Uncertainty Analysis Principles and Methods, RCC Document , September 2007 UNCERTAINTY ANALYSIS PRINCIPLES AND METHODS

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1 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 OCUMENT -7 TEEMETRY GROUP UNCERTAINTY ANAYSIS PRINCIPES AN METHOS WHITE SANS MISSIE RANGE REAGAN TEST SITE YUMA PROVING GROUN UGWAY PROVING GROUN ABEREEN TEST CENTER NATIONA TRAINING CENTER EECTRONIC PROVING GROUN HIGH ENERGY ASER SYSTEMS TEST FACIITY NAVA AIR WARFARE CENTER WEAPONS IVISION, PT. MUGU NAVA AIR WARFARE CENTER WEAPONS IVISION, CHINA AKE NAVA AIR WARFARE CENTER AIRCRAFT IVISION, PATUXENT RIVER NAVA UNERSEA WARFARE CENTER IVISION, NEWPORT PACIFIC MISSIE RANGE FACIITY NAVA UNERSEA WARFARE CENTER IVISION, KEYPORT 3TH SPACE WING 45TH SPACE WING AIR FORCE FIGHT TEST CENTER AIR ARMAMENT CENTER ARNO ENGINEERING EVEOPMENT CENTER BARRY M. GOWATER RANGE NATIONA NUCEAR SECURITY AMINISTRATION (NEVAA) ISTRIBUTION A: APPROVE FOR PUBIC REEASE ISTRIBUTION IS UNIMITE

2 Report ocmentaton Page Form Approved OMB No Pblc reportng brden for the collecton of nformaton s estmated to average hor per response, ncldng the tme for revewng nstrctons, searchng exstng data sorces, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths brden estmate or any other aspect of ths collecton of nformaton, ncldng sggestons for redcng ths brden, to Washngton Headqarters Servces, rectorate for Informaton Operatons and Reports, 5 Jefferson avs Hghway, Ste 4, Arlngton VA -43. Respondents shold be aware that notwthstandng any other provson of law, no person shall be sbject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a crrently vald OMB control nmber.. REPORT ATE SEP 7. REPORT TYPE 3. ATES COVERE --4 to TITE AN SUBTITE Uncertanty Analyss Prncples and Methods 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM EEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER TG-69 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AN ARESS(ES) Range Commanders Concl,5 Headqarters Avene,Whte Sands Mssle Range,NM,88 8. PERFORMING ORGANIZATION REPORT NUMBER SPONSORING/MONITORING AGENCY NAME(S) AN ARESS(ES). SPONSOR/MONITOR S ACRONYM(S). ISTRIBUTION/AVAIABIITY STATEMENT Approved for pblc release; dstrbton nlmted 3. SUPPEMENTARY NOTES. SPONSOR/MONITOR S REPORT NUMBER(S) 4. ABSTRACT efnes and catalogs standard mathematcal concepts and methods sed to estmate measrement ncertanty n test nstrmentaton systems. 5. SUBJECT TERMS Telemetry Grop; ncertanty estmaton; ncertanty combnaton; mltvarate ncertanty analyss; Bayesan analyss; system ncertanty analyss 6. SECURITY CASSIFICATION OF: 7. IMITATION OF ABSTRACT a. REPORT nclassfed b. ABSTRACT nclassfed c. THIS PAGE nclassfed Same as Report (SAR) 8. NUMBER OF PAGES 3 9a. NAME OF RESPONSIBE PERSON Standard Form 98 (Rev. 8-98) Prescrbed by ANSI Std Z39-8

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4 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 OCUMENT -7 UNCERTAINTY ANAYSIS PRINCIPES AN METHOS SEPTEMBER 7 Prepared by TEEMETRY GROUP Pblshed by Secretarat Range Commanders Concl U.S. Army Whte Sands Mssle Range New Mexco 88-5

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6 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 TABE OF CONTENTS IST OF FIGURES... v IST OF TABES... v PREFACE... v OVERVIEW OF THIS OCUMENT... x REVIEW OF KEY UNCERTAINTY ANAYSIS CONCEPTS... xv ACRONYMS... xx CHAPTER : BASIC CONCEPTS AN METHOS General efntons efnng the Measrement Process Errors and strbtons Error and Uncertanty Qantfyng Uncertanty Combnng Uncertantes Correlatng Error Sorces egrees of Freedom Confdence mts and Expanded Uncertanty Reportng Analyss Reslts CHAPTER : TYPE A UNCERTAINTY ESTIMATION General efntons Statstcal Sample Analyss CHAPTER 3: TYPE B UNCERTAINTY ESTIMATION General efntons Sbject and Measrng Parameter Bas Resolton Error Operator Bas gtal Samplng Error Comptaton Error Envronmental Factors Error Stress Response Error User efned Errors SpecMaster Type B egrees of Freedom Calclator CHAPTER 4: UNCERTAINTY COMBINATION General efntons Varance Addton... 4-

7 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September Error Sorce Correlatons Total Uncertanty and egrees of Freedom Pareto agrams Confdence mts CHAPTER 5: MUTIVARIATE UNCERTAINTY ANAYSIS General efntons Parameter Vale Eqaton Error Model Uncertanty Model Measrement Process Errors Measrement Process Uncertantes Error Sorce Correlatons Total Uncertanty and egrees of Freedom CHAPTER 6: SYSTEM UNCERTAINTY ANAYSIS General efntons System Model Modle Otpt Eqatons Modle Uncertantes System Otpt Uncertanty CHAPTER 7: UNCERTAINTY GROWTH ESTIMATION General efntons Basc Methodology Projected Uncertanty Relablty Models CHAPTER 8: BAYESIAN ANAYSIS (SMPC) General efntons SMPC Methodology Applcaton of SMPC CHAPTER 9: SOFTWARE VAIATION General efntons Software Valdaton Protocol Valdaton Examples REFERENCES APPENIX A: CYINER VOUME ANAYSIS... A- APPENIX B: OA CE ANAYSIS...B- APPENIX C: WINGBOOM AOA ANAYSIS... C- v

8 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 IST OF FIGURES Fgre -. The Normal strbton Fgre -. The ognormal strbton Fgre -3. The Exponental strbton... - Fgre -4. eft-handed Exponental strbton... - Fgre -5. The Qadratc strbton... - Fgre -6. The Cosne strbton... - Fgre -7. The U-Shaped strbton Fgre -8. The Unform strbton Fgre -9. The Tranglar strbton Fgre -. Stdent's t strbton Fgre -. Sgnal Qantzaton Fgre -. Random Error strbton Fgre 6-. Block agram for Example System Fgre 6-. Temperatre Measrement System agram Fgre 6-3. Block agram for Temperatre Measrement System Fgre 6-4. Schematc of Thermocople Sensor Modle (M) Fgre 6-5. Hysteress Effect Fgre 6-6. Schematc of Thermocople - owpass Flter Interface Fgre 6-7. Ampltde as a Fncton of Freqency for a Typcal Flter Fgre 6-8. Amplfer Errors Fgre 6-9. Samplng Rate Error Fgre 6-. Apertre Tme Error Fgre 6-. Implse Response Fgre 7-. Measrement Uncertanty Growth Fgre 7-. Measrement Relablty verss Tme Fgre 7-3. Sbject Parameter strbton Fgre 8-. Measrement Uncertanty Components Fgre 9-. Mltvarate Analyss Report for Cylnder Volme Example Fgre 9-. oad Cell Calbraton Setp Fgre 9-3. oad Cell Modle Report Fgre 9-4. Amplfer Modle Report Fgre 9-5. gtal Mltmeter Modle Report Fgre 9-6. oad Cell Calbraton System Report Fgre 9-7. Mltvarate Analyss Report for oad Cell Calbraton System Fgre 9-8 Block agram of Wngboom AOA Measrement System Fgre 9-9. Potentometer Modle Report Fgre 9-. Sgnal Condtoner Modle Report Fgre 9-. ata Processor Modle Report Fgre 9-. Wngboom AOA Measrement System Report v

9 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 IST OF TABES Table 6-. Parameters Used n Thermocople Modle Otpt Eqaton Table 6-. Parameters Used n Interface Modle Otpt Eqaton Table 6-3. Parameters Used n ow Pass Flter Modle Otpt Eqaton Table 6-4. Parameters Used n Interface Modle Otpt Eqaton Table 6-5. Parameters Used n Amplfer Modle Otpt Eqaton Table 6-6. Parameters Used n Interface3 Modle Otpt Eqaton Table 6-7. Parameters Used n A/ Converter Modle Otpt Eqaton Table 6-8. Parameters Used n ata Processor Modle Otpt Eqaton Table 9-. Comparson of Total Uncertanty and egrees of Freedom Table 9-. Uncertanty Estmates for Cylnder Volme sng Hand Calclatons Table 9-3. Comparson of oad Cell Total Uncertanty and egrees of Freedom Table 9-4. Spreadsheet Analyss Reslts for oad Cell Modle Table 9-5. Comparson of Amplfer Total Uncertanty and egrees of Freedom Table 9-6. Spreadsheet Analyss Reslts for Amplfer Modle Table 9-7. Comparson of gtal Mltmeter Total Uncertanty and egrees of Freedom Table 9-8. Spreadsheet Analyss Reslts for gtal Mltmeter Modle Table 9-9. Comparson of System Otpt Uncertanty and egrees of Freedom Table 9-. Spreadsheet Mltvarate Analyss Reslts for oad Cell System Table 9-. Comparson of Potentometer Total Uncertanty and egrees of Freedom Table 9-. Spreadsheet Analyss Reslts for Potentometer Modle Table 9-3. Comparson of Sgnal Condtoner Total Uncertanty and egrees of Freedom Table 9-4. Spreadsheet Analyss Reslts for Sgnal Condtoner Modle Table 9-5. Comparson of ata Processor Total Uncertanty and egrees of Freedom Table 9-6. Spreadsheet Analyss Reslts for ata Processor Modle v

10 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 PREFACE The Telemetry Grop of the Range Commanders Concl prepared ths docment to defne and catalog standard mathematcal concepts and methods sed to estmate measrement ncertanty n test nstrmentaton systems. Ths docment addresses detaled analyss procedres for se n dentfyng measrement process errors and n estmatng ther ncertanty. The docment s strctred so that specfc measrement methodologes of nterest can be easly accessed by the reader. ependng on the reader s knowledge and backgrond, he/she may want to revew the two p-front sectons mmedately followng ths preface. The frst secton s an overvew of ths docment offerng a descrpton of analyss methods dscssed. The second secton contans a revew of key ncertanty analyss concepts. Prmary contrbtor to ths report s shown below. Mr. Ray Falstch Member: Telemetry Grop (TG) CSC Range and Engneerng Servces 84-B Three Notch Road, exngton Park, M 653 Phone: (3) Fax: (3) E-Mal: rfalstch@csc.com Please drect any qestons to: Secretarat, Range Commanders Concl ATTN: TET-WS-RCC Bldg. 5 Headqarters Avene Whte Sands Mssle Range, New Mexco 88-5 Telephone:(575) 678-7, SN 58-7 E-mal: sarmy.wsmr.atec.lst.rcc@mal.ml v

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12 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 OVERVIEW OF THIS OCUMENT Ths docment has been prepared to defne and catalog standard mathematcal concepts and methods sed to estmate measrement ncertanty n test nstrmentaton systems. The analyss methods otlned n ths docment provde a comprehensve approach to estmatng measrement ncertanty. Basc gdelnes are presented for estmatng the ncertanty for the followng measrement processes: rect Measrements The vale of the qantty or sbject parameter s obtaned drectly by measrement and s not determned ndrectly by comptng ts vale from the measrement of other varables or qanttes. Mltvarate Measrements The vale of the qantty or sbject parameter s based on measrements of more than one attrbte or qantty. Measrement Systems The vale of the qantty or sbject parameter s measred wth a system comprsed of component modles arranged n seres. The strctred, step-by-step analyss procedres descrbed address the mportant aspects of dentfyng measrement process errors and estmatng ther ncertanty. Advanced topcs cover ncertanty growth over tme and the refnement of ncertanty estmates sng Bayesan methods. Approach The ncertanty that s determned and reported for a partclar measrement shold be the most realstc estmate possble. In ths regard, the person tasked wth condctng an ncertanty analyss mst be knowledgeable abot the measrement process nder nvestgaton. To facltate ths endeavor, the measrement process shold be descrbed n wrtten format. Ths wrte-p shold clearly specfy the measrement eqpment sed, the envronmental condtons drng measrement, and the procedre sed to obtan the measrement. The approach presented n ths docment provdes straghtforward and easy-to-nderstand prncples of measrement ncertanty analyss for drect and mltvarate measrements and measrement systems. Concepts and methods are consstent wth those fond n the ISO Gde to the Expresson of Uncertanty n Measrement (ISO GUM). Advanced measrement ncertanty analyss topcs that extend these methods and concepts are also presented. UncertantyAnalyzer, developed by the Integrated Scences Grop (ISG), mentoned throghot ths docment s a software tool for mplementng and demonstratng the methods and calclatons descrbed heren. Protocols, developed and mplemented by ISG, to valdate the UncertantyAnalyzer program are dscssed and examples are presented that compare UncertantyAnalyzer calclatons to vales obtaned by hand calclatons and from Excel spreadsheets. Concepts and methods presented heren are taken from materal developed by Integrated Scences Grop (see References). U.S. Gde to the Expresson of Uncertanty n Measrement, ANSI/NCS-Z x

13 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Other Methods Nmeros measrement ncertanty analyss standards and gdes have been pblshed over the past twenty years or so. Many of the ncertanty analyss references developed pror to the ISO GUM are based on technqes developed by r. Robert B. Abernathy and colleages. Examples of ncertanty analyss standards and other pblshed materal commonly sed n the U.S. engneerng commnty are: Test Uncertanty, ASME PTC (reaffrmed 4). Measrement Uncertanty for Fld Flow n Closed Condts, ANSI/ASME MFC- M-983 (reaffrmed ). Assessment of Wnd Tnnel ata Uncertanty, AIAA Standard S eck, R.H.: Measrement Uncertanty Methods and Applcatons, 3rd Edton, ISA. Coleman, H. W. and Steele, W. G.: Expermentaton and Uncertanty Analyss for Engneers, nd Edton, John Wley & Sons, 999. Althogh many of these references have been pdated or reaffrmed n recent years, the ncertanty analyss methods that they espose are dstnctly dfferent from those presented n the ISO GUM. Key dfferences are smmarzed below to llstrate how the methods and concepts presented n ths docment spplant technqes that are not based on the propertes of measrement error and the statstcal natre of measrement ncertanty.. Measrement Error. Pror to the ISO GUM, measrement errors were categorzed as ether random or systematc. 3 In ths context, random error s defned as the porton of the total measrement error that vares as a reslt of repeat measrements of a qantty. Systematc error s defned as the porton of the total measrement error that remans constant n repeat measrements of a qantty. The ISO GUM refers only to errors that can occr n a gven measrement process and does not dfferentate them as random or systematc. Measrement process errors can reslt from repeat measrements, operator bas, nstrment parameter bas, resolton, envronmental condtons, or other sorces. The key s to dentfy all sorces of error for a gven measrement process. Addtonally, each measrement error s consdered to be a random varable that can be characterzed by a statstcal dstrbton.. Uncertanty. Uncertanty de to random error s compted from the standard devaton, SX, of a sample of data S N ( X k X ) k N where N s the sample sze, X k s the k th measred vale, and X s the sample mean vale. If the mean vale s reported, then the standard devaton n the mean vale s gven by 3 The terms random and precson are often sed nterchangeably, as are the terms systematc and bas. x

14 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 s x S x. N Abernathy orgnally proposed that the ncertanty de to random error be compted by mltplyng the sample standard devaton by the stdent s t-statstc wth 95% confdence level. U ran t95s x or U ran t95s x Snce the pblcaton of the ISO GUM, some ncertanty analyss references have defned random ncertanty as beng eqal to the standard devaton. U ran S x or U ran Sx In all references, ncertanty de to systematc error, U bas, s based on past experence, manfactrer specfcatons, or other nformaton. In most cases, ths ncertanty estmate s assmed to be roghly eqvalent to a 95% confdence nterval or lmts for a systematc error that s normally dstrbted wth nfnte degrees of freedom. For example, f ± B represents the 95% confdence lmts for a normally dstrbted systematc error, then U bas ±B. The ISO GUM spplants systematc and random ncertantes wth standard ncertanty, 4 whch s a statstcal qantty eqvalent to the standard devaton of the error dstrbton. In ths regard, ncertanty s not consdered to be a ± lmt or nterval. The standard ncertanty of a measrement error s determned from Type A or Type B estmates. Type A ncertanty estmates are obtaned by the statstcal analyss of a sample of data. Type B ncertanty estmates are obtaned by herstc means. Wth a basc nderstandng of error dstrbtons and ther statstcs, the ncertanty, x, n a measrement, x x tre ε x, s the sqare root of the varance n the measrement error, ε x. var( x) var( x x tre ε ) x var( ε ) x Varance s a statstcal qantty, defned as the mean sqare dsperson of the error dstrbton abot ts mean or mode vale. strbton varance provdes a crcal lnk between measrement error and measrement ncertanty. To properly apply ths method, the dstrbton selected to estmate ncertanty for a gven error sorce shold provde the most realstc statstcal characterstcs. 3. Total Uncertanty. For decades, the qeston of how to combne random and systematc ncertantes has been a major sse, often sbject to heated debate. The vew spported by many data analysts and engneers was to smply add the ncertantes lnearly (A). 4 In ths docment, the terms standard ncertanty and ncertanty are sed nterchangeably. x

15 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 m B t U A 95 S N The vew spported by statstcans and measrement scence professonals was to combne them n root sm sqare (RSS). U RSS m S B t95 N In the late 97s, a compromse was proposed n whch ether method cold be sed as long as the followng constrants were met: a. The elemental random ncertantes and the elemental systematc ncertantes be combned separately. b. The total random ncertanty and total systematc ncertanty be reported separately. c. The method sed to combne the total random and total systematc ncertantes be stated. Ironcally, t was also recommended that the RSS method be sed to combne the elemental random ncertantes, S, and the elemental systematc ncertantes, B, as shown below. K S N / S K B B / Snce the pblcaton of the ISO GUM, most ncertanty analyss references now state that the total random and total systematc ncertantes be combned n RSS. In many nstances, the stdent s t-statstc, t 95, s set eqal to and U RSS s replaced by U 95. B S U 95 m N U 95 assmed to be eqvalent to 95% confdence lmts. In the ISO GUM, the ncertanty n the vale of an error s eqal to the sqare root of the varance of the error dstrbton. Conseqently, varance addton s sed to combne ncertantes from dfferent error sorces. To llstrate varance addton, let s consder the measrement of a qantty x that nvolves two error sorces ε and ε. x x tre ε ε x

16 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 The ncertanty n x s obtaned from x var( xtre ε ε ) var( ε ε ) var( ε ) var( ε ) cov( ε, ε ) where the covarance term, cov(ε, ε ), s the expected vale of the prodct of the devatons of ε and ε from ther respectve means. The covarance of two ndependent varables s zero. The covarance can be replaced wth the correlaton coeffcent, ρ,, whch s defned as where cov( ε, ε ) ρ, var( ε ) and var( ε ). Therefore, the ncertanty n x can be expressed as x ρ,. Snce correlaton coeffcents range from mns one to pls one, ths expresson provdes a more general and mathematcally rgoros method for combnng ncertantes. If, for example, ρ, (.e., statstcally ndependent errors), then the ncertantes are combned sng RSS. If ρ,, then the ncertantes are added. If ρ, -, then the ncertantes are sbtracted. Ths, varance addton 4. egrees of Freedom. When ncertantes are combned, t s mportant to estmate the degrees of freedom for the total ncertanty. Pror to the ISO GUM, there was no way to estmate the degrees of freedom for ncertantes de to systematc error. Conseqently, there was no way to compte the degrees of freedom for total ncertanty. Annex G of the ISO GUM provdes a relatonshp for comptng the degrees of freedom for a Type B ncertanty estmate v σ ( x) [ ( x) ] Δ( x) ( ) x where σ [(x)] s the varance n the ncertanty estmate, (x), and (x) s the relatve ncertanty n the ncertanty estmate. 5 5 Ths eqaton assmes that the nderlyng error dstrbton s normal. x

17 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Snce the pblcaton of the ISO GUM, a methodology for determnng σ [(x)] and comptng the degrees of freedom for Type B estmates has been developed by r. Howard T. Castrp. 6 In the ISO GUM, the effectve degrees of freedom, ν eff, for the total ncertanty, T, resltng from the combnaton of ncertantes and assocated degrees of freedom, ν, for n error sorces s estmated sng the Welch-Satterthwate formla 4 T* veff n 4 where the total ncertanty T* s compted assmng no error sorce correlatons. v 5. Confdence mts. Total ncertanty, T, and degrees of freedom, ν eff, can be sed to establsh the pper and lower lmts that contan the tre vale (estmated by the mean vale x ), wth some specfed confdence level, p. Confdence lmts are expressed as x t α /, v T trevale x tα / v eff, eff T where α - p and the t-statstc, t α/νeff, s a fncton of both the degrees of freedom and the confdence level. Pror to the ISO GUM, the total ncertanty U A, U RSS or U 95, was offered as type of confdence lmt. x U tre vale x 95 U 95 In some respects, these lmts are smlar to the expanded ncertanty, k, presented n the ISO GUM as an approxmate confdence lmt, n whch the coverage factor, k, s sed n place of the t-statstc. x k tre vale x k Unfortnately, the ntrodcton of an expanded ncertanty has served to perpetate confson abot what measrement ncertanty actally represents. The methods and concepts presented n ths docment are ntended to provde necessary clarfcaton abot ths and other ncertanty analyss sses. 6 See r. Castrp s works throghot the Reference secton. xv

18 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 REVIEW OF KEY UNCERTAINTY ANAYSIS CONCEPTS The followng paragraphs provde a bref overvew of key ncertanty analyss concepts and methods that are detaled n sbseqent chapters. The general ncertanty analyss procedre conssts of the followng steps:. efne the Measrement Process. Identfy the Error Sorces and strbtons 3. Estmate Uncertantes 4. Combne Uncertantes 5. Report the Analyss Reslts. efne the Measrement Process The frst step n any ncertanty analyss s to dentfy the physcal qantty whose vale s estmated va measrement. Ths qantty may be a drectly measred vale or ndrectly determned throgh the measrement of other varables. The former type of measrements are called drect measrements, whle the latter are called mltvarate measrements. For mltvarate measrements, t s mportant to develop an eqaton that defnes the mathematcal relatonshp between the qantty of nterest and the measred varables. S f ( x, y, z) where S sbject parameter or qantty of nterest f mathematcal fncton that relates S to measred qanttes x, y, and z. At ths stage of the analyss, t s also mportant to brefly descrbe the test setp, measrement procedres, envronmental condtons, nstrment specfcatons and other relevant nformaton that can help dentfy the measrement process errors.. Identfy the Error Sorces and strbtons In any gven measrement scenaro, each measred qantty s a potental sorce of error. The basc error model for a measred qantty,, s x meas x meas x tre where ε x s the measrement error. Measrement process errors are the basc elements of ncertanty analyss. Once these fndamental error sorces have been dentfed, we can begn to develop ncertanty estmates. Measrement errors most often encontered nclde, bt are not lmted to the followng: Measrement Bas Random or Repeatablty Error Resolton Error gtal Samplng Error ε x xv

19 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Comptaton Error Operator Bas Envronmental Factors Error Stress Response Error Another mportant aspect of the ncertanty analyss process s that measrement errors can be characterzed by statstcal dstrbtons. The statstcal dstrbton for a gven measrement error s a mathematcal descrpton that relates the freqency of occrrence of vales wth the vales themselves. In general, there are three error dstrbtons that have been fond to be relevant to most real world measrement applcatons: normal, lognormal, and Stdent s t. Measrement errors can also be characterzed by other dstrbtons sch as the nform, tranglar, qadratc, cosne, exponental, and U-shaped, althogh they are rarely applcable. The normal dstrbton shold be appled as the defalt dstrbton, nless nformaton to the contrary s avalable. The Stdent's t dstrbton s appled f the nderlyng dstrbton s normal, bt the ncertanty estmate s obtaned from a sample of measrements. The lognormal dstrbton shold be appled f t s sspected that the dstrbton of the vale of nterest s skewed. When sng the normal or lognormal dstrbton, some effort mst be made to estmate a contanment probablty. If % contanment has been observed, then the followng error dstrbton selecton crtera are recommended: Apply the cosne dstrbton f the vale of nterest has been sbjected to random sage or handlng stress, and s assmed to possess a central tendency. Apply ether the qadratc or half-cosne dstrbton, as approprate, f t s sspected that vales are more evenly dstrbted. The tranglar dstrbton may be applcable, nder certan crcmstances, when dealng wth parameters followng testng or calbraton. The tranglar dstrbton may also be applcable for errors de to lnear nterpolaton of tablated data. Apply the U-shaped dstrbton f the vale of nterest s the ampltde of a sne wave ncdent on a plane wth random phase. Apply the nform dstrbton for resolton error de to dgtal readot. Ths dstrbton s also applcable for qantzaton error and RF phase angle error. 3. Estmate Uncertantes As prevosly stated, all measrements are accompaned by error. Or lack of knowledge abot the sgn and magntde of measrement error s called measrement ncertanty. Ths leads s to Axom. Axom - The ncertanty n a measred vale s eqal to the ncertanty n the measrement error. Snce errors can be descrbed n sch a way that ther sgn and magntde have some defnable probablty of occrrence, we have Axom. Axom - Measrement errors follow statstcal dstrbtons. xv

20 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Wth a basc nderstandng of error dstrbtons and ther statstcs, we can estmate ncertantes. We begn wth the statstcal qantty called the varance, whch s defned as the mean sqare dsperson of the dstrbton abot ts mean or mode vale. var(x) Mean Sqare sperson n x If a qantty x s a random varable representng a poplaton of measrements, then the varance n x s jst the varance n the error n x, whch s expressed by the symbol ε x. var(x) var(x tre ε x ) var(ε x ). Fnally, we have Axom 3 to the crcal lnk between measrement error and measrement ncertanty. Axom 3 - The ncertanty,, n a measrement s the sqare root of the varance n the measrement error. Conseqently, the ncertanty n the measred vale x can be wrtten as x var( x) var( ε ) There are two approaches to estmatng varance and ncertanty. Type A estmates nvolve data samplng and analyss. Type B estmates se engneerng knowledge or recollected experence of measrement processes. The basc methods sed to estmate Type A and Type B ncertantes are presented n Chapters and 3, respectvely. 4. Combne Uncertantes Becase the ncertanty n the measrement error s eqal to the sqare root of the varance of the error dstrbton, we can se varance addton to combne ncertantes from dfferent error sorces. For prposes of llstraton, let s consder a qantty z that s obtaned ndrectly from the measrement of the qanttes x and y va the lnear fncton z ax by where the coeffcents a and b are constants. In ths case, we are nterested n the ncertanty n z n terms of the ncertantes n the measred qanttes x and y. Addtonally, measrement errors for x and y are composed of varos process errors (e.g., random, bas, resolton, envronmental, operator, etc.). The varance of z can be expressed n terms of the varances of the ndvdal varables, x and y var(z) var(ax by) a var(x) b var(y) ab cov(x,y) where the last term s the covarance between x and y. The covarance can be replaced by the correlaton coeffcent, ρ x,y, whch s defned as cov( x, y) ρ x, y. x y x xv

21 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Wth ths relatonshp and Axom 3, we can express the varance of z as z a x b y abρ x, y. The above eqaton can be generalzed to cases where there are n measred qanttes x, x,... x n. var( n a x ) n n n a var( x ) j> n a n n j> x j y a a ρ j j j a a ρ Wth varance addton, we have a logcal approach for combnng ncertantes that acconts for correlatons between error sorces. To assess the mpact of the correlated errors on combned ncertanty, let s consder the measrement of a qantty x that nvolves two error sorces ε andε. x z tre ε ε From Axoms and 3 and varance addton, the ncertanty n x s obtaned from j j x var( x ε ρ. tre ε var( ε ε ), The correlaton coeffcent, ρ,, for two error sorces can range n vale from - to. If the two error sorces are statstcally ndependent, then ρ, and x. Therefore, the ncertantes of statstcally ndependent error sorces are combned n a rootsm-sqare (RSS) manner. Conversely, f the two error sorces are strongly correlated then ρ, or -. If ρ,, then x, ( ) and the ncertantes are combned lnearly. If two error sorces are strongly correlated and compensate for one another, then ρ, - and x. ( ) Therefore, the combned ncertanty s the absolte vale of the dfference between the ndvdal ncertantes. Measrement process errors for a gven qantty aren t typcally correlated. Conseqently, t s safe to assme that there are no correlatons between the followng measrement process errors: Random Error and Parameter Bas (ρ ran,bas ) Random Error and Operator Bas (ρ ran,oper ) Parameter Bas and Resolton Error (ρ bas,res ) xv

22 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 Parameter Bas and Operator Bas (ρ bas,oper ) Operator Bas and Envronmental Factors Error (ρ oper,env ) Resolton Error and Envronmental Factors Error (ρ res, env ) gtal Resolton Error and Operator Bas (ρdres,oper ) In some nstances, the measrement process errors for dfferent qanttes may be correlated. Accontng for cross-correlatons s dscssed n Chapter 6. When ncertantes are combned, t s mportant to estmate the degrees of freedom for the total ncertanty. The effectve degrees of freedom, ν eff, for the total ncertanty, T, resltng from the combnaton of ncertantes and assocated degrees of freedom, ν, for n error sorces s estmated sng the Welch-Satterthwate formla Report the Analyss Reslts 4 T* veff n 4 When reportng the reslts of an ncertanty analyss, the followng nformaton shold be nclded:. The estmated vale of the qantty of nterest and ts combned ncertanty and degrees of freedom.. The mathematcal relatonshp between the qantty of nterest and the measred components. 3. The vale of each measrement component and ts combned ncertanty and degrees of freedom. 4. A lst of the measrement process ncertantes and assocated degrees of freedom for each component, along wth a descrpton of how they were estmated. 5. A lst of applcable correlaton coeffcents, ncldng any cross-correlatons between component ncertantes. It s also a good practce to provde a bref descrpton of the measrement process, ncldng the procedres and nstrmentaton sed, and addtonal data, tables and plots that help clarfy the analyss reslt v 7 Ths formla s based on the assmpton that there are no correlatons between error sorce ncertantes. xx

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24 Uncertanty Analyss Prncples and Methods, RCC ocment -7, September 7 ACRONYMS AC AOA AOP BOP AC MM EOP FS FSI FSO ISG ISO ISO GUM SBF MTE OOT PF RF RSS SU TME UUT VBA Vdc or VC Analog to gtal Converter Angle of Attack Average-over-Perod Begnnng-of-Perod gtal to Analog Converter gtal Mltmeter End-of-Perod Fll Scale Fll Scale Inpt Fll Scale Otpt Integrated Scences Grop Internatonal Standard for Organzaton ISO Gde to the Expresson of Uncertanty n Measrement east Sqares Best Ft Measrng and Test Eqpment Ot of Tolerance Probablty ensty Fncton Rado Freqency Root-sm-sqare method of combnng vales Resdal sm of sqares n regresson analyss Sbject Unt Test and Measrement Eqpment Unt Under Test Vsal Basc for Applcatons Volts rect Crrent xx

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26 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7. General CHAPTER BASIC CONCEPTS AN METHOS Ths chapter descrbes the basc concepts and methods sed to estmate measrement ncertanty. 8 The general ncertanty analyss procedre conssts of the followng steps:. efne the Measrement Process. Identfy the Error Sorces and strbtons 3. Estmate Uncertantes 4. Combne Uncertantes 5. Report the Analyss Reslts The followng sectons dscss these analyss steps n detal. A dscsson on sng ncertanty estmates to compte confdence ntervals and expanded ncertantes s also nclded.. efntons.. Combned Uncertanty. The ncertanty n the total error of a vale of nterest... Comptaton Error. The error n a qantty obtaned by comptaton. Normally de to machne rond-off error n vales obtaned by teraton. Sometmes appled to errors n tablated physcal constants...3 Confdence evel. The probablty that a set of error lmts or contanment lmts wll contan errors for a gven error sorce...4 Confdence mts. mts that bond errors for a gven error sorce wth a specfed probablty or "confdence."..5 Contanment mts. mts that are specfed to contan ether a parameter vale, devatons from the nomnal parameter vale, or errors n the measrement of the parameter vale...6 Contanment Probablty. The probablty that a parameter vale or errors n the measrement of ths vale wll le wthn specfed contanment lmts...7 Correlaton Analyss. An analyss that determnes the extent to whch two error sorces nflence one another. Typcally the analyss s based on ordered pars of vales of the two error sorce varables...8 Correlaton Coeffcent. A measre of the extent to whch two error sorces are lnearly related. A fncton of the covarance between the two error sorces. Correlaton coeffcents range from mns one to pls one. A postve correlaton coeffcent apples when ncreases n one sorce are accompaned by ncreases n the other. A negatve correlaton coeffcent apples when ncreases n one sorce are accompaned by decreases n the other. 8 The analyss procedre s based on tranng materals developed and presented by Integrated Scences Grop (see References). -

27 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7..9 Covarance. The expected vale of the prodct of the devatons of two random varables from ther respectve means. The covarance of two ndependent varables s zero... Coverage Factor. A mltpler sed to express an error lmt or expanded ncertanty as a mltple of the standard ncertanty... Cmlatve strbton Fncton. A mathematcal fncton whose vales F(x) are the probabltes that a random varable assmes a vale less than or eqal to x. Synonymos wth strbton Fncton... egrees of Freedom. A statstcal qantty that s related to the amont of nformaton avalable abot an ncertanty estmate. The degrees of freedom sgnfes how "good" the estmate s and serves as a sefl statstc n determnng approprate coverage factors and comptng confdence lmts and other decson varables...3 rect Measrements. Measrements n whch a measrng parameter X drectly measres the vale of a sbject parameter Y (.e., X measres Y). In drect measrements, the vale of the sbject parameter s obtaned drectly by measrement and s not determned by comptng ts vale from the measrement of other varables or qanttes...4 strbton Varance. The mean sqare dsperson of a dstrbton abot ts mean or mode vale. See also Varance...5 Effectve egrees of Freedom. The degrees of freedom for combned ncertantes compted from the Welch-Satterthwate formla...6 Error Eqaton. An algebrac expresson that defnes the total error n the vale of a sbject parameter n terms of all relevant component errors...7 Error mts. Bondng vales that are expected to contan the error from a gven sorce wth some specfed level of probablty or confdence...8 Error Sorce. A parameter, varable or constant that can contrbte error to the determnaton of the vale of a sbject parameter. Examples nclde: measrng parameter bas, random error, resolton error, operator bas, comptaton error and envronmental factors error...9 Error Sorce Correlaton. See Correlaton Analyss.. Error Sorce Uncertanty. The ncertanty n the error of a gven sorce... Expanded Uncertanty. A mltple of the standard ncertanty reflectng ether a specfed confdence level or arbtrary coverage factor... Herstc Estmate. An estmate resltng from accmlated experence and/or techncal knowledge concernng the ncertanty of an error sorce...3 Independent Error Sorces. Error sorces that are statstcally ndependent. Two error sorces are statstcally ndependent f one does not exert an nflence over the other or f both are not consstently nflenced by a common agency. See also Statstcal Independence...4 evel of Confdence. See Confdence evel...5 Mean Sqare Error. See Varance...6 Mean Vale. Sample Mean: The average vale of a measrement sample. Poplaton Mean: The expectaton vale for measrements sampled from a poplaton. -

28 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7..7 Measrand. Accordng to Annex B, Secton B..9 of the ISO GUM, the measrand s defned as "the partclar qantty sbject to measrement."..8 Measrement Error. The dfference between the measred vale of a parameter and ts tre vale...9 Measrement Process Errors. Measrement process errors refer to errors resltng from the measrement process (e.g., measrng parameter bas, operator bas, envronmental factors, ). Measrement process errors are the basc elements of ncertanty analyss. Once these fndamental error sorces have been dentfed, then we can begn to develop ncertanty estmates...3 Measrement Process Uncertantes. See Error Sorce Uncertanty...3 Measrement Uncertanty. The ncertanty n a measrement or n the error n the measrement...3 Measrng Parameter. Attrbte of a measrng devce that s sed to obtan nformaton that qantfes the vale of the sbject parameter...33 Medan Vale. () The vale that dvdes an ordered sample of data n two eqal portons. () The vale for whch the dstrbton fncton of a random varable s eqal to onehalf. (3) A pont of dscontnty sch that the dstrbton fncton mmedately below the pont s less than one-half and the dstrbton fncton mmedately above the pont s greater than one-half...34 Mode Vale. The vale of a parameter most often encontered or measred. Sometmes synonymos wth the nomnal vale or desgn vale of a parameter...35 Mltvarate Measrements. Measrements n whch the sbject parameter s a compted qantty based on measrements of two or more attrbtes or parameters...36 Nomnal Vale. The desgnated or pblshed vale of an artfact or parameter. It may also sometmes refer to the mode vale of an artfact or parameter...37 Parameter. Often thoght of as a specfed aspect or featre of an nstrment or tem. In general, however, a parameter does not have to be a toleranced qantty or vale. See also Attrbte...38 Poplaton. The total set of possble vales for a random varable nder consderaton...39 Poplaton Mean. It s the expectaton vale of a random varable descrbed by a poplaton dstrbton...4 Probablty. The lkelhood of the occrrence of a specfc event or vale from a poplaton of events or vales...4 Probablty ensty Fncton. A mathematcal fncton that descrbes the relatve freqency of occrrence of the vales of a random varable...4 Repeatablty. The closeness of the agreement between the reslts of sccessve measrements of the vale of a parameter carred ot nder the same measrement condtons. Repeatablty condtons nclde: the same measrement procedre, the same observer, the same measrng nstrment sed nder the same condtons, the same locaton, and repetton over a short perod of tme. -3

29 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Sample. A collecton of vales drawn from a poplaton. Typcally, nferences abot a poplaton are made from the sample. Therefore, the sample mst be statstcally representatve of the poplaton...44 Sample Hstogram. A bar chart showng the relatve freqency of occrrence of sampled data...45 Sample Mean. The arthmetc average of the measrements of a sample...46 Sample Sze. The nmber of measred vales that comprse a sample...47 Standard evaton. The sqare root of the varance of a sample or poplaton of vales. A qantty that represents the spread of vales abot a mean vale. In statstcs, the second moment of a dstrbton...48 Standard Uncertanty. A statstc representng spread or ncertanty n the vale of a parameter or error sorce. If determned statstcally from sampled data, the standard ncertanty s eqal to the sample standard devaton...49 Statstcal Independence. A property that descrbes two error sorces as beng ncorrelated. See also Independent Error Sorces...5 Sbject Parameter. An attrbte whose vale we seek to obtan from a measrement or set of measrements...5 Tolerance mts. mts that bond acceptable parameter vales...5 Tre Vale. The vale that wold be obtaned by a perfect measrement. Tre vales are by natre ndetermnate...53 Type A Estmates. Uncertanty estmates obtaned by the statstcal analyss of a sample of data...54 Type B Estmates. Uncertanty estmates obtaned by herstc means...55 Uncertanty. See Standard Uncertanty...56 Uncertanty Component. A contrbton to total combned ncertanty from an error sorce...57 Varance. () Poplaton: The expectaton vale for the sqare of the dfference between the vale of a varable and the poplaton mean. () Sample: A measre of the spread of a sample eqal to the sm of the sqared observed devatons from the sample mean dvded by the degrees of freedom for the sample. Also referred to as the mean sqare error.. efnng the Measrement Process The frst step n any ncertanty analyss s to dentfy the physcal qantty whose vale s estmated va measrement. Ths qantty, sometmes referred to as the measrand, may be a drectly measred vale, sch as the weght of a gm mass or the otpt of a voltage reference. Alternatvely, the qantty may be ndrectly determned throgh the measrement of other varables, as n the case of estmatng the volme of a cylnder by measrng ts length and dameter. The former type of measrements are called drect measrements, whle the latter are called mltvarate measrements. -4

30 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 For mltvarate measrements, t s mportant to develop an eqaton that defnes the mathematcal relatonshp between the qantty of nterest and the measred varables. For the cylnder volme example, ths mathematcal eqaton wold be expressed as where V π (-) V cylnder volme n nts of nterest cylnder dameter n approprate nts cylnder length n approprate nts π the rato of the crcmference of a crcle to ts dameter At ths stage of the analyss, t s also sefl to brefly descrbe the test setp, envronmental condtons, techncal nformaton abot the nstrments, reference standards, or other eqpment sed and the procedre for obtanng the measrement(s). Ths nformaton wll help dentfy the measrement process errors..3 Errors and strbtons In any gven measrement scenaro, each measred qantty s a potental sorce of error. For example, errors n the length and dameter measrements wll contrbte to the overall error n the estmaton of the cylnder volme. Therefore, the cylnder volme eqaton can be expressed as where V ε ε V π ( ε ) (-) V tre or nomnal cylnder volme tre or nomnal cylnder dameter tre or nomnal cylnder length ε V error n the cylnder volme measrement ε error n the cylnder dameter measrement ε error n the cylnder length measrement By rearrangng eqaton (-), we obtan an algebrac expresson for the cylnder volme error. -5

31 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ( ) ( ) 4 ) ( 4 ) ( ) ( V o V π ε ε ε ε ε ε ε π π ε ε ε π π ε ε π ε ε π ε (-3) The terms,, ε ε ε and, are referred to as second order terms and are consdered to be small compared to the other frst order terms n eqaton (-3). Neglectng these terms, we can express the cylnder volme error eqaton n a smpler form. ε ε ( ) 4 V π ε ε π ε (-4) Rearrangng eqaton (-4), we can frther smplfy the eqaton for. V ε V ε π ε π π π ε π ε π ε (-5) The coeffcents for ε and ε n eqaton (-5) are actally the partal dervatves of V wth respect to and. V π and V π Therefore, the cylnder volme error can be expressed as V V V ε ε ε (-6) where the partal dervatves are senstvty coeffcents that determne the relatve contrbton of the errors n length and dameter to the total error. The errors n length and dameter are the sm of the errors encontered drng the measrement process and can be expressed as n ε ε ε ε... n ε ε ε ε... -6

32 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where the nmbered sbscrpts sgnfy the dfferent measrement process errors. The errors most often encontered n makng measrements nclde, bt are not lmted to the followng: Measrement Bas Random or Repeatablty Error Resolton Error gtal Samplng Error Comptaton Error Operator Bas Envronmental Factors Error Stress Response Error Measrement process errors are the basc elements of ncertanty analyss. Once these fndamental error sorces have been dentfed, we can begn to develop ncertanty estmates. Another mportant aspect of the ncertanty analyss process s the fact that measrement errors can be characterzed by statstcal dstrbtons. The statstcal dstrbton for a type of measrement error s a mathematcal descrpton that relates the freqency of occrrence of vales wth the vales themselves. Error dstrbtons nclde, bt are not lmted to normal, lognormal, nform (rectanglar), tranglar, qadratc, cosne, exponental, -shaped, trapezodal, and stdent's t. Each dstrbton s characterzed by a set of statstcs. The statstcs most often sed n ncertanty analyss are the mean, or the mode, and the standard devaton. Wth the lognormal dstrbton, a lmtng vale and the medan vale are also sed. UncertantyAnalyzer atomatcally comptes the dstrbton statstcs n response to data entered by the ser. Alternatvely, the ser can select the desred error dstrbton from a dropdown lst. A bref descrpton of these dstrbtons s gven n the followng sbsectons..3. Normal strbton. When obtanng a Type A ncertanty estmate, we compte a standard devaton from a sample of vales. For example, we estmate ncertanty de to random error by comptng the standard devaton for a sample of repeated measrements of a gven vale. We also obtan a sample sze. The sample standard devaton s an estmate of the standard devaton for the poplaton from whch the sample was drawn. Except n rare cases, we assme that ths poplaton follows the normal dstrbton. Why do we assme a normal dstrbton? The prmary reason s becase ths s the dstrbton that ether represents or approxmates what we freqently see n the physcal nverse. It can be derved from the laws of physcs for sch phenomena as the dffson of gases and s applcable to nstrment parameters sbject to random stresses of sage and handlng. It s also often applcable to eqpment parameters emergng from manfactrng processes. -7

33 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre -. The Normal strbton. The probablty densty fncton for the normal dstrbton s gven n eqaton (-7). The varable µ s the poplaton mean and the varable σ s the poplaton standard devaton. ( x μ ) / ( ) σ f x e (-7) πσ In applyng the normal dstrbton, an ncertanty estmate s obtaned from contanment lmts and a contanment probablty. The se of the normal dstrbton s approprate n cases where the above consderatons apply and the lmts and probablty are at least approxmately known. The extent to whch ths knowledge s approxmate determnes the degrees of freedom of the ncertanty estmate. The degrees of freedom and the ncertanty estmate can be sed n conjncton wth the Stdent's t dstrbton to compte confdence lmts. et ± a represent the known contanment lmts and let p represent the contanment probablty. Then an estmate of the standard devaton of the poplaton of errors or devatons s obtaned from eqaton (-8). a (-8) p Φ The nverse normal dstrbton fncton, Φ - (), can be fond n statstcs texts and n most spreadsheet programs. If only a sngle contanment lmt s applcable, sch as wth sngle-sded tolerances, the approprate expresson s gven n eqaton (-9). a Φ ( p) (-9).3. ognormal strbton. The lognormal dstrbton can often be sed to estmate the ncertanty n eqpment parameter bas n cases where the tolerance lmts are asymmetrc. It s also sed n cases where a physcal lmt s present that les close enogh to the nomnal or mode -8

34 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 vale to skew the parameter bas probablty densty fncton n sch a way that the normal dstrbton s not applcable. Fgre -. The ognormal strbton. The probablty densty fncton for the lognormal dstrbton s gven n eqaton (-). The varable q s a physcal lmt for x, the varable m s the poplaton medan and the varable µ s the poplaton mode. The varable σ s not the poplaton standard devaton. Rather, σ s referred to as the "shape parameter." The accompanyng graphc shows a case where µ, q 9.67, σ.546, and m 3.7. The compted standard devaton for ths example s.376. x q ln ( μ q) m f ( x) exp (-) σ π x q σ Uncertanty estmates (standard devatons) for the lognormal dstrbton are obtaned by nmercal teraton..3.3 Exponental strbton. We sometmes enconter cases where there exsts a defnable pper or lower bond to the vales (or errors) attanable to a parameter wth a sngle-sded pper or lower tolerance lmt. In most nstances, the dstrbton to apply to these cases s the lognormal dstrbton. Ths dstrbton s characterzed by a nomnal or mode vale and, as ndcated, a bondng physcal lmt. It s possble that, for some parameters, the bondng lmt and the mode vale are eqal. If so, then the lognormal dstrbton sffers from a mathematcal dscontnty that makes t napproprate as the dstrbton of choce. To handle sch parameters, we employ the exponental dstrbton. A plot of ths dstrbton s shown below for a parameter whose mode vale μ s less than ts tolerance lmt. -9

35 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre -3. The Exponental strbton. The probablty densty fncton for the exponental dstrbton s gven n eqaton (-). f (x) λ e λ x μ,, x μ otherwse (-) The absolte vale for x μ s sed to accommodate cases where the mode s greater than the tolerance lmt, as depcted n Fgre -4. Fgre -4. eft-handed Exponental strbton. -

36 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The standard devaton or standard ncertanty estmate for the exponental dstrbton s obtaned from the sqare root of x. Employng the probablty densty fncton for the dstrbton, t can be shown that ths ncertanty s gven by eqaton (-). (-) λ.3.4 Qadratc strbton. A dstrbton that elmnates the abrpt change at the zero pont, does not exhbt nrealstc lnear behavor and satsfes the need for a central tendency s the qadratc dstrbton. Fgre -5. The Qadratc strbton. Ths dstrbton s defned by the probablty densty fncton gven n eqaton (-3). 3 f ( x) 4a [ ( x / a) ], a x a, otherwse Obtanng the mnmm bondng lmts ±a when a contanment probablty p and contanment lmts ± are known nvolves solvng the cbc eqaton (-4) a a p, a (-3) (-4) The solton can be obtaned nmercally. UncertantyAnalyzer contans a rotne that solves for a, along wth other parameters that represent more robst ncarnatons of the qadratc dstrbton. The teratve algorthm s gven n eqatons (-5) throgh (-7). a ' a F / F (-5) where F 3 3 a a 3 p (-6) -

37 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 and F ' 3 6a and a s the vale obtaned at the th teraton. The standard devaton, or standard ncertanty, estmate for the qadratc dstrbton s determned from eqaton (-8). p (-7) a (-8) Cosne strbton. Whle the qadratc dstrbton elmnates dscontntes wthn the bondng lmts, t rses abrptly at the lmts. And, even thogh the qadratc dstrbton has wder applcablty than ether the tranglar or nform dstrbton, ths featre nevertheless dmnshes ts physcal valdty. The cosne dstrbton overcomes ths shortcomng, exhbts a central tendency, and can be determned from mnmm contanment lmts. Fgre -6. The Cosne strbton. The probablty densty fncton for the cosne dstrbton s gven n eqaton (-9). f ( x) a πx cos, a,, a x a otherwse (-9) Solvng for a when a contanment probablty and contanment lmts ± are gven qres applyng nmercal teratve method to the expresson gven n eqaton (-). a sn(π / a) ap, π a (-) -

38 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The solton algorthm has been ncorporated nto UncertantyAnalyzer. The teratve algorthm s gven n eqatons (-) throgh (-3). a ' a F / F (-) where a F sn(π / a) ap (-) π and ' F sn( π / a) cos( π / a) p (-3) π a and a s the vale obtaned at the th teraton. The standard devaton or standard ncertanty estmate for the cosne dstrbton s obtaned from the expresson gven n eqaton (-4). 6 a (-4) 3 π The vale of for the cosne dstrbton translates to roghly 63% of the vale obtaned sng the nform dstrbton..3.6 U-Shaped strbton. The U dstrbton apples to snsodal RF sgnals ncdent on a load. Another applcaton for ths dstrbton wold be envronmental temperatre control n a laboratory or test chamber. The probablty densty fncton for the U-shaped dstrbton s gven n eqaton (-5). The parameter a represents the maxmm sgnal ampltde., a < x < a f ( x) π a x (-5), otherwse -3

39 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre -7. The U-Shaped strbton. If contanment lmts ± and a contanment probablty p are known, the parameter a can be compted from eqaton (-6). a, sn(π p / ) a The standard devaton or standard ncertanty for the U-shaped dstrbton s estmated from eqaton (-7). (-6) a (-7) The vale of for the U-shaped dstrbton translates to roghly % of the vale obtaned sng the nform dstrbton..3.7 Unform (Rectanglar) strbton. Ths nform dstrbton has % contanment lmts and the probablty of obtanng a vale wthn these lmts s eqally probable. -4

40 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre -8. The Unform strbton. The probablty densty fncton for the nform dstrbton s gven n eqaton (-8). f, a x a ( x) a (-8), otherwse The contanment lmts of the dstrbton are ± a. The probablty of lyng between -a and a s constant. The probablty of lyng otsde ± a s zero. The standard devaton, or standard ncertanty, for the nform dstrbton s obtaned from eqaton (-9). a (-9) Tranglar strbton. The tranglar dstrbton has been proposed for se n cases where the contanment probablty s %, bt there s a central tendency for vales of the varable of nterest. The tranglar dstrbton s the smplest dstrbton possble wth these characterstcs. The tranglar dstrbton sometmes apples to parameter vales mmedately followng test or calbraton. -5

41 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre -9. The Tranglar strbton. The probablty densty fncton for the tranglar dstrbton s gven n eqaton (-3). ( x a) / a f ( x) ( a x) / a,, a x, x a otherwse (-3) In cases where a contanment probablty p < can be determned for lmts ±, where < a, the lmts of the dstrbton are gven by eqaton (-3). a, p a Apart from representng post-test dstrbtons nder certan restrcted condtons, the tranglar dstrbton has lmted applcablty to physcal errors or devatons. Whle t does not sffer from the nform probablty crteron, as does the nform dstrbton, t nevertheless dsplays abrpt transtons at the bondng lmts and at the zero pont, whch are physcally nrealstc n most nstances. In addton, the lnear ncrease and decrease n behavor s somewhat fancfl for a probablty densty fncton. The standard devaton, or standard ncertanty, for the tranglar dstrbton s obtaned from eqaton (-3). (-3) a (-3) 6 ke the nform dstrbton, sng the tranglar dstrbton reqres the establshment of mnmm contanment lmts ± a. The same reservatons apply n ths regard to the tranglar dstrbton as to the nform dstrbton. -6

42 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Stdent s t strbton. If the nderlyng dstrbton s normal, and a Type A estmate and degrees of freedom are avalable, confdence lmts for measrement errors or parameter devatons may be obtaned sng the Stdent s t dstrbton. Ths dstrbton s avalable n statstcs textbooks and poplar spreadsheet applcatons. Fgre -. Stdent's t strbton. The probablty densty fncton for the stdent's t dstrbton s gven n eqaton (-33). The varable ν s the degrees of freedom and the parameter Γ() s the gamma fncton. The degrees of freedom qantfes the amont of knowledge sed n estmatng ncertanty. f ( x) v Γ ( x v πvγ / v) ( v )/ For Type A estmates the degrees of freedom s smply the sample sze, n, mns one, as shown n eqaton (-34). (-33) ν n (-34) The knowledge sed n estmatng ncertanty s ncomplete f the mnmm contan lmts ± a for the stdent's t dstrbton are approxmate and the contanment probablty p s estmated from recollected experence (.e., Type B). Therefore, the degrees of freedom assocated wth a Type B estmate s not nfnte. If the degrees of freedom varable s fnte bt nknown, the ncertanty estmate cannot be rgorosly sed to develop confdence lmts, perform statstcal tests or make decsons. Ths lmtaton has often preclded the se of Type B estmates as statstcal qanttes and has led to the msgded practce of sng fxed coverage factors. Fortnately, the ISO GUM provdes an expresson for obtanng the approxmate degrees of freedom for Type B estmates. However, the expresson nvolves the se of the varance n the ncertanty estmate, and a method for obtanng ths varance has been lackng ntl recently. A rgoros method for obtanng ths qantty has been mplemented nto UncertantyAnalyzer. -7

43 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The procedre s to frst estmate the ncertanty for the normal dstrbton and then estmate the degrees of freedom from the expresson gven n eqaton (-35). V B σ ( ) 3φ a φ ( Δa) πa e φ ( Δp), (-35) where φ Φ (-36) p The varables a and p represent "gve or take" vales for the contanment lmts and contanment probablty, respectvely. At frst glance, eqaton (-35) may seem to be anythng bt rgoros. However, several data npt formats have been ncorporated nto UncertantyAnalyzer that make the process of estmatng a and p thorogh. Once the degrees of freedom has been obtaned, the Type B estmate may then be combned wth other estmates, and the degrees of freedom for the combned ncertanty can be determned sng the Welch-Satterthwate formla otlned n Annex G, Secton 4. of the ISO GUM. If the nderlyng dstrbton for the combned estmate s normal, the t dstrbton can be sed to develop confdence lmts and perform statstcal tests. For a gven confdence or contanment lmts ± and correspondng degrees of freedom, the standard ncertanty can be estmated sng eqaton (-37). (-37) t The varable t α/,ν s the stdent's t statstc and the varable α - p, where p s the contanment probablty or confdence level. In UncertantyAnalyzer, the stdent's t statstc s compted teratvely for a gven set of α/ and ν..3. Choosng the Approprate strbton. In general, the three error dstrbtons that have been fond to be relevant to most real world measrement applcatons are the normal, lognormal, and Stdent s t dstrbtons. Other dstrbtons sch as the nform, tranglar, qadratc, cosne, exponental, and U-shaped are also possble, althogh they are rarely applcable. Some recommendatons for selectng the approprate dstrbton for a partclar error sorce are gven n the followng sbsecton. An addtonal sbsecton s nclded to llstrate the msgded applcaton of the nform dstrbton for Type B ncertanty estmates regardless of the error sorce. More specfc crtera for correctly selectng the nform dstrbton and example cases that satsfy ths crtera are gven n the fnal two sbsectons..3.. Recommendatons for Selectng strbtons. The normal dstrbton shold be appled as the defalt dstrbton, nless nformaton to the contrary s avalable. For Type B α, v -8

44 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 estmates, the data npt formats allded nder the dscsson of the Stdent's t dstrbton shold also be employed to estmate the degrees of freedom. Apply the lognormal dstrbton f t s sspected that the dstrbton of the vale of nterest s skewed. In sng the normal or lognormal dstrbton, some effort mst be made to estmate a contanment probablty. If % contanment has been observed, then the followng s recommended: Apply the cosne dstrbton f the vale of nterest has been sbjected to random sage or handlng stress, and s assmed to possess a central tendency. Apply ether the qadratc or half-cosne dstrbton, as approprate, f t s sspected that vales are more evenly dstrbted. The tranglar dstrbton may be applcable, nder certan crcmstances, when dealng wth parameters followng testng or calbraton. The tranglar dstrbton may also be applcable for errors de to lnear nterpolaton of tablated data. Apply the U-shaped dstrbton f the vale of nterest s the ampltde of a sne wave ncdent on a plane wth random phase. Apply the nform dstrbton f the vale of nterest s the resolton ncertanty of a dgtal readot. Ths dstrbton s also applcable to estmatng the ncertanty de to qantzaton error and the ncertanty n RF phase angle..3.. Blnd Acceptance of the Unform strbton. Applyng the nform dstrbton to obtanng Type B ncertanty estmates s a practce that has been ganng grond over the past few years. There are two man reasons for ths:. Applyng the nform dstrbton makes t easy to obtan an ncertanty estmate. If the lmts ±a of the dstrbton are known, the ncertanty estmate,, s smply compted from dvdng the contanment vale a by the sqare root of 3. It shold be sad that the "ease of se" advantage has been promoted by ndvdals who are gnorant of methods of obtanng ncertanty estmates for more approprate dstrbtons and by others who are smply lookng for a qck solton. In farness to the latter grop, they sometmes assert that the lack of specfcty of nformaton reqred to se other dstrbtons makes for crde ncertanty estmates anyway, so why not get yor crde estmate by ntentonally sng an napproprate dstrbton? At or present level of analytcal development, ths argment does not hold water. Snce the ntrodcton of the ISO GUM, methods have been developed to systematcally and rgorosly se dstrbtons that are physcally realstc. These methods have been ncorporated nto UncertantyAnalyzer.. It has been asserted by some that the se of the nform dstrbton s recommended n the ISO GUM. Ths s not tre. In fact, most of the methodology of the ISO GUM s based on the assmpton that the nderlyng error dstrbton s normal. Some of the belef that the nform dstrbton s called for n the ISO GUM stems from the fact that several ndvdals, who have come to be regarded as ISO GUM athortes, have been advocatng ts se. For clarfcaton on ths sse, the reader s referred to Secton 4.3 of the ISO GUM. -9

45 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Another sorce of confson s that some of the examples n the ISO GUM apply the nform dstrbton n statons that appear to be ncompatble wth ts se. It s reasonable to sppose that mch of ths s de to the fact that rgoros Type B estmaton methods and tools were not avalable at the tme the ISO GUM was pblshed, and the nform dstrbton was an "easy ot." As stated n tem above, the lack of sch methods and tools has snce been rectfed and ncorporated nto UncertantyAnalyzer. The phlosophy of ndscrmnately sng the nform dstrbton to compte Type B ncertanty estmates ndermnes efforts to estmate ncertantes that can be sed to perform statstcal tests, evalate measrement decson rsks, manage calbraton ntervals, develop meanngfl tolerances and compte vable confdence lmts. In other words, apart from provdng a nmber, the ncertanty estmate becomes a seless and potentally expensve commodty Crtera for Selectng the Unform strbton. The se of the nform dstrbton s approprate nder a lmted set of condtons. These condtons are smmarzed by three crtera. The frst crteron s that we mst know a set of mnmm bondng lmts for the dstrbton. Ths s the mnmm lmts crteron. Second, we mst be able to assert that the probablty of fndng vales between these lmts s nty. Ths s the % contanment crteron. Thrd, we mst be able to demonstrate that the probablty of obtanng vales between the mnmm bondng lmts s nform. Ths s the nform probablty crteron Mnmm mts Crteron. It s vtal that the lmts we establsh for the nform dstrbton are the mnmm bondng lmts. For nstance, f the lmts ± bond the varable of nterest, then so do the lmts ±, ±3, and so on. Snce the ncertanty estmate for the nform dstrbton s obtaned by dvdng the bondng lmt by the sqare root of three, sng a vale for the lmt that s not the mnmm bondng vale wll obvosly reslt n an nvald ncertanty estmate. Ths alone makes the applcaton of the nform dstrbton qestonable n estmatng bas ncertanty from sch qanttes as tolerance lmts, for nstance. It may be that ot-of-tolerances have never been observed for a partclar parameter (% contanment), bt t s nknown whether the tolerances are mnmm bondng lmts. Some years ago, a stdy was condcted nvolvng a voltage reference that showed that vales for one parameter were normally dstrbted wth a standard devaton that was approxmately / of the tolerance lmt. Wth -sgma lmts, t s nlkely that any ot-of-tolerances wold be observed. However, f the nform dstrbton were sed to estmate the bas ncertanty for ths tem, based on tolerance lmts, the ncertanty estmate wold be nearly sx tmes larger than wold be approprate. Some mght clam that ths s acceptable, snce the estmate can be consdered a conservatve one. That may be. However, t s also a seless estmate. Ths pont wll be elaborated later. A second dffclty we face when attemptng to apply mnmm bondng lmts s that sch lmts can rarely be establshed on physcal gronds. Ths s especally tre when sng parameter tolerance lmts. It s vrtally mpossble to magne a staton where desgn engneers have somehow been able to precsely dentfy the mnmm lmts that bond vales that are physcally attanable. If we add to ths the fact that tolerance lmts are often nflenced by marketng rather than engneerng consderatons, eqatng tolerance lmts wth mnmm bondng lmts becomes a very nfrtfl and msleadng practce. -

46 Uncertanty Analyss Prncples and Methods RCC ocment -7, September % Contanment Crteron. By defnton, the establshment of mnmm bondng lmts mples the establshment of % contanment. It shold be sad however, that an ncertanty estmate may stll be obtaned for the nform dstrbton f a contanment probablty less than % s appled. For nstance, sppose the contanment lmts are gven as ± and the contanment probablty s stated as beng eqal to some vale p between zero and one. Then, f the nform probablty crteron s met, the lmts of the dstrbton are gven by a, p a If the nform probablty crteron s not met, however, the nform dstrbton wold not be applcable, and we shold trn to other dstrbtons Unform Probablty Crteron. As dscssed above, establshng mnmm contanment lmts can be a challengng prospect. Harder stll s fndng real-world measrement error dstrbtons that demonstrate a nform probablty of occrrence between two lmts and zero probablty of occrrence otsde these lmts. Except n very lmted nstances, sch as ones dscssed n the next secton, assmng a nform probablty s jst not physcally realstc. Ths s tre even n some cases where the dstrbton wold appear to be applcable. For example, a conjectre has recently been advanced that the dstrbton of parameters mmedately followng test or calbraton can be sad to be nform. Whle ths seems reasonable at face vale, t trns ot not to be the case. Becase of false accept rsk (consmer's rsk), sch dstrbtons range from approxmately tranglar to havng a "hmped" appearance wth rolledoff sholders. As to whether we can treat parameter tolerance lmts as bonds that contan vales wth probablty, we mst magne that, not only has the nstrment manfactrer managed to mraclosly ascertan mnmm bondng lmts, bt has also jggled physcs to sch an extent as to make the parameter vale's probablty dstrbton nform between these lmts and zero otsde them. Ths wold be a trly amazng feat of engneerng for most toleranced qanttes, especally consderng the marketng nflence mentoned earler Cases that Satsfy the Crtera gtal Resolton Uncertanty. We sometmes need to estmate the ncertanty de to the resolton of a dgtal readot. For nstance, a three-dgt readot mght ndcate.5 V. If the devce employs the standard rond-off practce, we know that the dsplayed nmber s derved from a sensed vale that les between.45 V and.55 V. We also can assert to a very hgh degree of valdty that the vale has eqal probablty of lyng anywhere between these two nmbers. In ths case, the se of the nform dstrbton s approprate, and the resolton ncertanty s.5 V V. 9 V RF Phase Angle. RF power ncdent on a load may be delvered to the load wth a phase angle θ between - π and π. In addton, nless there s a compellng reason to beleve otherwse, the probablty of occrrence between these lmts s nform. Accordngly, the se of the nform dstrbton s approprate. Ths yelds a phase angle ncertanty estmate of -

47 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 π θ It s nterestng to note that, gven the above, f we assme that the ampltde of the sgnal s snsodal, the dstrbton for ncdent voltage s the U-shaped dstrbton Qantzaton Error. When an analog sgnal s dgtzed, the sampled sgnal ponts are qantzed n mltples of a dscrete step sze. The potental drop (or lack of a potental drop) sensed across each element of an analog to dgtal (A/) Converter sensng network prodces ether a "" or "" to the converter. Ths response constttes a "bt" n the bnary code that represents the sampled vale. For ladder-type networks, the poston of the bt n the code s determned by the locaton of ts orgnatng network element. Even f no errors were present n samplng and sensng the npt sgnal, errors wold stll be ntrodced by the dscrete natre of the encodng process. Sppose, for example, that the fll scale sgnal level (dynamc range) of the A/ Converter s a volts. If n bts are sed n the encodng process, then a voltage V can be resolved nto dscrete steps, each of sze a/. The error n the voltage V s ths a ε ( V ) V m, n where m s some nteger determned by the sensng fncton of the /A Converter. The contanment lmt assocated wth each step s one-half the vale of the magntde of the step. Conseqently, the contanment lmt nherent n qantzng a voltage V s (/)(a/ n ), or a/ n. Ths s emboded n the expresson a V qantzed Vsensed ±. n The ncertanty de to qantzaton error s obtaned from the contanment lmts and from the assmpton that the sensed analog vale has eqal probablty of occrrence between these lmts: -

48 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 n a / V. 3.4 Error and Uncertanty Fgre -. Sgnal Qantzaton. As prevosly stated, all measrements are accompaned by error. Or lack of knowledge abot the sgn and magntde of measrement error s called measrement ncertanty. To better nderstand the relatonshp between measrement error and measrement ncertanty, we wll dscss three mportant axoms that form the bass pon whch ncertantes can be estmated. We wll also revew the varance addton rle, whch provdes a method for correctly combnng ncertantes from dfferent error sorces. Axom - The ncertanty n a measred vale s eqal to the ncertanty n the measrement error. Ths statement can be shown to be tre from the followng steps:. By defnton, measrement error s the dfference between the measred vale and the tre vale. Conversely, the measred vale s eqal to the tre vale pls the measrement error. Measred Vale Tre Vale Measrement Error. We defne the fncton for ncertanty n the vale x as Uncertanty (x) Uncertanty n x 3. The ncertanty n the measred vale can then be expressed as Uncertanty (Measred Vale) Uncertanty (Tre Vale) Uncertanty (Measrement Error) Bt the ncertanty n the tre vale s zero, so -3

49 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Uncertanty (Measred Vale) Uncertanty (Measrement Error) Axom - Measrement errors follow statstcal dstrbtons. Ths statement bascally ndcates that errors can be descrbed n sch a way that ther sgn and magntde have some defnable probablty of occrrence. Wth a basc nderstandng of error dstrbtons and ther statstcs, we can estmate ncertantes. We begn wth the statstcal qantty called the varance, whch s defned as the mean sqare dsperson of the dstrbton abot ts mean or mode vale. var(x) Mean Sqare sperson n x If a varable x follows a probablty dstrbton, descrbed by a probablty densty fncton f(x), then the mean sqare dsperson or varance of the dstrbton s gven by var( x) ( x μx ) f ( x) dx (-38) where µ x s the mean of x. Becase of the form of ths defnton, the varance s also referred to as the mean sqare error. If a qantty x s a random varable representng a poplaton of measrements, then the varance n x s jst the varance n the error n x, whch s expressed by the symbol ε x. var( x) var( ε ) var( ε ) (-39) x tre Axom 3 - The ncertanty,, n a measrement s the sqare root of the varance n the measrement error. If x s a measred vale, then we can wrte x x x var( x) var( ε ) (-4) x Axom 3 provdes the crcal lnk between measrement error and measrement ncertanty..5 Qantfyng Uncertanty There are two approaches to estmatng varance and ncertanty. Type A estmates nvolve data samplng and analyss. Type B estmates se engneerng knowledge or recollected experence of measrement processes. The basc methods sed to estmate Type A and Type B ncertantes are presented heren. etals of how these and other advanced methods are ncorporated n the Integrated Systems Grop (ISG) Uncertanty Analyzer software are dscssed n Chapters and Type A Estmates. A Type A ncertanty estmate s defned as an estmate obtaned from a sample of data. ata samplng nvolves makng repeat measrements of the qantty of nterest. It s mportant that each repeat measrement s ndependent, representatve and taken randomly. Random samplng s a cornerstone for obtanng relevant statstcal nformaton. Ths, Type A estmates sally apply to the ncertanty n repeatablty or random error. Becase the data sample s drawn from a poplaton of vales, we make nferences abot the poplaton from certan sample statstcs and from assmptons abot the way the poplaton of -4

50 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 vales s dstrbted. A sample hstogram can ad n or attempt to pctre the poplaton dstrbton. Fgre -. Random Error strbton. The normal dstrbton s ordnarly assmed to be the nderlyng dstrbton for random errors. When samples are taken, the sample mean and the sample standard devaton are compted and assmed to represent the mean and standard devaton of the poplaton dstrbton. However, ths eqvalence s only approxmate. To accont for ths, the Stdent's t dstrbton s sed n place of the normal dstrbton n comptng confdence lmts arond sample mean. The sample mean can be thoght of as an estmate of the vale that we expect to get when we make a measrement. Ths "expectaton vale" s called the poplaton mean, whch s expressed by the symbol µ. The sample mean, x, s obtaned by takng the average of the sampled vales. The average vale s compted by smmng the vales sampled and dvdng them by the sample sze, n. x ( x n x... x ) n n x n (-4) The sample standard devaton provdes an estmate of how mch the poplaton s spread abot the mean vale. The sample standard devaton, s x, s compted by takng the sqare root of the sm of the sqares of sampled devatons from the mean dvded by the sample sze mns one. s x n n ( x x) (-4) The vale n- s the degrees of freedom for the estmate, whch sgnfes the nmber of ndependent peces of nformaton that go nto comptng the estmate. The greater the degrees of freedom, the closer the sample estmate wll be to ts poplaton conterpart. The degrees of freedom for an ncertanty estmate s sefl for establshng confdence lmts and other -5

51 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 decson varables. We have already stated that the sample standard devaton s an estmate of the ncertanty n a vale drawn randomly from ts poplaton. However, f the estmate s to represent the ncertanty n the mean vale rather than the ncertanty n a sngle measrement, then the ncertanty n the mean vale shold be sed. The ncertanty n the mean vale, s x, s eqal to the standard devaton dvded by the sqare root of the sample sze. sx sx (-43) n Once estmates of the sample mean and standard devaton have been obtaned, and the degrees of freedom have been noted, t becomes possble to compte lmts that bond the sample mean wth some specfed level of confdence. These lmts are called confdence lmts and the degree of confdence s called the confdence level. Confdence lmts can be expressed as mltples of the sample standard devaton. For normally dstrbted samples, ths mltple s called the t-statstc. The vale of the t-statstc s determned by the desred percent confdence level, C, and the degrees of freedom, ν, for the sample standard devaton. Confdence lmts arond the sample mean are gven by where α ( - C/) and ν n -. sx x ± tα /,v (-44) n.5. Type B Estmates. In some cases, we mst attempt to qantfy the statstcs of measrement error dstrbtons by drawng on or recollected experence concernng the vales of measred qanttes or on or knowledge of the errors n these qanttes. Estmates made n ths manner are called herstc or Type B estmates. Concevably, a Type B ncertanty estmate cold be obtaned by jst "wngng t." The problem s, that most of s do not have a pont of reference for abstract qanttes sch as standard devaton or ncertantes. At best, we have a range of vales that we have experenced or are able to srmse. The lmtng vales that bond these ranges are called contanment lmts. These lmts can be vewed as bondng ether measred vales or measrement errors. x les wthn ± In workng wth Type A estmates, we start wth sample statstcs and work toward developng confdence lmts that bond vales of a poplaton wth a specfed confdence level or probablty. In makng Type B estmates, we apply ths process n reverse. We begn wth contanment lmts and a contanment probablty, estmate the degrees of freedom, and se these qanttes to estmate the standard devaton or ncertanty. x les wthn ± wth C% confdence or probablty Contanment lmts may be estmated from experence or taken from some docmented reference, sch as manfactrer tolerance lmts, stated expanded ncertantes obtaned from calbraton records or certfcates, or statstcal process control lmts. Contanment probablty can be obtaned from servce hstory data, for example, as the nmber of observed n-tolerances, n n-tol, dvded by the nmber of calbratons, N. -6

52 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 C% % If a herstc estmate s obtaned solely from contanment lmts and contanment probabltes, then the degrees of freedom s sally taken to be nfnte. For example, f the measrement error s normally dstrbted, the ncertanty s compted from the contanment lmts, ±, the nverse normal dstrbton fncton, Φ - (), and the contanment probablty, p C/. The nverse normal dstrbton fncton can be fond n statstcs texts and n most spreadsheet programs. The approprate relaton between ncertanty and contanment lmts and contanment probablty s gven n eqaton (-45). nn N tol (-45) p Φ If there s an ncertanty n the contanment lmts (e.g., ± ± Δ) or the contanment probablty (e.g., ±p ± Δp), then t becomes mperatve to estmate the degrees of freedom. As wth Type A ncertanty estmates, the degrees of freedom qantfes the amont of nformaton that goes nto the Type B ncertanty estmate and s sefl for establshng confdence lmts and other decson varables. Annex G of the ISO GUM provdes a relatonshp for comptng the degrees of freedom for a Type B ncertanty estmate v ( x) σ [ ( x) ] Δ( x) ( ) x (-46) where σ [(x)] s the varance n the ncertanty estmate, (x), and (x) s the relatve ncertanty n the ncertanty estmate. 9 Hence, the degrees of freedom for a Type B estmate s nversely proportonal to the sqare of the rato of the ncertanty n the ncertanty dvded by the ncertanty. Whle ths approach s nttvely appealng, the ISO GUM offers no advce abot how to determne σ [(x)] or (x). Fortnately, snce the pblcaton of the ISO GUM, a methodology for determnng σ [(x)] and comptng the degrees of freedom for Type B estmates has been developed. Once the contanment lmts, contanment probablty and the degrees of freedom have been establshed, we can estmate the standard devaton or ncertanty of the dstrbton of nterest. For nstance, f the measrement errors are normally dstrbted, we can constrct a t-statstc based on the contanment probablty and degrees of freedom. The ncertanty estmate s then obtaned by dvdng the contanment lmt by the t-statstc, accordng to eqaton (-47). (-47) t α /,ν 9 Ths eqaton assmes that the nderlyng error dstrbton s normal. -7

53 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7.6 Combnng Uncertantes For prposes of llstraton, let s consder a qantty or parameter z that s obtaned ndrectly from the measrement of the qanttes x and y. We wll say that z s a lnear fncton of the qanttes x and y. z ax by where the coeffcents a and b are constants. In ths case, we are nterested n the ncertanty n z n terms of the ncertantes n the measred qanttes x and y. Addtonally, measrement errors for x and y are composed of varos process errors (e.g., random, bas, resolton, envronmental, operator, etc.). We recall that Axom 3 states that the ncertanty n the vale of an error s eqal to the sqare root of the varance of the error dstrbton. As a conseqence, we can apply the varance addton rle to obtan a method for correctly combnng ncertantes from dfferent error sorces..6. Varance Addton Rle. The varance of z can be expressed n terms of the varances of the ndvdal varables, x and y var( z) var( ax by) a var( x) b var( y) abcov( x, y) where the last term s the covarance between x and y. If we recall Axom 3 and eqaton (-4), we can express the varance of z as shown n eqaton (-48). z a x b y abcov( x, y) If two varables x and y are descrbed by a jont probablty densty fncton f(x,y), then the covarance of x and y s gven by cov( x, y) dx ( x μx )( y μ y ) f ( x, y) dy where µ x and µ y are mean vales for x and y, respectvely. The covarance s rarely sed explctly. Instead, we se the correlaton coeffcent, ρx,y, whch s defned as x y (-48) (-49) cov( x, y) ρ x, y (-5) Wth ths relatonshp, the varance n the sm of the two qanttes x and y s gven by eqaton (-5). z a x b y abρ x, y x y (-5) Eqaton (-5) can be generalzed to cases where there are n measred qanttes x, x,... x n. -8

54 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 var( n a x ) n n a a var( x ) n n n j> n j> a a ρ j a a ρ j j j j j (-5).7 Correlatng Error Sorces Wth the varance addton rle, we have a logcal approach for combnng ncertantes that acconts for correlatons between error sorces. To assess the mpact of the correlated errors on combned ncertanty, let s consder the measrement of a qantty x that nvolves two error sorces ε and ε. x x tre ε ε From Axoms and 3 and the varance addton rle, the ncertanty n x s obtaned from x var( x tre ε ) var( ε ε ) ε ρ, The correlaton coeffcent, ρ,, for two error sorces can range n vale from - to. If the two error sorces are statstcally ndependent, then ρ, and x. Therefore, ncertantes of statstcally ndependent error sorces are combned n a root-sm-sqare (RSS). Conversely, f the two error sorces are strongly correlated then ρ, or -. If ρ,, then x ( ). Therefore, the ncertantes are combned lnearly. When two error sorces are strongly correlated and compensate for one another, then ρ, - and x ( ). Therefore, the combned ncertanty s the absolte vale of the dfference between the ndvdal ncertantes..7. Correlatons between Measrement Process Errors. There typcally aren t any correlatons between measrement process errors for a gven qantty. In general t s safe to assme that there are no correlatons between the followng measrement process errors. Random Error and Parameter Bas (ρ ran,bas ) Random Error and Operator Bas (ρ ran,oper ) Parameter Bas and Resolton Error (ρ bas,res ) Parameter Bas and Operator Bas (ρ bas,oper ) Operator Bas and Envronmental Factors Error (ρ oper,env ) Resolton Error and Envronmental Factors Error (ρ res,env ) gtal Resolton Error and Operator Bas (ρ dres,oper ).7. Accontng for Cross-Correlatons. In some nstances, the measrement process errors for dfferent qanttes may be correlated. For the cylnder volme example, ths wold occr f the -9

55 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 same devce s sed to measre the cylnder length and dameter, and. In ths case, the ncertanty n the measrng parameter bas s the same for both qanttes and components. We accont for cross-correlatons by developng an expresson for the correlaton coeffcent, ρ,, between the total ncertantes for each component, and, n terms of the crosscorrelatons, ρ,j, between the measrement process ncertantes for each component, and j. n nj ρ ρ (-53), j, j Accontng for cross-correlatons wll be dscssed frther n Chapter 6..8 egrees of Freedom When ncertantes are combned, we need to know the degrees of freedom for the total ncertanty. As mght be expected, the degrees of freedom for a combned ncertanty estmate s not a smple sm of the degrees of freedom for each ncertanty component. The effectve degrees of freedom, ν eff, for the total ncertanty, T, resltng from the combnaton of ncertantes and assocated degrees of freedom, ν, for n error sorces can be estmated va the Welch-Satterthwate formla gven n Annex G of the ISO GUM. 4 T veff n 4 v j * (-54) The combned ncertanty T* s compted assmng no error sorce correlatons. Conseqently, the effectve degrees of freedom compted from eqaton (-54) s consdered a rogh estmate..9 Confdence mts and Expanded Uncertanty As prevosly stated, the ncertanty,, and degrees of freedom, ν, can be sed to establsh confdence lmts. These are the pper and lower lmts that contan the tre vale, µ (estmated by the mean vale x ), wth some specfed confdence level or probablty, p. Confdence lmts are expressed as x tα μ x t (-55) /, v α /, v where the mltpler s the t-statstc, t α/ν, and α - p. The ISO GUM defnes the term expanded ncertanty as "the qantty defnng an nterval abot the reslt of a measrement that may be expected to encompass a large fracton of the dstrbton of vales that cold reasonably be attrbted to the measrand." In less obtse langage, the expanded ncertanty s bascally defned as a set of lmts (± ) that are expected to contan the tre vale of the measrand. In ths context, the expanded ncertanty, k, s offered as an approxmate confdence lmt, n whch the coverage factor, k, s sed n place of Ths formla s based on the assmpton that there are no correlatons between error sorce ncertantes. -3

56 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 the t-statstc. x k tre vale x k (-56) The ntrodcton of the expanded ncertanty s confsng at best, snce t s conternttve to thnk of an ncertanty as havng a range. In actal practce, the terms expanded ncertanty and ncertanty are sed nterchangeably. Ths, of corse, can lead to ncorrect nferences and mscommncatons. To mtgate ths problem, the ISO GUM also ntrodced the term "standard ncertanty" to help dstngsh ncertanty from expanded ncertanty. Unfortnately, confson over and msapplcaton of these terms perssts.. Reportng Analyss Reslts When reportng the reslts of an ncertanty analyss, Secton 7 of the ISO GUM recommends that the followng nformaton be nclded:. The estmated vale of the qantty of nterest (measrand) and ts combned ncertanty and degrees of freedom.. The fnctonal relatonshp between the qantty of nterest and the measred components, along wth the senstvty coeffcents. 3. The vale of each measrement component and ts combned ncertanty and degrees of freedom. 4. A lst of the measrement process ncertantes and assocated degrees of freedom for each component, along wth a descrpton of how they were estmated. 5. A lst of applcable correlaton coeffcents, ncldng any cross-correlatons between component ncertantes. It s also a good practce to provde a bref descrpton of the measrement process, ncldng the procedres and nstrmentaton sed, and addtonal data, tables and plots that help clarfy the analyss reslts. -3

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58 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7. General CHAPTER TYPE A UNCERTAINTY ESTIMATION As prevosly stated, Type A ncertanty estmates are defned as estmates obtaned from a sample of data. For example, measrement ncertanty de to random or repeatablty error s estmated from a statstcal analyss of a sample of measrements. Ths chapter dscsses how key sample statstcs are compted and sed to estmate Type A ncertantes. In partclar, the methodology focses on ncertanty estmaton for the error de to random varatons n repeat measrements made by the sbject parameter or measrng parameter drng a measrement sesson.. efntons.. Adjsted Mean. The vale of a parameter or error sorce obtaned by applyng a correcton factor to a nomnal or mean vale... Between Sample Sgma. The standard devaton of the mean vales...3 Compted Mean Vale. The mean vale of a parameter determned from a sample of measrements...4 Confdence evel. The probablty that a set of error lmts or contanment lmts wll contan errors for a gven error sorce...5 Confdence mts. mts that bond errors for a gven error sorce wth a specfed probablty or "confdence."..6 Coverage Factor. A mltpler sed to express an error lmt of expanded ncertanty as a mltple of the standard ncertanty...7 egrees of Freedom. A statstcal qantty that s related to the amont of nformaton avalable abot an ncertanty estmate and serves as a sefl statstc n determnng approprate coverage factors and comptng confdence lmts and other decson varables. For Type A estmates, the degrees of freedom s the sample sze mns one...8 evaton from Nomnal. The dfference between a parameter's measred vale and ts nomnal vale...9 Hstogram. See Sample Hstogram... Krtoss. A measre of the "peakedness of the sample dstrbton. Normal dstrbtons have a peakedness vale of three... Mean evaton. The dfference between a sample mean vale and the nomnal vale. Statstcal methods presented heren can be fond n varos pblshed books ncldng, bt not lmted to: Mathematcal Statstcs and ata Analyss, Rce. J., xbry Press, 995 and Probablty and Statstcs, nd Edton, Spegel, M. R. et. al, McGraw-Hll,. -

59 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7.. Mean Vale Correcton. The correcton or adjstment of the compted mean vale for an offset de to parameter bas and/or envronmental factors...3 Measrng Parameter. Attrbte of a measrng devce that s sed to obtan nformaton that qantfes the vale of the sbject parameter...4 Nomnal Vale. The desgnated or pblshed vale of an artfact or parameter. It may also sometmes refer to the mode vale of an artfact or parameter...5 Parameter Bas. A systematc devaton of a parameter vale from ts nomnal or ndcated vale...6 Random Error. An error that appears as dfferences n the measred vales of a gven artfact or parameter drng a measrement sesson. Sometmes de to random flctatons n the sbject parameter vale, the measrng parameter vale, and the measrement process...7 Repeatablty. The closeness of the agreement between the reslts of sccessve measrements of the vale of a parameter carred ot nder the same measrement condtons...8 Repeatablty Error. See random error...9 Sample. A collecton of vales drawn from a poplaton. Typcally, nferences abot a poplaton are made from the sample. Therefore, the sample mst be statstcally representatve of the poplaton... Sample Hstogram. A bar chart showng the relatve freqency of occrrence of sampled data... Sample Mean. The average vale of a measrement sample... Sample Sze. The nmber of measred vales that comprse a sample...3 Skewness. A measre of the asymmetry of the sample dstrbton. A symmetrc dstrbton has zero skewness...4 Standard evaton. The sqare root of the varance of a sample or poplaton of vales. A qantty that represents the spread of vales abot a mean vale. In statstcs, the second moment of a dstrbton...5 Standard Uncertanty. A statstc representng spread or ncertanty n the vale of a parameter or error sorce. If determned statstcally from sampled data, the standard ncertanty s eqal to the sample standard devaton...6 Stdent s t-statstc. Typcally expressed as t α,ν, t denotes the vale for whch the area nder the t-dstrbton wth degrees of freedom, v, s eqal to α. A mltpler sed to express an error lmt or expanded ncertanty as a mltple of the standard ncertanty...7 Sbject Parameter. An attrbte whose vale we seek to obtan from a measrement or set of measrements...8 t strbton. A symmetrc, contnos dstrbton characterzed by the degrees of freedom parameter. Used to compte confdence lmts for normally dstrbted varables whose standard devaton s based on a fnte degrees of freedom. Also referred to as the Stdent s t dstrbton. -

60 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7..9 Uncertanty n the Mean Vale. The sample standard devaton dvded by the sqare root of the sample sze...3 Wthn Sample Sgma. An ndcator of the varaton wthn samples.. Statstcal Sample Analyss Repeat measrements are entered and analyzed n UncertantyAnalyzer s Measrement ata Entry Worksheet. The statstcal analyss methods employed n ths worksheet depend on the type of data entered... ata Entry Optons. There are for data sample types to choose from: Sampled Vales, Sampled Cells, Sampled Mean Vales, and Mxed. The measrements reslts data entry table changes dependng pon whch of these optons has been selected.... Sampled Vales. Ths data sample opton s selected when enterng ndvdal repeat measrements. The data can be entered as measred vales or devatons from the nomnal or specfed vale of the parameter.... Sampled Cells. Ths data sample opton s selected when enterng a sampled vale that has been observed one or more tmes. In ths case, the vale can be entered as a measred vale or a devaton from nomnal. The correspondng n mber of tmes that the vale was observed s entered nder the Nmber Sampled colmn....3 Sampled Mean Vales. Ths data sample opton s selected when enterng mean vales obtaned from sets of repeat measrements. The mean vales or devatons from the parameter nomnal vale are entered along wth the standard devaton and sample sze for each set of repeat measrements....4 Mxed Samples. Ths opton s selected when a pror sample of data has been sed to compte a standard ncertanty that characterzes the repeatablty ncertanty of a measrement process. ata are entered on the worksheet n the same way as wth the Sampled Vales opton. The sage of the pror standard ncertanty and the statstcs compted from the entered sample are n dscssed n sectons..3.3 and Importng ata ata can be mported drectly nto the Measrement Reslts table from the followng external applcatons: Mcrosoft Access dbase III or IV Mcrosoft Excel 3. throgh ots --3 WK throgh WK4 Hypertext Markp angage (HTM) elmted Text Fles Open atabase Connectvty (OBC) Importng s done va UncertantyAnalyzer s ata Import Profle Screen. -3

61 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7..3 Compted Statstcs. The compted statstcs vary slghtly dependng pon whch data entry or Samplng Opton has been chosen. The compted statstcs for all for samplng optons nclde the sample sze, sample mean, mean devaton, and standard ncertanty. The ser can npt a confdence level (%) or coverage factor to compte confdence lmts. The dsplay precson (e.g., decmal place, decmal places,...) for the compted statstcs can also be set by the ser. Addtonal statstcs for each samplng opton are descrbed below...3. Sampled Vales and Sampled Cells Optons. The coeffcent of skewness and coeffcent of krtoss are also compted for these data entry optons. Approxmate dstrbtons can be sed for the skewness and krtoss coeffcents compted from samples of sze n wth mean and standard devaton x n n x (-) s n n ( x x). (-) The skewness coeffcent s gven by b 3 n n ( x s 3 x) 3 (-3) For n, the skewness coeffcent follows an approxmately N(, σ) dstrbton. A poplaton mean µ and a standard devaton σ can be estmated from a smlated dstrbton for b 3, obtaned by generatng m samples of sze n. The vales of x are obtaned as random normal devates. By expermentaton, t has been fond that m prodces good reslts. Wth regard to the random nmber generator, the Rnd fncton of Vsal Basc s adeqate, bt the Ran fncton descrbed n Nmercal Recpes n FORTRAN seems to prodce better reslts. The coeffcent of krtoss s b 4 n n ( x s 4 x) For n, the coeffcent of krtoss follows an approxmate gamma dstrbton wth the followng probablty densty fncton: f ( x) ( x / b) c 4 ( x / b) e bγ( c) As wth skewness, a poplaton mean µ and a standard devaton σ can be estmated from a smlated dstrbton for b 4, obtaned by generatng m samples of sze n. The vales of x are obtaned as random normal devates. Agan, m, sed n conjncton wth the Ran fncton, prodces good reslts. (-4) (-5) -4

62 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The parameters b and c are obtaned from c ( μ / σ ) and b μ / c...3. Sampled Mean Vales Opton. When mean vales are entered nto the Measrement Reslts table, two addtonal statstcs are compted: ) Wthn Sample Sgma and ) Between Sample Sgma. Before we dscss these qanttes, let s frst revew what the sample mean opton entals. et s assme that there are k sampled mean vales entered nto the table. The th mean vale and standard devaton of the th sample have been compted va a spreadsheet or other program sng the followng eqatons x n n j x j (-6) where n sample sze for the th sample s n n j ( x j x ) x mean vale for the th sample (.e., th mean vale) s standard devaton of th sample the j th measrement of the th sample x j The Between Sample Sgma, as follows s b (-7), s the standard devaton of the mean vales and s compted s b n k n ( x x) where k nmber of samples (.e., nmber of mean vales entered) n total nmber of measrements (.e., cmlatve of all sample szes) x mean vale of all measrements (.e., overall mean vale), s an ndcator of the varaton wthn samples and s com- The Wthn Sample Sgma, pted as follows s w k n x j n j k n (-8) k n x n -5

63 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 s w n k ( n ) s (-9) The standard ncertanty, s, of the sample mean vales s gven as s s n n b k k s w n j ( x n ( x j x) x) n k ( n ) s (-)..3.3 Mxed Sample Opton. As prevosly mentoned, n the Mxed opton a pror sample of data has been obtaned, along wth a standard ncertanty that characterzes a measrement process. A sample of data s entered nto the Measrement Reslts table n the same way the Sampled Vales opton. et the sample sze for ths sample be desgnated n and the th measred vale be desgnated x. Then the sample mean and standard devaton (.e., standard ncertanty) are compted accordng to and x n n x (-) s n n ( x x) (-) et the pror sample sze and standard ncertanty be desgnated np and sp, respectvely. The pror standard ncertanty s entered nto the strbton Uncertanty Estmate feld and the assocated sample sze s entered nto the Applcable Sample Sze feld. Wth the Mxed opton, we se these varables to represent the repeatablty of the measrement process and to estmated confdence lmts for the compted mean vale wth a confdence level of ( α) % x tα /, v s ±, p where v n p and t α/,ν s the Stdent s t-statstc for a sgnfcance level of α/ and ν degrees of freedom Overrde Sample Sze and Uncertanty. In some cases, only one measrement or devaton may be avalable from a measrement process. If so, then a standard ncertanty wll not be compted. However, f a standard ncertanty has been obtaned elsewhere that characterzes the repeatablty ncertanty n a measrement process, t wold be benefcal to be -6

64 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 able to enter t nto the Compted Statstc secton of the Measrement ata Entry Worksheet, along wth the approprate sample sze. Bt, f data are entered nto ths secton of the Worksheet, UncertantyAnalyzer responds by deletng the measrement sample nformaton n the Measrement Reslts table. To prevent ths from happenng, the Overrde Sample Sze and Uncertanty box mst be checked Recalc Opton. Althogh UncertantyAnalyzer atomatcally comptes sample statstcs as data are entered the Recalc opton was added to provde the ser wth an alternatve means of ensrng that p-to-date comptatons have been completed Mean Vale Correcton. The Measrement ata Entry Worksheet also allows for correctng or adjstng the compted mean vale for an offset de to parameter bas and/or envronmental factors. The corrected mean vale s not dsplayed on the Measrement ata Entry Worksheet, bt s dsplayed for the Compted Mean Vale on the program Man Screen. If a bas (.e., systematc offset) from the parameter's nomnal vale has been npt n the Sbject Parameter Bas Uncertanty Worksheet, then the ser can select whether or not to nclde the parameter bas correcton factor to the compted mean vale. A fxed offset n the sbject (or measrng) parameter vale or readng can also reslt from envronmental factors (e.g., temperatre, hmdty, ar pressre,...). The ser can select whether or not to adjst the compted mean vale by an envronmental correcton. The envronmental correcton s estmated sng the Measrement Envronment Uncertanty Worksheet...4 Sample Statstcs. The Sample Statstcs screen also comptes statstcs for a ser-selected sbset of data entry cells. Compted statstcs for the sample sbset nclde the mean, mean devaton, standard devaton, skewness and krtoss. The Sample Statstcs screen s accessed va the Tools men of the Measrement ata Entry Worksheets or the Statstcs men of the Error Sorce Worksheets...5 Normalty Testng. UncertantyAnalyzer s Sample Statstcs screen also contans a normalty testng featre that can be sed to determne f the sampled data can be assmed to be normally dstrbted. Normalty testng n UncertantyAnalyzer s done sng three dfferent tests. A ch-sqared goodness of ft s performed for samples of sze 5 or more. For samples of sze or greater, tests are performed to evalate the skewness and krtoss of the sample n comparson to what s expected of samples from a normally dstrbted poplaton...6 Otler Removal. Statstcal otlers can be dentfed and removed from the data samples entered nto the Measrement ata Entry Worksheet or Error Sorce Uncertanty Worksheet by selectng the Otler Removal opton. Otlers are dentfed sng Chavenet s crteron, whch defnes acceptable scatter arond a mean vale x for a gven sample on n readngs and standard devaton s n. Chavenet s crteron specfes that all ponts shold be retaned that fall wthn a band arond the mean vale that corresponds to a probablty of / n. The normal dstrbton s sed to determne the nmber of sample standard devatons that relate to ths probablty. Ths coverage factor s obtaned sng the two-taled nverse normal fncton Φ (). Frther nformaton abot Chavenet crteron can be fond n Expermentaton and Uncertanty Analyss for Engneers, nd Edton, H. W. Coleman and W. G. Steele, John Wley & Sons,

65 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where Φ Pn n (-3) P n / n. Any ponts that le otsde x ± n s n are rejected...7 Sample Sze Evalaton. UncertantyAnalyzer s Sample Sze Evalator, whch s accessed from the Measrement ata Entry Worksheet or Error Sorce Uncertanty Worksheet, can be sed to determne f the sze of a sample of data s sffcent for obtanng an estmated sample mean that dffers from the tre (poplaton) mean by less than or eqal to some predetermned amont. For example, let x, x,, x n represent repeated ndependent nbased measrements from a dstrbton wth mean μ and standard devaton σ. Accordng to the law of large nmbers, the sample average for these measrements, X, converges to μ n probablty, and we can assme that X s a good estmate of µ f n s large. The central lmt theorem allows s to approxmate how close X s to μ. Sppose that we want to fnd the probablty P( X μ < c) for some predetermned constant c. To estmate ths probablty we frst standardze, sng E(X) μ, and Var(X) σ / n: P( X μ < c) P( c < X μ < c) c P σ / n c Φ σ / n X μ c < < σ / n σ / n where Φ() s the normal dstrbton fncton. Eqatng ths probablty to a confdence level β for the condton x μ < c, we have n ( σ / c) Φ. β In practce, we sally don t know the vale of σ. Accordngly, we se the best avalable estmate. In many cases, ths wll be the sample standard devaton of the sample s, especally f n s large. To obtan a relable estmate of the reqred vale of n, then, we cold take a large sample of, say ffty or more observatons and estmate n as above. Of corse, ths approach bears frt f we wsh to economze on takng ftre samples of measrements from the poplaton. If not, the vale of n obtaned above s at least a qantty that deserves consderaton...8 Estmated Uncertanty. The estmated ncertanty de to random or repeatablty error s eqal to the sample standard devaton. The sample standard devaton s an estmate of the ncertanty n the ncertanty n a vale drawn randomly from ts poplaton. -8

66 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7..9 Uncertanty n the Mean Vale. It s often desrable to estmate and report the ncertanty n the mean vale of a random sample. In ths case, the Use Uncertanty n the Mean opton s selected on the Measrement ata Entry Worksheet or Error Sorce Worksheet. The ncertanty n the mean vale s the sample standard devaton dvded by the sqare root of the sample sze. where s s s x (-4) n n n ( x x) and n sample sze x th measred vale x sample mean vale s sample standard ncertanty (.e., standard devaton) The standard ncertanty n the mean s compted somewhat dfferently for the Sampled Cells, Sampled Mean Vales and Mxed data entry optons...9. Sampled Cells Opton. When ths data entry opton has been selected, the standard ncertanty n the mean vale s compted as S X S where n total nmber of measred vales n sample sze for the th samplng cell k nmber of sampled cells s standard ncertanty for all measred vales..9. Sampled Mean Vales Opton. When mean vales are entered nto the Measrement Reslts table, the standard ncertanty n the mean vale s compted as k n n (-5) where S X n k n s n total nmber of measrements (.e., cmlatve of all sample szes) k nmber of samples (.e., nmber of mean vales entered) n sample sze for the th sample s standard devaton of th sample (-6) -9

67 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Mxed Opton. Wth ths opton, we se both the pror standard ncertanty estmate s p together wth the sample sze n of the crrent sample. The standard ncertanty n the mean vale s gven by s S p X n, (-7) where n sample sze of the crrent sample s p standard devaton of the pror sample. If we wsh to compte confdence lmts arond the sample mean for a confdence level of ( α) %, we employ the expresson where s p x ± α /,v (-8) n ν n p. and t α/,ν s the Stdent s t-statstc for a sgnfcance level of α / and ν degrees of freedom. -

68 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 3. General CHAPTER 3 TYPE B UNCERTAINTY ESTIMATION Uncertanty estmates for measrement process errors resltng from parameter bas, dsplay resolton, operator bas, comptaton, stress response and envronmental factors are typcally determned herstcally va contanment lmts and contanment probabltes. Ths chapter dscsses the concepts and methods sed to estmate Type B ncertantes. 3 In partclar, the methodology wll focs on ncertanty estmaton for the followng measrement process errors: Measrement or Parameter Bas Resolton Error gtal Samplng Error Comptaton Error Operator Bas Envronmental Factors Error Stress Response Error The error dstrbtons sed for Type B ncertanty estmates are also dscssed, along wth gdelnes for choosng approprate dstrbtons. 3. efntons 3.. % Confdence. See Confdence evel. 3.. % of Fll Scale. The contrbton of a parameter's tolerance lmt eqal to a stated percentage of the parameter's fll scale % n-tolerance. See Percent In-Tolerance % of Nomnal. The contrbton of a parameter's tolerance lmt eqal to a stated percentage of the parameter's nomnal vale % of Range. The contrbton of a parameter's tolerance lmt eqal to percentage of the parameter's range Ancllary Parameters. Parameters, other than the sbject and measrng parameters, that partcpate n a measrement or ndrectly nflence the sbject parameter or measrng parameter bas. An example of an ancllary parameter wold be an envronmental factor sch as ambent temperatre that s measred and sed to apply an envronmental correcton to the sbject or measrng parameter vales. Sometmes referred to as nflence qanttes Asymmetrc Tolerances. Two-sded tolerance lmts n whch the magntdes of the pper and lower lmts are neqal. 3 The methodology presented heren was developed by r. H. Castrp of Integrated Scences Grop (see References). 3-

69 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Bandwdth. The range of freqences that comprse the sampled sgnal Bas. A systematc dscrepancy between an ndcated or declared vale of an artfact and ts tre vale. 3.. Bas or Error strbton. A statstcal dstrbton that descrbes the relatve freqency of occrrence of vales of a measrement error. 3.. Bas Offset or Offset from Nomnal. The stated offset from nomnal of a calbrated artfact. 3.. Bas Uncertanty. The ncertanty n the bas of a parameter or artfact Confdence evel. The probablty that a set of error lmts or contanment lmts wll contan errors for a gven error sorce Confdence mts. mts that bond errors for a gven error sorce wth a specfed probablty or "confdence." 3..5 Contanment mts. mts that are specfed to contan ether a parameter vale, devatons from the nomnal parameter vale, or errors n the measrement of the parameter vale Contanment Probablty. The probablty that a parameter vale or errors n the measrement of ths vale wll le wthn specfed contanment lmts gtal Samplng. A process n whch sgnal ampltdes are perodcally sampled and converted to dgtal nmbers gtal Samplng Error. The error resltng from analog to dgtal (A/) converson and sbseqent dgtal to analog (/A) converson. Sorces of dgtal samplng error nclde: samplng rate error, qantzaton error (samplng fll scale and sgnfcant bts), apertre tme error, mplse response error, samplng nose error, samplng nose, sensng error and hysteress error Error mts. Bondng vales that are expected to contan the error from a gven sorce wth some specfed level of probablty or confdence. 3.. Expanded Uncertanty. A mltple of the standard ncertanty reflectng ether a specfed confdence level or arbtrary coverage factor. 3.. Fxed mt. A tolerance lmt for a parameter that does not depend on the nomnal or other vales assocated wth the parameter. 3.. Fll Scale. An dentfyng reference vale for readngs or otpts for a gven parameter. Often the hghest vale n a range Herstc. Pertanng to the se of general knowledge ganed by experence, sometmes expressed as "sng a rle-of-thmb." 3..4 Hysteress. The resstance of a varable to a change n stmls Hysteress Error. The resdal sgnal n a samplng event left over from the prevos samplng event Implse Response. The response of a samplng sensor to an nstantaneos change n npt. 3-

70 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Implse Response Error. The error de to the fnte tme reqred for a samplng sensor to respond to a sampled sgnal. In UncertantyAnalyzer, mplse response error ncldes hysteress error Inflence Qantty. See Ancllary Parameter Interacton Coeffcent. A nmber that converts the vale of an envronment or ancllary parameter to a change exerted by the parameter on the bas of a sbject parameter or measrng parameter. In error analyss, the nteracton coeffcent s the partal dervatve of the measred varable wth respect to the envronment parameter In-tolerance Probablty. The probablty that a parameter vale or the error n the vale s contaned wthn ts specfed tolerance lmts at the tme of measrement Measrement Bas. An error that perssts from measrement to measrement drng a measrement sesson Measrng and Test Eqpment. The devce or artfact featrng the Measrng Parameter Nomnal Vale. The desgnated or pblshed vale of an artfact or parameter. It may also sometmes refer to the mode vale of an artfact or parameter Nmber of p-n Jnctons. The nmber of postve to negatve jnctons n the samplng sensor. For example, f the sensor s a transstor, the nmber of jnctons s Operator Bas. Error de to the percepton or nflence of a hman operator or other agency. Operator bas s often referred to as a sorce of reprodcblty error Operatng Temperatre. The temperatre at whch the samplng sensor s operated Otpt Resstance. The real part of the mpedance of the samplng sensor Parameter. A dscrete fncton of a devce ths s assgned a tolerance specfcaton. For example, the C voltage readng wold be a parameter of a dgtal mltmeter Parameter Bas. A systematc devaton of a parameter vale from ts nomnal or ndcated vale Parameter rft. A systematc long-term varaton resltng from envronmental condtons encontered drng sage, calbraton or storage Percent In-Tolerance. The probablty, expressed n percent, that a parameter vale or the error n the vale s contaned wthn ts specfed tolerance lmts at the tme of measrement Precson. Precson corresponds to how many places past the decmal pont we can express a measrement reslt. Althogh hgher precson does not necessarly mean hgher accracy, the lack of precson n a measrement s a sorce of measrement error Qantzaton. The process of convertng a contnos, analog npt or sgnal to a dgtzed code comprsed of s and s Qantzaton Error. Error de to the granlarty of resolton n qantzng a sampled sgnal. 3-3

71 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Range. In a parameter specfcaton, a desgnated range of vales for whch specfed tolerances apply. In a calbraton or test procedre, a settng or desgnaton for a set of measrements Reference Standard. An artfact sed as a measrement reference whose vale and ncertanty have been determned by calbraton Reprodcblty. The closeness of the agreement between the reslts of sccessve measrements of the vale of a parameter carred ot nder changed measrement condtons. The changed condtons may nclde: prncple of measrement, method of measrement, observer, measrng nstrment(s), reference standard, locaton, condtons of se, tme Resolton. The smallest dscernble vale ndcated by a measrng parameter or sbject parameter Resolton Error. The error de to the fnteness of the precson of a measrement Response Tme Constant. The tme reqred for the otpt of a samplng sensor to make the change from an ntal vale to 63.% (or -/e) of the fnal steady-state vale resltng from a step-change n the npt RMS Crrent. The root mean sqare vale of the crrent throgh the samplng sensor Samplng Apertre. The tme over whch an ndvdal sample s taken. See also Apertre Tme Samplng Fll Scale. The maxmm peak-to-peak ampltde of a sampled sgnal Samplng Nose. Nose present n the samplng sensor. In electronc devces, nose may nclde thermal nose, shot nose, stray emf sgnals and so on. In mechancal and dmensonal devces, nose may nclde vbraton, temperatre flctatons, etc Samplng Rate. The rate at whch an npt sgnal s sampled Sgnfcant Bts. The nmber of bts sed n qantzng a sampled sgnal Sngle-sded Tolerance mt. A sngle-sded tolerance lmt n whch only the lower or pper lmt s specfed Stress mts. mts that are expected to contan stresses that may be encontered from a gven sorce Stress Response Coeffcent. A coeffcent ndcatng the response of a parameter to a stmls, sch as stress, expressed n nts approprate for the sorce of the stmls Stress Response Error. The error n a parameter vale de to appled stress Symmetrc Tolerances. Two-sded tolerance lmts n whch the magntdes of the pper and lower lmts are eqal Tolerance mts. mts that bond acceptable parameter vales Toleranced Parameter. A parameter wth a set of specfed tolerances or lmts that bond ts vale or the error n ths vale. 3-4

72 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Two-sded Tolerance mts. Upper and lower lmts that bond acceptable parameter vales. Stated pper and lower lmts that contan bases n parameter vale wth some specfed level of confdence. Two-sded tolerance lmts can be symmetrc or asymmetrc User efned Error. User specfed error sorce other than the measrement process error sorces ncorporated nto UncertantyAnalyzer. 3. Sbject and Measrng Parameter Bas The ncertanty n the systematc error or bas of a parameter s typcally estmated herstcally va contanment lmts and contanment probabltes. In some nstances, as wth reference standards, the stated parameter vale and ts assocated ncertanty estmate may be sppled by a hgher-level calbraton laboratory. In ths case, the parameter bas ncertanty mght have been obtaned from statstcal analyss of the calbraton data and assocated measrement process errors. In UncertantyAnalyzer, parameter bas ncertanty s compted sng the Sbject or Measrng Parameter Bas Uncertanty Worksheet. Parameter bas ncertanty estmates can be developed for reference standards, toleranced parameters, or from nformaton abot expected contanment lmts. Parameter tolerance specfcatons can be entered as a Fxed lmt, % of Nomnal or Readng, % of Fll Scale, and % of Range. The specfcatons can be entered as sngle-sded, two-sded asymmetrc or two-sded symmetrc tolerances and can be combned lnearly or by the root-sm-sqare (RSS) method. In addton, the blt-n SpecMaster Worksheet, descrbed n Secton 3., provdes a tool for determnng tolerances for parameters wth complex specfcatons. The ser can choose from an extensve lst of dstrbtons for descrbng the statstcal characterstcs of the parameter bases. Spported dstrbtons n UncertantyAnalyzer nclde: Normal, ognormal, Exponental, Qadratc, Cosne, U-Shaped, Unform (rectanglar), Tranglar, and Stdent's t. Plots of these dstrbtons are dsplayed n the Parameter Bas Uncertanty Worksheet to assst n developng parameter bas ncertanty estmates. The parameter bas ncertanty reported at the bottom of the Parameter Bas Uncertanty Worksheet s compted from opton selectons; tolerance lmts, expanded ncertanty or contanment lmts; and the confdence level or n-tolerance probablty at the tme of measrement. 3.3 Resolton Error The ncertanty resltng from the fnte resolton of a readng or otpt s sally determned from contanment lmts, whch are based on the resolton of the devce, and an estmated contanment probablty. In UncertantyAnalyzer, resolton ncertanty s compted sng the Parameter Resolton Error Worksheet. The dstrbton for resolton error depends on whether the readng or otpt s expressed n analog or dgtal format Analog splay Resolton. For analog dsplays, the resolton error s assmed to have a normal dstrbton. The resolton ncertanty s compted by settng the contanment lmts eqal to the smallest ncrement of resolton and applyng a contanment probablty that readngs can be dscerned wthn these lmts. 3-5

73 Uncertanty Analyss Prncples and Methods RCC ocment -7, September gtal splay Resolton. For dgtal dsplays, the resolton error s assmed to have a nform dstrbton. The contanment lmts are ± half the smallest dsplayed dgt and the contanment probablty s %. 3.4 Operator Bas Operator bas ncertanty resltng from error n the percepton of a hman operator may be determned statstcally, bt s sally estmated herstcally. The sal way s to lnk t to resolton ncertanty or some other aspect of a measrement that can nflence operator percepton. The nderlyng error dstrbton s typcally normal, bt n some cases, the nform dstrbton may apply. In UncertantyAnalyzer, operator bas ncertanty s compted sng the Operator Bas Uncertanty Worksheet. Operator bas ncertantes can be estmated for both the sbject parameter and measrng parameter readngs. 3.5 gtal Samplng Error gtal samplng ncertanty estmaton nvolves defnng a representatve "sgnal" to be sampled and specfyng a samplng rate, a samplng apertre tme, a qantzaton precson (samplng fll scale and sgnfcant bts), an mplse response and hysteress, a samplng nose level, and a sensor bas ncertanty. It also nvolves decdng on a model or methodology for evental converson of dgtzed data back to analog form. Ths mlt-faceted analyss process s dffclt wthot a strctred template approach. UncertantyAnalyzer contans specally desgned screens to evalate dgtal samplng ncertanty. Both the analog-to-dgtal and dgtal-to-analog converson processes are handled. Samplng fll scale, sgnfcant bts, and apertre tme are entered n the gtal Samplng Uncertanty Worksheet. The followng drll-down analyss screens and worksheets can be accessed from the gtal Samplng Worksheet: Inpt Sgnal Characterstcs Worksheet Samplng Implse Response Worksheet Samplng Nose Worksheet Samplng Sensor Worksheet 3.5. Inpt Sgnal Characterstcs Worksheet. The worksheet s sed to estmate ncertanty de to samplng rate error. Becase sgnal actvty occrs between samples, the rate at whch samples are collected can ntrodce error nto the samplng process. The magntde of ths error depends on the sgnal beng sampled, the converson of the contnos analog sgnal to dscrete dgtzed vales (A-), and on the converson from dgtal to analog (-A) followng dgtal sgnal processng. The npt sgnal ampltde, ampltde nts (e.g. V), and sgnal freqency nts (e.g., Hz) are entered n ths worksheet. The samplng rate or samplng freqency s also entered, along wth the samplng nts (e.g. khz). The npt s defned from p to components of the representatve sgnal. Each component s expressed n terms of a sgnal ampltde, freqency, and relatve phase that s expressed n degrees or radans. The sgnal components can be harmoncs of a base freqency. A lnear or qadratc splne model can be selected to represent the level of dgtal to analog converson. A 3-6

74 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 plot of the sgnal and the dgtal samplng ponts can also be dsplayed Samplng Implse Response Worksheet. Ths worksheet s sed to estmate the ncertanty de to mplse response error. Implse response error s the dscrepancy between the nstantaneos sgnal vale and the sampled vale resltng from the fnte tme reqred for the samplng sensor to respond to npt stml. The mplse response a(t) s modeled sng an exponental response fncton a( t) A ( A A )( e where A mplse response at tme t A nstantaneos sgnal vale t c response tme constant The response tme constant s the tme reqred for the sensor to acheve the vale A ( e )( A A ) t / t c At any gven tme t, wthn the apertre tme, the error n the sensed vale s ) (3-) (3-) a( t) A. (3-3) The shorter the sensor response tme constant, the closer the vale of a(t) gets to the nstantaneos sgnal vale A. Conseqently, the faster the sensor responds to sgnal changes, the smaller the mplse response error wll be. Conversely, as the sensor response tme constant approaches the apertre tme, the mplse response error ncreases sgnfcantly Samplng Implse Response Error. Assme that the response of a system to a sensed vale V s exponental: r( t) V ( e λt ) (3-4) wth error ε ( t) Ve λt (3-5) The nknowns here are the exact tme at whch the response fncton s appled, the response parameter λ, and the ampltde of the vale V. Snce we're focsng on hysteress ncertanty, we'll consder only the ncertanty de to the sampled tme. We'll assme that V can be represented by the average sgnal vale and that the ncertanty n λ s neglgble. The mplse response ncertanty s, accordngly, gven by and δε σ r ( t) σ δλ δε σ δt V λ σ e t λ t δε σ δv λt V δε σ δt t (3-6) 3-7

75 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 σ ( t) Vλσ e r t λt. (3-7) The ncertanty σ t s the ncertanty n the locaton of the sampled pont. If ths pont can be establshed wth confdence lmts, then the standard devaton of σ t can be determned as sal. If the locaton of the sampled pont s nknown, the samplng error s taken as an average over the apertre τ accordng to v V ε r τ τ e λt dt (3-8) etermnng σ r n these cases nvolves the sal defnton of the varance σ r τ [ ε r ( t) ε r ] dt τ τ ε r ( t) dt ε r τ V λτ V ( e ) λτ ( λτ ) V λτ ( e λτ λτ ( e ) ( e ) λτ ) λτ (3-9) The parameter λ s obtaned from the tme reqred for the response to reach /e the npt vale: λ (3-) Hysteress Error. Hysteress error arses from resdal vales left over from prevos samples. Represent ths qantty by the varable V h. If the samplng rate s ν s, then where V h V ( e τ e ) e λτ l( T τ ) v s (3-) T / (3-) The hysteress error s the sampled vale of ths resdal amont. Ths the hysteress error s gven by ε h λt ( t) V ( e ) If the sample pont t s nknown, we nstead se the average error (3-3) h λt ( e ) ε h Vh. (3-4) λτ The hysteress ncertanty can be obtaned from the hysteress error n the same way as the mplse response ncertanty was obtaned. If the sample pont s known, then 3-8

76 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 whereas, f the sample pont s nknown, σ h τ V h τ ε h λτ Vh λτ σ h ( t) ε h λt λvhe σ t λτ λτ ( e ) ( e ) λτ λτ λτ ( e ) ( e ) λτ ε h (3-5) (3-6) Samplng Nose Worksheet. Ths worksheet s sed to estmate the ncertanty n the sensed vale of a sampled pont de to nose. In electronc devces, nose may nclde thermal nose, shot nose, stray emf sgnals and so on. In mechancal and dmensonal devces, nose may nclde vbraton, temperatre flctatons, etc. The Samplng Nose Worksheet allows the ser to npt nformaton abot electronc nose and other nose sorces. The electronc nose porton of the worksheet provdes a means for comptng Thermal Nose and Shot Nose ncertanty estmates. Thermal nose s the nose de to random moton of crrent carrers (e.g., electrons) n a samplng sensor. Shot nose s the nose de to random flctatons n the nmber of carrers n a semcondctor devce. The ncertanty n the sampled vale de to thermal nose s estmated from ser npt vales for the Bandwdth, Operatng Temperatre, and Otpt Resstance. The ncertanty n the sampled vale de to shot nose s estmated from the RMS crrent and the nmber of sensor p-n jnctons, along wth the bandwdth and otpt resstance. The electronc sgnal to nose rato s also compted as spplemental nformaton. It can also be npt by the ser, f desred. Ths other nose porton of the Samplng Nose Worksheet allows the ser to npt error lmts and assocated confdence levels for other sorces of sgnal nose. The error lmts for a gven nose sorce shold be npt n ampltde nts assocated wth the sample sgnal that s defned n the Inpt Sgnal Characterstcs Worksheet. If the error lmts are npt n tolerance nts or nts from a dfferent measrement area, then the approprate converson factor mst be entered nto the Coeffcent data feld. The coeffcent data feld can also be sed to npt a mltplyng factor f the nose sorce ndergoes attenaton or amplfcaton by some external factor Samplng Sensor Worksheet. Ths worksheet s sed to estmate the ncertanty n the sampled vale de to the error n the samplng sensor. Ths worksheet allows the ser to npt sensor tolerance lmts and assocated confdence level. If a confdence level of % s entered, then the sensor error s assmed to be nformly dstrbted. Otherwse, a normal dstrbton s assmed for the sensor error. 3.6 Comptaton Error Comptaton ncertanty can reslt from rond-off or trncaton error and the applcaton of emprcal eqatons or the reslts of crve-ft or regresson analyss to compte parameter vales. Uncertanty resltng from rond-off error and the se of emprcal eqatons s sally determned herstcally. The nform dstrbton s applcable for rond-off or compter 3-9

77 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 trncaton error. The tranglar dstrbton s approprate for descrbng errors resltng from lnear nterpolaton. UncertantyAnalyzer s Comptaton Error Uncertanty Worksheet provdes a tool for estmatng comptaton ncertanty Rond-off of Trncaton. Snce compter or calclator precson s lmted, each step of a comptaton generates ncertanty. These ncertantes propagate throgh the comptaton, emergng as part of the reslt. Uncertanty resltng from rond-off error s sally determned herstcally. The nform dstrbton s applcable for rond-off or compter trncaton error. In most cases, the se of doble-precson arthmetc can redce comptaton error ncertanty. However, ncertanty de to rond-off or compter trncaton can become seros for qanttes based on a large nmber of calclatons, sch as n teratve processes, calclatons nvolvng matrx prodcts or nverses, calclatons nvolvng trgonometrc or lognormal fnctons, etc. Comptaton errors contrbte to the ncertanty n the resltng vale n varyng degrees. For example, f the calclated vale s the sm of two qanttes, the comptaton error ncertanty s the root sm sqare (RSS) of the rond-off ncertantes for each qantty. However, f the calclated vale s the prodct of two qanttes that are mltpled, the comptaton error ncertanty s the RSS of the second qantty tmes the rond-off error of the frst qantty and the frst qantty tmes the rond-off error of the second qantty. For llstraton, we wll evalate the comptaton error for the temperatre measrement of a sbstance sng a type J thermocople that gves otpt vales n C. The temperatre otpt of the thermocople s actally a compted vale based on thermocople reference tables that convert mllvolt otpt to C. The converson from mllvolt otpt to C s done va some external devce that s connected to the thermocople leads. The converson s ether obtaned from nterpolaton of tablated vales or from an eqaton that "best fts" the tablated data. In any event, some comptaton error s ntrodced nto the temperatre otpt vale. In ths example, we wll assme that the mllvolt (mv) otpt of the thermocople s converted to C by nterpolatng between tablated vales. In ths case, the nterpolaton eqaton wold be T T T otpt mvotpt (3-7) mv mv A nmber of comptatonal error sorces can be entered n the Rond-off error sorce table of the Comptaton Error Uncertanty Worksheet. In the thermocople example, each of the qanttes, m Votpt, T, T, mv and mv, n eqaton (3-7) represents a comptaton error sorce and mst be lsted as sch n the table. For nstance, we cold enter the followng error sorce descrptons and vales: Mllvolt Otpt.54 mv Temperatre Reference 4 C Temperatre Reference 5 C Mllvolt Reference.53 mv Mllvolt Reference.77 mv When enterng vales n the Typcal Vale data felds, care mst be taken to ensre that the 3-

78 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 resltng comptaton ncertanty for the error sorce s n the sbject parameter nomnal or tolerance nts that have been selected for reportng the ncertanty estmates. In the Approx. No. Calcs. data feld, the ser npts an estmated nmber of calclatons nvolved n obtanng the compted or measred vale of the error sorce. For the thermocople example, the error sorce vales are not actally compted, so we enter a vale of for each error sorce. In the ecmal gts data feld, the ser npts the precson of the devce that s makng the calclatons (.e. nmber of dgts to the rght of the decmal pont). In the thermocople example, we npt the nmber of dgts to the rght of the decmal place for each error sorce vale. However, we are nterpolatng to obtan the temperatre otpt and know that ts vale wll fall between 4 C and 5 C. Accordngly, we wll want to obtan an otpt temperatre wth a precson of at least decmal places and need to assme a precson of at least decmal places for the reference temperatres. The resltng decmal precson for the error sorces are lsted below. Error Sorce escrpton Typcal Vale ecmal gts Mllvolt Otpt.54 3 Temperatre Reference 4. Temperatre Reference 5. Mllvolt Reference.53 4 Mllvolt Reference.77 4 We mst now determne the partal dervatves of eqaton (3-7) wth respect to each of the comptaton error sorce qanttes, mv otpt, T, T, mv and mv. The nmercal vales of the partal dervatves are compted and entered nto the Senstvty Coeffcent data feld. The partal dervatve of the otpt temperatre eqaton wth respect to mvotpt s smply T mv otpt otpt T T mv mv (3-8) Enterng the compted or measred vales for T, T, mv and mv nto eqaton (3-8) gves s the vale of the Sorce Coeffcent for the Mllvolt Otpt error sorce. 4 5 Senstvty Coeffcent for mv otpt The vale of s then entered nto the Sorce Coeffcent data feld. The partal dervatve of the otpt temperatre eqaton wth respect to T s T mv otpt otpt T mv mv and enterng the compted or measred vales for m Vopt, mv and mv nto the above eqaton gves s the vale of the Sorce Coeffcent for the Temperatre Reference error sorce..54 T Sorce Coeffcent (3-9) 3-

79 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Smlarly, the partal dervatve of the otpt temperatre eqaton wth respect to T s T mv otpt otpt T mv mv and the Sorce Coeffcent for the Temperatre Reference error sorce s Senstvt y Coeffcent for T The partal dervatve of the otpt temperatre eqaton wth respect to mv s ( T T ) ( mv ) Totpt mvotpt mv mv and the Sorce Coeffcent for the Mllvolt Reference error sorce s Senstvty Coeffcent for mv ( 4 5) (.53.77) Smlarly, the partal dervatve of the otpt temperatre eqaton wth respect to mv s ( T T ) ( mv ) Totpt mvotpt mv mv and the Sorce Coeffcent for the Mllvolt Reference error sorce s Senstvty Coeffcent for T ( 4 5) (.53.77) UncertantyAnalyzer employs the nform dstrbton for estmatng ncertanty de to rond-off or other comptatonal error. The vale dsplayed n the Comptaton Uncertanty data feld s compted as follows:. The error lmts are assmed to be eqal to ± half of the last decmal place for the error sorce where s the nmber of decmal dgts. error lmts ± (.5 ) (3-) (3-) (3-) (3-3). The error lmts are mltpled by the sqare root of the nmber of calclatons, n, and dvded by the sqare root of 3 to obtan an estmate of the standard ncertanty for a nformly dstrbted error. n(.5 ) standard ncertanty (3-4) 3 3. The standard ncertanty s then mltpled by the senstvty coeffcent to obtan an estmate of the comptaton ncertanty for the error sorce. 3-

80 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 n(.5 ) Comptaton Uncertanty Senstvty Coeffcent (3-5) 3 For the thermocople example, the comptaton ncertanty for each error sorce s lsted n the Rond-Off data table as shown below. Error Sorce Compter or Approx. ecmal Sorce Comptaton escrpton Measred Vale No. Calcs gts Coeffcent Uncertanty Mllvolt Otpt Temperatre Reference Temperatre Reference Mllvolt Reference Mllvolt Reference The bottom porton of the Comptaton Error Uncertanty Worksheet dsplays the total ncertanty de to comptaton error. The total comptaton ncertanty s obtaned by combnng the comptaton ncertantes for the ndvdal error sorces n a root-sm-sqare (RSS) manner. For the thermocople example, the total comptaton ncertanty s calclated to be. C. Total Comptaton Uncertanty (.56) (.7).. (.35) 3.6. Emprcal Eqatons. In some nstances, we may se an eqaton to predct the vale of a varable y for any gven vales of one or more varables x. Sometmes, a physcal law or prncple connects the varables so that a mathematcal eqaton can be sed to express y as a fncton of the varables x. However, n complex measrement systems, there may be a more emprcal relatonshp that mst be estmated from expermental observaton. Emprcal eqatons are defned by estmated coeffcents whose vales have ncertanty. UncertantyAnalyzer s Emprcal Eqatons tab of the Comptaton Error Uncertanty Screen provdes a tool for estmatng the total ncertanty resltng from the se of emprcal eqatons. The eqaton and assocated varables and coeffcents are entered along wth ther typcal vale, error or contanment lmts and contanment probablty. The error dstrbton s selected from a drop down lst for each varable (where approprate). UncertantyAnalyzer then ses ths nformaton to compte the standard ncertanty for each varable. Alternatvely, the ser can enter the standard ncertanty for each varable Regresson Analyss. Regresson analyss, a trend lne s ft to the observed data. In UncertantyAnalyzer s Regresson Analyss Worksheet, sers have the opton of applyng st, nd, or 3 rd degree polynomals. The eqatons for the trend lnes are where m yˆ br x r r, m,,or 3 3-3

81 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ŷ represents a predcted dependent varable vale, and x represents the correspondng ndependent varable vale. In the dscsson to follow, the 3 rd degree (m 3) polynomal model s treated explctly. Extenson to st and nd degree models s straghtforward. The regresson or crve-ft process begns by mnmzng a qantty called the resdal sm of sqares wth respect to the polynomal model coeffcents br, r,,..., m. Next, the stage s set for predctng vales of gven vales of x. Each predcted vale of ŷ s accompaned by an ncertanty. Estmatng the ncertanty n a predcted vale reqres the development of a varance-covarance matrx Resdal Sm of Sqares. The regresson analyss conssts of fndng a polynomal ft to the data. Sampled data are arranged as follows: Independent ependent Nmber of Varable Varable Sampled Vales x y n x y n x k y k n k Gven the above, the resdal sm of sqares s wrtten k RSS n ( ) y y ˆ where y s the th observed ndependent varable vale, n, s the nmber of vales of y sampled at the pont x, and ŷ s compted from the vale x sng the applcable polynomal. The coeffcents of the model are solved for by mnmzng RSS Regresson Model Coeffcents. The soltons for the coeffcents are estmated sng matrces. The soltons for the coeffcents are gven by eqaton (3-7). ( X'WX) X' WY (3-6) b (3-7) To llstrate, the matrces for the 3 rd degree polynomal model are 3 b x x x n y b 3 x x x n y b X W and Y, b M M M M O y b 3 3 xn xn xn n k y3 and X s the transpose of X. Matrces for the st and nd degree polynomal models can be nferred from the above. 3-4

82 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Varance-Covarance Matrx. The varance-covarance matrx of the regresson coeffcents vector b s gven by V var( b) ( X'WX) s (3-8) where The qantty s s an estmate of the varance n the ndependent varable compted by s RSS, (3-9) n p and n k n, p m.. For the 3 rd degree model, the varance-covarance matrx s where var( cov V cov cov 3 cov b ) cov( b, b ) cov( b, b ) cov( b, b3 ) ( b, b ) var( b ) cov( b, b ) cov( b, b3 ) ( b, b ) cov( b, b ) var( b ) cov( b, b3 ) ( b, b ) cov( b, b ) cov( b, b ) var( b ) 3 ( b, b ) cov( b, b ),,,3 j,,, 3 j j j 3 3 (3-3) Predcted Vales and Uncertantes. A vale ŷ s compted for a gven vale of x accordng to where, for the 3 rd degree polynomal model, ( x) x' b ŷ (3-3) 3 ( x x x ) The varance n s estmated accordng to var yˆ ( x) x' X'WX x'vx where x s the transpose of x'. x' (3-3) [ ] ( ) xs (3-33) 3-5

83 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The qantty s y / x x'vx [ yˆ ( )] var x (3-34) s the standard error n a predcted mean vale of y for a gven vale of x. Snce the actal vale of y vares abot the tre mean vale wth a varance estmated by s, the varance of an ndvdal predcted vale s estmated by The qantty var [ yˆ ( x) ] s x' ( X'WX) xs s y / x (3-35) s var x' [ yˆ ( x) ] ( X'WX) x s (3-36) s called the standard error n the forecast. It represents the ncertanty n a projected ndvdal vale of y, gven a vale of x Regresson Model Testng. The method of regresson model testng dscssed n ths secton apples to m-order polynomal cases n whch n sampled vales are groped nto k cells, each correspondng to a sampled ndependent varable x,,,,k. We reqre that p < k < n. The nherent scatter n the sampled dependent varable vales or pre error s qantfed by the pre error varance gven by where and The qantty s k e ve y n k n n ( y y) (3-37) n y (3-38) v e n. (3-39) ( n y y ) (3-4) s the sm of sqares abot the mean for k-sample cases. For these cases, the varance de to lack of ft s wrtten where s V k j n ( yˆ y) (3-4) 3-6

84 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 V v e ( n p) p (3-4) m The qantty k n ( yˆ y) (3-43) s called the lack of ft sm of sqares for k-sample cases. If the lack of ft sm of sqares s close to or less than the sm of sqares abot the mean, we sspect that we have a good model. The confdence for rejectng the model, nvolves sng two χ varables V /σ s and v s /σ e e and formng a test varable F, gven by e s F (3-44) s Ths qantty s an F-dstrbted varable wth ν and νe degrees of freedom. The varable F s the rato of the varance de to scatter abot regresson to the varance de to pre error. If the former s smaller than the latter, then we can say that we have a vable model. That s, f we have a small vale of F, the confdence for rejectng the model shold be low. Accordngly, we se F as a test statstc and compte α f ( t; v, v ) F F where α s the rejecton confdence. The best model s consdered to be the model wth the lowest rejecton confdence of all avalable models. In Range Ft, the avalable models are polynomal models from frst to thrd degree. Conseqently, the best model s obtaned sng the followng algorthm: C mn 37 r For m to 3 Ft model y b... b m x Compte the rejecton confdence C m If C m < C mn then C mn C m r m End If Next Select model where m r. s s dt (3-45) Testng for Trend Sgnfcance. Ths secton dscsses testng whether the observed data follow a sgnfcant trend. For ths, a st degree regresson ft s performed. In st degree fts, the regresson model s 3-7

85 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ˆ b b x (3.46) y Testng for sgnfcance of a trend conssts of testng the magntde of b. If the slope b s sffcently small, t cannot be clamed that a sgnfcant trend exsts. Fortnately, the sgnfcance of a trend can be tested statstcally. The test statstc to be sed s where b t (3-47) ( ) var b s var( b) k (3-48) n ( x x) The applcable degrees of freedom s v n. Gven a sgnfcance level α, then f t < t α,ν, the slope s not consdered sgnfcant. Nonlnearty s defned as the error de to devaton of vales y(x) from a lnear relatonshp f(x) abx, where a and b are constants. Specfcally, Nonlnearty [ y( x ) ( a bx )] q q (3-49) where q s a nmber of vales to be determned by the method of nonlnearty comptaton. The constants a and b are obtaned as follows: et x lower and x pper be lower and pper ponts for a range of vales and let y lower and y pper be correspondng dependent varable vales. Then, for the range of vales n qeston, a x lower and b UncertantyAnalyzer comptes nonlnearty from two perspectves. One s the error de to devatons from observed vales. The other s the error de to devatons from projected vales. In the case of devaton from observed vales, q k and y(x ) y,,,..., k. Accordngly, y x pper pper y x lower lower Nonlnearty [ y( x ) ( a bx )] k k (3-5) In the case of devaton from projected vales, y(x) ŷ (x) where ŷ (x) s a projected vale at x, compted sng the regresson model. The nmber q can be any nmber that s sffcent to obtan a nonlnearty estmate. In Range Ft, q 5. 5 Nonlnearty [ ( x ) ( a bx )] 5 y (3-5) 3-8

86 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 In ths expresson where Δ x x xlower ( )Δ, (3-5) pper xlower xpper x q q lower (3-53) 3.7 Envronmental Factors Error Envronmental factors ncertanty s typcally estmated herstcally n a three-step process:. Estmate the ncertanty n each envronmental or ancllary process error.. Mltply each envronmental or ancllary ncertanty by ts respectve nteracton coeffcent. 3. Combne the envronmental or ancllary ncertantes, accontng for all correlatons. The nteracton coeffcent relates an envronmental or ancllary factor to an error sorce. For example, f the error sorce nder consderaton s the measrement of length and the envronmental factor s temperatre, then the nteracton coeffcent s the thermal expanson coeffcent for length. UncertantyAnalyzer contans specally desgned screens to evalate ncertanty resltng from envronmental factors error. The Envronmental Factors Uncertanty Worksheet contans an Envronment Parameter Table that allows p to envronment parameter error sorces for estmatng the total measrement ncertanty de to envronmental factors. The data for each envronmental error sorce can be npt drectly nto the Envronment Parameters table or the correspondng Tolerance Worksheet, whch s actvated by clckng the I Nmber btton to the left of the envronment parameter name. The drll-down Tolerance Worksheet provdes tool for estmatng two types of ncertantes: ) ncertanty n an envronmental/ancllary correcton or ) ncertanty de envronmental/ancllary varatons. The worksheet entry felds change slghtly dependng pon the ncertanty analyss opton that s chosen. UncertantyAnalyzer assmes that measrement ncertantes de to envronmental factors are normally dstrbted nless the assocated degrees of freedom are less than nfnte or the confdence level s %. If the degrees of freedom are less than nfnte and the confdence level s less than %, the stdent's t dstrbton s sed. If the confdence level s %, then the nform dstrbton s sed and the degrees of freedom are assmed to be nfnte. 3.8 Stress Response Error Stress response ncertanty resltng from shppng and handlng s sally estmated herstcally. The method of determnaton nvolves attemptng to estmate the ncertanty n the stresses encontered and mltplyng ths ncertanty by a stress response coeffcent (.e., the response of the parameter vale to stress). Stress response ncertanty shold be consdered, for example, when an tem or devce has been calbrated n an external laboratory to accont for any expected ncrease n the ncertanty 3-9

87 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 reported on the calbraton certfcate de to stresses ncrred drng transport back to the endser. If a calbraton laboratory already ncldes stress response ncertanty, t shold be ndcated n the ncertanty bdget (.e., the temzed lst of error sorces). In UncertantyAnalyzer, the Stress Response Uncertanty Worksheet provdes for p to 3 stress response error sorces n estmatng the total ncertanty n the vale of the sbject parameter de to shppng and handlng stress. UncertantyAnalyzer assmes that ncertantes de to shppng and handlng stresses are normally dstrbted nless the degrees of freedom are less than nfnte or the confdence level s %. If the degrees of freedom are less than nfnte and the confdence level s less than %, the stdent s t dstrbton s sed. If the confdence level s %, then the nform dstrbton s sed and the degrees of freedom are assmed to be nfnte. 3.9 User efned Errors UncertantyAnalyzer s Error Sorce Worksheet provdes a tool for developng Type A, Type B and Type A,B ncertanty estmates for other, ser specfed error sorces. Type A ncertanty estmates for ser defned error sorces can be estmated statstcally from measred vales or devatons from nomnal that are entered n the Type A Uncertanty data entry table. Enterng vales n the table reslts n the followng statstcs to be compted: Mean or average of the measred vales. Average or mean of the measred devatons from nomnal. Sample sze. Standard ncertanty or devaton of the data sample. Standard ncertanty n the mean vale. Uncertantes for ser defned error sorces can be estmated herstcally from bondng vales, or ± lmts, that are expected to contan the error wth some specfed probablty or confdence level. If approprate, a coverage factor can be entered nstead of a confdence level. The ± lmts can be specfed as a fxed tolerance lmt, % of Nomnal, % of Ref, % of Ref, % of Ref 3 or any lnear or root-sm-sqare combnaton of these qanttes. In UncertantyAnalyzer s Error Sorce Worksheet, Type B ncertanty analyss errors are typcally assmed to be normally dstrbted. However, n some cases, the Normal dstrbton may not apply. If so, then the approprate dstrbton can be selected from a drop-down strbton st that contans the Normal, Qadratc, Cosne, U-Shaped, Unform (Rectanglar), Tranglar, and Stdent s t. UncertantyAnalyzer assmes nfnte degrees of freedom for the Type B ncertanty estmate nless otherwse specfed. Snce, the degrees of freedom varable qantfes the amont of knowledge avalable for makng the ncertanty estmate, an nfnte degrees of freedom sgnfes complete certanty,.e., zero ncertanty. In most cases, t wold not be realstc to assme nfnte knowledge abot the ncertanty. The Type B egrees of Freedom Calclator, whch s accessed by clckng the egrees of Freedom btton, can be sed to provde addtonal nformaton abot the ncertanty n the ± lmts and assocated confdence level. 3-

88 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 3. SpecMaster The performance characterstcs of test and measrement eqpment can drft when sbjected to long-term exposre envronmental condtons drng storage or se. Parameter drft s common n electronc and mcrowave eqpment. Conseqently, some manfactrers may report eqpment specfcatons that vary wth tme and/or envronmental condtons. UncertantyAnalyzer s SpecMaster worksheet provdes a tool for developng tolerance lmts for parameters wth complcated specfcatons. SpecMaster s accessed from the Parameter Bas Uncertanty Worksheets. Eqatons or algorthms for comptng tolerance lmts are entered n the desgnated area of the SpecMaster Worksheet and the specfcaton data are entered n the table n the lower part of the worksheet. Tolerance algorthms are developed sng a smple bt powerfl scrptng langage called VBScrpt. A farly comprehensve gde to VBScrpt s contaned n a Help fle accessed from the SpecMaster Help men. Avalable mathematcal fnctons are lsted n the Math Fnctons secton of the Help fle. The Math Fnctons lst contans fnctons that are ntrnsc to VBScrpt and those that have been added for se n UncertantyAnalyzer. 3. Type B egrees of Freedom Calclator As prevosly dscssed n Chapter, f a Type B estmate s obtaned solely from contanment lmts and contanment probabltes, then the degrees of freedom s sally taken to be nfnte. For example, f the measrement error s normally dstrbted, the ncertanty s compted from the contanment lmts, ±, the nverse normal dstrbton fncton, Φ - (), and the contanment probablty, p C/. (3-54) p Φ However, f there s an ncertanty n the contanment lmts (e.g., ± ± Δ) or the contanment probablty (e.g., ±p ± Δp), then t s mportant that the degrees of freedom reflect ths fzzness or lack of knowledge n or estmate. UncertantyAnalyzer s blt-n Type B egrees of Freedom Calclator s a tool that allows the ser to provde addtonal nformaton abot the ncertanty n the ± lmts and assocated confdence level sed to make Type B ncertanty estmates. Ths nformaton s sed to estmate the ncertanty and the assocated degrees of freedom. The Type B egrees of Freedom Calclator can be accessed from all screens and worksheets that deal wth the estmaton of Type B ncertantes. 3.. Methodology. The approach sed to estmate the degrees of freedom for Type B estmates begns wth the relaton proposed n the ISO GUM. ( x) [ ( x) ] (3-55) σ v The method for comptng the varance n the ncertanty, σ [(x)], s otlned n the followng steps:. We generalze the eqaton for the Type B ncertanty estmate as 3-

89 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 B (3-56) ϕ ( p) where n the contanment lmt, p s the contanment probablty, and ϕ(p) s defned as and the fncton Φ [] ( p ) Φ [( p) / ] ϕ (3-57) s the nverse normal dstrbton fncton.. The error n the ncertanty,, B de to errors n and p s estmated sng a Taylor Seres expanson. ε B B B ε ε p p B B dϕ ε ε p ϕ dp ε dϕ ε p ϕ ϕ dp (3-58) where ε and ε are errors n and p, respectvely. 3. Assmng statstcal ndependence between these errors, the varance n B follows drectly: σ ( ) B var( ε ) b var ϕ dϕ ( ε ) var( ε ) 4 ϕ dp p (3-59) By defnton, the ncertanty, the error n x. x, of a qantty x s eqal to the sqare root of the varance n x var( ε x ) Therefore, the varance n ε and ε p can be expressed as var( ε ) var( ε p ) p Eqaton (3-59) can then be expressed as 3-

90 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 σ ( ) B var( ε ϕ b ) dϕ 4 ϕ dp p (3-6) vdng eqaton (3-6) by the sqare of eqaton (3-56), we get σ B ϕ B ( ) dϕ dp p (3-6) The dervatve n eqaton (3-6) s obtaned from eqaton (3-57). We frst establsh that where Φ[] p Φ[ ϕ] π ϕ ζ e / dζ s the probablty densty fncton for the normal dstrbton. (3-6) We next take the dervatve of both sdes of ths eqaton wth respect to p to get ϕ / dϕ e (3-63) π dp and, fnally, ϕ dp Sbstttng eqaton (3-64) n eqaton (3-6) yelds σ π d ϕ / e. (3-64) ( ) π B B ϕ e p (3-65) ϕ In applyng eqaton (3-65), we are mmedately confronted wth the problem of obtanng and p. These qanttes can be estmated sng any of the for formats descrbed below. Format : % of Vales. Ths format reads "Approxmately C% (±Δc%) of observed vales have been fond to le wthn the lmts ± (±Δ)." In ths format, a techncal expert s asked to provde ± error for both the contanment lmts (±Δ) and the contanment probablty (± Δc%). These lmts are sed to estmate A and p. The contanment probablty s p C / 3-3

91 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where C s the percentage of vales of y observed wthn ±. If we assme that the errors n the estmates of and p are approxmately nformly dstrbted wthn ±Δ and ±Δp ±Δc% /, respectvely, then we can wrte ( Δ ) (3-66) 3 ( Δ ) p p (3-67) 3 Use of the nform dstrbton s approprate here, snce the ranges ±Δ and ±Δp can be consdered analogos to "lmts of resolton," for whch the nform dstrbton s applcable. Ths obvates the need for estmatng confdence levels for Δ and Δp. Any lack of rgor ntrodced by ths tactc s felt as a thrd order effect and does not materally compromse the rgor of or fnal reslt. Sbstttng eqatons (3-66) and (3-67) n eqaton (3-65) gves σ ( B B ) ( Δ) π ( Δp) 3 ϕ e ϕ 3. (3-68) Usng eqaton (3-68) n eqaton (3-54) yelds an estmate for the degrees of freedom, ν, B for a Type B ncertanty estmate. v B σ ϕ ( ) B B 3ϕ ( ) ϕ Δ π e ( Δp) (3-69) If Δ and Δp are set to zero, then the Type B degrees of freedom becomes nfnte. Obvosly, n most cases, t s not realstc to have nfnte degrees of freedom for Type B ncertanty estmates. Therefore, t behooves s to attempt to apply whatever means we have at or dsposal to obtan a sensble estmate for p. Format : X ot of N. Ths format reads "Approxmately X ot of N observed vales have been fond to le wthn the lmts ± (±Δ)." In ths format, the contanment probablty s expressed as p X / N, where N s the nmber of observatons of a vale and X s the nmber of vales observed to fall wthn ± (± Δ). The varance n s obtaned the same as n Format. The varance n the contanment probablty p can be obtaned by takng advantage of the bnomal character of p. p p ( p) Sbstttng n eqatons (3-66) and (3-7) nto eqaton (3-65) gves N (3-7) 3-4

92 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ( ) ( ) N p p e B B Δ 3 ) ( ϕ π ϕ σ. (3-7) Usng eqaton (3-7) n eqaton (3-54) yelds an estmate for the degrees of freedom, ν B, for a Type B ncertanty estmate. B ( ) ( ) ( ) N p p e v B B B / 3 Δ ϕ π ϕ ϕ σ (3-7) Format 3: % of Cases. Ths format reads "Approxmately C% of N observed vales have been fond to le wthn the lmts ± (±Δ)." Ths format s a varaton of Format n whch the contanment probablty s stated n terms of a percentage C of the nmber of observatons N, wth p C /. The eqaton for estmatng the degrees of freedom s the same as for Format : ( ) ( ) N p p e v B / 3 Δ ϕ π ϕ ϕ (3-73 Format 4: % Range. Ths format reads "Between C% and C% of observed vales have been fond to le between the lmts ± (± )." Ths format s a varaton of Format n whch a range of vales s gven for the contanment probablty, p C/, where C (C C ) and ± c (C - C )/. The eqaton for estmatng the degrees of freedom s the same as for Format : ( ) ( ) 3 p e v B Δ Δ ϕ π ϕ ϕ. (3-74) 3-5

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94 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 4. General CHAPTER 4 UNCERTAINTY COMBINATION Ths chapter dscsses how ncertantes are combned and correlatons between error sorces are acconted for n an ncertanty analyss. 4. efntons 4.. Combned Uncertanty. The ncertanty n the total error of a vale of nterest. 4.. Correlaton Analyss. An analyss that determnes the extent to whch two error sorces nflence one another. Typcally the analyss s based on ordered pars of vales of the two error sorce varables Covarance. The expected vale of the prodct of the devatons of two random varables from ther respectve means. The covarance of two ndependent varables s zero Effectve egrees of Freedom. The degrees of freedom for combned ncertantes compted from the Welch-Satterthwate formla Error Sorce. A parameter, varable or constant that can contrbte error to the determnaton of the vale of a sbject parameter. Examples nclde: measrng parameter bas, random error, resolton error, operator bas, comptaton error and envronmental factors error Error Sorce Correlaton. See Correlaton Analyss 4..7 Error Sorce Uncertanty. The ncertanty n the error of a gven sorce Pareto Chart. See Pareto agram Pareto agram. A bar chart that ranks the relatve contrbton of component ncertantes to the total combned ncertanty. 4.. Total Uncertanty. See Combned Uncertanty. 4.. Varance. () Poplaton: The expectaton vale for the sqare of the dfference between the vale of a varable and the poplaton mean. () Sample: A measre of the spread of a sample eqal to the sm of the sqared observed devatons from the sample mean dvded by the degrees of freedom for the sample. Also referred to as the mean sqare error. 4. Varance Addton UncertantyAnalyzer ncorporates the varance addton rle, dscssed n Chapter, becase t provdes a logcal approach for combnng Type A and Type B ncertantes that acconts for correlatons between error sorces. The general form of the varance addton rle for the drect measrement of a qantty x that nvolves n error sorces x ε ε ε K ε x tre 3 n 4-

95 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 s expressed as var( n ε ) n n var ( ε ) n j> j> n n n j j ρ ρ j j (4-) where s the ncertanty n error sorce ε and ρ j s the correlaton coeffcent for error sorces ε and ε j. 4.3 Error Sorce Correlaton UncertantyAnalyzer s Correlaton Analyss Screen provdes a tool for correlatng error sorces for drect measrement, mltvarate measrement and system model analyses. Correlaton coeffcents can be entered for each selected par of error sorces. Alternatvely, the correlaton coeffcent can be compted from sample data pars entered n the Correlaton ata table. The correlaton coeffcent for pars of errors, εx and εy, s compted from where the covarance of εx and εy s ( ε, ε ) cov ρ ε (4-) x y x, ε y, x y n n ( x, ε y ) ( x x) ( cov ε y y). (4-3) n The Correlaton Analyss Screen ncldes a Compensatng Bases box that can be checked f the bas or error of one measred varable offsets the bas or error of another measred varable. For nstance, f the same measrng parameter s sed to measre the nsde dameter of a sleeve and the otsde dameter of a shaft that fts nto the sleeve, any error or bas n the two measrements wll not affect the qalty of ft. In other words, the measrement bases offset each other. 4.4 Total Uncertanty and egrees of Freedom The total or combned ncertanty, T, s compted by takng the sqare root of the varance. T n n n ρ (4-4) j> j j The effectve degrees of freedom, ν eff, for the total ncertanty, T, resltng from the combnaton of ncertantes wth assocated degrees of freedom, ν, for n error sorces s estmated sng the Welch-Satterthwate formla. 4 8 T veff n 4 v (4-5) 4-

96 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The combned ncertanty T* s compted assmng no error sorce correlatons. Conseqently, the effectve degrees of freedom s consdered an estmate. 4.5 Pareto agrams UncertantyAnalyzer s Pareto Chart screen dsplays a bar chart that ranks the relatve contrbton of component ncertantes to the total combned measrement ncertanty. Each error sorce ncertanty s weghted n accordance wth ts magntde and the extent that t s correlated wth other error sorce ncertantes. The Pareto Chart screen s accessed from screens and worksheets that accont for mltple error sorces. Some examples nclde dgtal samplng error, envronmental error, stress response error, and comptaton error. Becase of the fact that certan error sorces may be correlated wth others, the percent contrbton of an error sorce ncertanty to the total combned ncertanty cannot be obtaned by smply dvdng the error sorce ncertanty by the total ncertanty and mltplyng by. Instead, the percent contrbton s obtaned by extractng each ncertanty n trn and comptng the mpact on the total. 4.6 Confdence mts In UncertantyAnalyzer, the terms confdence lmts and expanded ncertanty are sed nterchangeably. Both are compted as pper and lower lmts that contan the tre vale µ (estmated) and expressed as x tα / s, v μ x tα / s, v (4-6) n whch the standard ncertanty,, s mltpled by a t-statstc, t α/,ν, that s compted for a gven degrees of freedom, ν, and probablty or confdence level, p, where α - p. 4-3

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98 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 5. General CHAPTER 5 MUTIVARIATE UNCERTAINTY ANAYSIS Ths chapter dscsses the approach sed to estmate the ncertanty of a qantty (or sbject parameter) that s compted from measrements of two or more attrbtes or parameters. The mltvarate ncertanty analyss procedre conssts of the followng steps: 4. evelop the Parameter Vale Eqaton 5. evelop the Error Model 6. evelop the Uncertanty Model 7. Identfy the Measrement Process Errors 8. Estmate Measrement Process Uncertantes 9. Accont for Error Sorce Correlatons. Combne Uncertantes The processes for developng error models and ncertanty models from the parameter vale eqaton are presented. Identfyng measrement process errors, estmatng ther ncertantes and accontng for cross-correlatons are also presented. The volme of a cylnder obtaned from length and dameter measrements s sed to llstrate the concepts and methods of condctng a mltvarate ncertanty analyss. 5. efntons 5.. Coeffcent Eqaton. An eqaton that expresses the partal dervatve of a parameter vale eqaton or modle otpt eqaton wth respect to a selected parameter or error sorce. Ths eqaton s sed to compte the senstvty coeffcent for the selected parameter or error sorce. 5.. Combned Uncertanty. The ncertanty n the total error of a vale of nterest Component Error. The error n the measrement of a gven component of a mltvarate measrement. For example, when expressng cylnder volme as a fncton of length and dameter components, the error n the cylnder volme measrement, εv, can be expressed n terms of the component errors, ε and ε Component Uncertanty. The prodct of the senstvty coeffcent and the standard ncertanty for a component error Compted Parameter Vale. The parameter vale compted on UncertantyAnalyzer's Mltvarate Analyss Screen. Based on ser specfed adjsted mean vales for root varables and the Parameter Vale Eqaton. 5-5

99 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Cross-correlaton. The correlaton between two error sorces for two dfferent components of a mltvarate analyss. For example, f the same person (operator) makes measrements of both component X and component Y, then there s a cross-correlaton between the operator bases for two components Compensatng Bases. Measrng parameter bases that offset or compensate one another. For example, f the same measrng parameter s sed to measre the nsde dameter of a sleeve and the otsde dameter of a shaft that fts nto the sleeve, any error or bas n the two measrements wll not affect the qalty of ft Effectve egrees of Freedom. The degrees of freedom for combned ncertantes compted from the Welch-Satterthwate formla Mltvarate Measrements. Measrements n whch the sbject parameter s a compted qantty based on measrements of two or more attrbtes or parameters. 5.. Nested Varables. Varables that are defned as a fncton of root varables or other nested varables. 5.. Nested Varables Eqatons. Eqatons that defne varables n terms of root varables or other nested varables. They are entered after the parameter vale eqaton. 5.. Parameter Vale Eqaton. A mathematcal expresson that defnes the vale of a measrement or parameter vale n terms of the vales of consttent root varables or error sorces Root Varables. Varables sed to compte the parameter vale that are not a fncton of other varables. Root varable nformaton s specfed by the ser va the Error Sorce Worksheets Senstvty Coeffcent. For a mltvarate analyss, the senstvty coeffcent of a gven root varable s the partal dervatve of the parameter vale eqaton wth respect to a root varable. For a system analyss, the senstvty coeffcent for a gven modle parameter s the partal dervatve of the modle otpt eqaton wth respect to a modle parameter System Eqaton. A mathematcal expresson that defnes the vale of a qantty n terms of ts consttent varables or components Total Uncertanty. See Combned Uncertanty. 5. Parameter Vale Eqaton The parameter vale eqaton s a mathematcal relatonshp between the qantty of nterest (sbject parameter) and the varables or components to be measred. The parameter vale eqaton s also referred to as the system or governng eqaton. For example, sppose we wsh to know the volme of a cylnder. The parameter vale eqaton for the cylnder volme s gven as V π (5-) 5-6

100 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where and are the cylnder length and dameter, respectvely. From ths eqaton, we see that, to determne the cylnder volme, we need to measre length and dameter components. In UncertantyAnalyzer, the parameter vale s compted based on ser specfed mean or nomnal vales for root varables nclded n the parameter vale eqaton and other nested varables eqatons. 5.3 Error Model In any gven measrement scenaro, each measred qantty s a potental sorce of error. For example, errors n the length and dameter measrements wll contrbte to the overall error n the estmaton of the cylnder volme. Therefore, the cylnder volme eqaton can be expressed as ( V V ε ε π ε ) (5-) where V tre or nomnal cylnder volme tre or nomnal cylnder dameter tre or nomnal cylnder length V ε error n the cylnder volme ε error n the cylnder dameter ε error n the cylnder length By rearrangng eqaton (5-), we obtan an algebrac expresson for the cylnder volme error. ( ) ( ) ( )( ) ( ) 4 4 V o V π ε ε ε ε ε ε ε π π ε ε π π ε ε π ε ε π ε (5-3) The terms, ε ε ε, and are referred to as second order terms and are consdered to be small compared to the other frst order terms. Neglectng these terms, the cylnder volme error eqaton can be expressed n a smpler form. ε ε ( ) 4 V π ε ε π ε (5-4) Rearrangng eqaton (5-4), we can frther smplfy the eqaton for ε V. 5-7

101 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 V ε π ε π π π ε π ε π ε (5-5) The coeffcents for ε and ε n eqaton (5-5) are actally the partal dervatves of V wth respect to and. V and V π π Therefore, the cylnder volme error model can be expressed as V V V ε ε ε (5-6) where the partal dervatves are senstvty coeffcents that determne the relatve contrbton of the errors n length and dameter to the total error. 5.4 Uncertanty Model Axom 3 provdes the drect lnk between error and ncertanty that we need to qantfy measrement ncertanty. It s restated here for convenence. Axom 3: The ncertanty n a qantty or varable s the sqare root of the varable's mean sqare error or varance. In mathematcal terms, ths s expressed as ( ) V V ε var (5-7) And, accordng to the varance addton rle, the varance n ε V can be expressed n terms of the varances n ε and ε. ( ) ( ) V V c c c c ρ ε ε ε var var (5-8) where ρ s the correlaton coeffcent for the ncertantes n the length and dameter component errors and c and c are the senstvty coeffcents. V c and V c (5-9) UncertantyAnalyzer atomatcally comptes partal dervatves once the ser has defned the parameter vale eqaton n mathematcal terms. Alternatvely, the ser has the opton of enterng a coeffcent eqaton for a component or varable. The protocol for enterng an eqaton n the Senstvty Coeffcent Eqaton feld s the same as for the Parameter Vale 5-8

102 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Eqaton feld Senstvty Coeffcents. The followng dscsson llstrates the method sed wthn UncertantyAnalyzer to compte partal dervatves of a fncton f(x, y, z, ) wth respect to a varable x ( ) ( ) x z y x f z y x x f m x z y x f x δ δ δ,,, ),,, (,,,, (5-) where the m symbol reads n the lmt as x δ goes to zero. As an example, consder the fncton ( ) xy y x f,. Usng the above defnton, we have ( ) ( ) ( ). ) ( ), ( ), (, y x y x m x xy y x xy m x xy y x x m x y x f y x x f m x y x f x x x x δ δ δ δ δ δ δ δ δ δ δ δ In ths case, the partal dervatve does not contan the varable x. However, t s easy to see that the senstvty coeffcent for x s the varable y. That s, the larger or smaller y s the more or less the mpact of changes n x. As another example, consder the fncton ( ) y x y x f,. In ths case, the partal dervatve wth respect to x s ( ) ( ). ) ( ) ( ) (, xy x xy m x y x x y x x y x m x y x y x x m x y x f x x x δ δ δ δ δ δ δ δ δ In ths nstance, the varable x s nclded n the partal dervatve. Bt how abot the varable y? For ths, we have 5-9

103 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 whch s fncton of x only. ( x, y) f y 5.5 Measrement Process Errors x ( y δy) x y m δy δy x, ( δy) x y x m δy δy The errors n the length and dameter components, ε and ε, can be expressed n terms of ther consttent process errors. bas ran res op x ε ε ε ε ε ε (5-) ε ε ε ε ε ε (5-) bas ran res op y env env The process error sorces are: Bas (bas) - the bas n the mcrometer readngs. Random (ran) - the error that prodces dfferent reslts from measrement to measrement. Resolton (res) - the error de to the fnte resolton of the mcrometer readngs. Operator (op) - the error de to any systematc bas on the part of the measrng techncan. Envronment (env) - the error n any thermal or other correcton de to a departre from nomnal condtons. Measrement process errors are the basc elements of ncertanty analyss. Once these fndamental error sorces have been dentfed, we can then begn to develop ncertanty estmates. 5.6 Measrement Process Uncertantes The ncertanty n each component s expressed n terms of the ncertantes n the error sorces obtaned sng Axom 3 and the varance addton rle. Operatng on ε above wth the varance operator gves, for the ncertanty n the length measrement, (5-3) bas ran kewse, applyng the varance operator toε above gves, for the ncertanty n the dameter measrement, bas ran res res op op env (5-4) env 5-

104 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Notce that there are no terms correlatng process ncertantes wthn each component expresson. Ths s becase the length measrement process errors are ndependent of one another, as are the dameter measrement process errors. However, some of the length measrement process errors may not be ndependent of some of the dameter measrement process errors. Therefore, we mst consder possble crosscorrelaton terms. Ths wll be addressed n the next secton. The methods of ncertanty estmaton are Bas (bas) - herstcally from tolerance lmts and n-tolerance probabltes. Random (ran) - statstcally from a measrement sample. Resolton (res) - herstcally from the measrng parameter resolton spec and assmptons abot contanment probablty. Operator (op) - herstcally as a fncton of measrng parameter resolton. Envronment (env) - herstcally from tolerances and n-tolerance probabltes for the envronment montorng eqpment. In UncertantyAnalyzer, measrement process ncertantes are estmated sng the Error Sorce Worksheets accessed from the Mltvarate Analyss Screen. The Error Sorce Worksheets provde a sefl tool for makng both Type A and Type B ncertanty estmates. 5.7 Error Sorce Correlatons Before we combne ncertantes, we mst consder f there are any possble cross-correlatons between process ncertantes for the two components. Frst, let s revew what we know abot the cylnder measrement process.. Both length and dameter are measred sng the same devce (.e., a mcrometer).. All measrements are made by the same person (operator). 3. All measrements were made n the same measrng envronment. Gven ths knowledge, we can assert that the followng process ncertantes are crosscorrelated between the length and dameter components: Measrement Bas - bas and bas Operator Bas - op and op Envronmental Factors - env and env Second, we need to wrte the an eqaton that expresses the correlaton coeffcent, ρ, for the component ncertantes, and, n terms of the correlaton coeffcents for the process ncertantes that are cross-correlated between components. The expresson s n xy j ( ε, ε ) n j ρ ρ (5-5) xy where and are the total component ncertantes and and j are the process ncertantes for the length and dameter components, respectvely. (sbscrpts don t match eqaton 5-5) Correlaton coeffcents range from mns one to pls one. A postve correlaton coeffcent apples when the error sorces are drectly related. A negatve correlaton coeffcent s sed when the error sorces are nversely related. x xj x xj 5-

105 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 UncertantyAnalyzer s Correlaton Analyss Screen allows yo to compte correlaton coeffcents from pars of observatons. Alternatvely, f yo have a pretty good estmate of a correlaton coeffcent, yo can enter t drectly. For the cylnder volme, we need only consder the cross-correlatons between component measrement bas ncertantes, operator bas ncertantes, and envronmental factors ncertantes. et s consder the vales for these three correlaton coeffcents Correlaton between Component Measrement Bases. Snce the same devce s sed to measre both length and dameter, the parameter bas for these measrements s the same. In ths nstance, the correlaton coeffcent, ρ bas,bas, s eqal to Correlaton between Component Operator Bases. Althogh the same operator makes both measrements, hman nconsstency prevents s from assgnng a correlaton coeffcent eqal to.. However, we also know that the correlaton coeffcent shold not be eqal to zero ether. Gven that ths s all we can say from herstc consderatons, we wll set the correlaton coeffcent between length and dameter operator bases, ρ op,op, eqal to Correlaton between Component Envronmental Factors Errors. Snce the length and dameter measrements are made n the same envronment, the correlaton coeffcent between the length and dameter envronmental factors, ρ env, env, s also eqal to Total Uncertanty and egrees of Freedom We can now expand the total ncertanty eqaton for cylnder volme, V, n terms of the process ncertantes. V c c ( bas ran res op env ) c ( bas ran res c c ( ρ ρ ρ ) c ρ bas, bas bas bas op, op op op env, env env env op env ) UncertantyAnalyzer atomatcally combnes the ncertantes sng the approprate senstvty coeffcents and accontng for correlatons between error sorce ncertantes. When ncertantes are combned, we need to know the degrees of freedom for the total ncertanty. We can compte the degrees of freedom for the ncertanty n the cylnder volme, V, from the Welch-Satterthwate formla (5-6) v v 4 V * v c c where V* s the total ncertanty compted wthot cross-correlatons between component process ncertantes. c c (5-7) V * The degrees of freedom for the component ncertantes are 5-

106 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 and v v 4 (5-8) bas ran res op env v v v v v bas bas ran ran res res op op env 4 (5-9) bas ran res op env v v v v v env In UncertantyAnalyzer, the effectve degrees of freedom for the total ncertanty s atomatcally compted sng the Welch-Satterthwate formla. 5-3

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108 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 6. General CHAPTER 6 SYSTEM UNCERTAINTY ANAYSIS Ths chapter dscsses the approach sed to estmate the ncertanty of a qantty (or sbject parameter) that s measred wth a system comprsed of component modles arranged n seres. The analyss process traces system ncertanty modle by modle from system npt to system otpt. Fgre 6-. Block agram for Example System. System ncertanty analyss follows a strctred procedre. Ths s necessary becase the otpt from any gven modle of a system may comprse the npt to another modle or modles. Snce each modle's otpt carres wth t an element of ncertanty, ths means that the same ncertanty may be present at the npt of some other modle. In analyzng lnear system models, we develop otpt eqatons for each modle. From these eqatons, we dentfy sorces of error for each modle. We then estmate the ncertanty n each error sorce and the combned ncertanty n the otpt of each modle. In dong ths, we make certan that the ncertanty n the otpt of each modle was nclded n the npt to the scceedng modle n the system. In ths respect, the system analyss reslts are compted somewhat dfferently that those prevosly dscssed for drect measrements and mltvarate measrements. The general system analyss procedre conssts of the followng steps:. evelop a Measrement System Model etermne the measrement system stages or modles nvolved n processng the measrement of nterest. Identfy the hardware and software sed.. evelop Modle Eqatons evelop the set of eqatons that descrbe modle otpts n terms of npts and dentfy the parameters that characterze these processes. 6-

109 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 3. Identfy Modle Error Sorces From the modle otpt eqatons, dentfy and descrbe fnctons or parameters that may contrbte to the error n the modle otpt vale. 4. Estmate Modle Uncertantes Compte otpt vales, ncertantes and assocated degrees of freedom for each modle, accontng for correlatons between error sorces. 5. Estmate Total System Uncertanty - Propagate modle otpt vales and ncertantes to determne the system otpt, ncertanty and assocated degrees of freedom. The processes for developng a system model and the correspondng modle otpt eqatons are presented. Procedres for dentfyng measrement process errors, estmatng ther ncertantes and accontng for correlatons are also presented sng a temperatre measrement system for llstraton. 6. efntons 6.. Modle Error Sorces. Sorces of error that accompany the converson of modle npt to modle otpt. 6.. Modle Inpt Uncertanty. The ncertanty n a modle s npt expressed as the ncertanty or standard devaton n the otpt of the precedng modle Modle Otpt Eqaton. The eqaton that expresses the otpt from a modle n terms of ts npt. The eqaton s characterzed by parameters that represent the physcal processes that partcpate n the converson of modle npt to modle otpt Modle Otpt Uncertanty. The total combned ncertanty n the otpt of a gven modle of a measrement system System Modle. A modle or stage of a system nvolved n processng a measrement System Otpt Uncertanty. The total ncertanty n the otpt of a measrement system Total Modle Uncertanty. See Modle Otpt Uncertanty Total System Uncertanty. See System Otpt Uncertanty. 6. System Model The frst step n the system analyss procedre s to develop a model that descrbes the modles nvolved n processng the measrement of nterest (.e., sbject parameter). The model shold nclde a dagram depctng the modles of the system and ther npts and otpts and dentfy the hardware and software sed. The system dagram can be a sefl gde for developng the eqatons that descrbe the modle otpts n terms of npts and dentfy the parameters that characterze these processes. It may also be benefcal to develop a fnctonal model that relates component errors to the overall system error. For llstratve prposes, we wll evalate a temperatre measrement system for convertng a tme-varyng analog vale to a dgtal representaton. A nmber of specalzed dscplnes are nvolved, so we wll go nto some detal abot the physcal processes. 4 4 A good dscsson of measrement systems can also be fond n Compter-Based ata Acqston Systems esgn Technqes, Taylor, J.., Instrment Socety of Amerca,

110 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The dgtal temperatre measrement system s desgned to pro vde the follong capabltes: Temperatre Range: Bandwdth: Mode of Heat Transfer: Measrement Sensor: 5 o C Hz (sne wave) Natral convecton from fld to probe, condcton from probe to thermocople. Type K Chromel-Almel Thermocople The system dagram shold be detaled enogh to depct key components or modles, ther npts and otpts, and dentfy the hardware and software sed. An example dagram for the dgtal temperatre measrement system s shown below. Fgre 6-. Temperatre Measrement System agram. A block dagram for the temperatre measrement system shold also be developed to dentfy and label each modle, as well as the npt and otpt for each modle. Interface modles are nclded to accont for any gan/losses as the sgnal passes from one system component to another. Fgre 6-3. Block agram for Temperatre Measrement System. 6-3

111 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 For nstance, there may be errors ntrodced de to sgnal attenaton, nose, or cross-talk at the connecton between the thermocople Reference Jncton otpt termnals and the npt termnals of the ow Pass Flter. The Interface modle has been nserted between the Thermocople and ow Pass Flter modles to accont for sch errors. Smlarly, the Interface and Interface3 modles have been added to accont for errors ntrodced at the nterfaces between the ow Pass Flter and Amplfer modles and between the Amplfer and the A/ Converter modles, respectvely. 6.3 Modle Otpt Eqatons Once a sffcently detaled block dagram has been establshed, we can develop the eqatons that relate the npts and otpts for each modle. The basc approach s to clearly descrbe the physcal processes and dentfy sorces of error that can affect the measred vale along ts path from modle to modle Thermocople Sensor Modle (M ). In developng the eqaton to compte the otpt vale, Y, as a fncton of the npt X, we need to consder the relevant varables or parameters that need to be nclded. The thermocople sensor modle conssts of the chromel-almel measrng jncton, a reference jncton and otpt copper leads, as shown n Fgre 6-4. Fgre 6-4. Schematc of Thermocople Sensor Modle (M). 6-4

112 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 When two wres of dfferent materals, sch as chromel and almel, are joned at one end and placed n a temperatre gradent, a thermoelectrc voltage s observed at the other end. Ths s the common operatng prncple of a thermocople and s descrbed by the Seebeck effect. Ideally, f the wres are made of thermoelectrcally homogenos materal, the temperatre of the measrng jncton (TM) can be obtaned from the measred voltage (V), knowledge of the Seebeck coeffcents of the wres, and knowledge abot the temperatre of the wres at the reference jncton. In realty, however, errors arse from nmeros avodable and navodable sorces that are lsted below. Electrcal shntng rft, agng, and hysteress Nose Calbraton or senstvty errors Reference jncton errors Electrcal Shntng. If the electrcal resstance of the nslaton between the thermocople wres degrades apprecably, then cross-condcton can reslt n the formaton of vrtal temperatre jnctons. The severty of electrcal shntng depends pon the propertes and thckness of the nslatng materal and the thckness of the thermocople wres. For the prposes of the temperatre measrement system example, we wll assme that the Type K thermocople modle does not experence any errors de to electrcal shntng rft and Agng. Thermocoples that are n servce for extended tme perods at elevated temperatres can change characterstcs. The changes depend on the ntal prty of the thermocople materal, contamnants ntrodced from the envronment, and temperatre. No general estmate of the rate of thermocople drft can be stated. However, we know that n some nstances, thermocoples have been fond to be stable for thosands of hors of contnos exposre to hgh temperatre, whle others nder smlar condtons drft apprecably wthn a few hors Hysteress. Hysteress s the varaton n the thermocople otpt voltage for a specfc temperatre npt when temperatre s approached from dfferent drectons, as llstrated below. When nclded as a performance specfcaton, t s stated as the maxmm dfference n the otpt voltage for the same temperatre drng a fll range traverse n each drecton. 6-5

113 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 6-5. Hysteress Effect. Hysteress error arses from resdal vales left over from prevos samples. Represent ths qantty by the varable Vh. If the samplng rate s ν s, then λτ λ ( T τ ) ( e ) e V V (6-) h where T /νs. The hysteress error s the sampled vale of ths resdal amont. Ths the hysteress error s gven by ε h λt ( t) V ( e ) If the sample pont t s nknown, we nstead se the average error ε (6-) h λτ ( e ) (6-3) λτ h V h The hysteress ncertanty can be obtaned from the hysteress error n the same way as the mplse response ncertanty was obtaned. If the sample pont s known, then whereas, f the sample pont s nknown, σ h τ V V h h τ λτ σ h ε ( t) dt ε h h λt λvhe σ t λτ λτ ( e ) ( e ) λτ λτ λτ ( e ) ( e ). λτ ε h (6-4) (6-5) 6-6

114 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Nose. Snce the thermocople leads are condctors, externally appled electromagnetc felds may ntrodce stray emfs that contamnate the voltage otpt de to the temperatre between the measrng jncton and reference jncton. Sch "nose" comprses an error. Nose s sally random n character. UncertantyAnalyzer acconts for thermal nose and shot nose n electronc crcts. Provson s also made for ser-estmated nose de to other effects. Note, that, snce nose s, n tself error, the ncertanty de to a gven sorce of nose s the sqare root of the varance of the nose dstrbton Thermal Nose. The mean sqare voltage de to nose prodced by delverng crrent to a load wth resstance R s gven by V thermal 4k TRΔf where k B Boltzmann s constant T temperatre of the load R resstance of the load f bandwdth of the sgnal Conseqently, the ncertanty de to thermal nose s thermal B (6-6) 4 k TRΔf (6-7) Shot Nose. Shot nose s the flctaton n the crrent of charge carrers passng throgh a srface at statstcally ndependent tmes. It has a nform spectral densty Vshot gven by where e I n Δf B V shot ei nδf electron charge crrent passed throgh one or more pn jnctons nmber of jnctons bandwdth of the sgnal The ncertanty de to shot nose s gven by shot (6-8) ei Δf (6-9) Total Electronc Nose. The ncertanty de to electronc nose s compted as the root-sm-sqare of thermal and shot (6-) e thermal Sgnal to Nose Rato. The total voltage de to electronc nose s compted as shot 6-7

115 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 V e V e thermal ettng Vsg represent the rms ampltde of the sgnal, the sgnal to nose rato s compted as V R sn V V sg e shot (6-) Other Nose. Users can accont for ncertanty de to nose from other sorces. Ths conssts of a Type B ncertanty analyss n whch the standard ncertanty s compted from error lmts ± and a confdence level c. Provson s made for fve nose sorces Normal Nose strbton. If the confdence level s less that %, UncertantyAnalyzer assmes a normal dstrbton. The nose contrbton of the th Other Nose sorce s gven by where (6-3) c Φ / error lmt c confdence level (%) Φ nverse normal dstrbton fncton Unform Nose strbton. If the confdence level s %, UncertantyAnalyzer assmes a nform dstrbton. The nose contrbton of the th Other Nose sorce s gven by. (6-4) Combned Other Nose. Each Other Nose ncertanty s weghted wth a sersppled coeffcent a,,,..., 5. Accordngly, the total ncertanty de to Other Nose s compted accordng to 5 a. (6-5) other Total Nose Uncertanty. The total ncertanty de to nose s compted n rss:. (6-6) nose Calbraton or Senstvty Errors. The relatonshp of the temperatre of the measrng jncton (T M ) and the measred voltage s tablated n thermocople calbraton tables for common thermocople types. These calbraton tables generally assme a reference temperatre (T R ) eqal to o C. If an ce bath s sed as the reference jncton or the thermocople has a means of compensatng for a reference jncton temperatre other than C, then the tempera- e other 6-8

116 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 tre of the measrng jncton for a gven voltage otpt can be obtaned drectly from these tables. However, t s mportant to note that these thermocople calbraton (or reference) tables provde the expected characterstcs for common thermocople types. Even f thermocoples are manfactred n conformance wth specfed standards, there s some dfference between the actal thermocople response characterstc (e.g., Seebeck Coeffcent) and that gven n the reference tables. Ths dfference ntrodces an error senstvty of the thermocople. Alternatvely, f the thermocople s ndvdally calbrated, errors arse from the fttng a calbraton crve to a fxed set of test ponts and from the accracy of the comparson thermometer. For ths analyss example, we wll assme that the Type K thermocople has been manfactred accordng to standard specfcatons and has not been ndvdally calbrated Reference Jncton Error. It s mportant to remember that the voltage otpt of a thermocople depends on the magntde of the temperatre dfference between the measrng jncton and the reference jncton. The larger the temperatre dfference, the larger the voltage otpt. If the reference jncton temperatre s allowed to warm p above C, whle the measrng jncton temperatre s held at a gven temperatre, the voltage otpt of the thermocople wll decrease. For ths analyss example, we wll assme that an ce bath has been sed to acheve a reference jncton temperatre of C. Based on or above assessment of the thermocople sensor modle error sorces and assmng that hysteress and drft can be expressed as fractons of the npt temperatre, X, the otpt eqaton for the thermocople modle M can be expressed as where Y ( p p p3)( p )( X p5 ) Table 6-. Parameters Used n Thermocople Modle Otpt Eqaton Parameter Nomnal Parameter escrpton Name Vale Error mts X Inpt vale (.e., Sbject Parameter). C p Senstvty (.e., Seebeck Coeffcent) μv/ o C ±.95 μv/ o C p Thermocople hysteress μv ± μv p 3 Thermocople drft μv ± μv p 4 Nose μv ± μv p 5 Reference jncton devaton from C C ±.7 C 4 6-9

117 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The otpt eqaton for a selected modle s entered nto the Modle Otpt Eqaton feld of UncertantyAnalyzer s System Model Screen. Modle otpt eqatons are entered n VB Scrpt format, whch s a smplfed form of VBA (Vsal Basc for Applcatons). The modle otpt eqaton can contan nmbers, constants, varables, fnctons, and mathematcal operators. The system npt, X, s the sbject parameter of nterest that we are measrng. In ths case, we are measrng temperatre. In UncertantyAnalyzer, the sbject parameter s defned by enterng descrptve nformaton and other specfcatons (as approprate) nto the Sbject Parameter Bas Uncertanty Worksheet Interface Modle (M ). The potental dfference at the Thermocople reference jncton otpt termnals s transmtted throgh copper condctors and appled across the npt termnals of a owpass Flter. The condctors and the flter termnals comprse an nterface between the Reference Jncton and the ata Acqston System, as shown n Fgre 6-6. The sorces of error are Interface oss Crosstalk Nose Fgre 6-6. Schematc of Thermocople - owpass Flter Interface Interface oss. The voltage appled across the termnals of the owpass Flter sffers a drop de to the resstance of the connectng leads from the Reference Jncton and of the owpass Flter contacts Crosstalk. eakage crrents between npt flter termnals may alter the potental dfference across the termnals Nose. Electromagnetc nose s a factor for the connectng leads, whle both the connectng leads and the owpass Flter termnals are sbject to thermal nose. 6-

118 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where The otpt eqaton for the Interface modle M s Y ( Y p p)( p) 3 Table 6-. Parameters Used n Interface Modle Otpt Eqaton Parameter Name Parameter escrpton Nomnal Vale Error mts Y Otpt from modle M p Interface loss μv ± μv p Crosstalk μv ± μv p 3 Nose μv ± μv owpass Flter Modle (M 3 ). The potental dfference that srvves the Reference Jncton - Flter nterface s altered by the flter tself. The flter attenates nose that may be present and provdes a "cleaned p" potental dfference to the system's amplfer. However, some nose gets throgh. Also, the flter attenates the sgnal somewhat and tself generates a small nose component. Fgre 6-7. Ampltde as a Fncton of Freqency for a Typcal Flter. The sorces of error are Sgnal Attenaton Nose Nonlnearty Sgnal Attenaton. Althogh the flter s ntended to attenate nwanted nose, some sgnal attenaton also occrs. 6-

119 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Nose. Not all the npt nose wll be fltered ot. The nose that remans wll be attenated by an amont that depends on the roll-off characterstcs of the flter. These characterstcs are sally assmed to be lnear and are expressed n terms of db per octave. Thermal nose s also generated wthn the flter tself Nonlnearty. The response of a flter over the range from ts ctoff freqency, fc, to ts termnatng freqency, fn, s sally consdered to be farly lnear. epartres from ths assmed lnearty consttte errors. The otpt eqaton for the lowpass flter modle M3 s where Y 3 ( p 3 ) Y p 3 ( p ( p p p 34 3 ) Y p p 3 33 ) Y 3 ( f p, f 3 f, f n 33 f c ) p 3, f c f f n Table 6-3. Parameters Used In ow Pass Flter Modle Otpt Eqaton Parameter Name Parameter escrpton Nomnal Vale Error mts Y Otpt from modle M p 3 Sgnal attenaton μv ±. μv p 3 Nose μv ± μv p 33 Ct-off freqency, f c Hz ± Hz p 34 Maxmm freqency otpt, f n N/A ± Hz Becase of the hgh ctoff freqency, only the pper eqaton for Y 3 apples Interface Modle (M4). The potental dfference otpt by the owpass Flter s fed to the Amplfer across an nterface comprsed of the leads from the owpass Flter and the npt termnals of the Amplfer. The sorces of error are Interface oss Crosstalk Nose Interface oss. The voltage at the Amplfer termnals sffers a drop de to the resstance of the connectng leads from the owpass Flter and of the npt termnal contacts Crosstalk. eakage crrents between npt Amplfer termnals may case a decrease n the potental dfference across the termnals Nose. Electromagnetc nose s a factor for the connectng leads, whle both the connectng leads and the Amplfer termnals are sbject to thermal nose. 6-

120 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where The otpt eqaton for the Interface modle M4 s Parameter Name Y ( Y p 4 p4)( p4) Table 6-4. Parameters Used n Interface Modle Otpt Eqaton Parameter escrpton 3 43 Nomnal Vale Error mts Y 3 Otpt from modle M 3 p 4 Interface loss μv ± μv p 4 Crosstalk μv ± μv p 43 Nose μv ± μv Amplfer Modle (M5). The Amplfer ncreases the potental dfference (and any nose receved from the owpass Flter) and otpts the reslt to an A/ Converter (Fgre 6-8). Fgre 6-8. Amplfer Errors. Several sorces of error are present. Hysteress cases an otpt tme lag, common-mode voltage rases the zero reference, normal mode voltage stretches the waveform, whle flterng compresses t. Rdng on top s spermposed nose. Key error sorces nclde: Gan Gan Instablty Normal Mode Range Offset Nonlnearty Common Mode Rejecton Rato Nose 6-3

121 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Gan. Amplfer gan s the rato of the otpt sgnal voltage to the npt sgnal voltage. Gan errors are those that lead to a nform shft n expected Amplfer otpt vs. actal otpt. Gan errors are composed of nherent (systematc) errors and temperatre ndced (random and systematc) errors Gan Instablty. If the Amplfer voltage gan s represented by G V, ts npt resstance by R and ts feedback resstance by R', then oscllatons are possble when RG V π. R R' These oscllatons appear as an nstablty n the amplfer gan Normal Mode Voltage. Normal mode voltages are dfferences n zero potental that occr when Amplfer npt (sgnal) lnes are not balanced. Normal mode voltages are essentally random n character Offset. Offset voltages and crrents are appled to the Amplfer npt termnals to compensate for systematcally nbalanced npt stages. The varos parameters nvolved n offset compensaton are the followng: Inpt Bas Crrent. A crrent sppled to compensate for neqal bas crrents n npt stages. Eqal to one-half the sm of the crrents enterng the separate npt termnals Inpt Offset Crrent. The dfference between the separate crrents enterng the npt termnals Inpt Offset Crrent rft. The rato of the change of npt offset crrent to a change n temperatre Inpt Offset Voltage. The voltage appled to acheve a zero Amplfer otpt when the npt sgnal s zero Inpt Offset Voltage rft. The rato of the change of npt offset voltage to a change n temperatre Otpt Offset Voltage. The voltage across the Amplfer otpt termnals when the npt termnals are gronded Power Spply Rejecton Rato (PSRR). The rato of the change n npt offset voltage to the correspondng change n a gven power spply voltage, wth all other power spply voltages held fxed Slew Rate. The maxmm tme rate of change of the Amplfer otpt voltage nder large-sgnal (sally sqare wave) condtons. Slew rate sally apples to the slower of the leadng edge and tralng edge responses Nonlnearty. As wth other components, actal Amplfer response may depart from the assmed otpt verss npt crve. Nonlnearty error conssts of the dsagreement between the characterstc sgnatre of an amplfer's response and ts expected characterstc Common Mode Rejecton Rato. CMRR s the rato of the amplfer sgnal voltage gan to the common mode voltage gan. CMRR s often sed n estmatng errors n amplfer otpt. 6-4

122 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Nose. Nose generated wthn the Amplfer that enters the sgnal path cases errors n the otpt sgnal. The otpt eqaton Amplfer modle M5 s where Y ( p 5 p5 p5 p55)( Y4 p53 p55 p56) 57 Parameter Name Table 6-5. Parameters Used n Amplfer Modle Otpt Eqaton Parameter escrpton Nomnal Vale Error mts Y 4 Otpt from modle M 4 p 5 Amplfer Gan % of Gan p 5 Gan Instablty μv ±.5% of Gan p 53 Normal Mode Voltage μv ± μv p 54 Offset μv ±. μv p 55 Nonlnearty μv ±.% of Gan p 56 Common Mode Reject Rato μv ±.% of common mode npt p 57 Nose μv ±.5 μv Interface3 Modle (M6). The amplfed potental dfference s appled across the A/ Converter npt termnals. The nterface between the Amplfer and the A/ Converter s prone to the followng error sorces: Interface oss Crosstalk Nose Interface oss. The voltage at the A/ Converter termnals sffers a drop de to the resstance of the connectng leads from the Amplfer Crosstalk. eakage between npt A/ Converter may case a decrease n the potental dfference across the termnals Nose. Electromagnetc nose s a factor for the connectng leads, whle both the connectng leads and the A/ Converter termnals are sbject to thermal nose. The otpt eqaton for the Interface3 modle M6 s Where Y ( Y p 6 p6)( p6)

123 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Table 6-6. Parameters Used n Interface3 Modle Otpt Eqaton Parameter Name Parameter escrpton Nomnal Vale Error mts Y 5 Otpt from modle M 5 p 6 Interface loss -.. p 6 Crosstalk μv ± μv p 63 Nose μv ± μv A/ Converter Modle (M 7 ). In dgtzng the analog sgnal, sgnal vales are sampled as potental dfferences appled across a network of analog components. The network otpts a coded plse consstng of ones and zeros. The locaton of these ones and zeros s a fncton of the npt sgnal level and the response of the network to ths sgnal level. The errors n the dgtzng process consttte a dscrepancy between the waveform emergng from /A converson and the orgnal npt waveform (pror to A/ converson). The sorces of error are Samplng Rate Error Apertre Tme Error Implse Response Error Alasng Error gtal Flterng Error Gan Error Nose Qantzaton Error Samplng Rate Error. When a sgnal s sampled for dgtal data processng, t s represented by a set of dscrete ponts. ater n the processng chan, these ponts need to be reconsttted nto a sgnal agan. When ths s done, the reconsttted sgnal dffers from the npt sgnal by some amont. What s done n the reconsttton process can be roghly descrbed as an electronc or mathematcal "connect the dots" procedre, as shown n Fgre

124 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 6-9. Samplng Rate Error. ow-tech /A converters reconsttte the sgnal sng a set of straght lnes from sampled pont to sampled pont. Hgh-end /A converters reconsttte the sgnal sng nvolved mathematcal operatons tlzng sophstcated crve fttng and otler detecton technqes. UncertantyAnalyzer provdes for both ends of the spectrm wth a "near" crve ft model and a "Qadratc Splne" crve ft model. The latter employs a second degree crve ft, bt mproves on ths by effectvely doblng the samplng rate. The reslt s somethng that s more or less representatve of hgh end /A converson. For example, let s consder a sgnal represented by f(t) that s dgtzed at dscrete ntervals Δt. The sgnal s later reconstrcted nto an approxmaton f ˆ( t ). The sgnal f (t) s composed of M components, each represented by a sne wave wth anglar freqency ω m and relatve phaseϕ m. M f ( t) a m sn( ω t φ ) m m m (6-7) The samplng rate varance conssts of the root mean sqare of f (t) - f ˆ( t ) over a complete wave form. where π M σ σ d n φm (6-8) π m nδ n σ [ f ( t) fˆ( t)] dt. (6-9) Δ ( n ) Δ 6-7

125 Uncertanty Analyss Prncples and Methods RCC ocment -7, September near Model. The lnear model s gven by f t f f ˆ ( Δ ) () ( t) f () t Δ (6-) where and f ( Δt) f () M m M a m m snφ, m a m sn( ω Δt ϕ ). m m (6-) (6-) The sample rate varance for the lnear model s where M σ a ( I I I ), (6-3) m m m m 3 m I, (6-4) m I m cosω t m ( cos ), ω Δt m (6-5) and I 3. (6-6) m Qadratc Splne Model. The qadratc splne model s gven by f ˆ( t) f () b t b t (6-7) where f () s gven above and b [4 f ( Δt / ) f ( Δt) 3 f ()], (6-8) Δt and b [ f ( Δt) f ( Δt / ) f ().] (6-9) Δt The samplng rate varance for the qadratc model s also gven by M σ a m( I I I3 ). (6-3) m m m m 6-8

126 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where and I, (6-3) m 4 I 3, 4,/ (,/,, cm cm sm sm ) (6-3) m ( ω Δ ) mδt m t ω I 3 cm,/ cm, ' (6-33) m The coeffcents n eqatons (6-3) and (6-33) are c m, m,/ s c s m, m,/ cos cos ( ωδt), ( ωδt / ), ( ωδt), sn sn( ωδt / ) Apertre Tme Error. A fnte amont of tme δt s reqred to sample the sgnal voltage V as shown n Fgre 6-. rng ths tme, the sgnal vale changes by an amont δv. For a sgnal comprsed of N components of freqency ω k and phase ϕ k, the average error n δv s gven by and N ak δ V ( ωkδt) sn( ωkt ϕk ), (6-34) 4 k δ V a k ω δt. (6-35) max N k k 6-9

127 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 6-. Apertre Tme Error. At ths pont, t s worth rememberng that, n the measrement system, the sampled data wll be qantzed n bnary code. As we wll see later, these errors are on the order of A δ V ) qantzaton ±, (6-36) n ( where n s the nmber of qantzng bts and A s the dynamc range of the qantzng apparats. If the qantzaton error s larger than the maxmm apertre error, then the apertre error can be gnored. For example, for the composte sne wave sgnal consdered here, apertre error can be gnored f N A akω kδt <, or δt < n N k k A a ω Implse Response Error. In samplng a sgnal, the samplng sensor mst be able to respond to and recover from sgnal changes. If the rse tmes and recovery tmes of the samplng sensor are not neglgble n comparson wth the samplng apertre tme, then mplse response errors occr. For example, magne that a sgnal s sampled n tme ncrements of T over a samplng apertre of draton τ, as shown n Fgre 6-. k k n. 6-

128 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 6-. Implse Error. Assme that the response of the sensor r(t) to a sgnal vale a ( ω t ϕ) the eqaton: dr dt V sn s governed by λ ( V r). (6-37) We rewrte ths eqaton n the followng form to facltate solton: dr λ r λv. (6-38) dt In the regon < t < τ, the homogeneos solton to eqaton () s λt r t) r e, (6-39) and the partclar solton s h( r p ( t) bsn( ω t ϕ) ccos( ωt ϕ). (6-4) Sbstttng r( t) r ( t) r ( t) h p n eqaton (6-37) gves so that and where ( ω b λc) cos( ωt ϕ) ( λb ωc)sn( ωt ϕ) λa sn( ωt ϕ) (6-4) ωb λc λb ωc λa b q q c q q ω / λ. a a, 6-

129 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Snce the vale of ϕ wll be an ndetermned varable, we can set r() wthot loss of generalty. Ths gves and a r (snϕ qcosϕ), (6-4) q a λt r( τ ) [sn( ωt ϕ) qcos( ωt ϕ) (snϕ qcosϕ) e ]. (6-43) q In theτ t T regon, eqaton (6-37) becomes wth the solton where r(τ) s taken from eqaton (6-43). dr λ r, (6-44) dt r( t) r( τ ) e In the T t T τ regon, the eqaton s agan dr dt λ ( t τ ) The homogeneos and partclar soltons are, respectvely,, (6-45) λ r λv. (6-46) r ( t) r e h λ ( t T ) (6-47) r p ( t) bsn( ω t ϕ) ccos( ωt ϕ). (6-48) Sbstttng n eqaton (6-46) gves The ntal condton s and λ ( t T ) a r( t) r e q r( T ) r( τ ) e [sn( ωt ϕ) qcos( ωt ϕ λ ( T t) so that λ ( t T ) a r( τ ) e r q )]. [sn( ωt ϕ) q cos( ωt ϕ)] λ ( t T ) a r r( τ ) e [sn( ωt ϕ) qcos( ωt q ϕ Sbstttng from eqaton (6-4) for r(τ) n eqaton (6-5) and plggng the reslt nto eqaton (6-49) gves for t T τ )]. (6-49) (6-5) (6-5) 6-

130 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 [ ] [ ] [ ] [ ] { } ϕ τ ω ϕ τ ω ϕ ω ϕ ω ϕ ϕ ϕ ωτ ϕ ωτ τ τ λ λτ ) ( cos ) ( sn ) cos( ) sn( ) cos (sn ) cos( ) sn( ) ( ) ( T q T q a T q T q a e e q q q a T r T (6-5) The dfference between the npt sgnal and the response at the pont t T τ s the mplse response error: ). ( ) ( τ τ ε T V T r r (6-53) Snce the phase ϕ s ndetermnate, the mplse response varance s gven by { }. ] ) ( )sn[ ( ) ( ] ) ( sn[ ) ( π π π ϕ ϕ τ ω τ π ϕ τ π ϕ ϕ τ ω τ π σ a d T T r a d T r d T a T r r (6-54) Sbstttng eqaton (6-5) nto ths expresson gves [. ) ( τ λ λ λτ σ T T r e e e q a ] (6-55) In general, ths expresson can be wrtten [. ) ( τ λ λ λτ σ T T k m m m rms r e e e q a V ] (6-56) Ths form permts the evalaton of mplse response error for sgnals composed of mltple sne wave components: k m m m a m t V ) sn( ϕ ω so that. k m rms a m V The parameter λ s obtaned from the tme t takes the sensor s response to reach of the npt sgnal: e /, / / / ) ( Ve e r e λτ τ 6-3

131 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 and λ τ / e Alasng Error. When two or more freqences are sampled that are ntegral mltples of one another, these freqences cannot be dstngshed from one another. Alasng error s avoded by settng the samplng rate hgher than twce the freqency of the hghest freqency component of the sampled sgnal and by employng low-pass flters n the npt of the A/ converter gtal Flterng Error. Followng A/ converson, becase of thermal nose n the flter and other factors, there s stll a chance that ndesrable hgh freqency contrbtons wll be present. These contrbtons are ordnarly removed by lowpass dgtal flters. gtal flterng of nwanted hgh freqences s done by smply averagng the dgtally encoded nformaton over a set of sampled plses. For example, sppose that the samplng freqency were doble the hghest sable freqency. Snce the samplng rate s twce the hghest sable freqency, changes n ampltde that transpre between sccessve coded plses are those that represent nwanted freqences. By averagng the ampltdes of sccessve coded plses or btstreams, these freqences are elmnated. Unfortnately, the elmnaton of nwanted freqences by dgtal flterng s not a free rde. The process ntrodces some error. Fortnately, for the freqences nvolved n the present example, these errors are neglgble and wll not be covered here. For cases where these errors are sgnfcant, the reader s encoraged to srvey the lteratre on dgtal flters Gan Error. One type of A/ Converter employs a ladder network of resstors. The confgraton of the network s sch that dfferent sgnal levels case dfferent dscrete responses. A major factor affectng the accracy of these responses s the error n the vale of the resstors n the network. Ths s becase the voltage drop (negatve gan) across each component resstor s a fncton of the sgnal level and the component's C resstance Nose. As expected, stray voltages are sensed along wth the sgnal voltage and contrbte to the voltage level appled to the network. In addton, thermal flctatons n components case flctatons n voltage drops Qantzaton Error. The potental drop (or lack of a potental drop) sensed across each element of the A/ Converter sensng network prodces ether a or to the converter. Ths response constttes a "bt" n the bnary code that represents the sampled vale. The poston of the bt n the code s determned by whch network element orgnated t. Even f no errors were present n samplng and sensng the npt sgnal, errors wold stll be ntrodced by the dscrete natre of the encodng process. Sppose, for example, 6-4

132 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 that the fll scale sgnal level (dynamc range) of the A/ Converter s A volts. If n bts are sed n the encodng process, then a voltage V can be resolved nto n dscrete steps, each of sze A/ n. The error n the voltage V s ths A ε V m, (6-57) V where m s some nteger determned by the sensng fncton of the /A Converter. The ncertanty assocated wth each step s one-half the vale of the magntde of the step. Conseqently, the ncertanty nherent n qantzng a voltage V s (/)(A/ n ), or A/ n. Ths s emboded n the expresson n A V V ±. (6-58) qantzed sensed n Wth UncertantyAnalyzer, the ncertantes de to samplng and other A/ converson error sorces are handled by the gtal Samplng Uncertanty Worksheet. If ths worksheet s sed, then the otpt eqaton for the /A Converter modle s smply where Parameter Name Y 7 Y6 S7 Table 6-7. Parameters Used n Interface3 Modle Otpt Eqaton Parameter escrpton Nomnal Vale Error mts Y 6 Otpt from modle M 6 S 7 ata Samplng Error μv The ncertanty de to data samplng error was compted sng UncertantyAnalyzer s gtal Samplng Uncertanty Worksheet by enterng the followng nformaton: Inpt Sgnal Freqency: Hz Inpt Sgnal Ampltde: (Compted by UncertantyAnalyzer) Samplng Rate: Hz Samplng Fll Scale: 5 mv Qantzaton Sgnfcant Bts: 6 Apertre Tme: µsec Sensor Response Tme Constant:. µsec Sgnal Bandwdth: Hz Operatng Temperatre: o C Sensor Otpt Resstance: kohm Sensor RMS Crrent:.5 ma Nmber of Sensor p-n Jnctons: Sensor Error mts:.5% of npt sgnal (99% confdence) ata Processor Modle (M 8 ). The qantzed (dgtal) otpt from the A/ Converter s npt to a ata Processor. Snce the otpt s dgtal, the nterface between the A/ Converter 6-5

133 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 and the ata Processor wll be assmed to not consttte an error sorce. The ata Processor converts bnary coded nmbers to vales, performs /A crve fttng and apples any correcton factors that may be approprate. The sorces of error are Correcton Factor Error ata Redcton Error ecodng Error Comptaton Error Correcton Factor Error. The correcton factor appled to the dgtally encoded voltage dfference attempts to correct for losses that occr between the Reference Jncton and the ata Processor. Uncertantes n estmatng these losses may lead to errors n the correcton factors ata Redcton Error. In convertng the corrected vale for the voltage dfference nto a temperatre dfference, the ata Processor attempts to solve the eqaton ΔV ( a b ) ΔT ( a b )( ΔT ) ( a3 b3 )( ΔT ) In arrvng at the solton, the seres s trncated at some polynomal order. Ths trncaton leads to a dscrepancy between the solved-for temperatre dfference and the actal temperatre dfference. For nstance, sppose that the seres s trncated to second order. Then the ata Processor solton for the temperatre dfference becomes a ΔT a * * b b VC VA b b a b a 4 a * * where the qanttes V C and VA are corrected vales for V and V C A, and O(3) represents the error de to neglectng thrd order and hgher terms. For the present example, data redcton error wll be assmed to be nearly zero and wll be gnored ecodng Error. The otpt of the ata Processor s a corrected reslt that s dsplayed as a decmal nmber. The followng error sorces are relevant n developng and dsplayng ths nmber: Sppose that the dgtal "resolton" of the bnary encoded sgnal s A/ n. Sppose frther that the fll-scale vale ata Processor readot s S and that m dgts are dsplayed. Then the resolton of the decmal dsplay of the ata Processor s S/ m. Another way of sayng ths s that the npt to the ata Processor s a mltple of steps of sze A h, whle the decmal encoded dsplay s presented n steps of sze b n S h. d m 3 What ths means s that a bnary encodng of a voltage V nto a representaton O3, 6-6

134 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 wll be translated nto a decmal representaton V ' x hb V" y hd where x and y are ntegers. The qantzaton error that reslts from expressng an analog vale frst as a bnary coded vale and second as a decmal coded vale s the sm of these two errors: ( hb hd ) A S / Qantzaton Error ± n m ± ± Comptaton Error. Convertng from a dgtal representaton of an npt sgnal to an analog (or at least decmal) representaton nvolves a sbstantal amont of nmercal comptaton. Each tme a comptaton s made, some rond off error s ntrodced. These errors accmlate throgh the comptng process. The comptaton error ncertanty was compted sng UncertantyAnalyzer s Comptaton Error Uncertanty Worksheet by enterng the followng nformaton: Error Sorce Compted or Approx. ecmal Sorce escrpton Measred Vale No. Calcs gts Coeffcent Crve Ft Fne Tnng 5 8. The otpt eqaton for the data processor modle M 8 s where Parameter Name Y 8 p8p83 Y7 p8 CE8 Table 6-8. Parameters Used n ata Processor Modle Otpt Eqaton Parameter escrpton Nomnal Vale Error mts Y 7 Otpt from modle M 7 p 8 Voltage to temperatre converson factor.45 C/µV. C/µV p 8 ecodng Error C. C p 83 Correcton factor. C. C CE 8 Comptaton Error.33E-7 C 6.4 Modle Uncertantes The next step n the system ncertanty analyss procedre s to npt nformaton abot the parameters for each modle and to estmate the ncertantes n the errors n these parameters. Ths can be done by enterng the reqste data for each parameter drectly nto the Modle Parameters table n the Modle Analyss Screen or sng the assocated Error Sorce Worksheets. 6-7

135 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Modle Parameters Table. The Modle Parameters table s sed to enter and store the parameter nformaton for each modle. For example, the Thermocople modle otpt eqaton has sx parameters, as lsted n Table 6-. As prevosly dscssed, the system npt (X) s defned va the Sbject Parameter Bas Uncertanty Worksheet. Bref descrptons of the modle parameters are entered nto the escrpton colmn of the table. For the thermocople modle, Senstvty wold be entered nto the escrpton feld for parameter p, and so on. The standard ncertanty for a parameter s ether compted from ser specfed ± Error mts and % Confdence, or entered drectly. The probablty that the error n the modle parameter wll be contaned wthn the specfed ± error lmts s entered nto the % Confdence colmn of the table. UncertantyAnalyzer assmes that the ncertantes n the parameter errors are normally dstrbted nless the assocated degrees of freedom are less than nfnte or the % Confdence s %. If the degrees of freedom are less than nfnte and the % Confdence s less than %, the Stdent's t dstrbton s sed. If the % Confdence s %, then the Unform dstrbton s sed and the degrees of freedom are assmed to be nfnte Error Sorce Worksheets. Alternatvely, the modle parameter ncertantes can be estmated sng the Error Sorce Worksheets accessed by clckng the Edt btton to the left of the desred parameter name. The Error Sorce Worksheets provde a sefl tool for makng both Type A and Type B ncertanty estmates. The Type A Uncertanty porton of the worksheet can be sed to statstcally estmate the ncertanty n the parameter error from measred vales or devatons from nomnal that are entered nto the data entry table. The Type B Uncertanty porton of the worksheet can be sed to develop a herstc ncertanty estmate from bondng vales, or ± lmts, that are expected to contan the error wth some specfed probablty or confdence level. UncertantyAnalyzer assmes nfnte degrees of freedom for the Type B ncertanty estmate nless otherwse specfed by the ser. Snce, the degrees of freedom qantfes the amont of knowledge avalable for makng the ncertanty estmate, an nfnte degrees of freedom sgnfes complete certanty,.e., zero ncertanty. In most cases, t wold not be realstc to assme nfnte knowledge abot the ncertanty. The Type B egrees of Freedom Calclator can be sed to provde addtonal nformaton abot the ncertanty n the ± lmts and assocated confdence level. The statstcal dstrbton for errors whose ncertantes are arrved at as a reslt of a Type B analyss can be selected from the drop-down strbton lst. In most cases, the Normal dstrbton wll be applcable. For cases where t s sspected or known that errors are bonded by fnte lmts, other dstrbtons may be more applcable. In cases where the degrees of freedom are fnte, the Stdent s t dstrbton s atomatcally selected. The dstrbton optons avalable from the Error Sorce Worksheet nclde: Normal Qadratc Cosne U-Shaped 6-8

136 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Unform Tranglar Stdent s t Senstvty Coeffcents. UncertantyAnalyzer atomatcally comptes the partal dervatves of the selected modle otpt eqaton wth respect to each modle parameter sng the method prevosly descrbed n Secton If a coeffcent eqaton has been entered for a gven modle parameter, then UncertantyAnalyzer comptes the Senstvty Coeffcent from ths eqaton and enters ts vale nto the Modle Parameters table Component Uncertanty. The component ncertanty for a modle parameter s the prodct of the parameter's senstvty coeffcent and standard ncertanty. It s a weghted vale that reflects the contrbton of the ncertanty n the parameter error to the overall ncertanty n the modle otpt vale. UncertantyAnalyzer atomatcally comptes component ncertantes for all varables n the Modle Parameters table Total Modle Uncertanty. After the ncertantes have been estmated or entered for each modle parameter and the assocated senstvty coeffcents and component ncertantes have been compted, selectng the AtoCalc All con on the System Model Screen toolbar comptes the modle otpt vale and total ncertanty. If the error sorces selected for nclson n the modle analyss are ndependent (.e., ncorrelated), then the error sorce ncertantes are combned n a root-sm-sqare (RSS) manner to obtan the Total Modle Uncertanty. Otherwse, the Total Modle Uncertanty wll reflect any correlatons between error sorces specfed by the ser va the Correlaton Analyss Screen. The System Modle Analyss table provdes a smmary table for the system modles. The smmary nformaton for or gtal Temperatre Measrement System analyss s shown below. Modle Name Inpt Otpt Uncertanty Unts Coeffcent Thermocople Sensor μv Thermocople - Flter Interface μv owpass Flter μv Flter - Amplfer Interface μv Amplfer , μv Amplfer - A/ Converter 8,889. 8, μv A/ Converter 8, , μv ata Processor 8, C. The last colmn of the table contans a coeffcent ndcatng the contrbton made by a gven modle to the total system otpt. For ths example, all modles shold have a coeffcent eqal to., except the last modle whch has a vale of System Otpt Uncertanty The system otpt vale s eqal to the compted otpt for the fnal modle n the seres. Each modle otpt calclaton s dependent pon ts precedng modle. Conseqently, the system 6-9

137 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 otpt vale wll change ntl all necessary parameter nformaton has been entered and assocated calclatons completed for each modle. Smlarly, the total system ncertanty s eqal to the otpt ncertanty for the fnal modle. The assocated degrees of freedom for the total system ncertanty s also eqal to the degrees of freedom for the fnal modle otpt ncertanty. The analyss reslts for or gtal Temperatre Measrement System analyss s shown below. System Inpt. System Otpt.8 C Total System Uncertanty.46 deg C egrees of Freedom nfnte The Total System Uncertanty of.46 deg C s the ncertanty of the system otpt at C. We can se ths ncertanty vale to tolerance the gtal Temperatre Measrement System. The tolerance lmts for the system are compted from the degrees of freedom and ser specfed confdence level. For example, f we wanted to tolerance the system to 95% confdence lmts, the correspondng tolerance lmts wold be ±.9 deg C. Alternatvely, f we wanted to tolerance the system sng the NIST conventon of mltplyng the ncertanty by a factor of, the tolerance lmts wold be ±.9 deg C wth a confdence level of 95.45%. Ether approach s acceptable. For ths example, we wll se the NIST approach, wrtng as a specfcaton for the measrements n the C range T ot T n ±.9 o C, coverage factor of.. The amont of tme that ths tolerance s applcable depends on the stablty of the varos parameters of the system. We can repeat the analyss process for other npt temperatres. For example, sppose that we were to repeat the or analyss for npt temperatres of C, 4 C, 6 C, 8 C, and C. All parameter specfcatons are the same as prevosly stated, except the thermocople senstvty (Seebeck Coeffcent), whch s specfed for each temperatre as follows: 6-3

138 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Inpt Temp. Senstvty (p) ± Error mts % Confdence C 4.33 µv/ C.76 µv/ C C µv/ C.56 µv/ C C 4.47 μv/ C.9 μv/ C C 4.54 µv/ C. µv/ C 99. C 4.37 µv/ C.3 µv/ C 99. The analyss reslts are smmarzed below. Inpt Temp. Otpt Temp. Otpt Uncertanty ± Tolerance mts C 9.8 C.4 C.84 C 4 C 4. C.8 C.6 C 6 C 6.9 C. C.44 C 8 C 8.4 C.63 C 3.6 C C.3 C. C 4.4 C 6-3

139 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Ths page ntentonally left blank. 6-3

140 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 7. General CHAPTER 7 UNCERTAINTY GROWTH ESTIMATION The error or bas n a sbject parameter may grow wth tme or may reman constant. In some cases, t may even shrnk. The ncertanty n ths error, however, always grows wth tme snce measrement or calbraton. Ths s the fndamental postlate of ncertanty growth. Ths chapter dscsses the methodology sed to project ncertanty growth n the sbject parameter bas. 5 Fgre 7- llstrates ncertanty growth over tme for a typcal measrement attrbte or parameter. The seqence shows the statstcal dstrbton at three dfferent tmes, wth the ncertanty growth reflected n the spreads n the crves. The ot-of-tolerance probabltes at the dfferent tmes are represented by the shaded areas nder the crves. Fgre 7-. Measrement Uncertanty Growth. The growth n ncertanty over tme corresponds to an ncrease n ot-of-tolerance probablty over tme. Conversely, t corresponds to a decrease n n-tolerance probablty, or measrement relablty over tme. Plottng ths qantty verss tme, as shown n Fgre 7-, sggests that measrement relablty can be modeled by a tme-varyng fncton. Once ths fncton s determned, the ncertanty n the bas of a parameter can be compted as a fncton of tme. 5 The methodology presented heren was developed by r. H. Castrp of Integrated Scences Grop (see References). 7-

141 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 7-. Measrement Relablty verss Tme. 7. efntons 7.. AOP Relablty. The n-tolerance probablty for a parameter averaged over ts calbraton or test nterval. The average-over-perod (AOP) measrement relablty s often sed to represent the n-tolerance probablty of a parameter for a measrng tem whose sage demand s random over ts test or calbraton nterval. 7.. Bas Reference. The tme to whch the n-tolerance probablty of a parameter s referenced BOP Relablty. The n-tolerance probablty for a parameter at the start or begnnng-of perod (BOP) of ts calbraton or test nterval Calbraton. Wth reference to ndstral and scentfc nstrments: ) To adjst the otpt of a devce, to brng t to a desred vale, wthn a specfed tolerance, for a partclar vale of an npt; ) to ascertan the error n the otpt of a devce by measrng or comparng aganst a standard Calbraton Interval. The elapsed tme between sccessve calbratons of a gven eqpment parameter or attrbte EOP Relablty. The n-tolerance probablty for a parameter at the end of ts calbraton or test nterval (.e., end-of-perod) Relablty Model. In ncertanty analyss measrement decson rsk analyss and calbraton nterval analyss, a relablty model s a mathematcal fncton whose characterstcs are based on a parameter's calbraton hstory. Used to project ncertanty growth over tme. 7-

142 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Uncertanty Growth. The ncrease n the ncertanty n the vale of a parameter or other attrbte over the tme elapsed snce measrement. 7. Basc Methodology UncertantyAnalyzer ses the combned ncertanty at the tme of measrement and the relablty model nformaton to estmate the ncertanty n the sbject parameter vale at the specfed tme snce measrement. The ncertanty, (t), n the vale of a sbject parameter at tme t elapsed snce measrement (t ) s compted sng the vale of the ntal measrement ncertanty, (), and the relablty model for the parameter poplaton. The basc concept s an extenson of the ergodc theorem that states that the dstrbton of an nfnte poplaton of vales at eqlbrm s dentcal to the dstrbton of vales attaned by a sngle member sampled an nfnte nmber of tmes. The relablty model predcts the n-tolerance probablty for the sbject parameter poplaton as a fncton of tme elapsed snce measrement. It can be thoght of as a fncton that qantfes the stablty of the poplaton. In ths vew, we begn wth a poplaton n- tolerance probablty at tme t (mmedately followng measrement) and extrapolate to the ntolerance probablty at tme t >. The relablty (n-tolerance probablty) of the sbject parameter at tme t s related to the parameter's ncertanty accordng to R ( t) f [ x ( t)] dx, (7-) where f [x (t)] s the probablty densty fncton for the parameter and x represents devatons from the parameter s desgn or nomnal vale. For prposes of dscsson, we wll assme for the moment that the sbject parameter pdf s gven by ( x μ ) / ( ) [ ( )] t f x t e. (7-) π ( t) We state that at a gven tme t, the sbject parameter s tre devaton from nomnal s gven by the relaton μ ( t ) μ b( t), (7-3) where b(). The relatonshp between, and µ s shown n Fgre 7-3, along wth the dstrbton of the poplaton of bases for the sbject parameter of nterest. 7-3

143 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 7-3. Sbject Parameter strbton. At the tme of measrement (t ), we estmate a vale for μ and label the ncertanty n ths estmate (). The remander of ths note dscsses a method for calclatng (t), gven (). 7.3 Projected Uncertanty If we had at or dsposal the relablty model for the ndvdal measred parameter, gven ts ntal ncertanty, we cold obtan the ncertanty (t) n eqaton (7-) drectly by teraton or other means. However, we sally have nformaton that relates only to the characterstcs of the relablty model to whch the sbject parameter belongs. We apply the relablty model for the poplaton to the ndvdal parameter nder consderaton. In UncertantyAnalyzer, sng a poplaton relablty model to estmate ncertanty growth for a parameter employs the followng set of premses:. The reslt of a parameter measrement s an estmate of a parameter s vale or bas. Ths reslt s accompaned by an estmate of the ncertanty n the parameter s bas.. The ncertanty of the measred parameter s bas or vale at tme t (mmedately followng measrement) s the estmated ncertanty of the measrement process. 3. The bas or vale of the measred parameter s ether normally dstrbted or t-dstrbted arond the measrement reslt. 4. The stablty of the parameter s eqated to the stablty of ts poplaton. Ths stablty s represented by the poplatons relablty model. 5. Therefore, the ncertanty n the parameter s vale or bas grows from ts vale at t n accordance wth the relablty model of the parameter s poplaton. The relablty fncton at tme t elapsed snce measrement s gven by R( t) f [ x μ ( t), ( t)] dx, (7-4) A where A s the "acceptance" regon for vales of x. For example, f the parameter s two-sded, wth tolerance lmts and, then A s comprsed of all vales of x sch that x, and R(t) s as gven n eqaton (7-). 7-4

144 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The relablty fncton at tme t s lkewse gven by R( ) f [ x μ, ()] dx. (7-5) A The frst step n estmatng (t) s to form the rato R( t) R() A A f [ x μ( t), ( t)] dx f [ x μ, ()] dx. (7-6) Next, we employ relabltes R(t) and R(), compted for the poplaton of tems represented by the dstrbton for x and solve for (t). However, for consstency on both sdes of the eqaton, we need to average the nmerator and denomnator on the rght hand sde over µ and µ, respectvely. Snce these qanttes are errors n x, ther expectaton vale s zero, and we have the general relaton R( t) R() A A f [ x ( t)] dx. f [ x ()] dx 7.3. Normally strbted Parameter Vales. Errors or bases are assmed to follow the normal dstrbton n cases where the degrees of freedom assocated wth the ncertanty estmate s nfnte. (7-7) General Two-Sded Cases. For parameters wth two-sded tolerance lmts, the relablty fncton at t s gven by R() () e π Φ x /, dx (7-8) where (). The relablty at tme t > s gven by R ( t) Φ Φ. ( ) ( ) (7-9) t t To solve for (t ), we take rato of eqaton (7-9) to eqaton (7-8) 7-5

145 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ) ( ) ( () ) ( Φ Φ Φ t t R t R (7-) where R() and R(t) are determned sng the relablty model. In ths eqaton,, R(t) and R() are known. To pt eqaton (7-) n a form that amenable to solton, we wrte. () ) ( ) ( ) ( R t R t t Φ Φ Φ Φ (7-) The ncertanty (t) s obtaned throgh teraton. The teraton process employs the Newton-Raphson method. In ths method, a fncton H and ts dervatve H' are defned accordng to ) ( ) ( x x H ρϕ ϕ and ), ( ' ' x H ϕ where ) ( ) ( ) ( Φ Φ x x x ϕ, [ ] / ) ( / ) ( ) '( x x e e x μ μ π ϕ, ) ( / t x, / x, () ) / ( R t R ρ. The teratons begn wth a startng vale for x of Φ ) ( ) ( x x ρϕ, where. Startng wth ths vale, the teraton proceeds accordng to ( ) /. ' / ) ( ) ( H H x x r r The teratons stop when the magntde of H / H shrnks to some predetermned vale ε << Symmetrc Two-Sded Cases. In cases where, eqaton (7-) becomes ρ Φ Φ ) ( t, (7-) 7-6

146 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 and ( t). (7-3) Φ { [ Φ( / ) ] } ρ Sngle-Sded Cases. In cases where tolerances are sngle-sded, (t) can be determned wthot teraton. In these cases, ether or s nfnte, and eqaton (7-) becomes Φ Φ ρ, (7-4) ( t) o where s eqal to - for sngle-sded lower cases and eqal to for sngle-sded pper cases. Solvng for (t) yelds ( t). (7-5) Φ Φ ρ For example, f R( t) λt R e, then ( t). Φ Φ λ { ( / ) e t } 7.3. Stdent's t-strbted Parameter Vales. For ncertanty estmates () whose degrees of freedom s fnte, the applcable dstrbton s the Stdent s t dstrbton General Tw-Sded Cases. The treatment s the same as for normally dstrbted vales, except that the dstrbton n eqaton (7-8) s the t dstrbton. where R( t) C C v / ( t) v / ( t ) ( ) ( v x / v ) / dx, (7-6) v Γ, (7-7) v Γ πv and v the degrees of freedom for the ncertanty estmate. In evalatng eqaton (7-6), we defne a fncton F accordng to F ( K) C v K ( x / v) v ( v ) / dx. (7-8) We wll also employ the fncton F ν (-K), whch, after a lttle manplaton, can be wrtten 7-7

147 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ) ( ) ( K F K F v v. (7-9) The Stdent's t dstrbton, denoted here by the fncton G v s sally expressed as the probablty of lyng wthn symmetrc lmts K ± (7-) ( ) ( ), ) ( / ) ( / K F dx v x C K G v K K v v v so that [ ) ( ) ( K G K F v v ]. (7-) Usng eqatons (7-) and (7-) n eqaton (7-6) gves () (). ) ( ) ( ) ( t G t G t F t F t R v v v v (7-) Note that, G ν ( ), so that sngle-sded cases are descrbed by ) ( ) ( t G t R v, (sngle-sded cases). (7-3) As n eqaton (7-), we have (), ) ( () ) ( G G t G t G R t R v v v v ρ and (). ) ( ρ G G t G t G v v v v (7-4) The Newton-Raphson solton for (t) s obtaned analogosly to the solton for normally dstrbted cases. For Stdent's t dstrbtons, ) ( ) ( x x H v v ρϕ ϕ and ) '( ' x H ϕ, where 7-8

148 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ( ) ( ) x G x G x v v ) ( ϕ, ) ( / t x, / x, and. ( ) ( ) ( ) ( ) [ ] / / / / ) '( v v v v x v x C x ϕ The ntal vale for x s set at ( ) [ ] ρ ) ( / G G x v v, or ( ) ) ( ρ / G F x v v, where s defned as before, and s the nverse t dstrbton fncton. F v Symmetrc Two-Sded Cases. In cases where, eqaton (7-4) becomes ρ ) ( G t F v v, (7-5) and o v v G F t ρ ) (. (7-6) Sngle-Sded Cases. In cases where tolerances are sngle-sded, (t) can be determned analytcally. In these cases, ether or n nfnte, and eqaton (7-4) becomes ρ ) ( G t F v v, (7-7) where s eqal to for sngle-sded lower cases and eqal to for sngle-sded pper cases. Solvng for (t) yelds ) ( G F t v v ρ. (7-8) 7-9

149 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Relablty Models In the ncertanty growth projecton process, we tlze nformaton abot the calbraton hstory of the sbject parameter to develop a relablty model. Ths relablty model provdes a means for determnng how the sbject parameter bas ncertanty grows wth tme snce calbraton. Ths means that the ncertanty srrondng the measred vale we report wll ncrease wth tme ntl the next calbraton. If we have access to a relablty modelng applcaton, we can dentfy the approprate relablty model and acqre the model s characterstcs and enter ths nformaton nto UncertantyAnalyzer s Sbject Parameter Relablty Model Worksheet. Alternatvely, we can enter an elapsed tme, a begnnng-of-perod (BOP) relablty and an end-of-perod (EOP) relablty for the calbraton nterval. For certan models, we mst also enter an average-over-perod (AOP) relablty. These vales apply to the sbject parameter's poplaton and are based on servce hstory records or engneerng knowledge. The Sbject Parameter Relablty Model Worksheet has eght relablty models to choose. Each model s defned by a mathematcal eqaton wth characterstc coeffcents. An applcable relablty model mst be chosen based on knowledge abot the stablty of the sbject parameter over tme. The applcaton of each of the relablty models avalable n UncertantyAnalyzer s descrbed below along wth nformaton needed to mplement each of them Exponental Model. The exponental relablty model s defned by the mathematcal eqaton R( t) ae bt (7-9) where R(t) s the n-tolerance probablty at tme t and a and b are the model coeffcents. The exponental model s sefl for parameters whose falre probablty s not a fncton of tme nterval T, begnnng at some tme t, s the same as the probablty of gong ot-of-tolerance n the same tme nterval T, begnnng at some other tme t'. To mplement the exponental model yo need to know ether of the followng:. The vale of the model coeffcents, a and b.. The begnnng of perod (BOP) n-tolerance probablty and the end of perod (EOP) n-tolerance probablty Mxed Exponental Model. The mxed exponental relablty model s defned by the mathematcal eqaton R( t) (7-3) b at b where R(t) s the n-tolerance probablty at tme t and a and b are the model coeffcents. The mxed exponental model s sefl for parameters whose ot-of-tolerance behavor depends on a nmber of consttent parameters, each of whch can be modeled wth the exponental model. 7-

150 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 To mplement the mxed exponental model yo need to know ether of the followng:. The vale of the model coeffcents, a and b.. The BOP and EOP n-tolerance probabltes Webll Model. The Webll relablty model s defned by the mathematcal eqaton R( t) ae where R(t) s the n-tolerance probablty at tme t and a, b and c are the model coeffcents. The Webll model s sefl for parameters that go ot-of-tolerance as a reslt of gradal wear or decay. To mplement the Webll model yo need to know ether of the followng: ( bt ) c. The vale of the model coeffcents, a, b and c.. The BOP and EOP n-tolerance probabltes and the average-over perod (AOP) ntolerance probablty Gamma Model. The gamma relablty model s defned by the mathematcal eqaton (7-3) bt ae R( t) 3 (7-3) ( bt) ( bt) bt 6 where R(t) s the n-tolerance probablty at tme t and a and b are the model coeffcents. The gamma model s sefl for parameters that go ot-of-tolerance n response to some nmber of events, sch as beng actvated and deactvated. To mplement the gamma model yo need to know ether of the followng:. The vale of the model coeffcents, a and b.. The BOP and EOP n-tolerance probabltes Mortalty rft Model. The mortalty drft relablty model s defned by the mathematcal eqaton ( ) ( bt ct R t ae ) (7-33) where R(t) s the n-tolerance probablty at tme t and a, b and c are the model coeffcents. The mortalty drft model s sefl for parameters that are characterzed by a slowly varyng ot-of-tolerance rate. To mplement the mortalty drft model yo need to know ether of the followng:. The vale of the model coeffcents, a, b and c.. The BOP, AOP, and EOP n-tolerances Warranty Model. The warranty relablty model s defned by the mathematcal eqaton R e (7-34) ( t) a( t b ) where R(t) s the n-tolerance probablty at tme t, and a and b are the model coeffcents. 7-

151 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 The warranty model s sefl for parameters that tend to stay n-tolerance ntl reachng a well-defned ct-off tme, at whch pont, they go ot-of-tolerance. To mplement the warranty model yo need to know ether of the followng:. The vale of the model coeffcents, a and b.. The BOP and EOP n-tolerance probabltes Random Walk Model. The random walk relablty model s defned by the mathematcal eqaton R( t) erf (7-35) a bt where R(t) s the n-tolerance probablty at tme t, and a and b are the model coeffcents. The random walk model s sefl for parameters whose vales flctate n a prely random way wth respect to magntde and drecton (postve or negatve). To mplement the random walk model yo need to know ether of the followng:. The vale of the model coeffcents, a and b.. The BOP and EOP n-tolerance probabltes Restrcted Random Walk Model. The restrcted random walk relablty model s defned by the mathematcal eqaton R( t) erf ct a b( e ) (7-36) where R(t) s the n-tolerance probablty at tme t, and a, b, and c are the model coeffcents. The restrcted random walk model s smlar to the random walk model, except that parameter flctatons are confned wthn a restrcted regon arond a mean or nomnal vale. To mplement the restrcted random walk model yo need to know ether of the followng:. The vale of the model coeffcents, a, b, and c.. The BOP, AOP, and EOP n-tolerances. 7-

152 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 CHAPTER 8 BAYESIAN ANAYSIS STATISTICA MEASUREMENT PROCESS CONTRO (SMPC) 8. General Ths chapter dscsses the prncples and methods sed n Bayesan or SMPC analyss. 6 SMPC s a way of estmatng the vale of a sbject parameter based on measrements made by a measrng parameter. SMPC also provdes an estmate of the vale of the measrng parameter, based on the same measrements. In dong so, SMPC formally recognzes that measrng parameters are not perfect. Uncertanty analyss n general acknowledges ths fact by attemptng to estmate the ncertanty assocated wth measrements. A typcal reslt of an ncertanty analyss s a statement of a measred vale accompaned by an estmate of the ncertanty n the vale. Whle the ncertanty estmate stands as an admsson that ncertanty exsts n a measrement, the measred vale or a statstcal mean of ths vale s sally taken at face vale. Ths apparent contradcton s especally nterestng when the sbject parameter s a toleranced qantty, as n the case of calbratng and testng. To see ths, magne that we arbtrarly measre a very accrate sbject parameter (.e., one wth tght tolerances and a hgh n-tolerance probablty) wth a moderately accrate measrng parameter. Obvosly, no reasonable person wold presme that the measred vale or the mean of a sample of measred vales provdes a good ndcaton of the bas n the sbject parameter. Instead, one wold be nclned to trn thngs arond and se the measrement reslt as an ndcaton of the bas n the measrng parameter. If we dd ths, we wold be takng a step toward nderstandng SMPC. 8. efntons 8.. a posteror vale. The vale calclated after takng measrements. 8.. a pror vale. The vale ndcated pror to takng measrements Estmated Tre Vale. The vale of a qantty obtaned by SMPC (Bayesan) analyss. 8. SMPC Methodology The fndamental prncple that s central to SMPC analyss states that: In measrement statons where we have a pror knowledge of measrng parameter and sbject parameter accraces, the roles of measrng parameter and sbject parameter are reversble. Ths s called the Prncple of Measrement Symmetry. A pror knowledge s the nformaton we have abot the measrng and sbject parameter before any measrements are made. A posteror estmates are estmates compted after takng measrements. As one wold expect, SMPC analyss s senstve to all measrement error 6 SMPC methods and concepts presented heren were developed by r. H. Castrp of Integrated Scences Grop (see References. 8-

153 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 sorces, not jst parameter bas. Therefore, SMPC analyss s most effectve when all measrement errors and ther assocated ncertantes are also estmated. By ncldng measrement ncertanty, yo are provdng addtonal nformaton from whch to revse or refne the SMPC estmates. The SMPC method derves n-tolerance probabltes and attrbte bases for both a nt nder test (UUT) and a set of ndependent test and measrng nstrments (TME). The dervaton of these qanttes s based on measrements of a UUT attrbte vale made by the TME set and on certan nformaton regardng UUT and TME attrbte ncertantes. The method accommodates arbtrary accracy ratos between TME and UUT attrbtes and apples to TME sets comprsed of any nmber of nstrments. To mnmze abstracton of the dscsson, the treatment focses on restrcted cases n whch both TME and UUT attrbte vales are normally dstrbted and are mantaned wthn twosded symmetrc tolerance lmts. Ths shold serve to make the mathematcs more concrete and more palatable. espte these mathematcal restrctons, the methodologcal framework s entrely general. Extenson to cases nvolvng one-sded tolerances and asymmetrc attrbte dstrbtons merely calls for more mathematcal brte force. 8.. Comptaton of UUT In-tolerance Probablty. Whether a UUT provdes a stmls, ndcates a vale, or shows an nherent property, the declared vale of ts otpt, ndcated vale, or nherent property, s sad to reflect some nderlyng tre vale. A freqency reference s an example of a stmls, a freqency meter readng s an example of an ndcated vale, and a gage block dmenson s an example of an nherent property. Sppose for example that the UUT s a voltmeter measrng a (tre) voltage of. mv. The UUT meter readng (. mv or 9.99 mv, or some sch) s the UUT s declared vale. As another example, consder a 5 cm gage block. The declared vale s 5 cm. The nknown tre vale (gage-block dmenson) may be 5. cm, or cm, or some other vale. The UUT declared vale s assmed to devate from the tre vale by an nknown amont. et Y represent the UUT attrbte s declared vale and defne a random varable ε as the devaton of Y from the tre vale. The varable ε s assmed a pror to be normally dstrbted wth zero mean and varance σ. The tolerance lmts for ε are labeled ±,.e., the UUT s consdered n-tolerance f - ε. A set of n ndependent measrements are also taken of the tre vale sng n TME. et Y be the declared vale representng the th TME s measrement. The observed dfferences between UUT and TME declared vales are labeled accordng to X Y Y,,, n (8-), where the qanttes X are assmed to be normally dstrbted random varables wth varances σ and mean ε. esgnatng the tolerance lmts of the th TME attrbte by ±, the th TME s consdered ntolerance fε X ε. In other words, poplatons of TME measrements are not expected to be systematcally based. Ths s the sal assmpton made when TME are chosen ether randomly from poplatons of lke nstrments or when no foreknowledge of TME bas s avalable. Indvdal nknown TME bases are assmed to exst. Accontng for ths bas s done by treatng ndvdal nstrment bas as a random varable and estmatng ts varance. In applyng SMPC methodology, we work wth a set of varables r, called dynamc accracy 8-

154 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ratos (or dynamc nverse ncertanty ratos) defned accordng to σ r,,,, n σ (8-) The adjectve dynamc wll dstngsh these accracy ratos from ther sal statc or nomnal conterparts, defned by /,,,, n. The se of the word dynamc nderscores the fact that each r defned by eqaton (8-) s a qantty that changes as a fncton of tme passed snce the last calbratons of the UUT and the th of the TME. The dynamc character exsts becase generally both UUT and TME poplaton standard devatons (bas ncertantes) grow wth the tme snce calbraton. et P be the probablty that the UUT s n-tolerance at some gve tme snce calbraton. Usng these defntons, we can wrte () ( ) P F( a ) F a, (8-3) where F s the dstrbton fncton for the normal dstrbton defned by and where F a ( a ) ± ± π e ζ / dζ, (8-4) X r r ± r a ±. (8-5) σ In these expressons and n others to follow, all smmatons are taken over,,, n. The dervaton of eqatons (8-3) and (8-5) s presented n Secton Note that the tme dependence of P s n the tme dependence of a and a -. The tme dependence of a and a - s, n trn, n the tme dependence of r. 8.. Comptaton of TME In-Tolerance Probablty. Jst as the random varables X, X,, X n are TME-measred devatons from the UUT declared vale, they are also UUT-measred devatons from TME declared vales. Therefore, t s easy to see that by reversng ts role, the UUT can act as a TME. In other words, any of the n TME can be regarded as the UUT, wth the orgnal UUT performng the servce of a TME. For example, focs on the th (arbtrarly labeled) TME and swap ts role wth that of the UUT. Ths reslts n the followng transformatons: X X X ' ' ' n X X X ' M X n X X M X X, 8-3

155 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where the prmes ndcate a redefned set of measrement reslts. Usng the prmed qanttes, the n-tolerance probablty for the th TME can be determned jst as the n-tolerance probablty for the UUT was determned earler. The process begns wth calclatng a new set of dynamc accracy ratos. Frst, we set ' ' ' ' ' σ σ, σ σ, σ σ,, σ σ, σ σ. Gven these label reassgnments, the needed set of accracy ratos can be obtaned sng eqaton (8-),.e., ' ' r σ / σ, ',,, n. Fnally, the tolerance lmts are relabeled for the UUT and the th TME accordng to ' ' and. If we desgnate the n-tolerance probablty for the th TME by P and we sbsttte the prmed qanttes obtaned above, eqatons (8-3) and (8-5) become and ' ' ( a ) F( ) P, F a ' ' X r ' r a. ' ' r ± ' ± ' σ Applyng smlar transformatons yelds n-tolerance probabltes for the remanng n- TME Varance n Parameter Bas. Comptng the ncertantes n UUT and TME attrbte bases nvolves establshng the relatonshp between attrbte ncertanty growth and tme snce calbraton. Several models have been sed to descrbe ths relatonshp (see Secton B.9). To llstrate the comptaton of bas ncertantes, the smple negatve exponental model wll be sed here. Wth the exponental model, f t represents the tme snce calbraton, then the correspondng n-tolerance probablty R(t) s gven by λt ( ) R( ) e R t n n, (8-6) where the parameter λ s the ot-of-tolerance rate assocated wth the nstrment n qeston, and R() s the n-tolerance probablty mmedately followng calbraton. Wth the exponental model, for a gven end-of-perod n-tolerance target, R *, the parameters λ and R() determne the calbraton nterval for a poplaton of nstrment attrbtes accordng to t R t exp ln λ T R * ( ) Rearrangng eqaton (8-7) and sbstttng n eqaton (8-6) gves. (8-7) 8-4

156 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 t R R ( t) R( ) exp ln. (8-8) T R * ( ) For an nstrment attrbte whose acceptable vales are bonded wthn tolerance lmts ±, the n-tolerance probablty can also be wrtten, assmng a normal dstrbton, as (, (8-9) ζ / σ b R t) e dζ πσ b where σ s the expected varance of the attrbte bas at tme t. Eqatng eqaton (8-9) to eqaton (8-8) and rearrangng yelds the attrbte bas standard devaton () σ b, (8-) * t R F R( ) exp ln ( ) T R where F s the nverse of the normal dstrbton fncton defned n eqaton (8-4). * Sbstttng, T,,, and R,,,, n, n eqaton (.) for, T, t, R(), and t ( ) R R * yelds the desred nstrment bas standard devatons. The varable calbraton of the UUT ( ) or of the th TME (,,, n). t s the tme passed snce 8..4 Accontng for Bas Flctatons. Each attrbte bas standard devaton s a component of the ncertanty n the attrbte s vale. Bas ncertanty represents long-term growth n ncertanty abot or knowledge of attrbte vales. Sch ncertanty growth arses from random and/or systematc processes exerted over tme. Another component of ncertanty stems from sch ntermedate-term processes as those assocated wth ancllary eqpment varatons, envronmental cycles, and drnal electrcal power level cycles. Uncertanty contrbtons de to ntermedate-term random varatons n attrbte vales sally mst be estmated herstcally on the gronds of engneerng expectatons. In the parlance of the ISO GUM, sch estmates are called Type B ncertantes. Yoden, for example, provdes a graphcal method for qaltatvely evalatng contrbtons from hman factors, laboratory processes, and reference standards. evelopment of a qanttatve method s a sbject of crrent research. For now, herstc estmates are sally the best avalable. Herstc estmates shold represent pper bond (.e., 3σ ) one-sded lmts for process ncertanty magntdes. Experenced metrologsts can often provde reasonable gesses for these lmts. If we denote pper bonds for herstcally estmated contrbtons by δ,,,..., n, the correspondng 3σ standard devaton s gven by σ δ / 3. (8-) δ Treatment of Mltple Measrements. In prevos dscssons, the qanttes X are treated as sngle measrements of the dfference between the UUT attrbte and the th TME s attrbte. Yet, n most applcatons, testng or calbraton of workload tems s not lmted to 8-5

157 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 sngle measrements. Instead, mltple measrements are sally taken. Instead of n ndvdal measrements, we wll ordnarly be dealng wth n sets or samples of measrements. In these samples, let n be the nmber of measrements taken sng the th TME s attrbte, and let X j Y Y j be the j th of these measrements. The sample mean and standard devaton are gven n the sal way: and s X n n n j X j n ( X j X ) j (8-). (8-3) The varance assocated wth the mean of measrements made sng the th TME s attrbte s gven by σ σ s n σ, b / δ where the varables σ b and σ δ are the long-term and ntermedate-term attrbte bas standard devatons, respectvely, as defned n Secton The sqare root of ths varance wll determne the qanttes r defned n eqaton (8-). Note that ncldng sample varances s restrcted to the estmaton of TME attrbte varances. UUT attrbte varance estmates contan only the terms σ b and σ δ. Ths nderscores what s soght n constrctng the pdf f ( ε X). What we seek are estmates of the n-tolerance probablty and bas of the UUT attrbte. In ths, we are nterested n the attrbte as an entty dstnct from process ncertantes nvolved n ts measrement. It s mportant to keep these consderatons n mnd when the UUT and the th TME swtch roles. What we are after n that event s nformaton on the attrbte of the th TME as a dstnct entty. Therefore, the stable transformatons are σ ' σ b σ δ σ ' ' ' n M σ M σ σ s b b bn σ s σ s n / n / n / n n σ σ δ δ σ δ n. (8-4) Other expressons are the same as those sed n treatng sngle measrement cases. The relatonshp of ncertanty varables to one another s shown n Fgre

158 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 8-. Measrement Uncertanty Components. The standard devaton σ b provdes an ndcaton of the ncertanty n the bas of the th nstrment s attrbte. The varable σ δ s a herstc estmate of the standard devaton assocated wth ntermedate-term random flctatons n ths bas. The varable s represents the short-term process ncertanty accompanyng measrements made wth the th nstrment s attrbte ervaton of Eqaton (8-3). et the vector X represent the random varables X, X,..., Xn obtaned from n ndependent TME measrements of ε. We seek the condtonal pdf for ε, gven X, that wll, when ntegrated over [-, ], yeld the condtonal probablty P that the UUT s n-tolerance. Ths pdf wll be represented by the fncton f (ε X). From basc probablty theory, we have where ) f ( ε ) ( X) f ( X ε f ( ε X), (8-5) f 8-7

159 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 ε / σ ( ε ) e. f (8-6) πσ ( In eqaton (8-5), the pdf f Xε ) s the probablty densty for observng the set of measrements X, X,, X n, gven that the bas of the UUT sε. The pdf f( ε ) s the probablty densty for UUT bases. Snce the components of X are statstcally ndependent, we can wrte where f f ( ε ) f ( X ε ) f ( X ε ) f ( X ) n ε X, (8-7), ( X ) ( X ) e ε / σ ε,,, n. (8-8) πσ Note that eqaton (8-8) states that, for the present dscsson, we assme the measrements ofε to be normally dstrbted wth a poplaton mean vale ofε (the UUT "tre" vale) and a standard devaton σ. At ths pont, we do not provde for an nknown bas n the th TME. 7 As we wll see, the SMPC methodology wll be sed to estmate ths bas, based on the reslts of measrement and on estmated measrement ncertantes. Combnng eqatons (8-5) throgh (8-8) gves f ( Xε ) f ( ε ) n ε C exp σ C exp ε σ Ce G ( X ε ) ( X ε ) ( ) exp ( ) X r X, r ε σ r n r σ where C s a normalzaton constant. The fncton G(X) contans noε dependence and ts explct form s not of nterest n ths dscsson. (8-9) The pdf f(x) s obtaned by ntegratng eqaton (8-9) over all vales ofε. To smplfy the notaton, we defne and α (8-) Σr ΣX r β. (8-) Σr 7 It can be readly shown that, f the bas of a TME s nknown, the best estmate for the poplaton of ts measrements s the tre vale beng measred,.e., zero bas. Ths s an mportant a pror assmpton n applyng the SMPC methodology. 8-8

160 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Usng eqatons (8-) and (8-) n eqaton (8-9) and ntegratng overε gves f G( X) α ( ε β ) ( X) Ce e Ce G ( X) πσ α. / σ dε (8-) vdng eqaton (8-) nto eqaton (8-9) and sbstttng n eqaton (8-5) yelds the pdf ( ε ) ( ε β ) / ( σ α ) / f X e. (8-3) π ( σ / α ) As we can see, ε condtonal on X s normally dstrbted wth mean β and standard devatonσ / α. The n-tolerance probablty for the UUT s obtaned by ntegratng eqaton (8-3) over [-, ]. Wth the ad of eqaton (8-5), ths reslts n P F π π ( a ) F( a ( σ / α ) ( β )/( σ / α ) ζ / ( β )/( σ / α ) F( a ) F e ) ( a ), e ( ε β ) / ( σ / α ) whch s eqaton (8-3) wth α and β as defned n eqatons (8-) and (8-) Estmaton of Bases. Obtanng the condtonal pdf f ( ε X) allows s to compte moments of the UUT attrbte dstrbton. Of partclar nterest s the frst moment, or dstrbton mean. The UUT dstrbton mean s the condtonal expectaton vale for the bas ε. Ths, the UUT attrbte bas s estmated by β E ( ε X) ε f dζ ( ε X) dε. Sbstttng from eqaton (8-3) and sng eqaton (8-) gves Σr dε (8-4) ΣX r β. (8-5) 8-9

161 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Smlarly, bas estmates can be obtaned for the TME set by makng the transformatons descrbed n Secton 8..; for example, the bas of TME s gven by ΣX β E( ε X). (8-6) Σ To exemplfy bas estmaton, let se consder a profcency adt qeston n whch three dfferent pressre measrng nstrments are sed to measre a, ps nomnal pressre sorce. All three nstrments have a specfed tolerance of ± ps. Instrment reads ps dfference from nomnal (Y ), nstrment reads 6 ps hgher than nomnal (Y 6), and nstrment 3 reads 5 ps hgher than nomnal (Y 5). In ths example, nstrment s desgnated as the UUT, nstrment as TME and nstrment 3 as TME. For smplcty, we set R(), and bas flctaton and process ncertantes eqal to zero. Ths, and X X Y 6 Y Y 5, Y r r. Unless otherwse shown, we can assme that the n-tolerance probabltes for all three nstrments are abot eqal to ther average-over-perod vales. For the nstrments sed n the profcency adt, t was determned that the poplaton ncertanty s managed to acheve an n-tolerance probablty of R *.7 at the end of the calbraton nterval. We assme that we can se average-over-perod n-tolerance probabltes for R(t) n ths example. Wth the exponental model, f R ( ), the average n-tolerance probablty s roghly eqal to the n-tolerance probablty halfway throgh the calbraton nterval. Ths, settng t T / n Eqaton (8-) yelds σ F F / ' ' r ' r exp ln.7 (.9) 8-

162 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Sbstttng n the expresson for a± above gves a ± ( m 7) ±.74. Ths, the n-tolerance probablty for the UUT (nstrment ) s (.75) ( 4.3) P F F To compte the n-tolerance probablty for TME (nstrment ), the UUT and TME swap roles. By sng the transformatons of Secton 8.., we have X X ' ' X 6 X 9 X ' n place of X and X n Eqaton (8-5). Recallng that σ a ' ± ( ) 3( ± ) ±.5. σ 6 9 ± ' σ ( ) n ths example gves Ths, by Eqaton (8-3), the n-tolerance probablty for TME (nstrment ) s P F(.4) F (.73) In comptng the n-tolerance probablty for TME, the UUT and TME swap roles. Ths X X ' ' X 5 9. X X ' Usng these qanttes n Eqaton (8-5) and settng σ σ gves ' a ±.49 ±

163 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Ths, by Eqaton (8-3), the n-tolerance probablty for TME (nstrment 3) s ( 4.47) (.5) P F F By smmarzng these reslts, we estmate a roghly 77% n-tolerance probablty for nstrment, a 99% n-tolerance probablty for nstrment, and a 69% n-tolerance probablty for nstrment 3. As prevosly stated, all three nstrments are managed to the same R * target, have the same tolerances, and are calbrated n the same way sng the same eqpment and procedres. Therefore, ther standard devatons when the measrements were made shold be abot eqal. By sng eqatons (8-5) and (8-6) and by recallng that σ σ σ, we get 6 5 Instrment (UUT) bas: β ( ) 7 Instrment (TME ) bas: Instrment 3 (TME ) bas: 6 9 β ( ) ( ) 5 9 β 8. If desred, these bas estmates cold serve as correcton factors for the three nstrments. If sed n ths way, the qantty 7 wold be added to all measrements made wth nstrment. The qantty wold be added to all measrements made wth nstrment. And, the qantty 8 wold be sbtracted from all measrements made wth nstrment 3. 8 Note that all bases are wthn the stated tolerance lmts (±) of the nstrments, whch mght encorage sers to contne to operate ther nstrments wth confdence. However, the compted n-tolerance probabltes showed only a 77% chance that nstrment was ntolerance and an even lower 69% chance that nstrment 3 was n-tolerance. Sch reslts tend provde valable nformaton from whch to make cogent jdgments regardng nstrment dsposton Bas Confdence mts. Another varable that can be sefl n makng decsons based on measrement reslts s the range of the confdence lmts for the estmated bases. Estmatng confdence lmts for the compted bases β and β,,,, n, means frst determnng the statstcal probablty densty fnctons for these bases. From eqaton (8-5) we can wrte 8 Snce all three nstrments are consdered a pror to be of eqal accracy, the best estmate of the tre vale of the measrand wold be the average of the three measred devatons: ε ( 6 5) / 3 7. Ths, a zero readng wold be ndcatve of a bas of -7, a 6 readng wold be ndcatve of a bas of -, and a 5 readng wold be ndcatve of a bas of 8. These are the same estmates we obtaned wth SMPC. Obvosly, ths s a trval example. Ths become more nterestng when each measrement has a dfferent ncertanty,.e., when σ σ σ. 8-

164 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 where β n c X, (8-7) r c. (8-8) Σr where Wth ths conventon, the probablty densty fncton of β can be wrtten: f ( β ) f ( Σc X ) f ( ΣΨ ), (8-9) Ψ c X. (8-3) Althogh the coeffcents c,,,..., n, are n the strctest sense random varables, to a frst approxmaton, they can be consdered fxed coeffcents of the varables X. Snce these varables are normally dstrbted (see eqaton (8-8)), the varables ψ are also normally dstrbted. The approprate expresson s where and ( ) ( Ψ η )/ σψ f Ψ e, (8-3) πσ Ψ σ σ (8-3) Ψ c η cε. (8-33) where Snce the varablesψ are normally dstrbted, ther lnear sm s also normally dstrbted: f ( Σψ ) e πσ e πσ f ( β ), ( ΣΨ η )/ σ ( β η ) / σ (8-34) σ Σσ ψ, (8-35) and η Ση. (8-36) 8-3

165 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Eqaton (8-34) can be sed to fnd the pper and lower confdence lmts for β. enotng these lmtsd by and, f the desred level of confdence s p x % then β β or β p f ( Β) dβ, β β β ) dβ ( p) / f ( Β) f ( Β dβ. Integratng eqaton (8-34) from β to and sng eqatons (8-35) and (8-36) yelds and Solvng for β gves β η F p σ β η F σ p ( )/ ( )/. p β η σf. (8-37) Solvng for the lower confdence for β n the same manner, we begn wth Ths yelds, wth the ad of eqaton (8-4), β f β ) dβ ( p) /. ( Usng the followng property of the normal dstrbton we can rewrte eqaton (8-38) as where β η F ( p) /. (8-38) σ F F ( x) F( x), β η ( p) / σ ( p) /, 8-4

166 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 p β η σf. (8-39) From eqaton (8-34), the parameter η s seen to be the expectaton vale for β. Or best avalable estmate for ths qantty s the compted UUT bas, namely β tself. We ths wrte the compted pper and lower confdence lmts for β as ± p β β ± σf. (8-4) In lke fashon, we can wrte down the soltons for the TME bases β,,, n :, where ± ' p β β ± σ F, (8-4) ' ' ' σ Σc σ, (8-4) and c ' ' r ' Σrj. (8-43) The varables r ' n ths expresson are defned as before. To llstrate the determnaton of bas confdence lmts, we agan trn to the profcency adt example. In ths example where and By eqatons (8-8) and (8-34), and σ σ σ 6.97, r r r. 3 σ ' c c, 3 σ 9 σ ' σ. σ 9 8-5

167 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Sbstttng n eqatons (8-4) and (8-4) yelds and ± p β β ± 3.9F, ± p β β ± 3.9F, ± p β β ± 3.9F. Sppose that the desred confdence level s 95%. Then p.95, and and F p F.96, (.975) p 3.9F 6.4. Snce β -7, β -, and β 8, ths reslt, when sbsttted n the above expressons, gves 95% confdence lmts for the estmated bases: 3.4 β β β Applcaton of SMPC As wth any analyss technqe, the reslts are only as good as the nformaton or data npt nto t. Or as the sayng goes: garbage n, garbage ot. To avod mssng SMPC, there are two provsons to keep n mnd.. At ts present stage of development, SMPC s strctly applcable when process errors and measrement bases are normally dstrbted. For process errors or parameter bas wth a contanment probablty (confdence level) less than %, UncertantyAnalyzer assmes a normal dstrbton.. In sng SMPC wth more than one measrement sample, the samples mst be based on measrements of the same qantty. SMPC needs to anchor measrements to a common reference for comparson and evalaton. 8-6

168 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 9. General CHAPTER 9 SOFTWARE VAIATION Ths chapter dscsses the protocols that Integrated Scences Grop (ISG) has developed and mplemented to valdate the UncertantyAnalyzer program. Examples are presented that compare UncertantyAnalyzer calclatons to vales obtaned by hand calclatons and from Excel spreadsheets. 9. efntons 9.. Protocol. A procedre or practce, set of rles, or code of behavor. 9.. Valdate. To confrm or prove to be vald or correct Valdaton. See Verfcaton Verfcaton. Establshment or confrmaton of the trth or accracy of a fact, theory, etc Verfy. To make certan, to check or test the accracy or correctness of, as by nvestgaton, comparson wth a standard, or reference to the facts. 9. Software Valdaton Protocol Crrently, there are no standards or gdelnes for testng and valdatng ncertanty analyss software. However, there are many common-sense protocols that Integrated Scences Grop has adopted for the valdaton and verfcaton of UncertantyAnalyzer. These protocols are descrbed below. Valdaton of mathematcal and statstcal methods. Verfcaton of nmercal approxmatons and calclatons. Verfcaton of program fnctonalty. 9.. Mathematcal and Statstcal Methods. In general, t s not sffcent to smply state that an ncertanty analyss applcaton ncorporates nternatonally accepted methods, sch as those descrbed n the ISO GUM. The software developer shold also pblsh papers and artcles that clearly descrbe the mathematcal and statstcal concepts that are ncorporated n the prodct. Ths serves two prposes:. It shows whether or not the developer has a sffcent techncal nderstandng of ncertanty analyss concepts and prncples.. The nformaton can be revewed and scrtnzed n the pblc doman. Snce 99, ISG has pblshed and presented papers at techncal conferences and symposms that specfcally dscss the prncples and methods of estmatng measrement ncertanty. These papers are avalable n conference proceedngs and va download from ISG s webste. ISG also actvely docments new ncertanty analyss methods and practces as they are developed. These artcles are avalable for download from ISG s webste for peer revew. 9-

169 Uncertanty Analyss Prncples and Methods RCC ocment -7, September Nmercal Approxmatons and Calclatons. ependng pon the sophstcaton of the nmercal algorthms, program calclatons can be verfed va hand calclatons, Excel spreadsheets, or math and statstcs applcatons sch as MathCA or Mathematca. Verfcaton of nmercal algorthms can be acheved n a nmber of ways ncldng: Alpha testng va nternal peer revew and verfcaton. Beta testng va external revew and verfcaton by selected cstomer base. Wdespread peer revew and verfcaton va dstrbton of freeware sbprograms and applets. arge-scale cstomer se and feedback. Ths apples to well establshed analyss programs that have been sed by cstomers for the past several years or more. Snce ntal prodct release n 994, ISG has condcted extensve nternal revew and verfcaton of UncertantyAnalyzer s sbrotnes and sbprograms. Internal revew and verfcaton s condcted to all new program featres and fnctons as they are added. ISG also perodcally releases key featres as freeware applcatons for external revew pror to fnal mplementaton nto UncertantyAnalyzer Program Fnctonalty. Another mportant aspect of software valdaton s the verfcaton that the program screens, templates, or worksheets fncton as ntended. For example, data entered nto a drll-down screen shold be properly stored and transferred to other screens as needed. The protocol for testng and valdatng program fnctonalty s the same as descrbed for nmercal algorthms. Of corse, software programs cannot flly elmnate ser npt error. However, UncertantyAnalyzer does contan error trappng sbrotnes to ensre that realstc nformaton and data are entered n the approprate felds and cells. UncertantyAnayzer also contans a comprehensve Help fle that s easly accessed from all screens, templates or worksheets. The Help topcs are wrtten n a concse manner that clearly conveys the approprate nformaton. 9.3 Valdaton Examples Three ncertanty analyses are nclded heren to llstrate how UncertantyAnalyzer calclatons are verfed and compared to vales obtaned from hand calclatons (where possble) and from calclatons performed va Excel spreadsheet Cylnder Volme Measrement. In ths example, the cylnder s a steel artfact wth nomnal desgn dmensons of.65 cm length by.4 cm dameter. The length and dameter are measred wth a mcrometer. The objectve s the estmate the ncertanty n the cylnder volme measrement. The mathematcal relatonshp between the cylnder volme n terms of length and dameter s gven as V π where and are the length and dameter components, respectvely. A comparson of the reslts from hand calclatons and those obtaned sng UncertantyAnalyzer are presented heren. etals of the ncertanty analyss are n Appendx A. 9-

170 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Table 9-. Comparson of Total Uncertanty and egrees of Freedom Hand Calclaton UncertantyAnalyzer Cylnder Volme.8 cm 3.8 cc Std Uncertanty.94 cm 3.94 cc eg of Freedom 7 69 The standard ncertantes and component ncertantes for the measrement process errors obtaned from hand calclatons are smmarzed n Table 9-. Component ncertanty s obtaned by mltplyng the standard ncertanty by the approprate senstvty coeffcent. The UncertantyAnalyzer mltvarate analyss report s shown n Fgre 9-. Table 9-. Uncertanty Estmates for Cylnder Volme sng Hand Calclatons Varable Name Standard Uncertanty % Confdence ± Error mts Senstvty Coeffcent Component Uncertanty Nomnal or Mean Vale.65 cm bas.45 cm mm cm 3 cm ran.9 cm cm 3.37 cm res.9 cm..5 mm cm 3 cm op.3 cm 9..5 mm cm 3 cm env.68 cm.63. cm 3 cm.4 cm bas.45 cm mm cm 3 cm ran.4 cm.649 cm 3.33 cm res.9 cm..5 mm.448 cm 3 cm op.3 cm 9..5 mm.464 cm 3 cm env.46 cm.6 cm 3 cm where nomnal cylnder length bas measrement bas n length measrement ran length repeatablty error res length resolton error op length operator bas env length envronmental factors error nomnal cylnder dameter bas measrement bas n dameter measrement ran dameter repeatablty error res dameter resolton error op dameter operator bas env dameter envronmental factors error 9-3

171 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 9-. Mltvarate Analyss Report for Cylnder Volme Example oad Cell Calbraton. In ths example, a load cell (.e., tenson transdcer) s calbrated sng a weght standard, as llstrated n Fgre 9-. The calbraton weght s extended from the load cell va a monoflament lne. Repeat measrements of C voltage are obtaned by addng and removng the calbraton weght. The C voltage otpt from an amplfer/sgnal condtoner s measred wth a dgtal mltmeter. 9-4

172 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 9-. oad Cell Calbraton Setp. The ncertanty n the load cell otpt voltage s estmated sng both a system model analyss approach and a more general mltvarate analyss method. Reslts obtaned from Excel spreadsheet analyses are compared to smlar analyses sng UncertantyAnalyzer. Analyss detals are presented n Appendx B System Model Reslts. Comparsons of the compted oad Cell Modle otpt, total ncertanty and degrees of freedom obtaned from Excel spreadsheet and UncertantyAnalyzer are lsted n Table 9-3. Table 9-3. Comparson of oad Cell Total Uncertanty and egrees of Freedom Excel Spreadsheet UncertantyAnalyzer oad Cell Otpt 8.88 mv 8.88 mv Std Uncertanty.5 mv. mv eg. Of Freedom Infnte Infnte The estmated ncertantes for the oad Cell Modle obtaned from Excel spreadsheet calclatons are lsted n Table 9-4. The UncertantyAnalyzer otpt report for the oad Cell Modle s shown n Fgre

173 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Table 9-4. Spreadsheet Analyss Reslts for oad Cell Modle Parameter Name Standard Uncertanty % Confdence ± Error mts Senstvty Coeffcent Component Uncertanty Nomnal or Mean Vale CalWeght.388 g , g Senstvty 95,.88 mv/g Nonlnearty.5 mv mv.5 mv mv Hysteress.5 mv mv.5 mv mv Nose.5 mv mv.5 mv mv Random.5 mv.5 mv mv ZeroOffset. mv mv. mv mv TempEffectOt. mv/ F mv/ F. mv mv/deg F TempEffectZero.5 mv/ F mv/ F. mv mv/deg F TempRange.776 F 99 F deg F where CalWeght Calbraton Weght Senstvty oad Cell Senstvty Nonlnearty oad Cell Nonlnearty Hysteress oad Cell Hysteress Nose Nose Random Error de to Repeat Measrements ZeroOffset Zero Balance TempEffectOt Temperatre Effect on Otpt TempEffectZero Temperatre Effect on Zero TempRange Temperatre Range 9-6

174 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 9-3. oad Cell Modle Report. Comparsons of the compted Amplfer Modle otpt, total ncertanty and degrees of freedom obtaned from Excel spreadsheet and UncertantyAnalyzer are lsted n Table 9-5. Table 9-5. Comparson of Amplfer Total Uncertanty and egrees of Freedom Excel Spreadsheet UncertantyAnalyzer Amplfer Otpt 4.44 V 4.44 V Std Uncertanty 5.6 mv 5.6 mv eg. Of Freedom Infnte Infnte Comparson of the estmated ncertantes for the Amplfer Modle obtaned from Excel spreadsheet calclatons are lsted n Table 9-6. The Uncertanty Analyzer otpt report for the Amplfer Modle s shown n Fgre

175 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Table 9-6. Spreadsheet Analyss Reslts for Amplfer Modle Parameter Name Standard Uncertanty % Confdence ± Error mts Senstvty Coeffcent Component Uncertanty Nomnal or Mean Vale oadcellotpt.5 mv 8.88 mv Gan V/mV GanAcc.5 mv mv.5 V V Stablty.5 mv mv.5 V V Nonlnearty.5 mv mv.5 V V Nose.65 mv 99 3 mv.65 V V BalStablty mv mv. V V TempCoeff mv/c mv/c V mv/c TempRange.47 C 99. C V 5.6 C where oadcellotpt Otpt from oad Cell Modle Gan Amplfer Gan GanAcc Amplfer Accracy Stablty Amplfer Stablty Nonlnearty Amplfer Nonlnearty Nose Amplfer Nose BalStablty Balance Stablty TempCoeff Temperatre Coeffcent TempRange Temperatre Range 9-8

176 Uncertanty Analyss Prncples and Methods RCC ocment -7, September 7 Fgre 9-4. Amplfer Modle Report. Comparsons of the compted gtal Mltmeter Modle otpt, total ncertanty and degrees of freedom obtaned from Excel spreadsheet and UncertantyAnalyzer are lsted n Table 9-7. These vales also represent to overall system otpt, total system ncertanty, and degrees of freedom. Table 9-7. Comparson of gtal Mltmeter Total Uncertanty and egrees of Freedom Excel Spreadsheet UncertantyAnalyzer MM Otpt 4.44 V 4.44 V Std Uncertanty 5.9 mv 5.9 mv eg. Of Freedom Infnte Infnte The estmated ncertantes for the gtal Mltmeter Modle obtaned from Excel spreadsheet calclatons are lsted n Table 9-8. The UncertantyAnalyzer otpt report for the gtal Mltmeter Modle s shown n Fgre 9-5. The UncertantyAnalyzer overall system analyss report s shown n Fgre