaty value ust ot be larger tha 1/3 of the Mau Tolerable rror (MT) of the auray lass of the albrated weght, [1,] U 1 ku MT (3 3

2 Where U The eaded uertaty k Coverage fator assoate to the ofdee level 3. Statstal otrol ass albrato A hek stadard s usually a weght of the sae lass ad the oal ass as the test weght to be albrated ad s luded the weghg desg as a "ukow" weght. The otrol roedure works best wth weghg desgs (subdvso odels) where the hek stadard a easly be ororated to the desg as a ukow weght. The urose of the hek stadard s to assure the goodess of albratos. A hstory of values o the hek stadard s requred for ths urose. The aeted value of the ass dfferee for the hek stadard (usually a average) s outed fro the hstoral data ad s based o at least 10 to 15 easureets ad the value of the hek stadard for ay ew albrato s tested for agreeet wth the aeted value usg a statstal otrol tehque. The test s based o the t-statst or t-test.[]. For dvdual albratos of weghts oe by oe the use of the hek stadard ay ot be reoeded. By other had the reso of the balae a also be otored usg a statstal otrol tehque. The resdual stadard devato fro weghg desg or a stadard devato of reeated easureets o a sgle weght s the bass for the test. The test reles o a ast hstory of stadard devatos o the sae balae. If there are stadard devatos of the balae fro hstoral data fro whh a ooled stadard devato s obtaed, the statstal test ossts oarg the value of F obtaed fro the ew "wth-grou stadard devato" ad the ooled stadard devato, versus the orresodg F rtal value (aordg to the degrees of freedo), F-test []. The use of F-test dates f the dserso of the dfferees betwee the weght stadard ad the weght ukow obtaed fro the balae, s wth the erssble lts (based hstoral data), but ths test s ot sesble to other ossble soures of error as, for eale, a ossble drft of the stadard. 4. Cosstey Test The osstey test ossts a albrato of a grou of weghts of a artular deade, (all together as a ukow weght), usg the orresodg weght stadard equvalet. The su of the dvdual values of the weghts should be equal to the value foud o the albrato of the grou of weghts betwee the uertaty values. For eale a set of weghts of oal values fro 100 g to 500 g, the weghts 100 g, 00 g, 00 g(*), ad 500 g are albrated oe by oe ad fally all weghts together as a grou of 1 kg as oal value s albrated versus a 1 kg weght stadard, see Table 1. Stadard Weght Ukow Weght 100 g Vs 100 g 00 g Vs 00 g 00 g Vs 00 g(*) 500 g Vs 500 g 1 kg Vs 100 g 00 g 00 g(*) 500 g Table 1. Shee of oarsos for 100 g to 500 g The dfferee betwee the ass value of the grou of weghts ad the su of the dvdual ass values obtaed should satsfy the followg rtero that s used to verfy equvalee of results easureet oarsos aog laboratores [5] ad [6], ( U ) 1 Foud ass values of the albrato are osstet, the albrato would be osdered orret. > 1 Foud ass values of the albrato are ot osstet betwee the, t s eessary take orretve atos. (4

3 Where, Noralzed error value U Covetoal Mass of the grou of the weghts (obtaed fro the equato 1, takg as ukow weght the set of weghts) Su of the ovetoal asses foud o ts dvdual albrato aded uertaty obtaed o the grou albrato aord wth the equato 3 U Su of the ass eaded uertates fro the dvdual albrato of the weghts, equato 5, U U k u (5 where the orrelato oeffet betwee the values of the ovetoal ass of the weghts s take as 1 [7]. 5. Crtero A eagful oarso of two results of easureet of the sae quatty requres the stateet of the uertates of the easureets se they haraterze the relablty of the easureet. The "oralzed error" ( ) takes to aout of ths. Here the su of asses rovdes a value to oare t wth the ass of the grou of weghts. The eaded uertates assoated wth the results of easureet are deoted wth U ad U result fro the ultlato of the stadard uertates by the sae fed overage fator k>1 that usually s take to be. For k, ofte the oatblty rtero 1 s roosed [5]. Ths s equvalet to the roedure for oarg aterals or roduts wth reset to average erforae develoed [8]. It s lear that a sgfat dfferee betwee ad dates the arorate treatet of the systeat effets by at least oe of the easureets. Thus, ths ase at least oe of the quoted results of easureets s ot reasoable estate of the value of the easurad but ay be vewed as the best estate of the value of a dfferet quatty. Therefore, f ad are take as best estates of the ukow values of the Covetoal Mass of the grou of the weghts M ad the Su of the Covetoal ( ) Masses ( M ), resetvely, they ay be regarded to eagfully rereset the sae quatty f the avalable data suort the hyothess that the dfferee D M M s zero. The best estate of D s d ad the stadard uertaty assoated wth d s 1 u d k u ( u ) ase of the assued deedet easureets. The values that a reasoably be attrbuted to D le wth the lts d ± ud. If the value zero s wth the lts the aetae of D 0 s ot oflt wth the avalable forato about M ad M, ad, thus, D. I ths ase the values of ad a reasoably be vewed as two dfferetly establshed estates of the value of the sae quatty ad, hee, as oatble. 6. Nuer ale O a set albrato are obtaed et ovetoal ass values o weghts fro 100 g to 500 g Noal Value Correto* g Uertaty (k) g 100 g 0,153 ±0,07 00 g -0,006 ±0,03 00 g -0,003 ±0, g -0,016 ±0,044 0,18 ±0,135 (1 kg) -0,01 ±0,174 Table. Covetoal Mass Values foud albrato a set weght for 100 g to 500 g *Correto s equal to the ovetoal ass less to the oal value of the weght

4 The oralzed error s obtaed fro equato 4, ( U ) 0,68 The value of s lower tha 1, t eas that the values of the albrato are osstet betwee the (Fg. 1). The value of dates that oe of the stadards used o the albrato have drfted (the oly ossblty s that all of the have drfted together), the alulato of the ovetoal ass ad the uertaty looks orret. Fgure 1. Coarso betwee ovetoal ass orretos of ad ad ts uertates resetvely Now let see other eale where the values obtaed the albrato are o table 3. Noal Value Correto Mg Uertaty (k) g 100 g 0,153 ±0,07 00 g 0,080 ±0,03 00 g -0,003 ±0, g -0,01 ±0,044 0,09 ±0,135 (1 kg) - 0,01 ±0,174 Table 3. Covetoal Mass Values foud albrato a set weght for 100 g to 500 g The oralzed error has et value, ( U ) 1,04 o ths eale s larger tha 1, ad t eas that values foud o the weghts albrato are ot osstet betwee the, although the uertaty bars touh betwee the (Fg ) Fgure. Coarso betwee ovetoal ass orreto of ad ad ts uertates resetvely Ths stuato gves us a alert essage about the results of the albrato. Ths offers us a hae to hek the albrato roess (data trasferrg, alulus, albrato of the stadards, sesors, et), before to subt a albrato ertfate. If the ass laboratory has a hstory of values for dfferet deades a statstal aalyss ould be doe fro these values ad establshg the tervals of albrato based o the. 7. Dsusso The osstey test ould be aled to ay auray lass of weght, but t s reoeded for lass ad lowers beause for ass albrato of weghts lass 1 are used odels of subdvso where hek stadards are trodued the odels for the statstal otrol. For auray lasses lowers tha 1 the osstey test ould offer ofdee to the ass laboratory o ther ertfates ad for the ustos for the serve reeved. It s eessary to say that ths test s ot sestve to a equal drft all stadards volved a albrato, beause the results obtaed would be osstet betwee the, but ay devates fro the real value; ths ossblty s avoded wth the albrato rogra of the stadard. O ass albrato of weght lasses 1 ad, orally the easureet of the volue of the weghts s requred. The volue s usually easured by hydrostat weghg ad the osstey test ould be adequate to volue easureets where the su of volues ould be oared versus the volue of the grou of weghts (orally a deade).

5 8. Colusos The use of the oralzed error ossts oarg the dfferee betwee the ovetoal ass of the grou of weghts ad the su of the ovetoal asses of dvdual weghts the rage of the obed uertaty of both alulatos. If the ass dfferee betwee both albrato s sall tha the obed uertaty of both values, the the ass albratos s aetable, but f the ass dfferee s large tha the obed uertates the there s a roble the albratos ad the values of the uertates obtaed do ot over t. The use the osstey test ould oleetary to the F-Test, beause the goal of the F-Test s to hek f the stadard devato of the balae was aetable. It s ortat to say that the uertaty has a lt (1/3 MT), ad fro ths ot of vew there s ot ossblty to elarge the uertaty order to over ths dfferee betwee the two values of ass. The osstey test ould be a useful tool for the etrologal assurae of the ass laboratores, beause wth the values the laboratory ould defe albrato tervals ad suort all values delared o ertfates or reorts of albrato. The ofdee of the ustos ould rease about the serve requred kowg that values obtaed o the albrato of ther weghts were less tha a hose value. [3] OIML, R33 Iteratoal Reoedato N 33 Covetoal Value of the result of weghg ar", 1979 [4] BIPM, IC, IFCC, ISO, IUPAC, IUPAP, OIML, "Gude to the eresso of uertaty easureet" Correted ad rerted, 1995 [5] Wolfag Wöger -Rearks o the - Crtero Used Measureet Coarso, PTB-Mttelge 109 1/99, Iteratoale Zusaearbet [6] uroea ooerato for Aredtato of Laboratores -AL Iterlaboratory Coarsos- (Marh 1996) [7] Bh W., 1990, "Varaes, Covaraes ad Restrats ass etrology", Metrologa 7, (1990) [8] Natrella G. Mary, -eretal Statsts; Hadbook Natoal Isttute of Stadards ad Tehology Cotat Perso for Paer: M. S. Lus Oar Beerra Mass ad Desty Dvso Address: k. 4,5 Carretera a los Cués Mo. l Marqués; Querétaro, Méo C.P Fa: (5) Phoe: (5) to 04 et. 360 e-al: lbeerra@ea. O evaluatos of tehal aablty of Mass laboratores ould be useful for the audtors hekg the values of obtaed o route albrato serves. 9. Referees [1] OIML, R111 Iteratoal Reoedato N 111 -Weghts of lasses 1,, F 1, F, M 1, M, M 3, 1994 [] OIML, Draft Iteratoal Reoedato N Weghts of lasses 1,, F 1, F, M 1, M, M 3,(ludg weghts for testg of hgh aaty weghg ahes) Part :1 Metrologal ad Tehal Requreets- February 000