Evaluating Measurement Accuracy: A Practical Approach

279 Pages·2010·3.58 MB·English

by  Semyon G. Rabinovich (auth.)

Evaluating Measurement Accuracy
Semyon G. Rabinovich
Evaluating
Measurement Accuracy
A Practical Approach
123
Dr.SemyonG.Rabinovich
142ManchesterDrive
BaskingRidge,NJ07920
USA
semyon [email protected]
ISBN978-1-4419-1455-2 e-ISBN978-1-4419-1456-9
DOI10.1007/978-1-4419-1456-9
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Preface
The goal of this book is to present methods for estimating the accuracy of real
measurements, that is, measurements performed in industry, trade, scientific re-
search–wherevertheproductionprocess,qualitycontroldecision,ortheinterpre-
tation of an experimentdependson measurementaccuracy.The necessity for this
book arises from the fact that the existing theory of measurement errors contains
significant gaps. In particular, the current theory focuses exclusively on multiple
measurements and overlooks single measurements. Meanwhile, single measure-
mentsarethe onesmostcommonlyused inpractice.Moreover,thecurrenttheory
isincompleteevenwithinthescopeofmultiplemeasurements.Forexample,itdoes
not provide answers to such fundamental questions as how to translate the inac-
curacy of a measuring instrument into the inaccuracy of a measurement utilizing
thisinstrument,orhowtofindthefulluncertaintyofameasurementresult,i.e.,the
uncertaintythatreflectsbothsystematicandrandomerrors.
The science of measurements – metrology – entered a period of rapid de-
velopment several decades ago, prompted by the growth in international trade,
globalizationofindustrialproduction,demandsofmedicineandpharmacology,the
increasedattentiontofoodqualityandenvironment,andotherneedsofthemodern
society.However,metrologywillnotfulfilltheseneedswithoutremovingtheblind
spots mentioned above. I devoted many years of research filling these gaps. This
bookgeneralizesandputsintoacoherentwholetheresultsofthiseffort.
Thebookdevelopsthegeneraltheoryofprocessingexperimentalmeasurement
data,whichaddressestheneedtoobtainthevalueofaquantitybeingmeasuredand
the accuracy of this estimate. For the first time, this book presents the postulates
ofthetheoryofmeasurements.Itintroducesthetermmeasurementinaccuracyasa
generaltermthatreflectsmeasurementuncertaintyinsomesituationsandlimitsof
error(orevenerrorsthemselves)inothers.Thebookshowstherelationshipbetween
the accuracy of measuring instruments and measurements utilizing these instru-
ments. It presents methodsof estimating the accuracyof both single and multiple
measurements. Moreover,it formulatesthese methodsin a systematic and unified
way by formulating and utilizing a new perspective that single measurementsare
the basic type of measurements and multiple measurements represent a series of
repeated single measurements. This approach,besides being logical and intuitive,
makesaccountingforthemeasuringinstrumentsinaccuracyaninherentpartofthe
v
vi Preface
calculationsoftheinaccuracyofthemeasurement.Thebookofferswell-grounded
and practicalmethodsfor combiningthe componentsof measurementinaccuracy.
Inparticular,itdescribeshowtocombinethelimitsofelementarysystematicerrors
and how to estimate the overallmeasurement uncertainty accounting for both the
systematicandrandomerrors.
Aspartofthegeneraltheoryofmeasurements,thebookdevelopsthetheoryof
indirectmeasurements.Fordependentindirectmeasurements,thebookproposesthe
methodof reductionin place of the traditionalmethodbased on the Taylorseries.
Thismethodismoreaccurate,simpler,andmostimportantlyallowstocalculatethe
confidencelimitsoftheinaccuracyofthesemeasurements,ratherthanjuststandard
deviationofthemeasurementresultasinthetraditionalmethods.Atthesametime
it removes the need to account for the correlation coefficient, which had been a
thornyissue in this area. The bookalso proposesa new methodof transformation
for independentindirectmeasurements.The bookfurther includesa discussion of
theapplicabilityoftheBayes’TheoremandMonteCarlomethodsinmeasurement
data processing, the topics that have been actively discussed in the metrological
researchpapers.
Asaresult,thisbookcanserveasacomprehensivereferencefordataprocessing
of all types of measurements, including single and multiple measurements, de-
pendentand independentindirect measurements, and combined and simultaneous
measurements.Itincludesmanyconcreteexamplesthatillustratetypicalproblems
encounteredin measurementpractice.Thus,thebookencompassestheentire area
ofmeasurementdataprocessing,fromgeneraltheorytopracticalapplications.
The book contains nine chapters. Chapter1 gives the general introduction to
measurementsand metrologyand outlinesmajor changesthatoccurredin metrol-
ogyduringthelasttwodecades.Althoughthischapterisofintroductorynature,it
presentssomeimportantgeneralperspectivesonthesubject.Inparticular,itincludes
aclassificationofmeasurementsandmeasurementinaccuracyandformulatespos-
tulatesofthetheoryofmeasurements.
Chapter2isdevotedtomeasuringinstruments.Itdescribesconventionalmethods
of representing their metrological characteristics as well as the methods of con-
trolling these characteristics through calibration or verification. The chapter also
analyzeserrorsoflargenumbersofinstrumentsofseveraltypesandshowsthatthe
distributionfunctionsoftheseerrorsareusuallynonstationary.
Chapter3 contains basic statistical methods of experimental data processing.
Thesemethodsaredirectlyapplicabletotheidealizedmultiplemeasurements.They
arealsonecessarywhenusingstatisticalmodelsofelementarymeasurementerrors
and for obtaining confidence intervals in the course of measurement uncertainty
calculations.
Chapter4 is devoted to direct measurements. It presents a step-by-step proce-
dureforthecalculationsoftheinaccuracyofsinglemeasurements.Thecalculation
of uncertainty of a multiple measurement is then derived as a summation of the
inaccuracyoftheunderlyingsinglemeasurementwiththerandomerrorofthemul-
tiplemeasurement,whichisestimatedfromtherepeatedsinglemeasurements.The
chapterdescribesanewsummationmethodandtheadvantagesofthenewmethod
Preface vii
overthe knownmethodsof summatingsystematic andrandomerrors.Finally,the
chapter briefly describes nonparametric and robust methods for processing direct
measurementdata.
Chapter5presentsthetheoryofindirectmeasurements.Inparticular,itdescribes
twonewmethods.Thefirstoneisthemethodofreduction,whichhandlesindirect
measurementswithdependentarguments.Thismethod,whichweproposedprevi-
ously but which is notyetwidely known,is the first to producereliable estimates
ofuncertaintyof these typesof measurements.Atthe same time,it eliminatesthe
need to calculate the correlation coefficients – a major stumbling block in these
measurements.Thesecondmethodisthemethodoftransformationforindirectmea-
surements with independent arguments, which compliments the known methods.
Thischapteralsoappliesthenewgeneralmethodforthesummationofsystematic
andrandomerrorsfromChap.4toindirectmeasurements,thusremovingtheneed
tousetheMonteCarlomethodwithitsknownlimitations(therelianceonunknown
distributionfunctionsandthecomplexityofimplementation)forthispurpose.
Chapter6 treats simultaneous and combined measurements, using the well-
known least-squares method, which is commonly applied for these measurement
types.
Chapter7 contains methods for combining measurement data or measurement
results.Thisproblemariseswhenthesamemeasurandismeasuredindifferentlab-
oratories, and the final result should reflect all these measurements. Along with a
traditionalsolution,whichtakesintoconsiderationonlyrandomerrors,Chap.7in-
cludesamethodaccountingforthesystematicerrorsaswell.
Chapter8includesanumberofconcreteexamplesofmeasurementdataprocess-
ingandevaluatingmeasurementaccuracy.Thebookistargetedforpracticaluse,and
theseexamplescanserveasspecificblueprintsforaddressingtypicalmeasurement
dataprocessingneedsfacedbyexperimenters.
Finally,Chap.9presentsconcludingremarks,includingourthoughtsonthenew
“InternationalVocabularyofMetrology–BasicandGeneralConceptsandAssoci-
atedTerms”[1]andanalysisofthedrawbacksofthe“GuidetotheExpressionsof
UncertaintyinMeasurement”[2].
This book is intended for anyone who is concerned with measurementsin any
fieldofscienceortechnology,whodesigntechnologicalprocessesandchoosesin-
struments with appropriate accuracy as part of their design, and who design and
testnewmeasuringdevices.Thisbookshouldalsobeusefultouniversitystudents
pursuingscienceandengineeringdegrees.Indeed,measurementsareofsuchfunda-
mentalimportanceformodernscienceandengineeringthateveryoneinthesefields
mustknowthebasicsofthetheoryofmeasurementsandespeciallyhowtoevaluate
theiraccuracy.
The book assumes reader’s familiarity with mathematical statistics, basic cal-
culus, and, in part of Sect.2.5, control theory. A reader without control theory
background can skip this part of Sect.2.5 without affecting the understanding of
therestofthebook.
Inconclusion,IwouldliketothankDr.AbramKagan,ProfessorattheUniver-
sityofMaryland,CollegePark,formanyyearsofcollaborationandfriendship.This
viii Preface
bookbenefitedfromourdiscussionsonvariousmathematicalproblemsinmetrol-
ogy. I would also like to thank Dr. Ilya Gertsbach, Professor at the Ben Gurion
University of Beersheva (Israel), for our discussions over the theory of indepen-
dentindirectmeasurements.Iwouldliketoexpressmyspecialgratitudetomyson,
Dr.MichaelRabinovich,ProfessoratCaseWesternReserveUniversity.Heprovided
supportandassistancethroughoutmyworkonthisbookfromeditingtheproposal
forpublicationto discussingnew resultsandthepresentationto editingthe whole
book.Thisbookwouldnotbepossiblewithouthishelp.
BaskingRidge,NJ SemyonG.Rabinovich
Contents
1 GeneralConceptsintheTheoryofMeasurements........................ 1
1.1 BasicConceptsandTerms.............................................. 1
1.2 TheBasicMetrologicalProblems...................................... 4
1.3 NewFormsofInternationalCooperationinMetrology............... 10
1.4 PostulatesoftheTheoryofMeasurements ............................ 12
1.5 ClassificationofMeasurements ........................................ 18
1.6 ClassificationofMeasurementsErrors................................. 22
1.7 GeneralApproachtoEvaluationofMeasurementInaccuracy........ 25
1.8 PresentationofMeasurementResults.................................. 26
2 MeasuringInstrumentsandTheirProperties.............................. 31
2.1 TypesofMeasuringInstruments ....................................... 31
2.2 MetrologicalCharacteristicsofMeasuringInstruments............... 33
2.3 RatingoftheErrorsofMeasuringInstruments........................ 36
2.4 DynamicCharacteristicsofMeasuringInstruments................... 49
2.5 CalibrationandVerificationofMeasuringInstruments ............... 53
2.6 DesigningaCalibrationScheme ....................................... 58
2.7 StatisticalAnalysisofMeasuringInstrumentErrors.................. 65
3 StatisticalMethodsforExperimentalDataProcessing.................... 71
3.1 MethodsforDescribingRandomQuantities........................... 71
3.2 RequirementsforStatisticalEstimates................................. 75
3.3 EvaluationoftheParametersoftheNormalDistribution............. 76
3.4 EliminationofOutlyingData........................................... 80
3.5 ConstructionofConfidenceIntervals .................................. 82
3.6 TestingHypothesesabouttheFormoftheDistributionFunction .... 87
3.7 TestingforHomogeneityofSamples .................................. 89
3.8 RobustEstimates........................................................ 95
3.9 ApplicationoftheBayes’Theorem.................................... 99
4 DirectMeasurements..........................................................101
4.1 RelationBetweenSingleandMultipleMeasurements................101
4.2 ClassificationofElementaryErrors....................................104
ix
x Contents
4.3 ModelingofElementaryErrors ........................................108
4.3.1 AbsolutelyConstantErrors.....................................109
4.3.2 ConditionallyConstantErrors..................................109
4.3.3 PurelyRandomErrors..........................................110
4.3.4 QuasirandomErrors ............................................110
4.4 CompositionofUniformDistributions.................................111
4.5 MethodsforPreciseMeasurements....................................117
4.6 AccuracyofSingleMeasurementsUsingMeasuring
InstrumentsUnderReferenceConditions..............................119
4.7 AccuracyofSingleMeasurementsUsingMeasuring
InstrumentsUnderRatedConditions...................................122
4.8 AccuracyofMultipleMeasurements...................................128
4.9 Comparisonof DifferentMethodsforCombining
SystematicandRandomErrors.........................................136
5 IndirectMeasurements .......................................................141
5.1 TerminologyandClassification.........................................141
5.2 CorrelationCoefficientandItsCalculation............................142
5.3 TheTraditionalMethodofExperimentalDataProcessing............145
5.4 MeritsandShortcomingsoftheTraditionalMethod..................149
5.5 TheMethodofReduction...............................................152
5.6 TheMethodofTransformation.........................................154
5.7 TotalUncertaintyofIndirectMeasurements...........................161
5.8 AccuracyofSingleIndirectMeasurements............................166
5.9 AccuracyofaSingleMeasurementwithaChainofInstruments.....167
5.10 ApplicationoftheMonteCarloMethod...............................169
6 CombinedandSimultaneousMeasurements...............................173
6.1 GeneralRemarksAbouttheMethodofLeastSquares................173
6.2 MeasurementswithLinearEquallyAccurate
ConditionalEquations ..................................................175
6.3 Measurements with Linear Unequally Accurate
ConditionalEquations ..................................................178
6.4 LinearizationofNonlinearConditionalEquations ....................180
6.5 ExamplesoftheApplicationoftheMethodofLeastSquares........181
6.6 GeneralRemarksonDeterminationoftheParameters
inFormulasfromEmpiricalData ......................................186
6.7 ConstructionofTransferFunctionsofMeasuringTransducers.......188
7 CombiningtheResultsofMeasurements...................................193
7.1 IntroductoryRemarks...................................................193
7.2 TheoreticalPrinciples...................................................193
7.3 EffectoftheErroroftheWeightsontheError
oftheWeightedMean...................................................197
Contents xi
7.4 Combining the Results of Measurements
withPredominatelyRandomErrors....................................199
7.5 Combining the Results of Measurements
ContainingBothSystematicandRandomErrors......................201
7.6 CombiningtheResultsofSingleMeasurements ......................204
8 ExamplesofMeasurementsandMeasurementDataProcessing.........209
8.1 VoltageMeasurementwithaPointer-TypeVoltmeter.................209
8.1.1 APrioriEstimationofMeasurementInaccuracy..............209
8.1.2 UniversalEstimationofMeasurementInaccuracy............212
8.1.3 IndividualEstimationofMeasurementInaccuracy ...........214
8.2 VoltageMeasurementwithaPotentiometerandaVoltageDivider...215
8.3 ComparisonofMassMeasures.........................................220
8.4 MeasurementofElectricPoweratHighFrequency...................221
8.5 AnIndirectMeasurementoftheElectricalResistanceofaResistor .222
8.5.1 ApplicationoftheTraditionalMethod.........................222
8.5.2 ApplicationoftheMethodofReduction ......................225
8.6 MeasurementoftheDensityofaSolidBody..........................226
8.6.1 ApplicationoftheTraditionalMethod.........................226
8.6.2 ApplicationoftheMethodofTransformation.................229
8.7 MeasurementofIonizationCurrent....................................233
8.8 MeasurementoftheActivityofaRadioactiveSource ................235
9 Conclusion......................................................................239
9.1 MeasurementDataProcessing:Past,Present,
andNextSteps...........................................................239
9.2 RemarksontheInternationalVocabularyofMetrology...............241
9.3 Drawbacks of the “Guide to the Expression
ofUncertaintyinMeasurement” .......................................244
Appendix............................................................................249
Glossary.............................................................................255
References...........................................................................261
Index.................................................................................265