WIXLIB

Definitions of base units Ampere:The ampere is that constant current which, if maintained in two straightparallel conductors of infinite length, of negligible circular cross-section,and placed 1 metre apart in vacuum, would produce between theseconductors a force equal to 2 10-7 newton per metre of length. NOTE 1: Direct realisation of the SI ampere is very difficult in practice NOTE 2: N kg · m · s-2 Depends on the prototype of kg Candela:The candela is the luminous intensity, in a given direction, of a source thatemits monochromatic radiation of frequency540 x 1012 hertz and that has a radiant intensity in that direction of 1/683watt per steradian.MH 201341

Need for redefining SI base units Drawbacks of the current SI are: Kilogram is defined with an artefact The definition of kelvin is based on material properties The definition of ampere limits the achievable accuracy andis not currently use (quantum standards of ohm and voltageare used instead through the Ohm’s law) Because of these, drift, non-uniqueness and limitations inaccuracy cannot completely be prevented.MH 201342

SI base units in futureGround state hyperfine splitting frequency of thecaesium 133 atomSpeed of light in vacuumAvogadro constantBoltzmann constantPlanck constantElementary chargeLuminous efficacyMH 201343

SI system – Derived units[2.1]MH 201344

SI system – Derived units with special names[2.1]MH 201345

SI system – Derived units with special names 2[2.1]MH 201346

Realisation of a derived unit The realisation forms a link between the measurements of aderived quantity and the SI base units. Characteristics of the realisation: Based on the definition of the quantity Measurements in terms of SI units (base or derived but not of thesame quantity) with traceability to national/internationalmeasurement standards Full uncertainty analysis Comparisons to other realisations of the same quantity National Standards Laboratories maintain realisations (orsecondary standards) for a set of derived quantitiesMH 201347

Realisation of the pascal Unit with a special name: 1 Pa 1 m-1 kg s-2Definition of pressure:pistonRealisation of the unit:Fp AForce lifting the piston F1:F1 pAcylinderForce lowering the piston F2:F2 mgIn equilibrium:Pressure balancemgF1 F2 p AMH 201348

Realisation of the unit of the dew-pointtemperature (K or C) Same unit as of the temperature but different meaning:Dew-point temperature (td):The temperature at which the gas would be saturated withrespect to a plane surface of pure liquid water or solid ice at thesame gas pressure.A saturator-based primary humidity standardAir or other gas is saturated with respect to plane water or ice surface: dew-point temperature (td) air/gas temperature in the saturator (ts)Saturated air/gas passes throughinternally polished stainless steeltubing .td ts ttubingRH 100 %td tsRH 100%td ts ΔtdP tchamberRH RH(td,tchamber)The air or gas enters a measurement chamber.Humidity in the chamber is determined by the temperaturesand pressures of the saturator and the chamberMH 201349

2.3Measurement standards andtraceability

Traceability in chemical measurementsSI unit: moleThe mole was added to SI in 1971.Chemical primary methods Isotope Dilution Mass SpectrometryGravimetryTitrimetryCoulometryFreezing Point Depression (Differential Scanning Calorimetry)Certified reference materialsISO Guides 30 – 35, 1989 - 2000Validated measurement methods andreference materials in laboratoriesMH 201351

Traceability to altitude measurements in amilitary ng instrumentPrimary standardSI – System of UnitsAltimeter 10 mThe pressure simulator of the brigade 30 Pa 2.6 mK013 pressure working standard 20 Pa 1.7 mK013 pressure reference standard 10 Pa 0.9 mMIKES Primary Pressure StandardMIKES Primary Mass and Length StandardsMH 201352

3 Measurement uncertainty – part 1:Introduction3.1 Terminology3.2 Importance of the measurement uncertainty

3.1 Terminology

TermsMEASUREMENT REPEATABILITY measurement precision under a set of repeatability (similar) conditionsof measurementMEASUREMENT REPRODUCIBILITY measurement precision under reproducibility (changed) conditions ofmeasurementRANDOM ERROR Component of measurement error that in replicate measurementsvaries in an unpredictable mannerSYSTEMATIC ERROR Component of measurement error that in replicate measurementsremains constant or varies in a predictable manner[1.1, 1.2]MH 201355

3.2 Importance of the measurementuncertainty

Measurement result and its uncertainty Estimated quality of a result is expressed as the uncertainty The uncertainty is an essential part of a measurement result:Measurement result Estimate uncertainty The uncertainty gives the limits of the range in which the “true”value of the measurand is estimated to be at a given probability. Too often the uncertainty is not presented explicitly with theestimate confusion, incorrect conclusions, non-equal users etc. If the uncertainty is not taken into account, incorrect conclusionsare drawn and the number of unsatisfactory products increases. When you know the uncertainty in a measurement, then you canjudge its fitness for purpose.MH 201357

Importance of uncertainty of measurements Measurements are never absolutely accurate, but there isalways some uncertainty in the measured values. Measurement result without uncertainty estimate ismeaningless. Estimation of measurement uncertainties is one of the mostimportant parts of practical metrology. A proper uncertainty budget indicates what parts of themeasurement should be developed to decrease the overalluncertainty most effectively.MH 201358

Certified range [U k ·u c (k 2)]: 1.226 - 1.294 01.3301.23-101.131.03-200.93-300.830.73-409 valuesbelow -50%0.63Deviation from middle of certified range in %503 valuesabove 50%-50Results from all participants.[Mittaustulosten epävarmuus, MIKES 26.6.2005]MH 201359

4 Measurement uncertainty – part 2:Methods4.1 Calculating uncertainty4.2 Calculations step by step4.3 Uncertainty calculation in practice

4.1 Calculating uncertainty

Calculating a measurement result A measurement result is calculated from input data. In additionto the measurement values, the data often include informationfrom earlier measurements, specifications, calibrationcertificates etc. The calculation method is described with an equation (or a setof equations) called measurement model (mathematical model) The model is used for both calculation of the estimate and theuncertainty of the results. The model should include all factors (input quantities) affectingsignificantly the estimate and/or the uncertainty. The model is never complete; approximations are needed.MH 201362

6 steps to evaluating uncertainty1) Measurement model:List essential input quantities (i.e. parameters xi having a significanteffect on the result) and build up a mathematical model (function)showing how they are related to the final result: y f(x1,, , xi)2) Standard uncertainty:Estimate the standard uncertainty of each input quantity (xi)3) Using the model in uncertainty calculations:Determine the uncertainty due to standard uncertainty of each inputquantity (xi).4) Correlation:Determine correlation between the input quantities (if relevant)5) Calculate the combined standard uncertainty6) Calculate the expanded uncertainty.MH 201363

4.2 Calculations step by stepStep 1: Measurement modelStep 2: Standard uncertaintyType A evaluation of standard uncertaintyType B evaluation of standard uncertaintyStep 3:Using the model in uncertainty calculationsStep 4: CorrelationStep 5: Combining the uncertainty componentsStep 6: Expanded uncertainty

Step 1: Measurement modelEquation which describes the measurement:MeasurandY f ( X 1,X 2 ,.,X n )Input quantities Xi The model should include:Measurement results, corrections, reference values, influence quantities. The magnitude of a correction can be zero but it can still have uncertainty The values and uncertainties of the input quantities should be determinedMeasurement result (y) is:y f ( x1,x2 ,.,xn ) xi is the value (estimate) of the input quantity XiExample :t x tind δtCal δt D δt resol δtGMH 201365

Step 2: Standard uncertainty All uncertainty components should be comparable standarduncertainty ui The variance of the sum of non-correlating random variables is the sumof their variances standard uncertainty is the square root of variance all uncertainty components should be expressed as standardsuncertainties For normal distribution standard uncertaintycorresponds to about 68%confidence levelMH 201366

Two methods for estimating the standarduncertainty of an input quantity Type A: Evaluated from a number of observations (usually 10) Tyyppi B: Evaluated from a single (or a small number of) data value(s) Often taken from data reported earlier or by othersMH 201367

Type A evaluation of standard uncertainty Uncertainty is evaluated by statistical analysis of a series of observations qi The spread of the results is assumed to be random1 nq qin i 1 An estimate for the value of the quantityis the arithmetic mean q An estimate for the variance of the probability distribution is s(x)2:1 n2()s (q) q q in 1 i 1 s(q) is termed the experimental standarddeviation22 An estimate for the variance of the mean s (q)(the experimental variance of the mean) is: The standard uncertainty of q equals theexperimental standard deviation of the mean:s 2 (q)s (q) n2n1u (q ) s(q) (qi q) 2 n(n 1) i 1If type A measurement uncertainty is based on few measurements the estimation of u(x)is not reliable and normal distribution can not be assumed.(unless other information on the distribution is available)MH 201368

Type B evaluation of standard uncertainty To be applied for estimates of input quantities that has not been obtained fromrepeated measurements Typical examples : uncertainties of values and drifts of reference standards uncertainties of environmental quantities uncertainties from specifications of instrument uncertainties from literature values uncertainty due to the method or calculation uncertainty due to staff uncertainties from calibration certificatesMH 201369

Rectangular distributionuncertainty distribution p All values in the range xi - a xi a have equal probability Standard uncertaintyu ( xi ) Examples: specifications, resolutiona 0,577a3 Applied if only limiting values are knownMH 201370

Triangular distributionxi - axi a Example: convolution of two rectangular distribution Standard uncertainty:u ( xi ) a 0,408 a6MH 201371

U-shape distribution2.50-101xi – axi a Example: sinusoidal variation between limits a Standard uncertainty:u ( xi ) a 0,707 a2MH 201372

Step 3:Using the model in uncertainty calculations The contribution of u(xi) to the uncertainty of y is determined bythe sensitivity coefficient ci The sensitivity coefficient can be determined from partial derivative of f(X1, X2, ,Xn) with Xii.e. ci f / Xi (at x1, x2, ,xn) by numerical methods ci Δy / Δxi experimentally by changing xi by Δxi and determining Δy ;ci Δy / Δxi The contribution of u(xi) to the uncertainty of y is:ui(y) ci u(xi)MH 201373

Step4: Correlation The covariance u(xi,xj) of two random variables is a measure of theirmutual dependence. If Xi F(Ql) and Xj G(Ql) depend on the same quantities Ql (l 1.n) F G 2thenu ( xi , x j ) u (ql )i ql ql Correlation coefficient :r ( xi , x j ) u ( xi , x j )u ( xi )u ( x j ) The covariance can increase or decrease uncertainty. If the correlation coefficient is r 1 the components will be added in alinear way.MH 201374

Step 5: Combining the uncertainty components Uncorrelated input quantities:2 δf 2 u ( xi )uc ( y ) u ( yi ) i 1i 1 δxi N 2iNCorrelated input quantities:2N 1 N δf 2δf δf u ( xi ) 2 u ( xi , x j )uc ( y ) i 1 δxi i 1 j i 1δxi δx jNuc(y) combined standard uncertainty; u(xi,xj) covarianceMH 201375

Step 6: Expanded uncertaintyOften the result of the measurement is reported with a higher level ofconfidence than given by the standard uncertainty. Expanded uncertainty U is the standard uncertainty multiplied by a coveragefactor k:U kuc (y) In calibration it is recommended to report 95 % level of confidence. For normal distribution this corresponds to k 2 (approximately).Normaldistribution:Coverage probabilityp68,27 %90 %95 %95,45%99 %99,73%Coverage factork1,001,651,962,002,583,00MH 201376

Normal distribution 1σ 2σMeasurement result is approximately normally distributed if- it is a combination of several random variables (independent of distribution)- none of the (non-normally distributed) components is dominating.MH 201377

Degrees of freedom and the coverage factor For a combined standard uncertainty, we can calculate the 4u ( y)effective number of degrees of freedom (νeff) :ν eff The figure shows that we need a coverage factorlarger than 2 to obtain 95 % confidence level if thenumber of degrees of freedom is small.N i 1u 4 ( xi )νip 95,45 % ; 2σ54,5Coverage 0lkmof freedomNumberVapausasteidenof effective degreesMH 201378

4.2 Uncertainty calculation in practice

Example: Measurement of SO2 content An analyzer with electrical current signal output was used formeasuring SO2 content in exhaust gas. The signal was measured with a DMM and the total error in thecurrent measurement was estimated to be within 0,1 mA. The arithmetic mean of the 15 recorded DMM readings is9,59mA and the corresponding standard deviation is 0,49 mA. An accredited laboratory has determined the calibration functionfor the analyzer:3mg/mfc (I m ) 7,64 mg/m 3 3,25 ImmAThe reported expanded uncertainty (k 2) is 6 mg/m3 When comparing two last calibrations, we can conclude that thedrift of the analyzer is less than 5 mg/m3/year (calibr. interval is1 year)MH 201380

Example: Measurement of SO2 content - continuing The measurement result is calculated as follows:CSO 2 f c ( I m δI m ) δf c δ Drift3mg/m 7,64 mg/m3 3,25 ( I m δI m ) δf c δ DriftmA The variables can be assumed independent on eachother; therefore we can calculate the uncertainty:u (CSO 2 ) [c1u ( I m )]2 [c2u (δI m )]2 [c3u (δf c )]2 [c4u (δ Drift )]2MH 201381

Example: Measurement of SO2 content - continuing The sensitivity coefficients are: CSO 2mg/m3c1 3,25 I mmA CSO 2c2 c1 δI m CSO 2 CSO 2 c4 1c3 δf c δ Drift Thus:u (CSO 2 ) c1 [u 2 ( I m ) u 2 (δI m )] u 2 (δf c ) u 2 (δ Drift )2MH 201382

Example: Measurement of SO2 content - continuing Standard uncertainties of the components:u ( I m ) 0,49 mAtype A, normal distribution0,1 mAtype B, rectangular distribution 0,06 mA36 mg/m3type B, normal distributionu (δf c ) 3 mg/m 325 mg/m 3u (δ Drift ) 2,9 mg/m 3 type B, rectangular distribution3u (δI m ) MH 201383

Example: Measurement of SO2 content - continuing Estimate for the SO2 content:mg/m3CSO 2 7,64 mg/m 3,25 (9,59 mA 0 mA) 0 mg/m 3 0 mg/m3mA 23,5 mg/m 33 and its combined standard uncertainty:2 mg/m 3 (0,49 mA)2 (0,06 mA)2 (3 mg/m 3 ) 2 (2,9 mg/m 3 ) 2u (CSO 2 ) 3,25mA [] 4,5 mg/m 3MH 201384

Example: Measurement of SO2 content - continuing Thus the expanded uncertainty at about 95 % confidencelevel (k 2) is:U k u 2 4,5 mg/m3 9,0 mg/m3 As a conclusion, the measurement result is :CSO2 (24 9) mg/m3MH 201385

References and literature[1.1] ISO/IEC Guide 99-12:2007, International Vocabulary of Metrology — Basic and General Concepts andAssociated Terms, VIM[1.2] International vocabulary of metrology — Basic and general concepts and associated terms (VIM), 3rd ed.,JCGM 200:2008(can be downloaded at http://www.bipm.org/en/publications/guides/)[2.1] http://www.bipm.org/en/si/si brochure/Metrology - in short, 3rd edition, EURAMET 2008, 84 p. (www.euramet.org)Literature about estimating uncertainty: Guide to the Expression of Uncertainty in Measurement. International Organization for Standardization,Geneva, Swizerland. ISBN 92-67-10188-9, First Edition, 1993. JCGM 100:2008, Evaluation of measurement data – Guide to the expression of uncertainty inmeasurement, First edition, JCGM 2008 (http://www.bipm.org/en/publications/guides/) European cooperation for Accreditation, EA-4/02 Expression of the Uncertainty of Measurement inCalibration, 1999. -02.pdf) UKAS M3003, The Expression of Uncertainty and Confidence in Measurement EURACHEM/CITAC Guide CG 4, Quantifying Uncertainty in Analytical mu/QUAM2000-1.pdf) S. A. Bell, A beginner's guide to uncertainty in measurement, Measurement Good Practice Guide No. 11 ,(Issue 2), National Physical Laboratory 2001, 41 p. (www.npl.co.uk)MIKES has published several guides in Finnish on the uncertainty estimations in different fields.MH 201386

Metrology (metrologia) metrology is: science of measurement and its application [1.1, 1.2] Metrology includes all theoretical and practical aspects of measurement, whatever the measurement uncertainty and field of application. definitions of the SI units system of fundamental constants dissem