3Contents Introduction GUM Basic Concepts Basic Statistics Evaluation of Measurement UncertaintyHow is Measurement Uncertainty estimated?Reporting ResultConclusions and Remarks
4IntroductionGuide to the Expression of Uncertainty in Measurement was published by the International Organization for Standardization in 1993 in the name of 7 international organizationsCorrected and reprinted in 1995Usually referred to simply as the “GUM”Last remark: This is a fairly brief overview of the GUM, since it should be becoming somewhat familiar to most of you by now.
5Guide to the Expression of Uncertainty in Measurement (1993)BIPM - International Bureau ofWeights and Measureshttp//:International OrganisationsIEC International ElectrotechnicalCommisionhttp//:IFCC - International Federation ofClinical Chemistryhttp//:IUPAP - International Union ofPure and Applied Physicshttp//:IUPAC - International Union ofPure and Applied Chemistryhttp//:ISO International Organisation forStandardisationhttp//:OIML - International Organisationfor legal metrologyhttp//:
6Basic concepts Every measurement is subject to some uncertainty. A measurement result is incomplete without a statement of the uncertainty.When you know the uncertainty in a measurement, then you can judge its fitness for purpose.Understanding measurement uncertainty is the first step to reducing it
7Introduction to GUMWhen reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability.Without such an indication, measurement results can not be compared, either among themselves or with reference values given in the specification or standard.GUM 0.1
8Stated PurposesPromote full information on how uncertainty statements are arrived atProvide a basis for the international comparison of measurement results
9Benefits Much flexibility in the guidance Provides a conceptual framework for evaluating and expressing uncertaintyPromotes the use of standard terminology and notationAll of us can speak and write the same language when we discuss uncertainty
10Uses of MU QC & QA in production Law enforcement and regulations Basic and applied researchCalibration to achieve traceability to national standardsDeveloping, maintaining, and comparing international and national reference standards and reference materialsGUM 1.1
11After uncertainty evaluation No uncertainty evaluation Are these results different?R1R2After uncertainty evaluationR1R210.511.511.012.012.5mg kg-1valueR1R2No uncertainty evaluation(only precision)
12En-score according to GUM “Normalized” versus ...propagated combined uncertaintiesPerformance evaluation:0 <|En|< 2 : good2 <|En|< 3 : warning preventive action|En|> 3 : unsatisfactory corrective action
13Measurement is What is Measurement? ‘Set of operations having the object of determining a value of a quantity.’( VIM 2.1 )Note: The operations may be performed automatically.
14Basic concepts Measurement the objective of a measurement is to determine the value of the measurand, that is, the value of the particular quantity to be measureda measurement therefore begins withan appropriate specification of the measurandthe method of measurement andthe measurement procedureGUM 3.1.1
15Principles of Measurement MethodofComparisonDUTResultStandard
16Basic concepts Result of a measurement is only an estimate of a true value and only complete when accompanied by a statement of uncertainty.is determined on the basis of series of observations obtained under repeatability conditionsVariations in repeated observations are assumed to arise because influence quantitiesGUM 3.1.2GUM 3.1.4Gum 3.1.5
17Influence quantity ( VIM 2.7 ) Quantity that is not the measurand but that affects the result of measurement.Example : temperature of a micrometer used to measure length.( VIM 2.7 )
18What is Measurement Uncertainty? “parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand” – GUM, VIMExamples:A standard deviation (1 sigma) or a multiple of it (e.g., 2 or 3 sigma)The half-width of an interval having a stated level of confidence
19The value is between 22.2 mg/kg and 23.2 mg/kg UncertaintyThe uncertainty gives the limits of the range in which the “true” value of the measurand is estimated to be at a given probability..Measurement result = Estimate ± uncertainty(22.7 ± 0.5) mg/kgThe value is between 22.2 mg/kg and 23.2 mg/kg
20Measurement ErrorMeasurement ErrorReal Number SystemMeasured ValueTrue ValueMeasured values are inexact observations of a true value.The difference between a measured value and a true value is known as the measurement error or observation error.
21Basic concepts The error in a measurement Measured value – True value.This is not known because:The true value for the measurandThis is not knownThe result is only an estimate of a true value and only complete when accompanied by a statement of uncertainty.GUM 2.2.4GUM 3.2.1
22Random & Systematic Errors Error can be decomposed into random and systematic partsThe random error varies when a measurement is repeated under the same conditionsThe systematic error remains fixed when the measurement is repeated under the same conditions
23Random errorResult of a measurement minus the mean result of a large number of repeated measurement of the same measurand.( VIM 3.13 )
24Random ErrorsRandom errors result from the fluctuations in observationsRandom errors may be positive or negativeThe average bias approaches 0 as more measurements are taken
25Random errorPresumably arises from unpredictable temporal and spatial variationsgives rise to variations in repeated observationsCannot be eliminated, only reduced.GUM 3.2.2
26Systematic Errors ( VIM 3.14 ) Mean result of a large number of repeated measurements of the same measurand minus a true value of the measurand.( VIM 3.14 )
27Systematic ErrorsA systematic error is a consistent deviation in a measurementA systematic error is also called a bias or an offsetSystematic errors have the same sign and magnitude when repeated measurements are made under the same conditionsStatistical analysis is generally not useful, but rather corrections must be made based on experimental conditions.
28Systematic errorIf a systematic error arises from a recognized effect of an influence quantitythe effect can be quantifiedcan not be eliminated, only reduced.if significant in size relative to required accuracy, a correction or correction factor can be applied to compensatethen it is assumed that systematic error is zero.GUM 3.2.3
29Basic concepts Systematic error It is assumed that the result of a measurement has been corrected for all recognised significant systematic effectsGUM 3.2.4
31Correcting for Systematic Error If you know that a substantial systematic error exists and you can estimate its value, include a correction (additive) or correction factor (multiplicative) in the model to account for itCorrection - Value that , added algebraically to the uncorrected result of a measurement , compensates for an assumed systematic error(VIM 3.15)Correction Factor - numerical factor by which the uncorrected result of a measurement is multiplied to compensate for systematic error. [VIM 3.16]
32UncertaintyThe result of a measurement after correction for recognized systematic effects is still only an estimate of the value of the measurand because of the uncertainty arising;from random effects andfrom imperfect correction of the result for systematic effectsGUM 3.3.1
33Classification of effects and uncertainties Random effectsUnpredictable variations of influence quantitiesLead to variations in repeated measurementsExpected value : 0Can be reduced by making many measurementSystematic effectsRecognized variations of influence quantitiesLead to BIAS in repeated measurementsExpected value : unknownCan be reduced by applying a correction which carries an uncertaintybunjob_ajchara
37Error versus uncertainty It is important not to confuse the terms error and uncertaintyError is the difference between the measured value and the “true value” of the thing being measuredUncertainty is a quantification of the doubt about the measurement resultIn principle errors can be known and correctedBut any error whose value we do not know is a source of uncertainty.
38BlundersBlunders in recording or analysing data can introduce a significant unknown error in the result of a measurement.Measures of uncertainty are not intended to account for such mistakesGUM 3.4.7
40Population and Sample Parent Population The set of all possible measurements.SampleA subset of the population - measurements actually made.SamplesHandful of marbles from the bagPopulationBag of MarblesSlide 7
41HistogramsWhen making many measurements, there is often variation between readings. Histogram plots give a visual interpretation of all measurements at once.The x-axis displays a given measurement and the height of each bar gives the number of measurements within the given region.Histograms indicate the variability of the data and are useful for determining if a measurement falls outside of “specification”.
42For a large number of experiment replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVECharacterised by:The mean value – xgives the center of the distributionThe standard deviation – smeasures the width of the distribution
43AverageThe most basic statistical tool to analyze a series of measurements is the average or mean value :“Sum of”Individual measurementNumber of measurementsThe average of the three values 10, 15and 12.5 is given by:
44Deviation Deviation = individual value – avg value Need to calculate an average or “standard” deviationTo eliminate the possibility of a zero deviation, we square di
45Standard DeviationThe average amount that each measurement deviates from the average is called standard deviation (s) and is calculated for a small number of measurements as:Sum of deviation squaredxi = each measurement= averagen = number of measurementsNote this is called root mean square: square root of the mean of the squares
47Standard DeviationFor example, calculate the standard deviation of the following measurements: 10, 15 and 12.5 (avg = 12.5)The values deviate on average plus or minus 2.5 :12.5 ± 2.5
48Other ways of expressing the precision of the data: VarianceVariance = s2Relative standard deviationPercent RSD or Coefficient of Variation (CV)
49Standard Deviation of the Mean The uncertainty in the best measurement is given by the standard deviation of the mean (SDOM)
50Gaussian Distribution Given a set of repeated measurements which have random error.For the set of measurements there is a mean value.If the deviation from the mean for all the measurements follows a Gaussian probability distribution, they will form a “bell-curve” centered on the mean value.Sets of data which follow this distribution are said to have a normal (statistical) distribution of random data.
51POPULATION DATA For an infinite set of data, n → ∞ x → µ and s → σ population mean population std. dev.The experiment that produces a small standard deviation is more precise .Remember, greater precision does not imply greater accuracy.Experimental results are commonly expressed in the form:mean standard deviation
52The Gaussian curve equation: = Normalisation factorIt guarantees that the area under the curve is unityThe Gaussian curve whose area is unity is called a normal error curve.µ = 0 and σ = 1
53Normal Error Curve m +3s -3s +2s -2s +1s -1s • 68.3% of measurements will fall within ± s of the mean.+3s-3s+2s-2s+1s-1sRelative frequency, dN / N95.5% of measurements will fall within ± 2s of the mean.99.7% of measurements will fall within ± 3s of the mean.xi
54EXAMPLEReplicate results were obtained for the measurement of a resistor. Calculate the mean and the standard deviation of this set of data.Replicateohms17522756347515760
55NB DON’T round a std dev. calc until the very end. Replicateohms17522756347515760NB DON’T round a std dev. calc until the very end.
57Student's t-Distribution If the sample size is not large enough, say n ≤ 30.Then the distribution of is not normal.It has a distribution called Student’s t-distribution.t = (x – )/(s/n).
58Student's t-Distribution The Student's t-distribution was discovered by W. S. Gosset in 1908.He used the pseudonym ‘Student’ to avoid getting fired for doing statistics on the job!!
59Student's t-Distribution The shape of the Student's t-distribution is very similar to the shape of the standard normal distribution.The Student's t-distribution has a (slightly) different shape for each possible sample size.They are all symmetric and unimodal.They are all centered at 0.
60Student's t-Distribution They are somewhat broader than normaldistribution, reflecting the additional uncertainty resulting from using s in place of .As n gets larger and larger, the shape of the t-distribution approaches the standard normal.
61Degrees of FreedomIf the sample size is n, then t is said to have n – 1 degrees of freedom.We use df to denote degrees of freedom.
62Student's t-Distribution for 95% Confident level
63Student's t-Distribution When s is estimated from the sample standard deviation , sThe distribution for the mean follows at- distribution with degrees of freedom, n − 1
64CONFIDENCE INTERVALThe confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean,The confidence interval is given by:Where t is the value of student’s t taken from the table
6595% confidence interval; n = 11 Use of t-Table95% confidence interval; n = 11Degrees of Freedom0.800.900.950.980.9913.07776.31412.70631.82163.65721.88562.92004.30276.96459.9250............101.37221.81252.22812.76383.1693............1001.29011.66041.98402.36422.62591.2821.64491.96002.32632.5758
68Example:The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content.Find x = 1.63s = 0.13150% confidence:t = for n-1 = 3There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm Hg.
6990% confidence: t = 2.353 for n-1 = 3 90% 50% 1.631.681.481.581.7890%50%90% confidence:t = for n-1 = 3There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm
70Evaluation of Measurement Uncertainty bunjob_ajchara
71Terms specific to the GUM Standard uncertainty,the uncertainty of the result of a measurement expressed as a standard deviationType A evaluation (of uncertainty)method of evaluation of uncertainty by the statistical analysis of a series of observationsType B evaluation (of uncertainty)method of evaluation of uncertainty by means other than the statistical analysis of series of observationsGUM 2.3.1GUM 2.3.2GUM 3.2.3
72Terms specific to the GUM Combined standard uncertaintythe standard deviation of the result of a measurement when the result is obtained from the values of a number of other quantities.It is obtained by combining the individual standard uncertainties (and covariances as appropriate), using the law of propagation of uncertainties, commonly called the "root-sum-of-squares" or "RSS method.GUM 2.3.4
73Terms specific to the GUM expanded uncertaintyquantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand.coverage factor, knumerical factor used as a multiplier of combined standard uncertainty in order to obtain expanded uncertaintyGUM 3.2.5GUM 3.2.6
74Process of Uncertainty Estimation Specify MeasurandIdentify all Uncertainty SourcesQuantify Uncertainty ComponentsCalculate Combined Uncertainty
76The measurand?Measurand = particular quantity subject to measurement [VIM 2.6 / GUM B.2.9]Example: the conventional mass of a 1kg weight.GUM 1.2
77Measurement ModelDefine the measurand – the quantity subject to measurementDetermine a mathematical model, with input quantities, X1,X2,…,XN, and (at least) one output quantity,Y.The values determined for the input quantities are called input estimates and are denoted by x1,x2,…,xN.The value calculated for the output quantity is called the output estimate and denoted by y.
78Identify all Uncertainty Sources 782. How is MU estimated?
79ISO/IEC 17025attempt to identify all the components of uncertaintyAll uncertainty components which are of importance shall be taken into account
80Sources of uncertainty ISO/IEC Note 1:Some sources contributing to the uncertainty:reference standardsreference materialsmethodsequipmentenvironmental conditionsproperties and condition of the item to be testedthe operator
81Sources of MU GUM 3.3.2 Incomplete definition of the measurand Imperfect realisation of the definition of the measurandNon-representative samplingEffects of environmental conditions on the measurementPersonal bias in reading analogue instrumentsFinite instrument resolution or discrimination thresholdInexact values of measurement standardsInexact values of constants obtained from external sourcesApproximations incorporated into the measurementVariations in repeated observations under apparently identical conditions812. How is MU estimated?
82Causes for uncertainty Measurement standardMeasuring methodsCalibration certificateSecular changeMeasuring instrumentMeasurement resultsMeasurerManufacturer’s specificationResolutionPeculiarities in readoutMeasurement environmentNumber of measurementsDispersions in repetition
83Sources of error and uncertainty in dimensional calibrations Reference standards and instrumentationThermal effectsElastic compressionCosine errorsGeometric errorsUKAS M3003 Dec 1999bunjob_ajchara
84Sources of error and uncertainty in electrical calibrations Instrument CalibrationSecular StabilityMeasurement ConditionsInterpolation of calibration dataResolutionLayout of apparatusThermal emfsLoading and lead impedanceRF mismatch errors and uncertaintyDirectivityTest port matchRF Connector repeatabilityUKAS M3003 Dec 1999bunjob_ajchara
85Sources of error and uncertainty in mass calibrations Reference weight calibrationSecular stability of reference weightsWeighing machine / weighing processAir buoyancy effectsEnvironmentUKAS M3003 Dec 1997bunjob_ajchara
86Quantify Uncertainty Components 862. How is MU estimated?
87The Measurement ModelUsually the final result of a measurement is not measured directly, but is calculated from other measured quantities through a functional relationshipThis is called function a “measurement model”The model might involve several equations, but we’ll follow the GUM and represent it abstractly as a single equation:
88Input and Output Quantities In the generic model Y = f(X1,…,XN), the measurand is denoted by YAlso called the output quantityThe quantities X1,…,XN are called input quantitiesThe value of the output quantity (measurand) is calculated from the values of the input quantities using the measurement model
89Input and Output Estimates When one performs a measurement, one obtains estimated values x1,x2,…,xN for the input quantities X1,X2,…,XNThese estimated values may be called input estimatesThe calculated value for the output quantity may be called an output estimate
90Measurement modelA measurand Y can be determined from N inputs quantities X1, X2, X3 … XNThe model is written abstractly as Y=f(X1,X2,…,XN) where X1,X2,…,XN are input quantities and Y is the output quantity
91Developing a Measurement model Decide what is the measurand Ythe quantity subject to measurementDecide what are the quantities X1, …, XN influencing the measurementobserved quantities, applied corrections, material properties, etcDecide the relationship between Y and X1, …, XNthe model of the measurementbunjob_ajchara
92Example: CALBRATION OF A HAND-HELD DIGITAL MULTIMETER AT 100 V DC The error of indication EX of the DMM to be calibrated is obtained fromwhereVi X - voltage, indicated by the DMM (index i means indication),VS - voltage generated by the calibrator,δ VI X - correction of the indicated voltage due to the finite resolutionof the DMM,δ VS - correction of the calibrator voltage due to(1) drift since its last calibration,(2) deviations resulting from the combined effect of offset, non-linearity and differences in gain,(3) deviations in the ambient temperature,(4) deviations in mains power,(5) loading effects resulting from the finite input resistance of the DMM to be calibrated.EA-4/02:1999
93Measurement modelAn estimate of Y, denoted by y, is obtained from x1, x2, x3 … xN, the estimates of the input quantities X1, X2, X3 … XN,Represent each input quantity Xi by1. Best estimate xi as mean of distribution, and2. Standard uncertainty u(xi) as s.d. of distributionbunjob_ajchara
94Measurement Model For each input quantity Obtain knowledge of that quantityAssign a probability distribution to each quantity consistent with that knowledgeOften a Gaussian (normal) or a rectangular distributionbunjob_ajchara
95Classification of uncertainty components Type A components: those that are evaluated by statistical analysis of a series of observationsType B components: those that are evaluated by other meansBoth based on probability distributionsstandard uncertainty of each input estimate is obtained from a distribution of possible values of input quantity: both based on the state of our knowledgeType A founded on frequency distributionsType B founded on a priori distributions
96Type A evaluations of uncertainty Type A evaluations of uncertainty are based on the statistical analysis of a series of measurements.
97Type A Evaluation of Standard Uncertainty For component of uncertainty arising from random effectApplied when multiple independent observations are made under the same conditionsData can be from repeated measurements, control charts, curve fit by least-squares method etcObtained from a probability density function derived from an observed frequency distribution (usually Gaussianbunjob_ajchara
98Type A Evaluation Arithmetic mean Best estimate of the expected value of a input quantity -
99Type A Evaluation Experimental standard deviation Distribution of the quantity
100Type A Evaluation Experimental standard deviation of the mean spread of the distribution of the means -
101Type A EvaluationType A standard uncertaintydegrees of freedom
102ExampleA digital multimeter is used to measure a high value resistor and the following readings are recorded.The standard uncertainty, u, is therefore kΩ.
103Type A Evaluation Pooled Experimental Standard Deviation For a well-characterized measurement under statistical control, a pooled experimental standard deviation Sp that characterizes the measurement may be available.The value of a measurand q is determined from n independent observations andThe standard uncertainty is
104Type A Evaluation Example: A previous evaluation of the repeatability of measurement process (10 comparisons between standard and unknown) gave an experimental standard deviationIf 3 comparisons between standard and unknown were made this time (using 3 readings on the unknown weight), this is the value of n that is used to calculate the standard uncertainty of the measurand
105Type B Evaluation of Standard Uncertainty Evaluation of standard uncertainty is usually based on scientific judgment using all relevant information available, which may include:previous measurement data,experience with, or general knowledge of the behavior and property of relevant materials and instruments,manufacturer's specifications,data provided in calibration and other reports, anduncertainties assigned to reference data taken from handbooks.GUM 4.3.1
106Type B Evaluations Normal distribution: Doc.: IEEE /0333r0March 2006Type B EvaluationsNormal distribution:Examples:expanded uncertainties from a calibration certificatewhere Ui is the expanded uncertainty of the contribution and k is the coverage factor (k = 2 for 95% confidence).March 2006Dr. Michael D. Foegelle, ETS-Lindgren
107Type B Evaluations Normal distribution Example A calibration certificate reports the measured value of a nominal 1kg OIML weight class F2 at approximately 95% confidence level as:
108“It is likely that the value is somewhere in that range” Rectangular distribution“It is likely that the value is somewhere in that range”Rectangular distribution is usually described in terms of: the average value and the range (±a)Certificates or other specification give limits where the value could be,without specifying a level of confidence (or degree of freedom).1/2a2a(= a)XThe value is between the limitsThe expectation
109Rectangular distribution Range = 2a ,Semi-range = Range /2 = aaaP=1/2aAB
111Example Example of Rectangular distribution From the previous example, if the Maximum Permissible Error (MPE) according to OIML class F2 (±16 mg) is used; then
112Example of Rectangular distribution HandbookA Handbook gives the value of coefficient of linear thermal expansion of pure copper at 20and the error in this value should not exceed,assuming rectangular distribution the standard uncertainty is:
113Example of Rectangular distribution Manufacturer’s SpecificationsA voltmeter used in the measurement process has the accuracy of ± 1 % of full scale on 100 V. rangesemi - range ( a ) = 1 V
114Example of Rectangular distribution Resolution of a digital indication123456If the resolution of the digital device is δx, the value of X can lie with equal probability anywhere in the interval X - δx /2 to X + δx /2 and thus described by a rectangular probability distribution with the width δx
115Example of Rectangular distribution Digital indicationA digital balance having capacity of 210g and the least significant digit 10 mg. The standard uncertainty contributed by this balance is:
116Example of Rectangular distribution HysteresisThe indication of instrument may differ by a fixed and known amount according to whether successive reading are rising or falling.If the range of possible readings from that is dx
117U-shaped distribution Doc.: IEEE /0333r0March 2006U-shaped distributionWhen the measurement result has a higher likelihood of being some value above or below the median than being at the median.Examples:Mismatch (VSWR)Distribution of a sine waveMarch 2006Dr. Michael D. Foegelle, ETS-Lindgren
118Example of U-Shaped distribution A mismatch uncertainty associated with the calibration of an RF power sensor has been evaluated as having semi-range limits of 1.3%. Thus the corresponding standard uncertainty will beUKAS M3003bunjob_ajchara
119Triangular distribution Distribution used when it is suggested that values near the centre of range are more likely than near to the extremes2a (=a)1/aAssumed standard deviation:X
120Example of Triangular distribution Values close to x are more likely than near the boundariesExample: A tensile testing machine is used in a testing laboratory where the air temperature can vary randomly but does not depart from the nominal value by more than 3°C. The machine has a large thermal mass and is therefore most likely to be at the mean air temperature, with no probability of being outside the 3°C limits. It is reasonable to assume a triangular distribution, therefore the standard uncertainty forits temperature is:UKAS M3003In case of doubt, use the rectangular distribution
121It should be recognized that a Which is better A or B?It should be recognized that aType B evaluation of a standard uncertainty can be as reliable as a Type A evaluation, especially in a measurement situation where a Type A evaluation is based on a comparatively small number of statistically independent observation.GUM 4.3.2
123combined standard uncertainty Components of standard uncertainty of measurand y=f(x1,x2,x3……xN) are combined using the “ Law of Propagation of Uncertainty” or “Root Sum of Square :RSS”bunjob_ajchara
124Combined Standard Uncertainty, uc The relationship between the measurand, Y, and A, B and C is written most generally as Y = f(A,B,C).u(a), u(b) and u(c) are the standard uncertainties of best estimates a, b and c respectively obtained through Type A or Type B evaluations.
126sensitivity coefficient Partial derivative with respect to input quantities Xi of functional relationship between measurand Y and input quantities Xi on which Y dependssensitivity coefficient formulabunjob_ajchara
127ExampleThe value of the resistance Rt, at the temperature t, is obtained from equation:Where:α is the temperature coefficient of the resistor in Ω / °ct is the temperature in °c , andR0 is the resistance in ohms at the reference temperature,The partial differentiation of Rt with respect to t is:bunjob_ajchara
128Correlation of Input Quantities SRefRefSUUTUUTScorrDifference (Correction Ref-UUT)bunjob_ajchara
131Uncorrelated input quantities For uncorrelated input quantities r (xi , xj) = 0For ci =1bunjob_ajchara
132Combinations of Uncertainties Addition/SubtractionFor independent variables, we have,
133Combinations of Uncertainties Multiplication/DivisionSimilar arguments would apply tothe expressionFor independent variables, we have,
134Worked exampleThe mass, m, of a wire is found to be g with a standard uncertainty of g. The length, l, of the wire is m with a standard uncertainty of m. The mass per unit length, , is given by:Determine the,a) best estimate of ,b) standard uncertainty in .
135Worked example continued The partial differentiation of µ with respect to m and l
136correlated input quantities For the very special case where all input estimates are correlatedThe combined standard uncertaintybunjob_ajchara
137Correlated input quantities Example1)Ri (R1,R2,R3,……,R10) each has nominal value 1000 ohms2)Each has been calibrated by direct comparison with negligible uncertainty3)Standard uncertainty of Rs is u(Rs) = 100 mohmsModel equation :R1R2R3R10Rref10 kWbunjob_ajchara
139Expanded Uncertainty expanded uncertainty quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand.GUM 3.2.5
140Expanded Uncertainty, U The Expanded Uncertainty, U, is a simple multiple of the standard uncertainty, given byU = kuc(y)k is referred to as the coverage factor.So we can write:Y = y U
141coverage factor, k coverage factor, k numerical factor used as a multiplier of combined standard uncertainty in order to obtain expanded uncertaintyGUM 3.2.6
142Coverage factor Coverage Factor - k Confidence Interval 1.00 68.27% 2.0095.45%2.5899.%3.0099.73%Most cal labs adopt 95.45% which gives k 2for effective degrees of freedom 30
143Coverage Factor of Combined Uncertainty Effective Degree of Freedomto determine the coverage factor of combined uncertainty, the effective degree of freedom must be first calculated from the Welch-Satterthwaite formula:Based on the calculated veff, obtain the t-factor tp(veff) for the required level of confidence p from the t-distribution tableThe coverage factor will be: kp = tp(veff)bunjob_ajchara
144Effective number of degrees of freedom Example -- A steel rod was measured 4 times. The calculated .The effective degree of freedom:For @ 95% confidence level and from “student’s t” table, we get k = 2.52bunjob_ajchara
145Effective number of degrees of freedom Therefore, the expanded uncertainty U is:bunjob_ajchara
146Relative standard uncertainty Relative standard uncertainty of input estimate ,Relative combined standard uncertainty, ythenbunjob_ajchara
147Relative standard uncertainty ExampleThe measurand:DescriptionValue,xStandard uncertainty,Relative standard uncertainty,repRepeatability1,00,0005Weight of KHP0,3888 g0,00013g0,00033Purity of KHP0,00029Molar mass of KHP204,2212 gmol-10,0038gmol-10,000019Volume of NaOH for KHP titration18,64 ml0,013ml0,0007bunjob_ajchara
148Relative standard uncertainty 1) Value of the measurand= 0,10214 mol l-12) Combined relative standard uncertaintyuc(CNaOH) = 0,00097 X mol l-1 = 0,00010 mol l-1bunjob_ajchara
150Reporting should include example of uncertainty statement result of measurementexpanded uncertainty with coverage factor and level of confidence specifieddescription of measurement method and reference standard useduncertainty budgetexample of uncertainty statemente.g.The expanded uncertainty of measurement is ± ____ , estimated at a level of confidence of approximately 95% with a coverage factor k = ____.
151Reporting ResultIt usually suffices to quote uc(y) and U [as well as the standard uncertainties u(xi) of the input estimates xi] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round-off errors in subsequent calculations.In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, uc(y) = 10,47 m might be rounded up to 11 m.However, common sense should prevail and a value such as u(xi) = 28,05 kHz should be rounded down to 28 kHz.Output and input estimates should be rounded to be consistent with their uncertainties.GUM 7.2.6bunjob_ajchara
152Reporting Conventions 1000 (30) mLDefines the result and the (combined) standard uncertainty1000 +/- 60 mLDefines the result and the expanded uncertainty (k=2)1000 +/- 60 mL at 95% confidence level.Defines the expanded uncertainty at the specified confidence interval
153The 9-steps GUM Sequence 1. Define the measurand 2. Build the model equation 3. Identify the sources of uncertainty 4. (If necessary) Modify the model 5. Evaluate of the input quantities and calculate the value of the result 6. Calculate the value of the measurand (using the equation model) 7.Calculate the combined standard uncertainty of the result 8. Calculate the expanded uncertainty (with a selected k) 9. Report resultbunjob_ajchara
155Some Important Practical Consequences … or a little common sense with errors!When several (independent) errors are to be added, addition in quadratureis much more realistic than addition.If one error ie less than one quarter of another error in the addition thenthe smaller error may be realistically ignored.There is little point in spending much time estimating small errors –concentrate on the large errors!The experimental procedure should minimise the dominant errors, This impliesthat these must be identified and estimated (usually in a pilot run) before thefinal data is taken.Try to bring the precision of each variable to a common level, if possible, byrepeated measurements.
156Basic concepts“…The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of measurement…”GUM
158Bibliography and acknowledgement ISO (1993) Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organisation for Standardisation).NIST Technical Note 1297 (1994) Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.M 3003, The Expression of Uncertainty and Confidence in Measurement, published by UKASEA-4/02 - December 1999• Expression of the Uncertainty of Measurement in CalibrationEURACHEM / CITAC Guide: Traceability in ChemicalMeasurement - A guide to achieving comparable results in chemical measurement 2003Assessment of Uncertainties of Measurement for Calibration and Testing Laboratories - Second Edition , c R R Cook 2002Published by National Association of Testing Authorities, AustraliaACN ISBN