Table Of ContentMeasurement Uncertainties
S. V. Gupta
Measurement Uncertainties
Physical Parameters and Calibration
of Instruments
123
Dr.S.V.Gupta
Sector-8,Rohini269/B-5
110085Delhi
India
[email protected]
ISBN978-3-642-20988-8 e-ISBN978-3-642-20989-5
DOI10.1007/978-3-642-20989-5
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011942600
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Dedicated tomywife, Mrs.Prem Gupta,
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as
Theyallinspiremeto livelonger
Preface
I have been associated with uncertainty calculations for quite sometime. Asia
Metrology Program (APMP) took interest in unification of expressing the mea-
surementresultsalongwith uncertainty.Inthose daystheuncertaintycomponents
were divided according to source of errors. The errors were named as random
and systematic errors. Hence components of uncertainty were named random
uncertainty and systematic uncertainty. Around 1980, BIPM took initiative by
circulating questioners to the countries who were members of Metre Convention.
Everymembershowedkeeninterestinexpressingtheuncertaintyinaharmonious
way.Asmallbutpath-breakingdocumentwasproducedin1980,whichemphasized
that uncertainty is not of different kinds; there are only the ways by which
one arrives at the uncertainty value, namely Type A and Type B evaluations of
uncertainty. Further the document emphasized that square root of variance or a
quantitysimilartoitisthestandarduncertainty.
Thebookisintendedtoserveasaguideforexpressingthemeasurementresult
along with uncertainty. The book conforms to “The Guide to the expression of
uncertainty in Measurement” jointly produced by International Organization for
Standardization (ISO), International Bureau of Weights and Measures (BIPM),
InternationalFederationofClinicalChemistry(IFCC),InternationalUnionofPure
andAppliedChemistry(IUPAC),InternationalUnionofPureandAppliedPhysics
(IUPAP) and International Organization of Legal Metrology (OIML). The book
differsfromISOGuideinexplainingthebasictheorybehindrelationsprovidedby
the Guide. Lots of examples are provided to supportthe theoretical formulations.
All technical and scientific terms used have been explained in the first chapter
itself. Various distributions used in uncertainty calculations have been explained
in Chaps. 2 and 3. The stress has been given on the properties of Gaussian
(Normal)probabilitydistribution.Evaluationofdataweatherprimaryorsecondary
is one specialtopic discussed in Chap. 4. For each statistical parameterlike mean
or standard deviation lots of practical examples have been cited. The chapter is
highly useful for the nodal laboratories involved in international measurement
programmes. Propagation of uncertainty has been discussed by first explaining
the Taylor expansion and highlighting the need for the function to be linear. The
vii
viii Preface
processofcalibrationofthemeasuringinstrumentsatafewpointsandexpressing
the whole calibration result in the form of the function of the input quantity has
been discussed by citing several examples in Chap. 6. The functions discussed
are linear, exponential, power and polynomials types. Detailed steps for arriving
at the uncertainty, starting from the modelling of the measurand as a function of
inputquantities,havebeengiveninChap.7.TheadvantagesandlimitationsofISO
GUM method havebeen given.Monte Carlo and Bayesian methodsof arrivingat
the uncertainty have been mentioned. The detailed procedure for calibrating the
surfaceplateandtheoreticaldeductionsofheightatvariouspointsalongthevarious
designated lines has been given in Chap. 8. The chapter includes the uncertainty
calculations between the points on the same line and also on different lines. The
uncertaintycalculationasperNABL(NationalAccreditationBoardforTestingand
CalibrationLaboratories)requirementhasalsobeengivenincalibrationofsurface
Chap. 8. The uncertainty calculations in mass measurement have been dealt with
in Chap. 9. While discussing various sources of errors such as that of buoyancy
correction, uncertainty requirements in measurement of various environmental
parametershave also been cited. Uncertainty in volume measurementsby various
methodshas been detailed. Uncertaintyin the calibration of volumetricglassware
bygravimetricmethod,largercapacitymeasuresbyvolumetricmethodandstorage
tanks by dimensional measurements have been discussed in detail in Chap. 10.
The uncertaintycalculation in the calibration hydrometersby comparison method
has also been given. Chapter 11 deals with the uncertainty calculation in the
measurementofandcalibrationofmeasuringinstrumentsforlength,pressure,tem-
peratureandluminousflux.Chapter12dealswithelectricalparameters;uncertainty
in measurement of and calibration of measuring instruments has been detailed.
Uncertaintycalculationsofvectormeasurandswhicharethefunctionofdependent
inputquantitieshavebeendiscussed.SomeimportantTablesforNormal(Gaussian)
probability distributions, Student’s t distribution, (cid:2)2 and Fisher’s F values for
different percentage points have been given. The limits for mean and standard
deviationsforvariousdegreesoffreedomhavealsobeentabulated.Abibliography
of recent papers, books and documents on uncertainty in measurement has been
appended.
I wouldlike tothankMr. VivekBaggafordiscussingthecalibrationofsurface
plates.MyearnestthanksareduetoMrs.ReetaGupta,ScientistNationalPhysical
Laboratory, for logistic support. I wish to record my profound appreciation for
the keen interest and strenuous efforts put in by Dr. Habil. Claus Ascheron, the
ExecutiveEditorPhysicsinmakingitpossibletobringthemanuscripttothedesired
levelandtoSpringerVerlagGmbH,Germany,inbringingthisbooktolight.
NewDelhi S.V.Gupta
July2011
Contents
1 SomeImportantDefinitions............................................... 1
1.1 Introduction........................................................... 1
1.2 TermsPertainingtoQuantity........................................ 2
1.2.1 Quantity..................................................... 2
1.2.2 SystemofBaseQuantities ................................. 2
1.2.3 DerivedQuantity ........................................... 2
1.2.4 QuantityEquation .......................................... 3
1.2.5 DimensionofaQuantity ................................... 3
1.2.6 Measurand .................................................. 4
1.2.7 TrueValueofaQuantity ................................... 4
1.2.8 ConventionalTrueValueofaQuantity.................... 4
1.2.9 MeasuredValue............................................. 5
1.2.10 RelationinBetweenMeasuredValue
andTrueorConventionalTrueValue...................... 5
1.3 TermsPertainingtoMeasurement................................... 5
1.3.1 Measurement................................................ 5
1.3.2 MethodofMeasurement ................................... 5
1.3.3 SubstitutionMethod........................................ 5
1.3.4 DifferentialMethod ........................................ 6
1.3.5 NullMethod ................................................ 6
1.3.6 MeasurementProcedure.................................... 6
1.3.7 ResultofMeasurement..................................... 6
1.3.8 Error......................................................... 6
1.3.9 SpuriousError .............................................. 7
1.3.10 RelativeError............................................... 7
1.3.11 RandomError............................................... 7
1.3.12 SystematicError............................................ 7
1.3.13 AccuracyofMeasurement ................................. 8
1.3.14 PrecisionofMeasurementResult.......................... 8
1.3.15 Repeatability................................................ 8
ix
x Contents
1.3.16 Reproducibility(ofMeasurementResults)................ 9
1.3.17 Correction................................................... 9
1.4 TermsPertainingtoStatistics........................................ 9
1.4.1 Observation................................................. 9
1.4.2 IndependentObservations.................................. 9
1.4.3 Population................................................... 10
1.4.4 Sample ...................................................... 10
1.4.5 Measurement................................................ 10
1.4.6 PopulationofMeasurement................................ 10
1.4.7 SampleofMeasurements................................... 11
1.4.8 Frequency/RelativeFrequency............................. 11
1.4.9 Mean ........................................................ 11
1.4.10 SampleMean ............................................... 11
1.4.11 PopulationMean............................................ 11
1.4.12 MeritsandDemeritsofArithmeticMean[3] ............. 12
1.4.13 Median...................................................... 12
1.4.14 Quartiles..................................................... 12
1.4.15 Dispersion................................................... 13
1.4.16 StandardDeviation ......................................... 13
1.4.17 Variance..................................................... 13
1.4.18 SampleStandardDeviation ................................ 14
1.4.19 PopulationStandardDeviation............................. 14
1.4.20 EstimateofPopulationStandardDeviation............... 14
1.4.21 EstimateofPopulationandSampleStandard
Deviations-Relation ........................................ 15
1.4.22 IndependentVariable....................................... 15
1.4.23 DependentVariableorResponseVariable................. 15
1.4.24 Correlation.................................................. 15
1.4.25 CorrelationCoefficient..................................... 15
1.4.26 Covariance.................................................. 16
1.4.27 RandomVariable ........................................... 16
1.4.28 DiscreteRandomVariable ................................. 16
1.4.29 ContinuousRandomVariable.............................. 16
1.4.30 Probability .................................................. 16
1.4.31 ProbabilityDistribution .................................... 17
1.4.32 NormalDistribution........................................ 17
1.4.33 PropertiesofNormalDistribution ......................... 18
1.4.34 ProbableError .............................................. 18
1.4.35 Range........................................................ 18
1.4.36 ConfidenceLevel........................................... 18
1.4.37 ConfidenceInterval......................................... 19
1.4.38 Outlier....................................................... 19
1.4.39 Parameter ................................................... 19
1.4.40 RandomSelection .......................................... 19
1.4.41 SampleStatistic............................................. 19
