Objective of a measurementTo determine a value of the measurand To sample one value out of a universe of possible values Measurement Uncertainty

Variability in Measurementwhen a measurement is repeated many times, one will obtain many different answersThis observed variability in the results of repeated measurements arises because influence quantities that can affect the measurement result are not held constant. In general, there are many- if not infinitely many- influence quantities affecting a measurement result Measurement Uncertainty

Measurement UncertaintySome concepts related to uncertainty

True value of a measurand is indeterminate, except when known in terms of theory.

What is obtained from the measurement process an estimate of or approximation to the true value.

Accuracy of measurementthe closeness of agreement between a test result and the accepted reference value

A statement of results of measurement is complete only if it contains both the values attributed to the measurand and the uncertainty in measurement associated with that value.

Without such an indication, measured results can not be compared, either among themselves or with reference values given in a specification or standardMeasurement Uncertainty

Systematic and random components of uncertainty affecting the observed results

Random components arise from unpredictable and spatial variations of influence quantities, like:the way connections are made or the measurement method employeduncontrolled environmental conditions inherent instability of the measuring equipmentpersonal judgement of the operator, etc.

These cannot be eliminated totally, but can be reduced by exercising appropriate controls.Measurement Uncertainty

Systematic componentsthose reported in the calibration certificate of the reference standards/ instruments useddifferent influence conditions at the time of measurement compared with those prevalent at the time of calibration of the standard etc.Measurement Uncertainty

Uncertainty of Measurementa parameter characterizing the range of values within which the value of the measurand can be said to lie within a specified level of confidence

Uncertainty (of measurement) Parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand.Measurement Uncertainty

ISO/ IEC 17025 Measurement Uncertainty - Requirements5.4.6.1 Calibration laboratories and Testing laboratories carrying their own calibrations shall estimate measurement uncertainty of calibrations

The method of estimation of measurement uncertainty for calibrations where traceability is not applicable/ feasible becomes less rigorous

Measurement Uncertainty

ISO/ IEC 17025 Measurement Uncertainty - Requirements5.4.6.2 - Testing laboratories shall have and apply procedures for estimating uncertainty of measurement

Nature of the test method may preclude rigorous, meteorologically and statistically valid calculation of uncertainty of measurement

Laboratory shall at least attempt to identify all uncertainty components, make a reasonable estimation and ensure that the form of reporting of result does not give a wrong impression of uncertaintyMeasurement Uncertainty

ISO/ IEC 17025 Measurement Uncertainty - Requirements Note 1 - Degree of rigor depends on:Requirements of the methodRequirements of the clientExistence of narrow limits on specification conformance

Note 2 - In cases where a well-recognized test method specifies limits to the values of the major sources of uncertainty and specifies the form of presentation of calculated results, the laboratory is considered to have satisfied this clause by following the test method and reporting instructionsMeasurement Uncertainty

ISO/ IEC 17025 Measurement Uncertainty - Reporting Requirements5.10.3.1 c) Information on uncertainty is needed in test reports when it is relevant to the validity or application of the test results, when a customers instruction so requires, or when the uncertainty affects compliance to a specification limit

5.10.1 - In the case of a written agreement with the customer, results may be reported in a simplified wayMeasurement Uncertainty

Five categories of test methodsQualitative No uncertainty calculations required Examples: Ignitability, Microbiological Screening

Well recognized methods that specify limits to uncertainty contributionsNo further uncertainty calculations requiredExamples: Flash point, Hardness, Vicat Softening Temperature, Mooney Viscosity Hardness (Brinell, Vickers), Tension & Compression Proof, pH of Water Extract Measurement Uncertainty

Five categories of test methods contd.Published methods that do not specify limits to uncertainty sources and/ or reporting Format Uncertainty estimated using standard deviation of laboratory control samples Examples: Alloy analysis, Medical testing

Problems: Normal process for analyzing control samples may lead to an underestimate of uncertaintyMeasurement Uncertainty

Five categories of test methods, continuedMethods requiring identification of major uncertainty components and reasonable estimate of uncertainty Examples: One-off tests

Methods requiring full uncertainty analysis consistent with ISO Guide to the Expression Of Uncertainty in MeasurementExample: Reference material value assignment, Dimensional InspectionMeasurement Uncertainty

Estimation of measurement uncertainty needs

Understanding the objective of the measurement Identification of the factors influencing the measurement result

It depends on detailed knowledge of the nature of the measurand and of the measurement. Measurement Uncertainty

The evaluation of uncertainty is neither a routine task nor a purely mathematical one Even with industry standard test methods that set tolerances on all the various parts and features of a testing machine, places limits on the environmental conditions, and specifies the method of preparation of the samples, the nature of the material itself may be the major source of variability in test resultsMeasurement Uncertainty

Since uncertainty evaluation is neither a purely mathematical task nor a merely routine task, the details of its procedures. as applied to a given test, can never be codified to the point of complete unambiguousnessEven after correcting for known systematic effects, the corrected measurement result is still only an estimate of the value of the measurand because of random effects and because our knowledge of the magnitudes of the corrections is itself only an estimate uncertainty of a measurement must not be confused with the remaining unknown errorMeasurement Uncertainty

Sources of uncertainty in measurementincomplete definition of the measurandimperfect realization of the definition of the measurandnonrepresentative sampling inadequate knowledge of the effects of environmental conditions and their imperfect measurementpersonal bias in reading analog instruments, including the effects of parallaxMeasurement Uncertainty

Sources of uncertainty in measurementfinite resolution or discrimination thresholdinexact values of measurement standards and reference materialsinexact values of constants and other parameters obtained from external sources and used in the data-reduction algorithmapproximations and assumptions incorporated in the measurement method and procedurevariations in repeated observations of the measurand under apparently identical conditions Measurement Uncertainty

If all of the quantities on which the result of a measurement depend could be varied, then its uncertainty could be evaluated based purely on a statistical treatment of experimental data. But this is seldom possible in practice because of the time and expense involved in such an exhaustive experimental evaluation of uncertainty Whenever possible, the use of check standards and control charts (often called measurement assurance) should be used to establish that a measurement system is under statistical controlThis data should be used as part of the effort to obtain a reasonable estimate of the measurement uncertainty Measurement Uncertainty

Measurement UncertaintyIf the input quantities are designated as x1, x2, , xn, then we can write the functional relationship between the measurement result y and the input quantities asy = f (x1, x2,....... xn )This function becomes very complicated in many testing measurementsIf a laboratory has sufficient data, analysis by regorous method is not necessary to satisfy the measurement uncertainty requirements of ISO/IEC 17025: 1999

Measurement UncertaintyTypes of Uncertainty EstimationType A uncertainty estimate is an estimate derived from the statistical analysis of experimental data Some uncertainty contributors cannot be evaluated statistically, or else a statistical evaluation would be impractical, or a statistical evaluation may simply be unnecessary. In these cases, the magnitude and associated uncertainty of an influence quantity has to be estimated based on past experience, taken from a handbook, extracted from a calibration report, etc. Estimates obtained in this way are called Type B estimates

Measurement UncertaintyIt will usually be the case that the best estimate of the value of a measurand will be the average of several test results The experimental standard deviation s characterizes the variability or spread, of the observed values xi It is given by the equations = [(x1 x0)2 + (x2 x0)2 + . (xn x0)2] / (n-1)It is best to use a calculator or spreadsheet program like Excel for these calculations Standard Uncertainty = s/ n

Sensitivity coefficients are essentially conversion factors that allow conversion of the units of an input quantity into the units of the measurand Mathematically, sensitivity coefficients are obtained from partial derivatives of the model function f with respect to the input quantities Measurement UncertaintySensitivity coefficients can also be evaluated experimentally. In cases where the model function is not known, obviously it is not possible to determine the sensitivity coefficients mathematically

The ISO Guide for Uncertainty of Measurement relies on identifying and quantifying the uncertainties of the input quantities and expressing those uncertainties as one standard deviation. The combined standard uncertainty is consequently a standard deviation and for a normal distribution u = (ua )2 + [(ub1 )2 + (ub2 )2 + . (ubn )2]Ua Type A component of uncertaintyUb1, ub2 Ubn Type B components of uncertaintyMeasurement Uncertainty

Measurement UncertaintyX0X0 + s (68.3 %)X0 + 2s (95.5 %)X0 + 3s (99.73 %)X0 MeanS Standard DeviationNormal Distribution

Measurement UncertaintyOne standard deviation encompasses approximately 68% of possible values of the measurandFor 95% confidence limits coverage factor of 2 is used Two standard deviations

SPECIFYING THE MEASURANDAny uncertainty analysis must begin with a clear specification of the measurand In complex tests, it is not necessarily clear what is being measured and what is influencing the measurement result.Level of detail in the definition of the measurand depends on the required level of accuracy of the measurement. The specification of a measurand may require statements about quantities such as time, temperature, and pressure. Measurement Uncertainty

MODELING THE MEASUREMENT Sources of uncertainty in measurementincomplete definition of the measurandimperfect realization of the definition of the measurandnonrepresentative sampling (the sample measured may not represent the defined measurand)inadequate knowledge of the effects of environmental conditions on the measurement or imperfect measurement of environmental conditionspersonal bias in reading analog instruments, including the effects of parallaxfinite resolution or discrimination thresholdMeasurement Uncertainty

Sources of uncertainty in measurement contd.inexact values of measurement standards and reference materialsinexact values of constants and other parameters obtained from external sources and used in the data-reduction algorithmapproximations and assumptions incorporated in the measurement method and procedurevariations in repeated observations of the measurand under apparently identical conditionsMeasurement Uncertainty

Usually the mathematical model is always incomplete Relevant input quantities should be varied to the extent possible so that the uncertainty estimate can be based, as much as possible, on experimental data the use of check standards and control charts (often called measurement assurance) to ensure that the measurement system is under statistical control, these data should be used as part of the effort to obtain a reasonable estimate of the measurement uncertainty. When the observed data shows that the mathematical model is incomplete, then the model should be revised Measurement Uncertainty

TYPE A EVALUATION OF STANDARD UNCERTAINTYType A uncertainty estimate is an estimate derived from the statistical analysis of experimental data. Type A does not refer to the nature of the uncertainty contributor itselfType A uncertainty estimates are not necessarily random components of uncertaintyMeasurement Uncertainty

It will usually be the case that the best estimate of the value of a measurand will be the average of several test results The experimental standard deviation s characterizes the variability or spread, of the observed values xi It is given by the equations = [(x1 x0)2 + (x2 x0)2 + . (xn x0)2] / (n-1)It is best to use a calculator or spreadsheet program like Excel for these calculations Standard Uncertainty = s/ nMeasurement Uncertainty

TYPE B EVALUATION OF STANDARD UNCERTAINTYSome uncertainty contributors cannot be evaluated statisticallyelse a statistical evaluation would be impracticala statistical evaluation may simply be unnecessary.

past experiencetaken from a handbook,extracted from a calibration report etc. In these cases, uncertainty of an influence quantity has to be estimated based on

Estimates obtained in this way are called type B estimates.Measurement Uncertainty

SENSITIVITY COEFFICIENTSSensitivity coefficients are essentially conversion factors that allow one to convert the units of an input quantity into the units of the measurand. Sensitivity coefficients are also measures of how much change is produced in the measurand by changes in an input quantity. Mathematically, sensitivity coefficients are obtained from partial derivatives of the model function f with respect to the input quantities. Measurement Uncertainty

Combined UncertaintyType A Component of uncertainty UaType B Components Ub1, Ub2, Ub3 ...... Type B estimate = Ub = Ub12 + Ub22 + Ub32 ....

Combined Uncertainty (Uc) = Ua2 + Ub2

If U1, U2, U3 ........ are the uncertainty components and corresponding sensitivity coeficients are C1, C2, C3 ........ the the combined uncertainty is

Uc = C1 U12 + C2 U22 + C3 U32 ....Measurement Uncertainty

Although the combined standard uncertainty can be used to express the uncertainty of a measurement result, in many commercial, industrial, or regulatory applications, it is often necessary to give a measure of uncertainty that defines an interval about the measurement result that may be expected to encompass a larger fraction of the values (95%)

The expanded uncertainty U is obtained by multiplying the combined standard uncertainty by a coverage factor k. U = KUcK is chosen from Student t distribution (for df < 30) or Normal distribution (for df > 30) and confidence level (95%)Measurement Uncertainty

ReasonabilityEvery uncertainty estimate should be subjected to a Reasonability checkUncertainty estimates that look strange- either too big or too small- should be re-evaluated Engineering tolerancesLong experience with the mechanical propertiesLook for Mathematical blunders, Uncertainty contributors which have been poorly estimated or completely neglected. Finally revise the mathematical model if needed Measurement Uncertainty

UNCERTAINTY BUDGETSA well-documented uncertainty evaluation contains Identification and value of each input estimate Its standard uncertainty A description of how they were obtainedDegrees of freedom for the standard uncertainty of each input estimate How they were obtained Functional relationship between the measurand and the input quantities Sensitivity coefficientsMeasurement Uncertainty

Uncertainty Budget

Commonly used DistributionsMeasurement Uncertainty

DistributionCoverage factor @ 95 % CLNormal1.96 (or 2)Rectangular3Triangular6U shaped2

SUMMERY OF THE METHODSpecify the measurandDerive the mathematical modelQuantify the influence quantitiesEvaluate the standard uncertainty of each influence quantityEvaluate sensitivity coefficients and covariancesCalculate the measurement resultDetermine the combined standard uncertaintyDetermine the expanded uncertaintyReport the measurement result and associated uncertainty estimateMeasurement Uncertainty

Measurement UncertaintyExample: Volume of pipette is estimated by weighing water dispensed by the pipette and dividing by the density of water, taken from standard tables at the measurement temperaturev = w/ dSensitivity coefficient for weight (Csw)v/ w = 1/d = v/ wSensitivity coefficient for density (Csd)v/ d = -w/d2 = -v/ dUncertainty in volume measurementUv = (Csw)2Uw2 + (Csd)2Ud2

Measurement UncertaintyWe measure 5 (n = 5) times weights and note down corresponding temperatures Weight TemperatureMean value W0 T0Standard deviation sw st Type A component sw / n st / nType B componentsCalibration Ubw1 Ubt1Resolution Ubw2 Ubt2Combined Uw = Uaw2 + Ubw12 + Ubw22uncertainty Ut = Uat2 + Ubt12 + Ubt22

Measurement UncertaintyDensity of water at T0 = d0 from the tablesUncertainty of density measurement corresponding to Ut uncertainty measurement in temperature = UdVolume of pipette V0 = W0/ d0

Uncertainty estimate in volume measurementUv = (Csw)2Uw2 + (Csd)2Ud2

Volume of pipette = V0 + KUvK is the coverage factor = 2 @ 95 % Confidence limits