Draft EURACHEM/CITAC Guide Quantifying Uncertainty in ......Quantifying Uncertainty Contents DRAFT: EURACHEM WORKSHOP, HELSINKI 1999 Page i Contents Foreword to the Second Edition

DRAFT: EURACHEM WORKSHOP, HELSINKI 1999

Draft EURACHEM/CITAC Guide

Quantifying Uncertainty in Analytical Measurement

Second Edition

Draft: June 1999

Prepared by the EURACHEM Measurement Uncertainty Working Group

in collaboration with members of CITAC and AOAC International

Quantifying Uncertainty Contents

DRAFT: EURACHEM WORKSHOP, HELSINKI 1999 Page i

Contents

Foreword to the Second Edition 1

1. Scope 3

2. Uncertainty 4

3. Analytical Measurement and Uncertainty 7

4. Principles of Measurement Uncertainty Estimation 11

5. Step 1. Specification 13

6. Step 2. Identifying Uncertainty Sources 14

7. Step 3 - Quantifying Uncertainty 16

8. Step 4. Calculating the Combined Uncertainty 24

9. Reporting uncertainty 28

Appendix A. Examples - Introduction 31

Example A1. Preparation of a calibration standard 33

Example A2. Standardising a sodium hydroxide solution 39

Example A3. An acid/base titration 49

Example A4. Using validation data. Organophosphorus pesticides in bread. 59

Example A5. Determination of cadmium release from ceramic ware 70

Example A6. Using collaborative study data. Crude Fibre in Animal Feed 80

Example A7. Lead in water using Double Isotope Dilution ICP-MS 88

Appendix B - Definitions 95

Appendix C – Uncertainties in Analytical Processes 99

Appendix D: Analysing uncertainty sources 100

Appendix E - Useful Statistical Procedures 102

Appendix F - Common sources and values of uncertainty 112

Appendix G - Bibliography 119

Quantifying Uncertainty Foreword to the Second Edition

DRAFT: EURACHEM WORKSHOP, HELSINKI 1999 Page 1

Foreword to the Second EditionMany important decisions are based on the results of chemical quantitative analysis; the results are used, forexample, to estimate yields, to check materials against specifications or statutory limits, or to estimatemonetary value. Whenever decisions are made on the basis of analytical results, it is important to havesome indication of the quality of the results, that is, the extent to which they can be relied on for the purposein hand. Users of the results of chemical analysis, particularly in those areas concerned with internationaltrade, are coming under increasing pressure to eliminate the replication of effort frequently expended inobtaining them. Confidence in data obtained outside the user’s own organisation is a prerequisite tomeeting this objective. In some sectors of analytical chemistry it is now a formal (frequently legislative)requirement for laboratories to introduce quality assurance measures to ensure that they are capable of andare providing data of the required quality. Such measures include: establishing traceability of themeasurements, the use of validated methods of analysis, the use of defined internal quality controlprocedures, participation in proficiency testing schemes, and becoming accredited to an InternationalStandard, normally the ISO/IEC Guide 25 [G.1]. In analytical chemistry there has been in the past greateremphasis on precision of results obtained using a specified method rather than traceability to a definedstandard or SI unit. Consequently this has led the use of “official methods” to fulfil legislative and tradingrequirements. However as there is now a formal requirement to establish the confidence of results it isessential that a measurement is traceable to defined standard such as a SI unit, reference material or whereapplicable a defined, or empirical, (sec. 5.2.) method. Internal quality control procedures, proficiencytesting and accreditation can be an aid in establishing traceability to a given standard.

As a consequence of these requirements, chemists are, for their part, coming under increasing pressure todemonstrate the quality of their results, i.e. to demonstrate their fitness for purpose by giving a measure ofthe confidence that can be placed on the result, including the degree to which a result would be expected toagree with other results, normally irrespective of the methods used. One useful measure of this ismeasurement uncertainty.

Although the concept of measurement uncertainty has been recognised by chemists for many years it wasthe publication in 1993 of the “Guide to the Expression of Uncertainty in Measurement” [G.2] by ISO incollaboration with BIPM, IEC, IFCC, IUPAC, IUPAP and OIML, which formally established general rulesfor evaluating and expressing uncertainty in measurement across a broad spectrum of measurements. Thisdocument shows how the concepts in the ISO Guide may be applied in chemical measurement. It first givesan introduction to the concept of uncertainty and the distinction between uncertainty and error. This isfollowed by a description of the steps involved in the evaluation of uncertainty with the processes illustratedby worked examples in Appendix A.

This Guide assumes that the evaluation of uncertainty requires the analyst to look closely at all the possiblesources of uncertainty. It recognises that, although a detailed study of this kind may require a considerableeffort, it is essential that the effort expended should not be disproportionate. It suggests that in practice apreliminary study will quickly identify the most significant sources of uncertainty, and as the examplesshowed, the value obtained for the total uncertainty is almost entirely controlled by the major contributions.It recommends that a good estimate can be made by concentrating effort on the largest contributions andthat once evaluated for a given method applied in a particular laboratory, the uncertainty estimate obtainedmay be reliably applied to subsequent results obtained by the method in the same laboratory provided thatthis is justified by the relevant quality control data. No further effort should be necessary unless the methoditself or the equipment used is changed, in which case the estimate would be reviewed as part of the normalre-validation.

The first edition of the EURACHEM Guide for “Quantifying Uncertainty in Analytical Measurement [G.3]was published in 1995 based on the ISO Guide.

The first edition of the EURACHEM Guide has now been revised in the light of experiences gained in itspractical application in chemistry laboratories and the even greater awareness of the need to introduceformal quality assurance procedures by laboratories. The second edition stresses that the proceduresintroduced by a laboratory to estimate its measurement uncertainty must be integrated with its existingquality assurance measures, with these measures themselves frequently providing much of the informationrequired to evaluate the measurement uncertainty. It attempts to correct the impression gained within the

Quantifying Uncertainty Foreword to the Second Edition


wider Analytical Community that it is only the so-called component-by-component approach to theestimation of measurement uncertainty that is acceptable to the customers of providers of analytical data.

NOTE Worked examples are given in Appendix A. A numbered list of definitions is given at Appendix B. Terms aredefined, upon their first occurrence in the main body of the text, via a reference to one of these lists. Theconvention is adopted of printing defined terms in bold face upon their first occurrence: a reference to thedefinition immediately follows, enclosed in square brackets. The definitions are, in the main, taken from theInternational vocabulary of basic and general standard terms in Metrology (VIM) [G.4], the Guide [G.2] andISO 3534 (Statistics - Vocabulary and symbols) [G.5] Appendix C shows, in general terms, the overallstructure of a chemical analysis leading to a measurement result. Appendix D describes a general procedurewhich can be used to identify uncertainty components and plan further experiments as required; Appendix Edescribes some statistical operations used in uncertainty estimation in analytical chemistry, and Appendix Flists many common uncertainty sources and methods of estimating the value of the uncertainties. Abibliography is provided at Appendix G.

Quantifying Uncertainty Scope


1. ScopeThis Guide gives detailed guidance for the evaluation and expression of uncertainty in quantitative chemicalanalysis, based on the approach taken in the ISO “Guide to the Expression of Uncertainty in Measurement”.It is applicable at all levels of accuracy and in all fields - from routine analysis to basic research and toempirical and rational methods [see section 5.3.]. Some common areas in which chemical measurementsare needed and in which the principles of this Guide may be applied are:

• Quality control and quality assurance in manufacturing industries.

• Testing for regulatory compliance.

• Testing utilising an agreed method

• Calibration of standards and equipment.

• Development and certification of reference materials.

• Research and development.

As formal quality assurance measures have to be introduced by laboratories in a number of sectors thissecond EURACHEM Protocol is now able to illustrate how data from the following procedures may be usedfor the estimation of measurement uncertainty:

• Evaluation of the effect on the analytical result of the identified sources of uncertainty for a singlemethod in a single laboratory.

• Results from defined internal quality control procedures in a single laboratory.

• Results from collaborative trials used to validate methods of analysis in a number of competentlaboratories.

• Results from proficiency test schemes used to assess the analytical competency of laboratories.

Quantifying Uncertainty


2. Uncertainty

2.1. Definition of Uncertainty

2.1.1. The word uncertainty means doubt, andthus in its broadest sense uncertainty ofmeasurement means doubt about the validity ofthe result of a measurement as well as doubt as tothe exactness of the result.

2.1.2. In this guide, the word uncertainty withoutadjectives refers both to the general concept andto any or all measures of that concept. When aspecific measure is intended, appropriateadjectives are used.

2.1.3. The definition of the term uncertainty (ofmeasurement) used in this protocol and takenfrom the current version adopted for theInternational Vocabulary of Basic and GeneralTerms in Metrology G.4 is “A parameterassociated with the result of a measurement, thatcharacterises the dispersion of the values thatcould reasonably be attributed to the measurand”

Note 1 The parameter may be, for example, astandard deviation B.24 (or a given multipleof it), or the width of a confidence interval.

NOTE 2 Uncertainty of measurement comprises, ingeneral, many components. Some of thesecomponents may be evaluated from thestatistical distribution of the results of seriesof measurements and can be characterised bystandard deviations. The other components,which also can be characterised by standarddeviations, are evaluated from assumedprobability distributions based on experienceor other information. The ISO Guide refers tothese different cases as Type A and Type Bestimations respectively.

2.1.4. In many cases in chemical analysis themeasurand [B.6] will be the concentration of ananalyte. However chemical analysis is used tomeasure other quantities, e.g. colour, pH, etc., andtherefore the general term "measurand" will beused.

2.1.5. The definition of uncertainty given abovefocuses on the range of values that the analystbelieves could reasonably be attributed to themeasurand.

2.2. Uncertainty Sources

2.2.1. In practice the uncertainty on the result mayarise from many possible sources, including

examples such as incomplete definition, sampling,matrix effects and interferences, environmentalconditions, uncertainties of weights andvolumetric equipment, reference values,approximations and assumptions incorporated inthe measurement method and procedure, andrandom variation (a fuller description ofuncertainty sources will be found at section 6.6.)

2.3. Uncertainty Components

2.3.1. In estimating the overall uncertainty, it maybe necessary to take each source of uncertaintyand treat it separately to obtain the contributionfrom that source. Each of the separatecontributions to uncertainty is referred to as anUncertainty Component. When expressed as astandard deviation, an uncertainty component isknown as a standard uncertainty [B.14]. Ifthere is correlation between any components thenthis has to be taken into account by determiningthe covariance. However, it is often possible toevaluate the combined effect of severalcomponents. This may reduce the overall effortinvolved and, where components whosecontribution is evaluated together are correlated,there may be no additional need to take account ofthe correlation.

2.3.2. For a measurement result y, the totaluncertainty, termed combined standarduncertainty [B.15] and denoted by uc(y), is anestimated standard deviation equal to the positivesquare root of the total variance obtained bycombining all the uncertainty components,however evaluated, using the law of propagationof uncertainty (see section 8.).

2.3.3. For most purposes in analytical chemistry,an expanded uncertainty [B.16] U, should beused. The expanded uncertainty provides aninterval within which the value of the measurandis believed to lie with a particular level ofconfidence. U is obtained by multiplying uc(y),the combined standard uncertainty, by a coveragefactor [B.17] k. The choice of the factor k isbased on the level of confidence desired. For anapproximate level of confidence of 95%, k is 2.

NOTE The coverage factor k should always be statedso that the combined standard uncertainty ofthe measured quantity can be recovered foruse in calculating the combined standard

Quantifying Uncertainty Uncertainty


uncertainty of other measurement results thatmay depend on that quantity.

2.4. Error and Uncertainty

2.4.1. It is important to distinguish between errorand uncertainty. “Error” [B.20] is defined as thedifference between an individual result and thetrue value [B.3] of the measurand. As such, erroris a single value.

NOTE Error is an idealised concept and errors cannotbe known exactly.

2.4.2. Uncertainty, on the other hand, takes theform of a range, and, if estimated for an analyticalprocedure and defined sample type, may apply toall determinations so described. No part ofuncertainty can be corrected for.

2.4.3. To illustrate further the difference, theresult of an analysis after correction may bychance be very close to the value of themeasurand, and hence have a negligible error.However, the uncertainty may still be very large,simply because the analyst is very unsure of howclose that result is to the value.

2.4.4. The uncertainty of the result of ameasurement should never be interpreted asrepresenting the error itself, nor the errorremaining after correction.

2.4.5. An error is regarded as having twocomponents, namely, a random component and asystematic component.

2.4.6. Random error [B.21] typically arises fromunpredictable variations of influence quantities.These random effects, give rise to variations inrepeated observations of the measurand. Therandom error of an analytical result cannot becompensated by correction but it can usually bereduced by increasing the number of observations.

NOTE 1 The experimental standard deviation of thearithmetic mean [B.23] or average of a seriesof observations is not the random error of themean, although it is so referred to in somepublications on uncertainty. It is instead ameasure of the uncertainty of the mean due tosome random effects. The exact value of therandom error in the mean arising from theseeffects cannot be known.

2.4.7. Systematic error [B.22] is defined as acomponent of error which, in the course of anumber of analyses of the same measurand,remains constant or varies in a predictable way.It is independent of the number of measurements

made and cannot therefore be reduced byincreasing the number of analyses under constantmeasurement conditions.

2.4.8. Constant systematic errors, such as failingto make an allowance for a reagent blank in anassay, or inaccuracies in a multi-point instrumentcalibration, are constant for a given level of themeasurement value but may vary with the level ofthe measurement value.

2.4.9. Effects which change systematically inmagnitude during a series of analyses, caused, forexample by inadequate control of experimentalconditions, give rise to systematic errors that arenot constant.

EXAMPLES:

1. A gradual increase in the temperature of a setof samples during a chemical analysis can leadto progressive changes in the result.

2. Sensors and probes that exhibit ageing effectsover the time-scale of an experiment can alsointroduce non constant systematic errors.

2.4.10. The result of a measurement should becorrected for all recognised significant systematiceffects.

NOTE Measuring instruments and systems are oftenadjusted or calibrated using measurementstandards and reference materials to correctfor systematic effects; however, theuncertainties associated with these standardsand materials and the uncertainty in thecorrection must still be taken into account.

2.4.11. A further type of error is a spurious erroror blunder. Errors of this type invalidate ameasurement and typically arise through humanfailure or instrument malfunction. Transposingdigits in a number while recording data, an airbubble lodged in a spectrophotometer flow-through cell, or accidental cross-contamination oftest items are common examples of this type oferror.

2.4.12. Measurements for which errors such asthese have been detected should be rejected andno attempt should be made to incorporate theerrors into any statistical analysis. Howevererrors such as digit transposition can be corrected(exactly), particularly if they occur in the leadingdigits.

2.4.13. Spurious errors are not always obviousand, where a sufficient number of replicatemeasurements is available, it is usuallyappropriate to apply an outlier test to check forthe presence of suspect members in the data set.

Quantifying Uncertainty Uncertainty


Any positive result obtained from such a testshould be considered with care and, wherepossible, referred back to the originator forconfirmation. It is generally not wise to reject avalue on purely statistical grounds.

2.4.14. Uncertainties estimated using this guideare not intended to allow for the possibility ofspurious errors/blunders.

Quantifying Uncertainty Analytical Measurement and Uncertainty


3. Analytical Measurement and Uncertainty

3.1. Method validation

3.1.1. In practice, the fitness for purpose ofanalytical methods applied for routine testing ismost commonly assessed through methodvalidation studies. Such studies produce data onoverall performance and on individual influencefactors which can be applied to the estimation ofuncertainty associated with the results of themethod in normal use.

3.1.2. Method validation studies rely on thedetermination of overall method performanceparameters. These are obtained during methoddevelopment and interlaboratory study orfollowing in-house validation protocols.Individual sources of error or uncertainty aretypically investigated only when significantcompared to the overall precision measures inuse. The emphasis is primarily on identifying andremoving (rather than correcting for) significanteffects. This leads to a situation in which themajority of potentially significant influencefactors have been identified, checked forsignificance compared to overall precision, andshown to be negligible. Under thesecircumstances, the data available to analystsconsists primarily of overall performance figures,together with evidence of insignificance of mosteffects and some measurements of any remainingsignificant effects.

3.1.3. Validation studies for quantitativeanalytical methods typically determine some orall of the following parameters:

Precision. The principal precision measuresinclude repeatability standard deviation sr,reproducibility standard deviation sR, (ISO 3534-1) and intermediate precision, sometimes denotedsZi, with i denoting the number of factors varied(ISO 5725-3:1994). The repeatability sr indicatesthe variability observed within a laboratory, overa short time, using a single operator, item ofequipment etc. sr may be estimated within alaboratory or by inter-laboratory study.Interlaboratory reproducibility standard deviationsR for a particular method may only be estimateddirectly by interlaboratory study; it shows thevariability obtained when different laboratoriesanalyse the same sample. Intermediate precisionrelates to the variation in results observed whenone or more factors, such as time, equipment andoperator, are varied within a laboratory; different

figures are obtained depending on which factorsare held constant. Intermediate precisionestimates are most commonly determined withinlaboratories but may also be determined byinterlaboratory study. The observed precision ofan analytical procedure is an essential componentof overall uncertainty, whether determined bycombination of individual variances or by studyof the complete method in operation.

Bias. The bias of an analytical method is usuallydetermined by study of relevant referencematerials or by spiking studies. Bias may beexpressed as analytical recovery (value observeddivided by value expected). Bias is expected to benegligible or otherwise accounted for, but theuncertainty associated with the determination ofthe bias remains an essential component ofoverall uncertainty.

Linearity. Linearity of response to an analyte isan important property where methods are used toquantify at a range of concentrations. Thelinearity of the response to pure standards and torealistic samples may be determined. Linearity isnot generally quantified, but is checked for byinspection or using significance tests for non-linearity. Significant non-linearity is usuallycorrected for by non-linear calibration oreliminated by choice of more restricted operatingrange. Any remaining deviations from linearityare normally sufficiently accounted for by overallprecision estimates covering severalconcentrations, or within any uncertaintiesassociated with calibration (Appendix E.3).

Detection limit. During method validation, thedetection limit is normally determined only toestablish the lower end of the practical operatingrange of a method. Though uncertainties near thedetection limit may require careful considerationand special treatment (Section ###), the detectionlimit, however determined, is not of directrelevance to uncertainty estimation.

Robustness or ruggedness. Many methoddevelopment or validation protocols require thatsensitivity to particular parameters beinvestigated directly. This is usually done by apreliminary ‘ruggedness test’, in which the effectof one or more parameter changes is observed. Ifsignificant (compared to the precision of theruggedness test) a more detailed study is carriedout to measure the size of the effect, and apermitted operating interval chosen accordingly.



Ruggedness test data can therefore provideinformation on the effect of important parameters.

Selectivity/specificity. Though loosely defined,both terms relate to the degree to which a methodresponds uniquely to the required analyte.Typical selectivity studies investigate the effectsof likely interferents, usually by adding thepotential interferent to both blank and fortifiedsamples and observing the response. The resultsare normally used to demonstrate that thepractical effects are not significant. However,since the studies measure changes in responsedirectly, it is possible to use the data to estimatethe uncertainty associated with potentialinterferences, given knowledge of the range ofinterferent concentrations.

Conduct of validation studies

3.1.4. The detailed design and execution ofmethod validation studies is covered extensivelyelsewhere [G.6] and will not be repeated here.However, the main principles as they affect therelevance of a study applied to uncertaintyestimation are pertinent and are consideredbelow.

3.1.5. Representativeness is essential. That is,studies should, as far as possible, be conducted toprovide a realistic survey of the number and rangeof effects operating during normal use of themethod, as well as covering the concentrationranges and sample types within the scope of themethod. Where a factor has been representativelyvaried during the course of a precisionexperiment, for example, the effects of that factorappear directly in the observed variance and needno additional study unless further methodoptimisation is desirable.

3.1.6. In this context, representative variationmeans that an influence parameter must take adistribution of values appropriate to theuncertainty in the parameter in question. Forcontinuous parameters, this may be a permittedrange or stated uncertainty; for discontinuousfactors such as sample matrix, this rangecorresponds to the variety of types permitted orencountered in normal use of the method. Notethat representativeness extends not only to therange of values, but to their distribution.

3.1.7. In selecting factors for variation, it isimportant to ensure that the larger effects arevaried where possible. For example, where day today variation (perhaps arising from recalibrationeffects) is substantial compared to repeatability,two determinations on each of five days will

provide a better estimate of intermediateprecision than five determinations on each of twodays. Ten single determinations on separate dayswill be better still, subject to sufficient control,though this will provide no additional informationon within-day repeatability.

3.1.8. It is generally simpler to treat data obtainedfrom random selection than from systematicvariation. For example, experiments performed atrandom times over a sufficient period will usuallyinclude representative ambient temperatureeffects, while experiments performedsystematically at 24-hour intervals may be subjectto bias due to regular ambient temperaturevariation during the working day. The formerexperiment needs only evaluate the overallstandard deviation; in the latter, systematicvariation of ambient temperature is required,followed by adjustment to allow for the actualdistribution of temperatures. Random variation is,however, less efficient; a small number ofsystematic studies can quickly establish the sizeof an effect, whereas it will typically take wellover 30 determinations to establish an uncertaintycontribution to better than about 20% relativeaccuracy. Where possible, therefore, it is oftenpreferable to investigate small numbers of majoreffects systematically.

3.1.9. Where factors are known or suspected tointeract, it is important to ensure that the effect ofinteraction is accounted for. This may beachieved either by ensuring random selectionfrom different levels of interacting parameters, orby careful systematic design to obtain bothvariance and covariance information.

3.1.10. In carrying out studies of overall bias, it isimportant that the reference materials and valuesare relevant to the materials under routine test.

3.1.11. Any study undertaken to investigate andtest for the significance of an effect should havesufficient power to detect such effects before theybecome practically significant, that is, significantcompared to the largest component ofuncertainty.

Relevance of prior studies

3.1.12. When uncertainty estimates are based atleast partly on prior studies of methodperformance, it is necessary to demonstrate thevalidity of applying prior study results. Typically,this will consist of

• Demonstration that a comparable precision tothat obtained previously can be achieved



• Demonstration that the use of the bias dataobtained previously is justified, typicallythrough determination of bias on relevantreference materials (see, for example, ISOGuide 33), by appropriate spiking studies, orby satisfactory performance on relevantproficiency schemes or other laboratoryintercomparisons

• Continued performance within statisticalcontrol as shown by regular QC sampleresults and the implementation of effectiveanalytical quality assurance procedures.

3.1.13. Whether carrying out measurements orassessing the performance of the measurementprocedure, effective quality assurance and controlmeasures should be in place to ensure that themeasurement process is stable and in control.Such measures normally include, for example,appropriately qualified staff, proper maintenanceand calibration of equipment and reagents, use ofappropriate reference standards, documentedmeasurement procedures and use of appropriatecheck standards and control charts

3.1.14. Where the conditions above are met, andthe method is operated within its scope and fieldof application, it is normally acceptable to applythe data from prior validation studies directly touncertainty estimates in the laboratory inquestion.

3.2. Traceability

3.2.1. Traceability is intimately linked touncertainty and is an important concept in allbranches of measurement. It provides the basisfor establishing the uncertainty on a particularresult and for judging whether results agree orwhether a result is above or below someprescribed limit. In order to compare resultseither with each other or with a limit, it isnecessary for the results and the limit to betraceable to a common reference.

3.2.2. Traceability is formally defined [G.4] as:

“The property of the result of a measurementor the value of a standard whereby it can berelated to stated references, usually nationalor international standards, through anunbroken chain of comparisons all havingstated uncertainties.”

Thus it is a property of the result of ameasurement. The uncertainty on a result whichis traceable to a particular reference, will be theuncertainty on that reference together with the

uncertainty on making the measurement relativeto that reference.

3.2.3. The stated references, as well as beingnational or international standards can, accordingto VIM [G.4], be a material measure, measuringinstrument, reference material or measuringsystem, where a measuring system is furtherdefined as:

“The complete set of measuring instrumentsand other equipment assembled to carry outspecific measurements”

with a note that the system may include materialmeasures and chemical reagents.

3.2.4. Utilising these definitions it is possible toexamine how traceability of the results might beestablished for the measurement procedurescommonly used in analytical chemistry. For allsuch analytical measurement it is straightforwardto ensure that the results from such operations asweighing, volume determination, temperaturemeasurement are traceable to SystemeInternationale (SI) units. The evaluation of theuncertainty on the final result arising from suchoperations is described in example 1.

3.2.5. In all cases the calibration of the measuringequipment used must be traceable to appropriatestandards. The quantification stage of theanalytical procedure is often calibrated usingeither pure samples of the analyte or appropriatereference materials, whose values are traceable tothe SI. This practice provides traceability of theresults to SI for this part of the procedure.Because operations prior to the finalquantification frequently introduce large effects,however, traceability of the results obtained fromthe complete measurement procedure is moredifficult to establish.

3.2.6. Typical ways in which the traceability ofthe result of the complete analytical proceduremight be established are

1. By using a primary method

2. By using the analytical procedure to makemeasurements on a quantified pure sample ofthe analyte

3. By using the analytical procedure to makemeasurements on an appropriate CertifiedReference Material (CRM)

4. By making measurements using a definedprocedure.

It may also be necessary to use a combination ofthese methods. Each is discussed in turn below.



3.2.7. Measurements using Primary Methods

A primary method is currently described asfollows:

“A primary method of measurement is amethod having the highest metrologicalqualities, whose operation is completelydescribed and understood in terms of SI unitsand whose results are accepted withoutreference to a standard of the same quantity.”

with notes that

“…a primary direct method results in a valueof an unknown quantity without reference to astandard of the same quantity.”

“.. a primary ratio method results in a value ofthe ratio of two values of the samew quantitywithout reference to a standard of the samequantity.”

It is normally understood that a Primary methodis traceable directly to the SI, and is of thesmallest achievable uncertainty. It is usually amethod used to define the base units. It isaccodingly traceable to the SI by definition, butrarely available to routine testing or calibrationlaboratories.

3.2.8. Measurements on a pure sample of theanalyte.

In principle traceability can be achieved bymeasurement of a quantified sample of the pureanalyte, for example by spiking or by standardadditions. However it may be difficult toestablish the relative response of themeasurement system to the quantified sample ofthe analyte and the sample being analysed. Thisis a particular example of a problem common toall areas of measurement; it is always necessaryto evaluate the difference in response of themeasurement system to the standard used and thesample under test. In many areas ofmeasurement, particularly in the physicalsciences, the causes of any potential differencehave been investigated and well understood, anynecessary corrections can be applied and theuncertainty on these corrections can bequantified. Unfortunately the same is not true formany chemical analyses and in the particular caseof spiking or standard additions both thecorrection for the difference in response and itsuncertainty may be large. Thus, although thetraceability of the result to SI units can inprinciple be established, in practice, in all but the

most simple cases, the uncertainty on the resultmay be unacceptably large or evenunquantifiable. If the uncertainty isunquantifiable then traceability has not beenestablished

3.2.9. Measurement on a Certified ReferenceMaterial (CRM)

Measurement on a CRM can reduce theuncertainty compared to the use of a pure sample,providing that there is a suitable CRM available.If the value of the CRM is traceable to SI, thenthese measurements could provide traceability toSI units and the evaluation of the uncertaintyutilising reference materials is discussed insection 7.9.1. However, even in this case, theuncertainty on the result may be unacceptablylarge or even unquantifiable.

3.2.10. Measurement by means of definedprocedure.

This is often used in combination with either ofthe ways 1 or 2 described above, since theuncertainty of the result obtained on the basis ofthe defined procedure will be less than if itstraceability is back to SI units. This smalleruncertainty will only apply for comparison withresults utilising the same procedure and forsamples that are within the scope of the definedmethod, but this could be sufficient for examplewhen carrying out analyses for production controlor for certain regulatory purposes, if theregulations specify the measurement procedure tobe used. This technique of utilising a definedprocedure is not unique to analytical chemistry,for example for many years voltagemeasurements were made relative to standardcells prepared using a standard procedure becauseof the large uncertainty on realising the SI volt.

In addition the defined procedure is utilised,when the method defines the analyte, e.g.measurement of fat or fibre content of food.Measurements on CRMs or in house referencematerials may also be carried out for QCpurposes, but the traceability of the result is to thedefined procedure. Again the results can only becompared with limits or other results obtainedutilising the same procedure and for samples thatare within the scope of the method. Theevaluation of the uncertainty in this case isdiscussed in section 7.5.1.

Quantifying Uncertainty


4. Principles of Measurement Uncertainty Estimation

4.1. Uncertainty estimation is simple in principle.The following paragraphs summarise the tasksthat need to be performed in order to obtain anestimate of the uncertainty associated with ameasurement result. Subsequent chapters provideadditional guidance applicable in differentcircumstances, particularly relating to the use ofdata from method validation studies and the useof formal uncertainty propagation principles. Thesteps involved are:

Step 1 SpecificationWrite down a clear statement of what isbeing measured, including the relationshipbetween the measurand and the parameters(e.g. measured quantities, constants,calibration standards etc.) upon which itdepends. Where possible, include correctionsfor known systematic effects. Thespecification information, if it exists, isnormally given in the relevant StandardOperating Procedure (SOP) or other methoddescription.

Step 2 Identify Uncertainty SourcesList the possible sources of uncertainty. Thiswill include sources that contribute to theuncertainty on the parameters in therelationship specified in Step 1, but mayinclude other sources and must includesources arising from chemical assumptions.A general procedure for forming a structuredlist is suggested at Appendix [CE].

Step 3 Quantify Uncertainty ComponentsMeasure or estimate the size of theuncertainty component associated with eachpotential source of uncertainty identified. Itis often possible to estimate or determine asingle contribution to uncertainty associatedwith a number of separate sources. It is alsoimportant to consider whether available dataaccounts sufficiently for all sources ofuncertainty, and plan additional experimentsand studies carefully to ensure that allsources of uncertainty are adequatelyaccounted for.

Step 4 Calculate Total UncertaintyThe information obtained in step 3 willconsist of a number of quantifiedcontributions to overall uncertainty, whetherassociated with individual sources or withthe combined effects of several sources. Thecontributions have to be expressed asstandard deviations, and combined accordingto the appropriate rules, to give a combinedstandard uncertainty. The appropriatecoverage factor should be applied to give anexpanded combined uncertainty.

Figure 1 shows the process schematically.

4.2. The following chapters provide guidanceon the execution of all the steps listed above andshows how the procedure may be simplifieddepending on the information that is availableabout the combined effect of a number of sources.

Quantifying Uncertainty Principles of Measurement Uncertainty Estimation


Figure 1: The Uncertainty Estimation Process

SpecifyMeasurand

IdentifyUncertainty

Sources

Is methodperformance

data available

?

Yes

Simplify bygrouping

componentscovered by this

data

Quantify thegrouped

components

Are allcomponents

included?

Quantify eachcomponent

Quantify eachremaining

component

Convert allcomponents to

StandardDeviations

No

Calculatecombined

uncertainty

Do thesignificant

components needre-evaluating

No End

Re-evaluate thesignificant

components

Yes

Yes

No

Quantifying Uncertainty Step 1. Specification


5. Step 1. Specification

5.1. In the context of uncertainty estimation,“specification” requires both a clear andunambiguous statement of what is beingmeasured, and a quantitative expression relatingthe value of the measurand to the parameters onwhich it depends. These parameters may be othermeasurands, quantities which are not directlymeasured or constants. All of this informationshould be in the SOP.

5.2. In analytical measurement, it is particularlyimportant to distinguish between measurementsintended to produce results which areindependent of the method used, and those whichare not so intended. The latter are often referredto as empirical methods. The following examplesmay clarify the point further.

EXAMPLES:

1. Methods for the determination of the amountof nickel present in an alloy are normallyexpected to yield the same result, in the sameunits, usually expressed as a mass or molefraction. In principle, any systematic effect dueto method bias or matrix would need to becorrected for, though it is more usual to ensurethat any such effect is small. Results would notnormally need to quote the particular methodused, except for information.

2. Determinations of “extractable fat may differsubstantially, depending on the extractionconditions specified. Since “extractable fat” isentirely dependent on choice of conditions; the

method used is empirical. It is not meaningfulto consider correction for bias intrinsic to themethod, since the measurand is defined by themethod used. Results are generally reportedwith reference to the method, uncorrected forany bias intrinsic to the method.

3. In circumstances where variations in thesubstrate, or matrix, have large andunpredictable effects, a systematic procedure isoften developed with the sole aim of achievingcomparability between laboratories measuringthe same material. The method may then beadopted as a local, national or internationalstandard on which trading or other decisions aretaken; with no intent to obtain an absolutemeasure of the true amount of analyte present.Corrections for method bias or matrix effect areignored by convention (whether or not theyhave been minimised in method development).Results are normally reported uncorrected formatrix or method bias.

5.3. The distinction between empirical and non-empirical (sometimes called rational) methods isimportant because it affects the estimation ofuncertainty. In examples 2 and 3 above, becauseof the conventions employed, uncertaintiesassociated with some quite large effects are notrelevant in normal use. Due consideration shouldaccordingly be given to whether the results areexpected to be dependent upon, or independentof, the method in use and only those effectsrelevant to the result as reported should beincluded in the uncertainty estimate.

Quantifying Uncertainty Step 2. Identifying Uncertainty Sources


6. Step 2. Identifying Uncertainty Sources

6.1. A comprehensive list of possible sources ofuncertainty should be assembled. At this stage, itis not necessary to be concerned about thequantification of individual components; the aimis to be completely clear about what should beconsidered. In Step 3, the best way of treatingeach source will be considered.

6.2. In forming the required list of uncertaintysources it is usually convenient to start with thebasic expression used to calculate the measurandfrom intermediate values. All the parameters inthis expression may have an uncertaintyassociated with their value and are thereforepotential uncertainty sources. In addition theremay be other parameters that do not appearexplicitly in the expression used to calculate thevalue of the measurand but nevertheless effectpart or the whole measurement process andcontribute to the uncertainty on the value of themeasurand, e.g. extraction time or temperature.These are also potential sources of uncertainty.All these different sources should be included.Additional information is given in Appendix C(Structure of Analytical Procedures) and inAppendix D (Analysing uncertainty sources).

6.3. The measurand has a relationship to thevalues p,q,r.. of these parameters which, inprinciple can be expressed algebraically asy=f(p,q,r, ...),. The expression then forms acomplete model of the measurement process interms of all the individual factors affecting theresult. This function may be very complicated andit may not be possible to write it down explicitly.It is introduced as a convenient way of indicatingthat the uncertainty on the value of y is dependenton the uncertainty in values of the parameters p,q, r etc.

6.4. It may additionally be useful to consider ameasurement procedure as a series of discreteoperations (sometimes termed unit operations),each of which may be assessed separately toobtain estimates of uncertainty associated withthem. This is a particularly useful approach wheresimilar measurement procedures share commonunit operations. The separate uncertainties foreach operation then form contributions to theoverall uncertainty..

6.5. In practice, it is more usual in analyticalmeasurement to consider uncertainties associatedwith elements of overall method performance,such as observable precision and bias measuredwith respect to appropriate reference materials.These contributions generally form the dominantcontributions to the uncertainty estimate, and aregenerally modelled as separate effects on theresult. It is then necessary to evaluate otherpossible contributions only to check theirsignificance, quantifying only those that aresignificant. Further guidance on this approach,which applies particularly to the use of methodvalidation data, is given in section 7.2.1.

6.6. Typical sources of uncertainty are

• Sampling

Where sampling forms part of the specifiedprocedure, effects such as random variationsbetween different samples and any potentialfor bias in the sampling procedure formcomponents of uncertainty affecting the finalresult.

• Storage Conditions

Where test items are stored for any periodprior to analysis, the storage conditions mayaffect the results. The duration of storage aswell as conditions during storage shouldtherefore be considered as uncertaintysources.

• Instrument effects

Instrument effects may include, for example,the limits of accuracy on the calibration of ananalytical balance; a temperature controllerthat may maintain a mean temperature whichdiffers (within specification) from itsindicated set-point; an auto-analyser thatcould be subject to carry-over effects.

• Reagent purity

The molarity of a volumetric solution will notbe known exactly even if the parent materialhas been assayed, since some uncertaintyrelated to the assaying procedure remains.Many organic dyestuffs, for instance, are not

Quantifying Uncertainty Step 2. Identifying Uncertainty Sources


100% pure and can contain isomers andinorganic salts. The purity of suchsubstances is usually stated by manufacturersas being not less than a specified level. Anyassumptions about the degree of purity willintroduce an element of uncertainty.

• Measurement conditions

For example, volumetric glassware may beused at an ambient temperature different fromthat at which it was calibrated. Grosstemperature effects should be corrected for,but any uncertainty in the temperature ofliquid and glass should be considered.Similarly, humidity may be important wherematerials are sensitive to possible changes inhumidity.

• Sample effects

The recovery of an analyte from a complexmatrix, or an instrument response, may beaffected by other elements of the matrix.Analyte speciation may further compoundthis effect.

The stability of a sample/analyte may changeduring analysis as a result of a changingthermal regime or photolytic effect.

When a ‘spike’ is used to estimate recovery,the recovery of the analyte from the samplemay differ from the recovery of the spike,introducing an uncertainty which needs to beevaluated.

• Computational effects

Selection of the calibration model, e.g. usinga straight line calibration on a curvedresponse, leads to poorer fit and higheruncertainty.

Truncation and round off can lead toinaccuracies in the final result. Since theseare rarely predictable, an uncertaintyallowance may be necessary.

• Blank Correction

There will be an uncertainty on both the valueand the appropriateness of the blankcorrection. This is particularly important intrace analysis.

• Operator effects

Possibility of reading a meter or scaleconsistently high or low.

Possibility of making a slightly differentinterpretation of the method.

• Random effects

Random effects contribute to the uncertaintyin all determinations. This entry should beincluded in the list as a matter of course.

NOTE: These sources are not necessarilyindependent.

Quantifying Uncertainty Step 3 - Quantifying Uncertainty


7. Step 3 - Quantifying Uncertainty

7.1. Introduction

7.1.1. The uncertainty on the value y of themeasurand can be obtained

• by evaluating the uncertainty on theparameters p, q, r etc and calculating theircontribution to the uncertainty on y

• by determining directly the combinedcontribution to the uncertainty on y fromsome or all of these parameters using, e.g.method performance data.

• or by using a combination of these

7.1.2. Whichever of these approaches is used,most of the information needed to evaluate theuncertainty is likely to be already available fromthe results of validation studies, from QA/QCdata and from other experimental work that hasbeen carried out to check the performance of themethod. However data may not be available toevaluate the uncertainty from all of the sourcesand it may be necessary to carry out furtherexperimental work as described in section 7.8. orthe uncertainty may be evaluated based on priorexperience or judgement as described in section7.10.

7.1.3. It is important to recognise that not all ofthe components will make a significantcontribution to the combined uncertainty; indeedin practice it is likely that only a small numberwill. Unless there is a large number of them,components that are less than one third of thelargest need not be evaluated in detail. Apreliminary estimate of the contribution of eachcomponent or combination of components to theuncertainty should be made and those that are notsignificant eliminated.

7.2. Uncertainty evaluation usingmethod performance data

7.2.1. The stages in estimating the overalluncertainty using existing data about the methodperformance are:

• Reconcile the information requirementswith the available data

First the list of uncertainty sources should beexamined to see which sources of uncertaintyare accounted for by the available data,whether by explicit study of the particularcontribution or by implicit variation withinthe course of whole-method experiments.These sources should be checked against thelist prepared in step 2 and any remainingsources listed to provide an auditable recordof the which contributions to the uncertaintyhave been included.

• Obtain further data as required

For sources of uncertainty not adequatelycovered by existing data, either obtainadditional information from the literature orstanding data (certificates, equipmentspecifications etc.), or plan experiments toobtain the required additional data.Additional experiments may take the form ofspecific studies of a single contribution touncertainty, or the usual method performancestudies conducted to ensure representativevariation of important factors.

7.2.2. The following sections provide guidance onthe coverage and limitations of data acquired inparticular circumstances and on the additionalinformation required for an estimate of overalluncertainty.

7.3. Uncertainty estimation using priorcollaborative method developmentand validation study data

7.3.1. A collaborative study carried out, forexample according to AOAC/IUPAC or ISO 5725standards, to validate a published method, is avaluable source of data to support an uncertaintyestimate. How this data can be utilised dependson the factors taken into account when the studywas carried out. During the ‘reconciliation’ stageindicated above, the sources which needparticular consideration are:

• Sampling. Studies rarely include a samplingstep; if the method used in house involvessub-sampling, or the measurand (see



Specification) is estimating a bulk propertyfrom a small sample, then the effects ofsampling should be investigated and theireffects included.

• Pre-treatment. In most studies, samples arehomogenised, and may additionally bestabilised, before distribution. It may benecessary to investigate and add the effects ofthe particular pre-treatment proceduresapplied in-house.

• Method bias. Method bias is often examinedprior to or during interlaboratory study, wherepossible by comparison with referencemethods or materials. Where the bias itself,the uncertainty in the reference values used,and the precision associated with the biascheck, are all small compared to sR, noadditional allowance need be made for biasuncertainty. Otherwise, it will be necessary tomake additional allowances.

• Variation in conditions. Laboratoriesparticipating in a study may tend towards themean of allowed ranges of experimentalconditions, resulting in an underestimate ofthe range of results possible within themethod definition. Where such effects havebeen investigated and shown to beinsignificant across their full permitted range,however, no further allowance is required.

• Changes in sample matrix. The uncertaintyarising from matrix compositions or levels ofinterferents outside the range covered by thestudy will need to be considered.

7.3.2. For methods operating within their definedscope, when the reconciliation stage shows thatall the identified sources have been included inthe validation study or when the contributionsfrom any remaining the sources such as thosediscussed in section 7.3.1. have been shown to benegligible, then the reproducibility standarddeviation sR, adjusted for concentration ifnecessary, may be used as the combined standarduncertainty.

7.3.3. Where additional factors apply, theseshould be evaluated in the form of standarduncertainties and combined with thereproducibility standard deviation in the usualway (section 8.)

7.4. Uncertainty estimation during in-house development and validationstudies

7.4.1. In-house development and validationstudies consist chiefly of the determination of themethod performance parameters indicated insection 3.1.3. Uncertainty estimation from theseparameters requires:

• The best available estimate of overallprecision

• The best available estimate(s) of overall biasand its uncertainty

• Quantification of any uncertainties associatedwith effects incompletely accounted for in theabove overall performance studies.

Precision study

7.4.2. The precision contribution should beestimated as far as possible over an extended timeperiod, and chosen to allow natural variation ofall factors affecting the result. Typicalexperiments include

• Distribution of results for a typical sampleanalysed several times over a period of time,using different analysts and equipment wherepossible (A QC check sample may providesufficient information)

• The distribution of replicate analysesperformed on each of several samples.

NOTE: Replicates should be performed at materiallydifferent times to obtain estimates ofintermediate precision; within-batchreplication provides estimates of repeatabilityonly.

• Formal multi-factor experimental designs,analysed by ANOVA to provide separatevariance estimates for each factor.

Bias study

7.4.3. Overall bias is best estimated by repeatedanalysis of a relevant CRM, using the completemeasurement procedure. Where this is done, andthe bias found to be insignificant, the uncertaintyassociated with the bias is simply the combinationof the uncertainty in the CRM value and thestandard deviation associated with the bias check(adjusted for number of determinations).

NOTE: Bias estimated in this way combines bias inlaboratory performance with any bias intrinsicto the method in use. Special considerations



may apply where the method in use isstandardised; see section 7.5.1..

• When the reference material is onlyapproximately representative of the testmaterials, additional factors should beconsidered, including (as appropriate)differences in homogeneity; referencematerials are frequently more homogeneousthat test samples

• Any effects following from differentconcentrations of analyte; for example, it isnot uncommon to find that extraction lossesdiffer between high and low levels of analyte

7.4.4. Bias for a method under study is frequentlydetermined against a reference method, bycomparison of the results of the two methodsapplied to the same samples. In suchcircumstances, given that the bias is notstatistically significant, the uncertainty is that forthe reference method (if applicable; see section7.5.1.), combined with the uncertainty associatedwith the measured difference between methods.The latter contribution to uncertainty commonlyappears as the standard deviation term used in thesignificance test applied to decide whether thedifference is statistically significant.

EXAMPLE

A method (method 1) for determining theconcentration of Selenium is compared with areference method (method 2). The results (inmg kg-1) from each method are as follows:

x s n

Method 1 5.40 1.47 5

Method 2 4.76 2.75 5

The standard deviations are pooled to give apooled standard deviation sc

( ) ( )( )cs =

× − + × −

+ −

=

1471 5 1 2 750 5 15 5 2

2 2052 2. .

.

and a corresponding value of t:

( )t =−

+

⋅5 40 4 76

2 20515

15

. .

.

=0.641 4

= 0.46

tcrit is 2.3 for 8 degrees of freedom, so there isno significant difference between the means ofthe results given by the two methods. But thedifference (0.64) is compared with a standarddeviation term of 1.4 above. This value of 1.4 is

the standard deviation associated with thedifference, and accordingly represents therelevant contribution to uncertainty associatedwith the measured bias.

7.4.5. Overall bias is also commonly studied byaddition of analyte to a previously studiedmaterial. The same considerations apply as forthe study of reference materials (above). Inaddition, the differential behaviour of addedmaterial and material native to the sample shouldbe considered and due allowance made. Such anallowance can be made on the basis of

• studies of the distribution of error observedfor a range of matrices and levels of addedanalyte

• comparison of result observed in a referencematerial with the recovery of added analyte inthe same reference material

• judgement on the basis of specific materialswith known extreme behaviour. For example,oyster tissue, a common marine tissuereference, is well known for a tendency to co-precipitate some elements with calcium saltson digestion, and may provide an estimate of‘worst case’ recovery on which an uncertaintyestimate can be based (e.g. by treating theworst case as an extreme of a rectangular ortriangular distribution)

• judgement on the basis of prior experience

7.4.6. Bias may also be estimated by comparisonof the particular method with a value determinedby the method of standard additions, in whichknown quantities of the analyte are added to thetest material, and the correct analyteconcentration inferred by extrapolation. Theuncertainty associated with the bias is thennormally dominated by the uncertaintiesassociated with the extrapolation, combined(where appropriate) with any significantcontributions from the preparation and additionof stock solution.

NOTE: To be directly relevant, the additions shouldbe made to the original sample, rather than aprepared extract.

7.4.7. It is a general requirement of the ISO Guidethat corrections should be applied for allrecognised and significant systematic effects.Where a correction is applied to allow for asignificant overall bias, the uncertainty associatedwith the bias is estimated as paragraph 7.4.4.describe in the case of insignificant bias



7.4.8. Where the bias is not insignificant, but isnonetheless neglected for practical purposes, theuncertainty associated with bias should beincreased by addition of a term equal to themeasured bias.

NOTE The inclusion of a term equal to a measuredbias is justifiable only to the extent thatneglecting a significant bias is justifiable. It isbetter practice to report the bias and itsuncertainty separately. Where this is not done,increasing the uncertainty estimate byincluding such a term simply avoidsmisleading any user of the reported result anduncertainty.

Additional factors

7.4.9. The effects of factors held constant duringprecision studies should be estimated separately,either by experimental variation or by predictionfrom established theory. Where practicable, theuncertainty associated with such factors should beestimated, recorded and combined with othercontributions in the normal way.

7.4.10. Typical experiments include

• Study of the effect of a variation of a singleparameter on the result. This is particularlyappropriate in the case of continuous,controllable parameters, independent of othereffects, such as time or temperature. The rateof change of the result with the change in theparameter can then be combined directly withthe uncertainty in the parameter to obtain therelevant uncertainty contribution.

NOTE: The change in parameter should be sufficientto change the result substantially compared tothe precision available in the study (e.g. byfive times the standard deviation of replicatemeasurements)

• Robustness studies, systematically examiningthe significance of moderate changes inparameters. This is particularly appropriate forrapid identification of significant effects, andcommonly used for method optimisation. Themethod can be applied in the case of discreteeffects, such as change of matrix, or smallequipment configuration changes, which haveunpredictable effects on the result. Where afactor is fond to be significant, it is normallynecessary to investigate further. Whereinsignificant, the associated uncertainty is (atleast for initial estimation) that associated withthe robustness study.

• Systematic multifactor experimental designsintended to estimate factor effects andinteractions.

7.4.11. Where the effect of an additional factor isdemonstrated to be negligible compared to theprecision of the study (i.e. statisticallyinsignificant), it is recommended that anuncertainty contribution equal to the standarddeviation associated with the relevantsignificance test be associated with that factor.

EXAMPLE

The effect of a permitted 1-hour extraction timevariation is investigated by a t-test on fivedeterminations each on the same sample, for thenormal extraction time and a time reduced by 1hour. The means and standard deviations were:Standard time: mean 1.8, standard deviation0.21; alternate time: mean 1.7, standarddeviation 0.17. A t-test uses the pooled varianceof

((5 - 1) 0.212 + (5 - 1) 0.172 ) / ((5 - 1) + (5 - 1))

=

× ×

0 037.

to obtain

)5/15/1(037.0/)7.18.1( +⋅−=t =0.82; notsignificant compared to tcrit = 2.3. But note thatthe difference (0.1) is compared with acalculated standard deviation term, of

)5/15/1(037.0 +⋅ =0.3. This value is thecontribution to uncertainty associated with theeffect of permitted variation in extraction time.

7.4.12. Where an effect is detected and isstatistically significant, but remains sufficientlysmall to neglect in practice, it is recommendedthat an uncertainty contribution equal to themeasured effect combined with its statisticaluncertainty be associated with the effect.

NOTE: See the note to section 7.4.3.

7.5. Empirical methods

7.5.1.An ‘empirical method’ is a method agreedupon for the purposes of comparativemeasurement within a particular field ofapplication where the measurandcharacteristically depends upon the method inuse. The method accordingly defines themeasurand. Examples include methods forleachable metals in ceramics and dietary fibre infoodstuffs.

7.5.2. Where such a method is in use within itsdefined field of application, the bias associated



with the method is defined as zero. In suchcircumstances, bias estimation need relate only tothe laboratory performance and should notadditionally account for bias intrinsic to themethod. This has the following implications.

7.5.3. Reference material investigations, whetherto demonstrate negligible bias or to measure bias,should be conducted using reference materialscertified using the particular method, or for whicha value obtained with the particular method isavailable for comparison.

7.5.4. Where reference materials so certified areunavailable, overall control of bias is associatedwith the control of method parameters affectingthe result; typically such factors as times,temperatures, masses, volumes etc. Theuncertainty associated with these input factorsmust accordingly be assessed and either shown tobe negligible or quantified. Section 7.3.1. thenapplies.

7.6. Ad-hoc methods

7.6.1. Ad-hoc methods are methods established tocarry out exploratory studies in the short term, orfor a short run of test materials. Such methods aretypically based on standard or well-establishedmethods within the laboratory, but are adaptedsubstantially (for example to study a differentanalyte) and will not generally justify formalvalidation studies for the particular material inquestion.

7.6.2. Since limited effort will be available toestablish the relevant uncertainty contributions, itis necessary to rely largely on the knownperformance of related systems or blocks withinthese systems. Uncertainty estimation shouldaccordingly be based on known performance on arelated system or systems, combined with anyspecific study necessary to establish relevance ofthose studies. The following recommendationsassume that such a related system is available andhas been examined sufficiently to obtain areliable uncertainty estimate, or that the methodconsists of blocks from other methods and thatthe uncertainty in these blocks has beenestablished previously.

7.6.3. As a minimum, it is essential that anestimate of overall bias and an indication ofprecision be available for the method in question.Bias will ideally be measured against a referencematerial, but will in practice more commonly beassessed from spike recovery. The considerations

of section 7.4.3. then apply, except that spikerecoveries should be compared with thoseobserved on the related system to establish therelevance of the prior studies to the ad-hocmethod in question. The overall bias observed forthe ad-hoc method, on the materials under test,should be comparable to that observed for therelated system, within the requirements of thestudy.

7.6.4. A minimum precision experiment consistsof a duplicate analysis. It is, however,recommended that as many replicates as practicalare performed. The precision should be comparedwith that for the related system; the standarddeviation for the ad-hoc method should becomparable.

NOTE: It recommended that the comparison be basedon inspection; statistical significance tests(e.g. an F-test) will generally be unreliablewith small numbers of replicates and will tendto lead to the conclusion that there is ‘nosignificant difference’ simply because of thelow power of the test.

7.6.5. Where the above conditions are metunequivocally, the uncertainty estimate for therelated system may be applied directly to resultsobtained by the ad-hoc method, making anyadjustments appropriate for concentrationdependence and other known factors.

7.6.6. Where these conditions are not met, it isnot recommended that an uncertainty estimate beprovided. Where an indication of reliability isnonetheless required, it is suggested that themeasured bias and 95% confidence interval of thereplicated results are reported, with the caveatthat systematic effects on the result were not fullyinvestigated.

7.7. Estimation based on other resultsor data

7.7.1. It is often possible to estimate some of thestandard uncertainties using whatever relevantinformation is available about the uncertainty onthe quantity concerned. The following paragraphssuggest some sources of information.

7.7.2. Proficiency Testing schemes A laboratory’sresults from participation in such schemes can beused as a check on the evaluated uncertainty,since the uncertainty should be compatible withthe spread of results obtained by that laboratoryover a number of proficiency test rounds. Further,in the special case where



• the compositions of samples used in thescheme cover the full range analysedroutinely

• the assigned value is traceable, and

• the uncertainty on the assigned value is smallcompared to the observed spread of results

then the standard deviation of the results obtainedfrom repeated rounds would provide a goodestimate of the uncertainty arising from thoseparts of the measurement procedure within thescope of the scheme. Of course, systematicdeviation from traceable assigned values and anyother sources of uncertainty (such as those notedin section 7.3.1.) must also be taken into account.

7.7.3. Quality Assurance (QA) data. As notedpreviously it is necessary to ensure that thequality criteria set out in standard operatingprocedures are achieved, and that measurementson QA samples show that the criteria continue tobe met. Where reference materials are used in QAchecks, section 7.9.1. shows how the data can beused to evaluate uncertainty. Where any otherstable material is used, the QA data provides anestimate of intermediate precision (Section7.4.2.). QA data also forms a continuing check onthe value quoted for the uncertainty. Clearly, thecombined uncertainty arising from random effectscannot be less than the standard deviation of theQA measurements.

7.7.4. Suppliers' information. For many sourcesof uncertainty, calibration certificates or supplierscatalogues provide information. For example, thetolerance of all volumetric glassware may beobtained from the manufacturer’s catalogue or acalibration certificate relating to a particular itemin advance of its use.

7.8. Quantification from repeatedobservations

7.8.1. The standard uncertainty arising fromrandom effects is typically measured fromrepeatability experiments and is quantified interms of the standard deviation of the measuredvalues. In practice, no more than about fifteenreplicates need normally be considered, unless ahigh precision is required.

7.8.2. By varying all parameters on which theresult of a measurement is known to depend, itsuncertainty could be evaluated by statistical

means, but this is rarely possible in practice dueto limited time and resources.

7.8.3. However, in many cases it will be foundthat just a few components of the uncertaintydominate. Where it is realistic to do so theseparameters should be varied to the fullestpracticable extent so that the evaluation ofuncertainty is based as much as possible onobserved data.

7.9. Using Reference Materials

7.9.1. Measurements on reference materialsprovide very good data for the assessment ofuncertainty since they provide information on thecombined effect of many of the potential sourcesof uncertainty. (ISO Guide 33 [G.7]gives a usefulaccount of the use of reference materials inchecking method performance). The sources thatthen need to be taken into account are:

• the uncertainty on the assigned value of thereference material.

• the reproducibility of the measurements madeon the reference material.

• any difference between the measured value ofthe reference material and its assigned value.

• differences between the composition of thereference material and the sample.

• differences in the response of themeasurement system to the reference materialand the sample, e.g. due to interferences ormatrix effects.

• operations that are carried out on the samplebut not on the reference material e.g. taking ofthe original sample and its subdivision in thelaboratory.

7.9.2. It is much easier to evaluate the abovesources of uncertainty than to worksystematically through an assessment of the effectof every potential source and thereforemeasurements on reference materials shouldalways be carried out, even if only in housereference materials are available. This isdiscussed further in section 7.4.3.

7.10.Estimation based on judgement

7.10.1. The evaluation of uncertainty is neither aroutine task nor a purely mathematical one; itdepends on detailed knowledge of the nature ofthe measurand and of the measurement method



and procedure used. The quality and utility of theuncertainty quoted for the result of ameasurement therefore ultimately depends on theunderstanding, critical analysis, and integrity ofthose who contribute to the assignment of itsvalue.

7.10.2. Most distributions of data can beinterpreted in the sense that it is less likely toobserve data in the margins of the distributionthan in the centre. The quantification of thesedistributions and their associated standarddeviations is done through repeatedmeasurements.

7.10.3. However, other assessments of intervalsmay be required in cases when repeatedmeasurements cannot be performed or do notprovide a meaningful measure of a particularuncertainty component.

7.10.4. There are numerous instances inanalytical chemistry when the latter prevails, andjudgement is required. For example:

• An assessment of recovery and its associateduncertainty cannot be made for every singlesample. One then makes such assessment forclasses of samples (e.g. grouped by type ofmatrix) and applies them to all samples ofsimilar type. The degree of similarity isitself an unknown, thus this inference (fromtype of matrix to a specific sample) isassociated with an extra element ofuncertainty that has no frequentisticinterpretation.

• The model of the measurement as defined bythe specification of the analytical procedureis used for converting the measured quantityto the value of the measurand (analyticalresult). This model is - like all models inscience - subject to uncertainty. It is onlyassumed that nature behaves according to thespecific model, but this can never be knownwith ultimate certainty.

• The use of reference materials is highlyencouraged, but there remains uncertaintyregarding not only the true value, but alsoregarding the relevance of a particularreference material for the analysis of aspecific sample. A judgement is required ofthe extent to which a proclaimed standardsubstance reasonably resembles the nature ofthe samples in a particular situation.

• Another source of uncertainty arises when themeasurand is insufficiently defined by theprocedure. Consider the determination of"permanganate oxidizable substances" thatare undoubtedly different whether oneanalyses ground water or municipal wastewater. Not only factors such as oxidationtemperature, but also chemical effects such asmatrix composition or interference, mayhave an influence on this specification.

• A common practice in analytical chemistrycalls for spiking with a single substance, suchas a close structural analogue or isotopomer,from which either the recovery of therespective native substance or even that of awhole class of compounds is judged. Clearly,the associated uncertainty is experimentallyassessable provided one is ready to study thisrecovery at all concentration levels and ratiosof measurands to the spike, and all "relevant"matrices. But frequently this experimentationis avoided and substituted by judgements on

• the concentration dependence ofrecoveries of measurand,

• the concentration dependence ofrecoveries of spike,

• the dependence of recoveries on (sub)typeof matrix,

• the identity of binding modes of nativeand spiked substances.

7.10.5. Judgement of this type is not based onimmediate experimental results, but rather on asubjective (personal) probability, an expressionwhich here can be used synonymously with"degree of belief", "intuitive probability" and"credibility" [G.4]. It is also assumed that adegree of belief is not based on a snap judgement,but on a well considered mature judgement ofprobability.

7.10.6. Although it is recognised that subjectiveprobabilities vary from one person to another, andeven from time to time for a single person theyare not arbitrary as they are influenced bycommon sense, expert knowledge, by earlierexperiments and observations.

7.10.7. This may appear to be a disadvantage, butneed not lead in practice to worse estimates thanthose from repeated measurements particularly ifthe true, real-life, variability in experimentalconditions cannot be simulated and the resulting



variability in data thus does not give a realisticpicture.

7.10.8. A typical problem of this nature arises iflong-term variability needs to be assessed whenno round-robin data are available. A scientistwho dismisses the option of substitutingsubjective probability for an actually measuredone (when the latter is not available) is likely toignore important contributions to combineduncertainty thus being ultimately less objective,than one who relies on subjective probabilities.

7.10.9. For the purpose of estimation of combineduncertainties two features of degree of beliefestimations are essential:

• degree of belief is regarded as interval valuedwhich is to say that a lower and an upperbound similar to a classical probabilitydistribution is provided,

• the same computational rules apply incombining 'degree of belief' contributions ofuncertainty to a combined uncertainty as forstandard deviations derived by other methods

Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty


8. Step 4. Calculating the Combined Uncertainty

8.1. Standard Uncertainties

8.1.1. Before combination all uncertaintycontributions must be expressed as standarduncertainties, that is, as standard deviations. Thismay involve conversion from some other measureof dispersion. The following rules give someguidance for converting an uncertaintycomponent to a standard deviation.

8.1.2. Where the uncertainty component wasevaluated experimentally from the dispersion ofrepeated measurements, then it can readily beexpressed as a standard deviation. For thecontribution to uncertainty in singlemeasurements, the standard uncertainty is simplythe observed standard deviation; for resultssubjected to averaging, the standard deviation ofthe mean [B.25] is used.

8.1.3. Where an uncertainty estimate is derivedfrom previous results and data it may already beexpressed as a standard deviation. Howeverwhere a confidence interval is given with a levelof confidence, (in the form ±a at p%) then dividethe value a by the appropriate percentage pointof the Normal distribution for the level ofconfidence given to calculate the standarddeviation.

EXAMPLE

A specification states that a balance reading iswithin ±0.2 mg with 95% confidence. Fromstandard tables of percentage points on thenormal distribution, a 95% confidence intervalis calculated using a value of 1.96σ. Using thisfigure gives a standard uncertainty of (0.2/1.96) ≈ 0.1.

8.1.4. If limits of ±a are given without aconfidence level and there is reason to expect thatextreme values are not likely, it is normallyappropriate to assume a rectangular distribution,with a standard deviation of a/√3 (see AppendixE).

EXAMPLE

A 10 ml Grade A volumetric flask is certified towithin ±0.2 ml. The standard uncertainty is0.2/√3 ≈ 0.11 ml.

8.1.5. If limits of ±a are given without aconfidence level, but there is reason to expect thatextreme values are unlikely, it is normallyappropriate to assume a triangular distribution,with a standard deviation of a/√6 (see AppendixE).

EXAMPLE

A 10 ml Grade A volumetric flask is certified towithin ±0.2 ml, but routine in-house checksshow that extreme values are rare. The standarduncertainty is 0.2/√6 ≈ 0.08 ml.

8.1.6. Where an estimate is to be made on thebasis of judgement, then it may be possible toestimate the component directly as a standarddeviation. If this is not possible then an estimateshould be made of the maximum deviation whichcould reasonably occur in practice (excludingsimple mistakes). If a smaller value is consideredsubstantially more likely, this estimate should betreated as descriptive of a triangular distribution.If there are no grounds for believing that a smallerror is more likely than a large error, theestimate should be treated as characterising arectangular distribution.

8.1.7. Conversion factors for the most commonlyused distribution functions are given in AppendixE.

8.2. Combined standard uncertainty

8.2.1. Following the estimation of individual orgroups of components of uncertainty andexpressing them as standard uncertainties, thenext stage is to calculate the combined standarduncertainty using one of the procedures describedbelow.

8.2.2. The general relationship between theuncertainty u(y) of a value y and the uncertaintyof the independent parameters x1, x2, ...xn onwhich it depends is

u(y(x1,x2,...) = ∑= ni

ii xuc,1

22 )(

where y(x1,x2,..) is a function of severalparameters x1,x2..., and ci is a sensitivitycoefficient evaluated as ci=∂y/∂xi, the partial



differential of y with respect to xi. Each variable'scontribution is just the square of the associateduncertainty expressed as a standard deviationmultiplied by the square of the relevant sensitivitycoefficient. These sensitivity coefficients describehow the value of y varies with changes in theparameters x1, x2 etc.

NOTE: Sensitivity coefficients may also be evaluateddirectly by experiment; this is particularlyvaluable where no reliable mathematicaldescription of the relationship exists.

8.2.3. Where variables are not independent, therelationship is more complex:

∑∑≠==

⋅+=

kinki

kini

iiji ikxccxucxyu,1,,1

22..., ),()())(( s

where s(x,ik) is the covariance between xi andxk.and ci and ck the sensitivity coefficients asdescribed and evaluated in 8.2.2. In practice, thecovariance is often related to the correlationcoefficient rik using

s(x,ik) = u(xi).u(xk).rik

where -1 ≤ rik ≤ 1.

8.2.4. These general procedures apply whetherthe uncertainties are related to single parameters,grouped parameters or to the method as a whole.However, when an uncertainty contribution isassociated with the whole procedure, it is usuallyexpressed as an effect on the result. In such cases,or when the uncertainty on a parameter isexpressed directly in terms of its effect on y, thecoefficient ∂y/∂xi is equal to 1.0.

EXAMPLE

A result of 22 mg l-1 shows a measured standarddeviation of 4.1 mg l-1. The standarduncertainty u(ε) associated with precision underthese conditions is 4.1 mg l-1. The implicitmodel for the measurement, neglecting otherfactors for clarity, is

y = (Calculated result) + ε

where ε represents the effect of randomvariation under the conditions of measurement.∂y/∂ε is accordingly 1.0

8.2.5. Except for the case above when thesensitivity coefficient is equal to one and for thespecial cases given in Rule 1 and Rule 2 below,the general procedure, requiring the generation ofpartial differentials or the numerical equivalentmust be employed. Appendix E gives details of anumerical method, suggested by Kragten, which

makes effective use of standard spreadsheetsoftware to provide a combined standarduncertainty from input standard uncertainties anda known measurement model [G.5]. It isrecommended that this method be used for all butthe simplest cases.

8.2.6. In some cases, the expressions forcombining uncertainties, reduce to much simplerforms. Two simple rules for combining standarduncertainties are given here.

Rule 1

For models involving only a sum or difference ofquantities, e.g. y=k(p+q+r+...) where k is aconstant, the combined standard uncertainty uc(y)is given by

.....)()(..)),(( 22 ++⋅= qupukqpyuc

Rule 2

For models involving only a product or quotient,e.g. y=k(pqr...), where k is a constant, thecombined standard uncertainty uc(y) is given by

.....)()(

)(22

+

+

⋅⋅=

q

qu

p

pukyyuc

where )/)(( ppu etc. are the uncertainties in theparameters, expressed as relative standarddeviations.

NOTE Subtraction is treated in the same manner asaddition, and division in the same way asmultiplication.

8.2.7. For the purposes of combining uncertaintycomponents, it is most convenient to break theoriginal mathematical model down to expressionswhich consist solely of operations covered by oneof the rules above. For example, the expression

)()(

rqpo

++

should be broken down to the two elements (o+p)and (q+r). The interim uncertainties for each ofthese can then be calculated using rule 1 above;these interim uncertainties can then be combinedusing rule 2 to give the combined standarduncertainty.

8.2.8. The following examples illustrate the use ofthe above rules:



EXAMPLE 1

y = m.(p-q+r) The values are m=1, p=5.02,q=6.45 and r=9.04 with standard uncertaintiesdeviations u(p)=0.13, u(q)=0.05 and u(r)= 0.22.

y = 5.02 - 6.45 + 9.04 = 7.61

26.022.005.013.01)( 222 =++×=yu

NOTE Since the value of y is only calculated to 2decimal places then the final uncertainty valueshould not be quoted to more than 3 decimalplaces.

EXAMPLE 2

y = (op/qr). The values are o=2.46, p=4.32,q=6.38 and r=2.99, with standard uncertaintiesof u(o)=0.02, u(p)=0.13, u(q)=0.11 and u(r)=0.07.

y=( 2.46 × 4.32 ) / (6.38 × 2.99 ) = 0.56

22

22

99.207.0

38.611.0

32.413.0

46.202.0

56.0)(

+

−

+

+

×=yu

⇒ u(y) = 0.56 × 0.043 = 0.024

8.2.9. There are many instances in whichcomponents of uncertainty vary with the level ofanalyte. For example, uncertainties in recoverymay be smaller for high levels of material, orspectroscopic signals may vary randomly on ascale approximately proportional to intensity(constant coefficient of variance). In such cases itis important to take account of the changes in thecombined standard uncertainty with level ofanalyte. Approaches include:

• Restricting the specified procedure oruncertainty estimate to a small range ofanalyte concentrations.

• Providing an uncertainty estimate in the formof a relative standard deviation.

• Explicitly calculating the dependence andrecalculating the uncertainty for a givenresult.

Appendix [E1] gives additional information onthese approaches.

8.3. Combined expanded uncertainty

8.3.1. The final stage is to multiply the combinedstandard uncertainty by the chosen coveragefactor in order to obtain an expanded uncertainty.The expanded uncertainty is required to providean interval which may be expected to encompassa large fraction of the distribution of valueswhich could reasonably be attributed to themeasurand.

8.3.2. In choosing a value for the coverage factork, a number of issues should be considered. Theseinclude:

• The level of confidence required

• Any knowledge of the underlyingdistributions

• Any knowledge of the number of values usedto estimate random effects; small samplesmay lead to optimistic estimates of expandeduncertainty.

8.3.3. For most purposes it is recommended that kis set to 2. However, this may be an under-estimate where the uncertainty is based onstatistical observations with relatively fewdegrees of freedom (less than about six). Thechoice of k then depends on the effective numberof degrees of freedom.

8.3.4. Where the combined standard uncertaintyis dominated by a single contribution with fewerthan six degrees of freedom, it is recommendedthat k be set equal to the two-tailed value ofStudent’s t for the number of degrees of freedomassociated with that contribution, and for thelevel of confidence required (normally 95%).

Table 1(page 27) gives a short list of values for t.

EXAMPLE:

A combined standard uncertainty for a weighingoperation is formed from contributionsucal=0.01 mg arising from calibrationuncertainty and sobs=0.08 mg based on thestandard deviation of five repeatedobservations. The combined standarduncertainty uc is equal to

mg081.008.001.0 22 =+ . This is clearlydominated by the repeatability contribution sobs,

which is based on five observations, giving 5-1=4 degrees of freedom. k is accordingly basedon Student’s t. The two-tailed value of t for fourdegrees of freedom and 95% confidence is,from tables, 2.8; k is accordingly set to 2.8 and



the combined expanded uncertaintyUc=2.8×0.081=0.23 mg.

8.3.5. The Guide [G.1] gives additional guidanceon choosing k where a small number ofmeasurements is used to estimate large randomeffects, and should be referred to when estimatingdegrees of freedom where several contributionsare significant.

8.3.6. Where the distributions concerned arenormal, a coverage factor of 2 (or chosenaccording to paragraphs 8.3.3.-8.3.5. using a levelof confidence of 95%) gives an intervalcontaining approximately 95% of the distributionof values. It is not recommended that this intervalis taken to imply a 95% confidence intervalwithout a knowledge of the distributionconcerned.

Table 1: Student’s t for 95% confidence (2-tailed)

Degrees of freedomνν

t

1 12.7

2 4.3

3 3.2

4 2.8

5 2.6

6 2.5

Quantifying Uncertainty Reporting uncertainty


9. Reporting uncertainty

9.1. General

9.1.1. The information necessary to report theresult of a measurement depends on its intendeduse. The guiding principles are:

• present sufficient information to allow theresult to be re-evaluated if new informationor data become available

• it is preferable to err on the side of providingtoo much information rather than too little.

9.1.2. At different levels of chemicalmeasurement from primary reference materialcharacterisation to routine testing, successivelymore of the information required may beavailable in the form of published reports,national or international standards, methoddocumentation and test and calibrationcertificates. When the details of a measurement,including how the uncertainty was determined,depend on references to publisheddocumentation, it is imperative that thesepublications are kept up to date and consistentwith the methods in use.

9.2. Information required

9.2.1. A complete report of a measurement resultshould include or refer to documentationcontaining,

• a description of the methods used tocalculate the measurement result and itsuncertainty from the experimentalobservations and input data

• the values and sources of all corrections andconstants used in both the calculation andthe uncertainty analysis

• a list of all the components of uncertaintywith full documentation on how each wasevaluated

9.2.2. The data and analysis should be presentedin such a way that its important steps can bereadily followed and the calculation of the resultrepeated if necessary.

9.2.3. Where a detailed report includingintermediate input values is required, the reportshould

• give the value of each input value, itsstandard uncertainty and a description ofhow each was obtained

• give the relationship between the result andthe input values and any partial derivatives,covariances or correlation coefficients usedto account for correlation effects

• state the estimated number of degrees offreedom for the standard uncertainty of eachinput value (methods for estimating degreesof freedom are given in the Guide [G.2]).

NOTE: Where the functional relationship isextremely complex or does not existexplicitly (for example, it may only exist as acomputer program), the relationship may bedescribed in general terms or by citation ofappropriate references. In such cases, it mustbe clear how the result and its uncertaintywere obtained.

9.2.4. When reporting the results of routineanalysis, it may be sufficient to state only thevalue of the expanded uncertainty.

9.3. Reporting standard uncertainty

9.3.1. When uncertainty is expressed as thecombined standard uncertainty uc (that is, as asingle standard deviation), the following form isrecommended:

"(Result): x (units) [with a] standard uncertaintyof uc (units) [where standard uncertainty is asdefined in the International Vocabulary of Basicand General terms in metrology, 2nd ed., ISO1993 and corresponds to one standarddeviation.]"

NOTE The use of the symbol ± is not recommendedwhen using standard uncertainty as thesymbol is commonly associated withintervals corresponding to high levels ofconfidence.

Terms in parentheses [] may be omitted orabbreviated as appropriate.



EXAMPLE:

Total nitrogen: 3.52 %w/w

Standard uncertainty: 0.07 %w/w *

*Standard uncertainty corresponds to onestandard deviation.

9.4. Reporting expanded uncertainty

9.4.1. Unless otherwise required, the result xshould be stated together with the expandeduncertainty U calculated using a coverage factork=2 (or as described in section 8.3.3.). Thefollowing form is recommended:

"(Result): x ± U (units)

[where] the reported uncertainty is [an expandeduncertainty as defined in the InternationalVocabulary of Basic and General terms inmetrology, 2nd ed., ISO 1993,] calculated usinga coverage factor of 2, [which gives a level ofconfidence of approximately 95%]"

Terms in parentheses [] may be omitted orabbreviated as appropriate. The coverage factorshould, of course, be adjusted to show the valueactually used.

EXAMPLE:

Total nitrogen: 3.52 ± 0.14 %w/w *

*The reported uncertainty is an expandeduncertainty calculated using a coverage factorof 2 which gives a level of confidence ofapproximately 95%.

9.5. Numerical expression of results

9.5.1. The numerical values of the result and itsuncertainty should not be given with anexcessive number of digits. Whether expandeduncertainty U or a standard uncertainty u isgiven, it is seldom necessary to give more thantwo significant digits for the uncertainty. Resultsshould be rounded to be consistent with theuncertainty given.

9.6. Compliance against limits

9.6.1. Regulatory compliance often requires thata measurand, such as the concentration of a toxicsubstance, be shown to be within particularlimits. Measurement uncertainty clearly has

implications for interpretation of analyticalresults in this context. In particular:

• The uncertainty in the analytical result mayneed to be taken into account when assessingcompliance.

• The limits may have been set with someallowance for measurement uncertainties.

Consideration should be given to both factors inany assessment. The following paragraphs giveexamples of common practice.

9.6.2. Assuming that limits were set with noallowance for uncertainty, four situations areapparent for the case of compliance with anupper limit (see figure 6.1):

i) The result exceeds the limit value plus theestimated uncertainty.

ii) The result exceeds the limiting value by lessthan the estimated uncertainty.

iii) The result is below the limiting value byless than the estimated uncertainty

iv) The result is less than the limiting valueminus the estimated uncertainty.

Case i) is normally interpreted as demonstratingclear non-compliance. Case iv) is normallyinterpreted as demonstrating compliance. Casesii) and iii) will normally require individualconsideration in the light of any agreements withthe user of the data. Analogous arguments applyin the case of compliance with a lower limit.

9.6.3. Where it is known or believed that limitshave been set with some allowance foruncertainty, a judgement of compliance canreasonably be made only with knowledge of thatallowance. An exception arises wherecompliance is set against a stated methodoperating in defined circumstances. Implicit insuch a requirement is the assumption that theuncertainty, or at least reproducibility, of thestated method is small enough to ignore forpractical purposes. In such a case, provided thatappropriate quality control is in place,compliance is normally reported only on thevalue of the particular result. This will normallybe stated in any standard taking this approach.



Figure 2: Uncertainty and compliance limits

UpperControlLimit

( i )Result plus uncertainty above limit

( iv )Result minus uncertainty below limit

( ii )Result

above limit but limit within

uncertainty

( iii )Result below limit but limit

within uncertainty

Quantifying Uncertainty Appendix A. Examples


Appendix A. Examples

Introduction

General introduction

These examples illustrate how the techniques forevaluating uncertainty, described in section 7, canbe applied to some typical chemical analyses.They all follow the procedure shown in the flowdiagram (Figure 1) The uncertainty sources areidentified and set out on a cause and effectdiagram (see appendix D). This helps to avoiddouble counting of sources and also assists inwith the grouping together of components whosecombined effect can be evaluated. Each examplehas an introductory summary page, giving anoutline of the analytical method, a list of theuncertainty sources, the values derived for theuncertainty components and the combineduncertainty.

Examples 1-3 illustrate the evaluation of theuncertainty by the quantification of theuncertainty arising from each source separately.Each gives a detailed analysis of the uncertaintyassociated with the measurement of volumesusing volumetric glassware and masses fromdifference weighings. The detail is for illustrativepurposes, and should not be taken as a generalrecommendation as to the level of detail required.For many analyses the uncertainty associatedwith these operations will not be significant andsuch a detailed evaluation will not be necessary.It would be sufficient to use typical values forthese operations with due allowance being madefor the actual values of the masses and volumesinvolved.

Example 1

Example 1 deals with the very simple case of thepreparation of a calibration standard of Cadmiumin HNO3 for AAS. Its purpose is to show how toevaluate the components of uncertainty arisingfrom the basic operations of volume measurementand weighing and how these components arecombined to determine the overall uncertainty.

Example 2

This deals with the preparation of a standardisedsolution of sodium hydroxide (NaOH) which is

standardised against the titrimetric standardpotassium hydrogen phthalate (KHP). It includesthe evaluation of uncertainty on simple volumemeasurements and weighings, as described inexample 1, but also examines the uncertaintyassociated with the end-point determination.

Example 3

This example expands on example 2 by includingthe standardisation of the NaOH against atitrimetric standard of KPH.

Example 4

This illustrates the use of in house validationdata, as described in section 7.4., and shows howthe data can be used to evaluated the uncertaintyarising from combined effect of a number ofsources. It also shows how to evaluate theuncertainty associated with method bias.

Example 5

This shows how to evaluate the uncertainty onresults obtained using a standard or “empirical”method to measure the amount of heavy metalsleached from ceramic ware using a definedprocedure, as described in section 7.2.-7.5.. Itspurpose is to show how, in the absence ofcollaborative trial data or ruggedness testingresults, it is necessary to consider the uncertaintyarising from the range of the parameters e.g.temperature, etching time and acid strength,allowed in the method definition. This process isconsiderably simplified when collaborative studydata is available as is shown in the next example.

Example 6

The sixth example is based on an uncertaintyestimate for a crude (dietary) fibre determination.Since the analyte is defined only in terms of thestandard method, the method is empirical. In thiscase, collaborative study data, in-house QAchecks and literature study data were available,permitting the approach of section 7.3. The in-house studies verify that the method isperforming as expected on the basis of thecollaborative study. The example shows how theuse of collaborative study data backed up by in-

Quantifying Uncertainty Appendix A. Examples


house method performance checks cansubstantially reduce the number of differentcontributions required to form an uncertaintyestimate under these circumstances.

Example 7

This gives a detailed description of the evaluationof uncertainty on the measurement of the leadcontent of a water sample using IDMS. In

addition to identifying the possible sources ofuncertainty and quantifying them by statisticalmeans the examples shows how it is alsonecessary to include the evaluation ofcomponents based on judgement as described insection 7.10.(Use of judgement is a special caseof Type B evaluation as described in the ISOGuide [G.2])

Quantifying Uncertainty Example A1: Preparation of a calibration standard


Example A1: Preparation of a calibration standard

Summary

Goal

Preparation of a 1000 mg·l–1 calibration standarda high purity metal (Cadmium).

Measurement procedure

The surface of the high purity metal is cleaned toremove any metal-oxide contamination.Afterwards the metal is weighed and thendissolved in nitric acid in a volumetric flask. Thestages in the procedure are show in the followingflow chart.

Clean metalsurface

Weigh metal

Dissolve and dilute

RESULT

Figure A1.1: Preparation of Cadmiumstandard

Measurand

VPm

cCd⋅⋅

=1000

[mg l-1]

where the parameters are those in Table A1.1below. The factor of 1000 is a conversion factorfrom ml to l

Identification of the uncertainty sources:

The relevant uncertainty sources are shown in thecause and effect diagram below:

PurityV

m

Repeatability

Calibration

Temperature

c(Cd)

m(tare)

m(gross)

Repeatability

Repeatability

Calibration

Linearity

Sensitivity

Calibration

Linearity

Sensitivity

Quantification of the uncertainty components

The values and their uncertainties are shown inthe Table below.

Combined Standard Uncertainty

The combined standard uncertainty for thepreparation of a 1002.7 mg/l Cd calibrationstandard is 0.9 mg/l

The different contributions are showndiagramatically in Figure A1.1.

Table A1.1: Values and uncertainties

Description Value Standard

uncertainty

Relative standarduncertainty*

P Purity of the metal 0.9999 0.000058 0.000058

m Weight of the metal 100.28 mg 0.05 mg 0.0005

V Volume of the flask 100.0 ml 0.07 ml 0.0007

cCd concentration of thecalibration standard

1002.7 mg/l 0.9 mg/l 0.0009

*u(x)/x



Figure A1.1: Uncertainty contributions in Cadmium standard preparation

0 0.0002 0.0004 0.0006 0.0008 0.001

c(Cd)

V

m

Purity

relative standard uncertainty



Example A1: Detailed description

A1.1 Introduction

This introductory example discusses thepreparation of a calibration standard for atomicabsorption spectroscopy (AAS) from thecorresponding high purity metal (in this example≈ 1000 mg/l Cd in HNO3 0.5 mol/l). Even thoughthe example does not represent an entireanalytical measurement, the use of calibrationstandards is part of nearly every determination,because modern routine analytical measurementsare relative measurements, which need areference standard to provide traceability to theSI.

A1.2 Step 1: Specification

The goal of this first step is to write down a clearstatement of what is being measured. Thisspecification includes a description of thepreparation of the calibration standard and themathematical relationship between the measurandand the parameters upon which it depends.

Procedure

The specific information on how to prepare acalibration standard is normally given in aStandard Operating Procedure (SOP). Thepreparation consists of the following stages

Clean metalsurface

Weigh metal

Dissolve and dilute

RESULT

Figure A1.2: Preparation of Cadmiumstandard

The surface of the high purity metal is treatedwith an acid mixture to remove any metal-oxidecontamination. The cleaning method is providedby the manufacture of the metal and needs to becarried out to obtain the purity quoted on thecertificate.

The volumetric flask (100 ml) is weighed withoutand with the purified metal inside. The balanceused has a resolution of 0.01 mg.

1 ml of nitric acid (65%) and 3 ml of ion-freewater are added to the flask to dissolve theCadmium (approximately 100 mg). Moderateheat is applied to speed up the dissolutionprocess. Afterwards the flask is filled with ion-free water up to the mark and mixed by invertingthe flask at least thirty times.

Calculation:

The measurand in this example is theconcentration of the calibration standard solution,which depends upon the weighing of the highpurity metal (Cd), its purity and the volume of theliquid in which it is dissolved. The concentrationis given by

VPm

cCd⋅⋅

=1000

[mg/l]

wherecCd :concentration of the calibration standard

[mg/l]1000 :conversion factor from [ml] to [l]m :weight of the high purity metal [mg]P :purity of the metal given as mass fraction

[]V :volume of the liquid of the calibration

standard [ml]

A1.3 Step 2: Identifying and analysinguncertainty sources

The aim of this second step is to list all theuncertainty sources for each of the parameterwhich effect the value of the measurand.



Purity

The purity of the metal (Cd) is quoted in thesupplier's certificate as 99.99 ±0.01%. P istherefore 0.9999 ±0.0001. These values dependon the effectiveness of the surface cleaning of thehigh purity metal. If the manufacturer's procedureis strictly followed no additional uncertainty dueto the contamination of the surface with metal-oxide needs to be added to the value given in thecertificate.

m

The second stage of the preparation involvesweighing the high purity metal. A 100 ml quantityof a 1000 mg/l Cadmium solution is to beprepared.

The relevant weighings are:

container and high purity metal56.76325 gobserved

container less high purity metal56.66297 gobserved

high purity metal0.10028 g(calculated)

As these numbers show, the final weight is aweight by difference. The uncertainty sources foreach of the two weighings are the run to runvariability and the contribution due to theuncertainty in the calibration function of thescale. This calibration function has two potentialuncertainty sources: The sensitivity of the balanceand its linearity. The sensitivity can be neglectedbecause the weight by difference is done on thesame balance over a very narrow range.

V

§ The volume of the solution contained in thevolumetric flask is subject to three majorsources of uncertainty:

§ The uncertainty in the certified internalvolume of the flask.

§ Variation in filling the flask to the mark.

§ The flask and solution temperatures differingfrom the temperature at which the volume ofthe flask was calibrated.

The different effects and their influences areshown as a cause and effect diagram in FigureA1.3 (see Appendix D for description).

Figure A1.3: Uncertainties in Cd Standardpreparation

PurityV

m

Repeatability

Calibration

Temperature

c(Cd)m(tare)

m(gross)

Repeatability

RepeatabilityCalibration

Linearity

Sensitivity

Calibration

Linearity

Sensitivity

A1.4 Step 3: Quantifying the uncertaintycomponents

In step 3 the size of each identified potentialsource of uncertainty is either directly measured,estimated using previous experimental results orderived from theoretical analysis.

Purity

The purity of the Cadmium is given on thecertificate as 0.9999 ±0.0001. Because there is noadditional information about the uncertaintyvalue a rectangular distribution is assumed. Toobtain the standard uncertainty u(P) the value of0.0001 has to be divided by 3 (see section ...)

000058.03

0001.0)( ==Pu

m

The weight of the Cadmium is obtained fromdifference weighings which consist of twoindependent measurements. If the weighingprocedure is performed on the same scale, in thesame narrow weight region and within a shortperiod of time, the sensitivity contribution to thecalibration influence quantity cancels itself out.

Repeatability: The run to run variabilities aregiven in the manufacturer's handbook. Thestandard deviation (s) for weight measurementsfrom 50 g to 200 g is quoted as 0.04 mg.According to the manufacture's information thevalue for the repeatability was determined by aseries of always ten measurements of a tare andgross weight. The difference of each of the tenmeasurement pairs was calculated and then thestandard deviation of these differences. The same



value for the standard deviation of the differenceswas obtained independently in the laboratoryusing check weights. The repeatabilitycontribution has to be taken into account onlyonce, because the standard deviation of thedifferences was directly determined in the givenexperimental design.

Linearity: The specification of the balance quotesthat the difference from the actual weight on thescale pan and the reading of the scale is withinthe limits of ±0.03 mg. This value is only validfor a range of up to 10 g from the tare weight tothe gross weight. According to the manufacturer'sown uncertainty evaluation a rectangulardistribution has to be assumed. Hence the valuefor the linearity contribution needs to be dividedby 3 to give the component of uncertainty as astandard uncertainty

mg017.03mg03.0

=

This component has to be taken intoaccount twice because of the weight bydifference.

Finally the two components, repeatability andlinearity, are combined by taking the square rootof the sum of their squares (see Appendix: 1) togive the standard uncertainty u(m)

mg05.0)017.0(204.0)( 22 =⋅+=mu

Notes:-The given value for the repeatabilityreflect only pure repeatability conditions.Especially they do not account for differencesbetween operators or measurement conditions.

-There is no need for a buoyancycorrection, because the density of cadmium(8642 kg/m3) and of the calibration weight(8006 kg/m3 since 1997) are nearly the same.

V

Calibration: The manufacturer quotes a volumefor the flask of 100 ml ±0.1 ml measured at atemperature of 20°C. The value of the uncertaintyis given without a confidence level, therefore theappropriate standard uncertainty is calculatedassuming a triangular distribution, since theactual volume is more likely to be at the centrethan at the extremes of the range.

ml04.06ml1.0

=

Repeatability: The uncertainty due to variationsin filling can be estimated from a repeatabilityexperiment on a typical example of the flaskused. A series of ten fill and weight experimentson a typical 100 ml flask gave a standarduncertainty of 0.02 ml. This standard uncertaintycan be used directly.

Temperature: According to the manufacturer theflask has been calibrated at a temperature of20°C, whereas the laboratory temperature variesbetween the limits of ±4°C. The uncertainty fromthis effect can be calculated from the estimate ofthe temperature range and the coefficient of thevolume expansion. The volume expansion of theliquid is considerably larger than that of the flask,therefore only the former needs to be considered.The coefficient of volume expansion for water is2.1·10–4 °C–1, which leads to a volume variationof

ml084.0C101.2C4ml100 14 ±=⋅⋅±⋅ − -oo

The standard uncertainty is calculated using theassumption of a rectangular distribution for thetemperature variation i.e.

ml05.03

ml084.0=

The three contribution are combined to thestandard uncertainty u(V) of the volume V

ml07.005.002.004.0)( 222 =++=Vu

A1.5 Step 4: Calculating the combinedstandard uncertainty

cCd is given by

]lmg[1000V

PmcCd

⋅⋅=

The intermediate values, their standarduncertainties and their relative standarduncertainties are summarised overleaf (TableA1.2)

Using those values the concentration of thecalibration standard is

1lmg7.10020.100

9999.028.1001000 −⋅=⋅⋅

=Cdc



Table A1.2: Values and Uncertainties

Description Value x u(x) u(x)/x

Purity of themetal P

0.9999 0.000058 0.000058

Weight ofthe metal m(mg)

100.28 0.05 mg 0.0005

Volume ofthe flaskV (ml)

100.0 0.07 ml 0.0007

For this simple multiplicative expression theuncertainties associated with each component arecombined as follows.

222 )()()()(

+

+

=

V

Vu

m

mu

P

Pu

c

c

Cd

Cdcu

0009.00007.00005.0000058.0 222

=++=

1

1

lmg9.0

0009.0lmg7.10020009.0)(−

−

⋅=

⋅⋅=⋅= CdCdc ccu

However it is preferable to derive the combinedstandard uncertainty (uc(cCd)) using thespreadsheet method given in Appendix E, since

this can be utilised even for complex expressions.The completed spreadsheet is show below. Thevalues of the parameters are entered in the secondrow from C2 to E2. Their standard uncertaintiesare in the row below (C3-E3). The spreadsheetcopies the values from C2-E2 into the secondcolumn from B5 to B7. The result (c(Cd)) usingthese values is given in B9. The C5 shows thevalue of P from C2 plus its uncertainty given inC3. The result of the calculation using the valuesC5-C7 is given in C9. The columns D and Efollow a similar procedure. The values shown inthe row 10 (C10-E10) are the differences of therow (C9-E9) minus the value given in B9. In row11 (C11-E11) the values of row 10 (C10-E10) aresquared and summed to give the value shown inB11. B13 gives the combined standarduncertainty, which is the square root of B11.

The relative contributions of the differentparameters are shown Figure A1.1. Thecontribution of the uncertainty on the volume ofthe flask is the largest and that from the weighingprocedure is similar, whereas the uncertainty onthe purity of the Cadmium has virtually noinfluence on the overall uncertainty.

The expanded uncertainty U(cCd) is obtained bymultiplying the combined standard uncertaintywith a coverage factor of 2 giving.

11 lmg8.1lmg9.02)( −− ⋅=⋅⋅=CdcU

A B C D E

1 P m V

2 Value 0.9999 100.28 100.00

3 Uncertainty 0.000058 0.05 0.07

4

5 P 0.9999 0.999958 0.9999 0.9999

6 m 100.28 100.28 100.33 100.28

7 V 100.0 100.00 100.00 100.07

8

9 c(Cd) 1002.69972 1002.75788 1003.19966 1001.99832

10 0.05816 0.49995 -0.70140

11 0.74529 0.00338 0.24995 0.49196

12

13 u(c(Cd)) 0.9

Table A1.3: Spreadsheet calculation of uncertainty

Quantifying Uncertainty Example A2


Example A2 - Standardising a sodium hydroxide solution

Summary

Goal

A solution of sodium hydroxide (NaOH) isstandardised against the titrimetric standardpotassium hydrogen phthalate (KHP).

Measurand:

]lmol[1000 1−⋅

⋅⋅⋅

=TKHP

KHPKHPNaOH VF

Pmc

wherecNaOH :concentration of the NaOH solution [mol.l–

1]1000 :conversion factor [ml] to [l]mKHP :weight of the titrimetric standard KHP [g]PKHP :purity of the titrimetric standard given asmass fraction []FKHP :molecular weight of KHP [g.mol–1]VT :titration volume of NaOH solution [ml]


The titrimetric standard (KHP) is dried andweighed. After the preparation of the NaOHsolution the sample of the titrimetric standard(KHP) is dissolved and then titrated using theNaOH solution. The stages in the procedure areshown in the flow chart

Weighing KHP

Preparing NaOH

Titration

RESULT


The relevant uncertainty sources are shown as acause and effect diagram in Figure A2.1.


The different uncertainty contributions are givenin Table A2.1, and shown diagramatically inFigure A2.2.

The combined standard uncertainty for the0.10214 mol/l NaOH solution is 0.00009 mol/l

Table A2.1: Values and uncertainties in NaOH standardisation

Description Value x Standard uncertainty u Relative standarduncertainty*

mKHP weight of KHP 0.3888 g 0.00015 g 0.00036

PKHP Purity of KHP 1.0 0.00029 0.00029

FKHP Formula weight of KHP 204.2212 g mol-1 0.0038 g mol-1 0.000019

VT Volume of NaOH for KHPtitration

18.64 ml 0.015 ml 0.0008

cNaOH NaOH solution 0.10214 mol l-1 0.00009 mol l-1 0.00092

*u(x)/x



Figure A2.1: Cause and effect diagram for titration

c(NaOH)

P(KHP)

F(KHP)V(T)

repeatability

calibration

temperature

end-point

biasrepeatability

m(KHP)

m(gross)m(tare)

repeatability

repeatability

calibration

linearity

sensitivity

calibration

linearity

sensitivity

Figure A2.2: Contributions to Titration uncertainty

0 0.0002 0.0004 0.0006 0.0008 0.001

c(NaOH)

V (T)

F(KHP)

P(KHP)

m(KHP)




Example A2: Detailed description

A2.1 Introduction

This second introductory example discusses anexperiment to determine the concentration of asolution of sodium hydroxide (NaOH). TheNaOH is titrated against the titrimetric standardpotassium hydrogen phthalate (KHP). It isassumed that the NaOH concentration is known tobe of the order of 0.1 mol.l–1. The end-point of thetitration is determined by an automatic titrationsystem using a combined pH-electrode to measurethe shape of the pH-curve. The functionalcomposition of the titrimetric standard potassiumhydrogen phthalate (KHP), which is the numberof titratable protons in relation to the overallnumber of molecules, provides traceability of theconcentration of the NaOH solution to the SIunits of measurement.


The aim of the first step is to describe themeasurement procedure. This description consistsof a listing of the measurement steps and amathematical statement of the measurand and theparameters upon which it depends.

Procedure:

The measurement sequence to standardise theNaOH solution has the following stages.

Weighing KHP

Preparing NaOH

Titration

RESULT

Figure A2.3: Standardisation of a solution ofsodium hydroxide

The separate stages are:

i) The primary standard potassium hydrogenphthalate (KHP) is dried according to thedescription of the supplier. This descriptionis given in the supplier's catalogue, whichalso states the purity of the titrimetricstandard and its uncertainty. A titrationvolume of approximately 19 ml of 0.1 mol/lsolution of NaOH entails weighing out anamount as close as possible to

g388.00.11000

191.02212.294=

⋅⋅⋅

The weighing is carried out on a balance with aresolution of 0.1 mg.

ii) A 0.1 mol/l solution of sodium hydroxide isprepared. In order to prepare 1 l of solution,it is necessary to weight out ≈ 4 g NaOH.However, since the concentration of theNaOH solution is to be determined by assayagainst the primary standard KHP and not bydirect calculation, no information on theuncertainty sources connected with themolecular weight or the weight taken isrequired.

iii) The weighed quantity of the titrimetricstandard KHP is dissolved with ≈ 50 ml ofion-free water and then titrated using theNaOH solution. An automatic titrationsystem controls the addition of NaOH andrecords the pH-curve. It also determines theend-point of the titration from the shape ofthe recorded curve.

Calculation:

The measurand is the concentration of the NaOHsolution, which depends on the weight of KHP,its purity, its molecular weight and the volume ofNaOH at the end-point of the titration

]lmol[1000 1−⋅

⋅⋅⋅

=TKHP

KHPKHPNaOH VF

Pmc

wherecNaOH :concentration of the NaOH solution

[mol.l–1]1000 :conversion factor [ml] to [l]mKHP :weight of the titrimetric standard KHP

[g]



PKHP :purity of the titrimetric standard givenas mass fraction []

FKHP :molecular weight of KHP [g.mol–1]VT :titration volume of NaOH solution [ml]


The aim of this step is to identify all majoruncertainty sources and to understand their effecton the measurand and its uncertainty. This hasbeen shown to be one of the most difficult step inevaluating the uncertainty of analyticalmeasurements. Because there is a risk of

neglecting uncertainty sources on the one handand an the other of double-counting them. Theuse of a cause and effect diagram (Appendix CE)is one possible way to help prevent thishappening. The first step in preparing the diagramis to draw the four parameters of the equation ofthe measurand as the main branches.

Afterwards, each step of the method is consideredand any further influence quantity is added as afactor to the diagram working outwards from the

main effect. This is carried out for each branchuntil effects become sufficiently remote, that is,until effects on the result are negligible.

m (KHP)

Approximately 388 mg of KHP are weighed tostandardise the NaOH solution. The weighingprocedure is a weight by difference. This meansthat a branch for the determination of the tare(mtare) and another branch for the gross weight(mgross) have to be drawn in the cause and effectdiagram. Each of the two weighings is subject torun to run variability and the uncertainty of thecalibration of the balance. The calibration itselfhas two possible uncertainty sources: thesensitivity and the linearity of the calibrationfunction. If the weighing is done on the samescale and over a small range of weight then thesensitivity contribution can be neglected.

All these uncertainty sources are added into thecause and effect diagram .

P (KHP)

The purity of KHP is quoted in the supplier'scatalogue to be within the limits of 99.95% and100.05%. PKHP is therefore 1.0000 ±0.0005. Thereis no other uncertainty source if the dryingprocedure was performed according to thesuppliers specification.

F (KHP)

Potassium hydrogen phthalate (KHP) has theempirical formula

C8H5O4KThe uncertainty in the formula weight of thecompound can be determined by combining the

c(NaOH)

m(KHP)P(KHP)

F(KHP)V(T)

Figure A2.4: First step in setting up a causeand effect diagram

c(NaOH)

m(KHP)P(KHP)

F(KHP)V(T)

m(gross)m(tare)

repeatability

repeatability

calibration

linearity

sensitivity

calibration

linearity

sensitivity

Figure A2.5:Cause and effect diagram with added uncertainty sourcesfor the weighing procedure



uncertainty in the atomic weights of itsconstituent elements. A table of atomic weightsincluding uncertainty estimates is publishedbiennially by IUPAC in the Journal of Pure andApplied Chemistry. The formulae weight can becalculated directly from these; the cause andeffect diagram (Figure A2.6) omits the individualatomic masses for clarity

V (T)

The titration is accomplished using a 20 ml pistonburette. The delivered volume of NaOH from thepiston burette is subject to the same threeuncertainty sources as the filling of the volumetricflask in the previous example. These uncertaintysources are the repeatability of the deliveredvolume, the uncertainty of the calibration of thatvolume and the uncertainty resulting from thedifference between the temperature in thelaboratory and that of the calibration of the pistonburette. In addition there is the contribution of theend-point detection, which has two uncertaintysources.

1. The repeatability of the end-point detection.Which is independent of the repeatability ofthe volume delivery.

2. The possibility of a systematic differencebetween the determined end-point and theequivalence point (bias), due to carbonateabsorption during the titration and inaccuracyin the mathematical evaluation of the end-point from the titration curve.

These items are included in the completed causeand effect diagram shown in Figure A2.6.

A2.4 Step 3: Quantifying uncertaintycomponents

In step 3 uncertainty from each source identifiedin step 2 has to be quantified and then convertedto a standard uncertainty.

m (KHP)

The relevant weighings are:container and KHP: 60.5450 g(observed)container less KHP: 60.1562 g(observed)KHP 0.3888 g(calculated)

1. Repeatability: The quality control log shows astandard uncertainty of 0.05 mg for checkweighings of weights up to 100 g. This valuefor the repeatability was determined by aseries of ten measurements of the tare andgross weight, followed by the calculation ofthe difference of each of the ten measurementpairs and the evaluation of the standarddeviation of these differences. Thisrepeatability contribution has to be taken intoaccount only once, because the standarddeviation of the differences was directlydetermined in the given experimental design.

2. Calibration/Linearity: The calibrationcertificate of the balance quotes ±0.15 mg forthe linearity. This value is the maximumdifference between the actual weight on thepan and the reading of the scale. The balancemanufacture's own uncertainty evaluationrecommends the use of a rectangulardistribution to convert the linearitycontribution to a standard uncertainty.

The balance linearity contribution is accordingly

c(NaOH)

P(KHP)

F(KHP)V(T)

repeatability

calibration

temperature

end-point

biasrepeatability

m(KHP)

m(gross)m(tare)

repeatability

repeatability

calibration

linearity

sensitivity

calibration

linearity

sensitivity

Figure A2.6: Final cause and effect diagram



mg09.03mg15.0

=

This contribution has to be counted twice, oncefor the tare and once for the gross weight.

Combining the two contributions gives for thestandard uncertainty u(mKHP) of the mass mKHP, avalue of

mg14.0)()09.0(205.0)( 22

=⇒

⋅+=

KHP

KHP

m

m

u

u

NOTE 1: Within the framework of titration experimentsbuoyancy corrections are very rarely made,because density differences between analysedsamples are often smaller than 1000 kg/m3.There is also no need for an additionaluncertainty contribution due to neglecting thebuoyancy correction, because its contributionis smaller than the repeatability or linearitycontributions.

NOTE 2:-There are other difficulties weighing atitrimetric standard: A temperature differenceof only 1°C between the standard and thebalance causes a drift in the same order ofmagnitude than the repeatability contribution.The titrimetric standard has been completelydried, but the weighing procedure is carriedout at a humidity of around 50 % relativehumidity. Therefore adsorption of somemoisture is expected.

P (KHP)

PKHP is 1.0000 ±0.0005. The supplier gives nofurther information concerning the uncertainty inthe catalogue. Therefore this uncertainty is takenas having a rectangular distribution, so thestandard uncertainty u(PKHP) is

00029.030005.0 = .

F (KHP)

From the latest IUPAC table, the atomic weightsand listed uncertainties for the constituentelements of KHP (C8H5O4K) are:

Element Atomicweight

Quoteduncertainty

Standarduncertainty

C 12.0107 ±0.0008 0.00046H 1.00794 ±0.00007 0.000040O 15.9994 ±0.0003 0.00017K 39.0983 ±0.0001 0.000058

For each element, the standard uncertainty isfound by treating the IUPAC quoted uncertainty

as forming the bounds of a rectangulardistribution. The corresponding standarduncertainty is therefore obtained by dividingthose values by 3 .

The separate element contributions to the formulaweight, together with the uncertainty contributionfor each, are:

Calculation Result Standarduncertainty

C8 8·12.0107 96.0856 0.0037H5 5·1.00794 5.0397 0.00020O4 4·15.9994 63.9976 0.00068K 1·39.0983 39.0983 0.000058

The uncertainty in each of these values iscalculated by multiplying the standard uncertaintyin the previous table by the number of atoms.

This gives a formula weight for KHP of

1molg2212.204

0983.399976.630397.50856.96−⋅=

+++=KHPF

As this expression is a sum of independent values,the standard uncertainty u(FKHP) is a simplesquare root of the sum of the squares of thecontributions:

1

2

222

molg0038.0)(

000058.0

00068.00002.00037.0)(

−⋅=⇒

+

++=

KHP

KHP

F

F

u

u

NOTE: Since the element contribution to KHP aresimply the sum of the single atomcontributions, it might be expected from thegeneral rule for combing uncertaintycontributions that the uncertainty for eachelement contribution would be calculated fromthe sum of squares of the single atomcontributions, that is, for carbon ,

001.000037.08)( 2 =⋅=CFu . Recall,however, that this rule applies only toindependent contributions, that is,contributions from separate determinations ofthe value. In this case, since the totalcontributions is obtained by multiplying thevalue from a single value by 8. Notice that thecontributions from different elements areindependent, and will therefore combine in theusual way.



V (T)

1. Repeatability of the volume delivery: The useof a piston burette, unlike that of a volumetricflask, does not in general involve the completedelivery of its contents. When estimating thevariability of the delivery it will be necessaryto take into account the graduation marksbetween which discharge occurs. For a givenburette, several such sets of delivery limitsshould be investigated and recorded; e.g. 0-10,10-20, 5-15 etc. Similarly, it may be necessaryto investigate the repeatability of differentvolumes delivered, such as 5, 10, 15 ml etc. Inthis example, the repeatability of theanticipated deliveries of 19 ml were checked,giving a sample standard deviation of0.004 ml, used directly as a standarduncertainty.

2. Calibration: The limits of accuracy of thedelivered volume are indicated by themanufacturer as a ± figure. For a 20 ml pistonburette this number is typically ±0.03 ml.Assuming a triangular distribution gives astandard uncertainty of ml012.0603.0 = .

Note: The ISO Guide (F.2.3.3) recommendsadoption of a triangular distribution ifthere are reasons to expect values in thecentre of the range being more likelythan those near the bounds. Thereforeassuming unconditionally a rectangulardistribution in all the cases where limitsare given as its maximum bounds(without a confidence level) can hardlybe justified.

3. Temperature: The uncertainty due to the lackof temperature control is calculated in thesame way as in the previous example, but thistime taking a possible temperature variation of±3°C (with a 95% confidence). Again usingthe coefficient of volume expansion for wateras 2.1.10–4 °C–1 gives a value of

ml006.096.1

3101.219 4

=⋅⋅⋅ −

Thus the standard uncertainty due to incompletetemperature control is 0.006 ml.

NOTE: When dealing with uncertainties arising fromincomplete control of environmental factorssuch as temperature, it is essential to takeaccount of any correlation in the effects ondifferent intermediate values. In this example,the dominant effect on the solutiontemperature is taken as the differential heatingeffects of different solutes, that is, thesolutions are not equilibrated to ambienttemperature. Temperature effects on eachsolution concentration at STP are thereforeuncorrelated in this example, and areconsequently treated as independentuncertainty contributions.

4. Repeatability of the end-point detection: Therepeatability of the end-point detection wasthoroughly investigated during the methodvalidation. Under the given conditions astandard uncertainty of 0.004 ml isappropriate.

5. Bias of the end-point detection: The titration isperformed under a layer of Argon to excludeany bias due to the absorption of CO2 in thetitration solution. This approach follows theguidelines, that it is better to prevent any biasinstead of correcting for it. There are no otherindications, that the end-point determined fromthe shape of the pH-curve does not correspondto the equivalence-point, because a strong acidis titrated with a strong base. Therefore it isassumed that the bias of the end-pointdetection and its uncertainty are negligible.

VT is found to be 18.64 ml and combining the fourremaining contributions to the uncertainty u(VT)of the volume VT gives a value of

ml015.0)(004.0006.0012.0004.0)( 2222

=⇒+++=

T

T

V

V

u

u




cNaOH is given by

]lmol[1000 1−⋅

⋅⋅⋅

=TKHP

KHPKHPNaOH VF

Pmc

The values of the parameters in this equation,their standard uncertainties and their relativestandard uncertainties are collected in table TableA2.1

Using the values given above:

1lmol10214.064.182212.204

0.13888.01000 −⋅=⋅

⋅⋅=NaOHc

In order to combine the uncertainties associatedwith each component of a multiplicativeexpression (as above) the standard uncertaintiesare used in the following way

00092.0

0008.0

000019.000029.000036.0

)()(

)()(

)(

2

222

22

22

=

+

++=

+

+

+

=

T

T

KHP

KHP

KHP

KHP

KHP

KHP

NaOH

NaOHc

VVu

FFu

PPu

mmu

ccu

( ) 1lmol00009.000092.0 −⋅=⋅=⇒ NaOHNaOH cccu

A standard spreadsheet is used to simplify theabove calculation of the combined standarduncertainty. A comprehensive introduction intothe method is given in Appendix E. Thespreadsheet filled in with the appropriate values isshown as Table A2.2.

The values of the parameters are given in thesecond row from C2 to F2. Their standarduncertainties are entered in the row below (C3-F3). The spreadsheet copies the values from C2-F2 into the second column from B5 to B8. Theresult (c(NaOH)) using these values is given inB10. The C5 shows the value of m(KHP) from C2plus its uncertainty given in C3. The result of thecalculation using the values C5-C8 is given inC10. The columns D and F follow a similarprocedure. The values shown in the row 11 (C11-F11) are the differences of the row (C10-F10)minus the value given in B10. In row 12 (C12-F12) the values of row 11 (C11-F11) are squaredand summed to give the value shown in B12. B14gives the combined standard uncertainty, which isthe square root of B12.

At the end it is instructive to examine the relativecontributions of the different parameters. Theshare of each contribution can easily be visualisedusing a histogram displaying the relative standarduncertainties:

The contribution of the uncertainty of the titrationvolume VT is be far the largest. The weighingprocedure and the purity of the titrimetric

Table A2.1: Values and uncertainties for titration

Description Value x Standard uncertaintyu(x)

Relative standarduncertainty u(x)/x

mKHP weight of KHP 0.3888 g 0.00014 g 0.00036


FKHP Formula weight of KHP 204.2212 g/mol 0.0038 g/mol 0.000019

VT Volume of NaOH for KHPtitration

18.64 ml 0.015 ml 0.0008

0 0.0002 0.0004 0.0006 0.0008 0.001

c(NaOH)

V(T)

F(KHP)

P(KHP)

m(KHP)

relative s tandard uncertainty

Figure A2.7: Uncertainty contributions inNaOH standardisation



standard show the same order of magnitude fortheir relative standard uncertainties, whereas theuncertainty in the formula weight is again nearlyan order of magnitude smaller.

The expanded uncertainty U(cNaOH) is obtained bymultiplying the combined standard uncertainty bya coverage factor of 2.

1lmol0002.0200009.0) −⋅≅⋅=NaOHcU(

Thus the concentration of the NaOH solution is0.1021 ±0.0002 mol.l–1.

A2.6 Step 5: Re-evaluate the significantcomponents

The contribution of V(T) is the largest one. Thevolume of NaOH for titration of KHP (V(T)) itselfis affected by four influence quantities, which arethe repeatability of the volume delivery, thecalibration of the piston burette, the differencebetween the operation and calibration temperatureof the burette and the repeatability of the end-point detection. Checking the size of eachcontribution the calibration is by far the largestone. Therefore this contribution needs to beinvestigated more thoroughly.

The standard uncertainty of the calibration ofV(T) was calculated from the data given by the

manufacturer assuming a triangular distribution.The influence of the choice of the shape of thedistribution is shown in Table A2.3. 1) Accordingto the ISO Guide 4.3.9 Note 1: "For a normaldistribution with expectation µ and standarddeviation σ, the interval µ ±3σ encompassesapproximately 99.73 percent of the distribution.Thus, if the upper and lower bounds a+ and a-

define 99.73 percent limits rather than 100percent limits, Xi can be assumed to beapproximately normally distributed rather thanthere being no specific knowledge about Xi

between the bounds as in 4.37, then u2(xi) = a2/9.By comparison, the variance of a symmetricrectangular distribution of the half-width a is a2/3[equation (7)] and that of a symmetric triangulardistribution of the half-width a is a2/6 [equation(9b)]. The magnitudes of the variances of thethree distributions are surprisingly similar in viewof the differences the assumptions upon whichthey are based."

Thus the choice of the distribution function ofthis influence quantity has little effect on thevalue of the combined standard uncertainty(uc(cNaOH)) and it is adequate to assume that it istriangular.

Table A2.2: Spreadsheet calculation of titration uncertainty

A B C D E F1 m(KHP) P(KHP) F(KHP) V(T)2 Value 0.3888 1.0 204.2212 18.643 Uncertainty 0.00014 0.00029 0.0038 0.01545 m(KHP) 0.3888 0.38894 0.3888 0.3888 0.38886 P(KHP) 1.0 1.0 1.00029 1.0 1.07 F(KHP) 204.2212 204.2212 204.2212 204.2250 204.22128 V(T) 18.64 18.64 18.64 18.64 18.6559

10 c(NaOH) 0.102136 0.102173 0.102166 0.102134 0.10205411 0.000037 0.000030 -0.000002 -0.00008212 9.0E-9 1.37E-9 9E-10 4E-12 6.724E-91314 u(c(NaOH)) 0.000095



Table A2.3: Effect of different distribution asumptions

Distribution factor u(V(T;cal))(ml)

u(V(T))(ml)

uc(cNaOH)

rectangular 3 0.017 0.019 0.00011 mol.l-1

triangular 6 0.012 0.015 0.00009 mol.l-1

normaldistribution1 9 0.010 0.013 0.000085 mol.l-1



Example A3 - An acid/base titration

Summary

Goal

A solution of hydrochloride acid (HCl) isstandardised against a solution of sodiumhydroxide (NaOH) with known content.


A solution of hydrochloride acid (HCl) is titratedagainst a solution of sodium hydroxide (NaOH),which before has been standardised against thetitrimetric standard potassium hydrogen phthalate(KHP), to determine its concentration. The stagesof the procedure are shown in Figure A3.1.

Measurand:

cm P V

V F VHClKHP KHP T

T KHP HCl

=⋅ ⋅ ⋅

⋅ ⋅1000 2

1

where the symbols are as given in Table A3.1 andthe value of 1000 is a conversion factor from mlto l.


The relevant uncertainty sources are shown inFigure A3.2.


The final uncertainty is estimated as0.00016 mol l-1. Table A3.1 summarises the

values and their uncertainties; Figure A3.3 showsthe values diagrammatically.

Figure A3.1: Titration procedure

weighing of KHP

titration of KHP with

NaOH

titration of HCl with

NaOH

RESULT

aliquot of HCl

Figure A3.2: Cause and Effect diagram for acid-base titration

c(HCl)

V(T2)

V(HCl)

Repeatability

Calibration

Temperature

Repeatability

Calibration

Temperature

End-point

BiasRepeatability

V(T1) F(KHP)

Repeatability

Calibration

Temperature

End-point

BiasRepeatability

m(KHP)

m(gross)m(tare)

repeatability

repeatability

calibration

linearity

sensitivity

calibration

linearity

sensitivity

same balance

P(KHP)



Table A3.1: Acid-base Titration values and uncertainties

Description Value x Standarduncertainty u(x)


mKHP Weight of KHP 0.3888 g 0.00013 g 0.00033


VT2 Volume of NaOH for HCl titration 14.89 ml 0.015 ml 0.0010

VT1 Volume of NaOH for KHP titration 18.64 ml 0.016 ml 0.00086


VHCl HCl aliquot for NaOH titration 15 ml 0.011 ml 0.00073

cNaOH HCl solution 0.10139 mol l-1 0.00016 mol l-1 0.0016

Figure A3.3: Uncertainties in acid-base titration

0 0.0005 0.001 0.0015

c(HCl)

V(HCl)

F(KHP)

V(T1)

V(T2)

P(KHP)

m(KHP)




Example A3 - An acid/base titration. Detailed discussion

A3.1 Introduction

This example discusses a sequence ofexperiments to determine the concentration of asolution of hydrochloride acid (HCl). In additiona number of special aspects of the titrationtechnique are highlighted. The HCl is titratedagainst solution of sodium hydroxide (NaOH),which was freshly standardised with potassiumhydrogen phthalate (KHP). As in the previousexample (A2) it is assumed that the HClconcentration is known to be of the order of0.1 mol.l–1 and that the end-point of the titrationis determined by an automatic titration systemusing the shape of the pH-curve. This evaluationgives the measurement uncertainty in terms of theSI units of measurement.


A detailed description of the measurementprocedure is given in the first step. Itcompromises a listing of the measurement stepsand a mathematical statement of the measurand.

Procedure

The determination of the concentration of theHCl solution consists of the following stages(Figure A3.4). The separate stages are:

i) The titrimetric standard potassium hydrogenphthalate (KHP) is dried to obtain the purity,which is quoted in the supplier's certificate.Afterwards approximately 0.388 g of thestandard is weighed to achieve a titrationvolume of 19 ml NaOH.

ii) The KHP titrimetric standard is dissolvedwith ≈ 50 ml of ion free water and thentitrated using the NaOH solution. A titrationsystem controls automatically the addition ofNaOH and samples the pH-curve. The end-point is evaluated from the shape of therecorded curve.

iii) 15 ml of the HCl solution is transferred bymeans of a volumetric pipette. The HClsolution is diluted with ion free water tohave ≈ 50 ml solution in the titration vessel.

iv) The same automatic titrator performs themeasurement of HCl solution.

weighing of KHP

titration of KHP with

NaOH

titration of HCl with

NaOH

RESULT

aliquot of HCl

Figure A3.4: Determination of theconcentration of a HCl solution

Calculation:

The measurand is the concentration of the HClsolution, given by

]lmol[1000 1

1

2 −⋅⋅⋅

⋅⋅⋅=

HClKHPT

TKHPKHPHCl VFV

VPmc

wherecHCl :concentration of the HCl solution [mol.l–1]

1000 :conversion factor [ml] to [l]mKHP :weight of KHP taken [g]

PKHP :purity of KHP given as mass fraction []

VT2 :volume of NaOH solution to titrate HCl[ml]

VT1 :volume of NaOH solution to titrate KHP[ml]

FKHP : formula weight of KHP [g.mol–1]

VHCl :volume of HCl titrated with NaOHsolution [ml]




The different uncertainty sources and theirinfluence on the measurand are best analysed byvisualising them first in a cause and effectdiagram (Figure A3.5).

The cause and effect diagram will not be furthersimplified in the first part of the example. Such asimplification could be achieved by combiningthe different run to run variabilities to one overallrepeatability contribution (cf. second part of thisexample). The influence quantities of theparameter VT2, VT1, mKHP, PKHP and FKHP havebeen discussed extensively in the previousexample, therefore only the new influencequantities of VHCl will be dealt with in more detailin this section.

V(HCl)

15 ml of the investigated HCl solution is to betransferred by means of a volumetric pipette. Thedelivered volume of the HCl form the pipette issubject to the same three sources of uncertainty asall the volumetric measuring devices.

1. The variability or repeatability of the deliveredvolume

2. The uncertainty in the stated volume of thepipette

3. The solution temperature differing from thecalibration temperature of the pipette.


The goal of this step is to quantify eachuncertainty source analysed in step 2. Thequantification of the branches or rather of thedifferent components was described in detail inthe previous two examples. Therefore only asummary for each of the different contributionswill be given.

m(KHP)

1. Repeatability: The quality control log shows astandard uncertainty of 0.05 mg for checkweighings in the range of the balance of 0 g upto 100 g. The employed check weights are inthe same order of magnitude as the amount ofthe titrimetric standard.

2. Calibration/linearity: The balancemanufacturer quotes ±0.15 mg for the linearitycontribution. This value represents themaximum difference between the actualweight on the pan and the reading of the scale.The linearity contribution is assumed to showa rectangular distribution and is converted to astandard uncertainty:

mg087.0315.0

=

The contribution for the linearity has to beaccounted for twice, once for the tare and oncefor the gross weight.

Combining the two contributions to the standarduncertainty ( )u mKHP of the mass mKHP givesa value of

Figure A3.5: Final cause and effect diagram

c(HCl)

V(T2)

V(HCl)

Repeatability

Calibration

Temperature

Repeatability

Calibration

Temperature

End-point

BiasRepeatability

V(T1) F(KHP)

Repeatability

Calibration

Temperature

End-point

BiasRepeatability

m(KHP)

m(gross)m(tare)

repeatability

repeatability

calibration

linearity

sensitivity

calibration

linearity

sensitivity

same balance

P(KHP)



mg13.0)()087.0(205.0)( 2

=⇒

⋅+=

KHP

KHP

m

m

u

u

Note: There is no need for a buoyancy correction,because the density difference between thetitrimetric standard and the HCl solution isonly ≈600 kg m-3 leading to a correction orotherwise uncertainty contributionconsiderably smaller than the linearitycomponent.

P(KHP)

P(KHP) is given in the supplier's certificate as100% ±0.05%. The quoted uncertainty is taken asa rectangular distribution, so the standarduncertainty u(PKHP) is

00029.03

0005.0)( ==KHPPu .

V(T2)

1. Repeatability of the volume delivery: Samplestandard deviation of 0.004 ml obtained fromcheck weighing of the delivered volume.

2. Calibration: Figure given by the manufacturer(±0.03 ml) and approximated to a triangulardistribution 0 03 6 0 012. .= ml .

3. Temperature: The possible temperaturevariation is within the limits of ±4°C andapproximated to a rectangular distribution

ml007.034101.215 4 =⋅⋅⋅ − .

4. Repeatability of the end-point detection:Evaluations during the method validationprovided a standard uncertainty of 0.004 ml.

5. Bias of the end-point detection: A biasbetween the determined end-point and theequivalence-point can be prevented byperforming the titration under a layer of Argongas.

VT2 is found to be 14.89 ml and combining the

four contributions to the uncertainty ( )u VT2 of

the volume VT2 gives a value of

( )( ) ml015.0

004.0007.0012.0004.0

2

22222

=⇒+++=

T

T

V

V

u

u

V(T1)

All contributions except the one for thetemperature are the same as for VT2

1. Repeatability of the volume delivery: 0.004 ml

2. Calibration: ml012.0603.0 =

3. Temperature: The approximate volume for thetitration of 0.3888 g KHP is 19 ml NaOH,therefore its uncertainty contribution is

ml009.034101.219 4 =⋅⋅⋅ − .

4. Repeatability of the end-point detection:0.004 ml

5. Bias: Negligible

VT1 is found to be 18.64 ml with a standard

uncertainty ( )u VT1 of

( )( ) ml016.0

004.0009.0012.0004.0

1

22221

=⇒+++=

T

T

V

V

u

u

F(KHP)

Atomic weights and listed uncertainties (fromIUPAC tables) for the constituent elements ofKHP (C8H5O4K) are:

Element Atomicweight

Quoteduncertaint

y

Standarduncertaint

yC 12.0107 ±0.0008 0.00046H 1.00794 ±0.00007 0.000040O 15.9994 ±0.0003 0.00017K 39.0983 ±0.0001 0.000058

For each element, the standard uncertainty isfound by treating the IUPAC quoted uncertaintyas forming the bounds of a rectangulardistribution. The corresponding standarduncertainty is therefore obtained by dividingthose values by 3 .

The formula weight FKHP for KHP and itsuncertainty u(FKHP)are, respectively:

1molg2212.204

0983.399994.15400794.150107.128

−⋅=

+⋅+⋅+⋅=KHPF

1

22

22

molg0038.0)(

000058.0)00017.04(

)00004.05()00046.08()(

−⋅=⇒

+⋅+

⋅+⋅=

KHP

KHP

F

F

u

u

Note: The single atom contributions are notindependent, therefore the uncertainty forthe atom contribution is calculated bymultiplying the number of atoms with thestandard uncertainty of the single atomdirectly.



V(HCl)

1. Repeatability: Standard uncertainty of0.0037 ml obtained from a replicate weighingexperiment of the delivered volume.

2. Calibration: Uncertainty stated by themanufacturer for a 15 ml pipette as ±0.02 mland approximated with a triangulardistribution: ml008.0602.0 = .

3. Temperature: The temperature of thelaboratory is within the limits of ±4°C. Usinga rectangular temperature distribution gives astandard uncertainty of 34101.215 4 ⋅⋅⋅ − =0.007 ml.

Combining those contributions to the standarduncertainty ( )HClVu give a figure of

( )( ) ml011.0

007.0008.000370 222

=⇒

++=

HCl

HCl

V

.V

u

u


cHCl is given by

HClKHPT

TKHPKHPHCl VFV

VPmc

⋅⋅⋅⋅⋅

=1

21000

All the intermediate values of the two stepexperiment and their standard uncertainties arecollected in Table A3.2. Using these values:

1-lmol10139.0152212.20464.18

89.140.13888.01000⋅=

⋅⋅⋅⋅⋅

=HClc

The uncertainties associated with eachcomponent are combined accordingly:

0016.0

00073.0000019.000086.0

001.000029.000033.0

)()()(

)()()(

)(

222

222

222

1

1

2

2

2

22

=

++

+++=

+

+

+

+

+

=

HCl

HCl

KHP

KHP

T

T

T

T

KHP

KHP

KHP

KHP

HCl

HClc

V

Vu

F

Fu

V

Vu

V

Vu

P

Pu

m

mu

c

cu

1lmol00016.00016.0)( −⋅=⋅=⇒ HClHClc ccu

A standard spreadsheet method is employed tosimplify the above calculation of the combinedstandard uncertainty. A comprehensiveintroduction into the method is given in AppendixE. The spreadsheet filled in with the appropriatevalues is shown in Table A3.3.

The values of the parameters are given in thesecond row from C2 to H2. Their standarduncertainties are entered in the row below (C3-H3). The spreadsheet copies the values from C2-H2 into the second column from B5 to B10. Theresult (c(HCl)) using these values is given in B12.The C5 shows the value of m(KHP) from C2 plusits uncertainty given in C3. The result of thecalculation using the values C5-C10 is given inC12. The columns D and H follow a similarprocedure. The values shown in the row 13 (C13-H13) are the differences of the row (C12-H12)minus the value given in B12. In row 14 (C14-H14) the values of row 13 (C13-H13) are squaredand summed to give the value shown in B14. B16gives the combined standard uncertainty, which isthe square root of B14.

Table A3.2: Acid-base Titration values and uncertainties (2-step procedure)

Description Value x StandardUncertainty u(x)










The size of the different contributions can be bestcompared using a histogram showing theirrelative standard uncertainty (Figure A3.6).

Figure A3.6: Uncertainties in acid-basetitration

0 0.0005 0.001 0.0015

c(HCl)

V (HCl)

F(KHP)

V (T1)

V (T2)

P(KHP)

m(KHP)


The expanded uncertainty U(cHCl) is calculated bymultiplying the combined standard uncertainty bya coverage factor of 2

-1lmol0003.020016.0)( ⋅=⋅=HClcU

The concentration of the HCl solution is 0.1014±0.003 mol l–1

A3.6 Special aspects of the titration example

Three special aspects of the titration experimentwill be dealt with in this second part of theexample. It is quite interesting to see what effect

changes in the experimental set up or in theimplementation of the titration would have on thefinal result and its combined standard uncertainty.

Influence of a mean room temperature of 25°C

Analytical chemist rarely correct for thesystematic effect of the current temperature in thelaboratory on the volume. The question shouldone do so, as is explained in the rest of thissection, or should the uncertainty for each of thevolumes be increased accordingly. Thevolumetric measuring devices have beencalibrated at a temperature of 20°C. But rarelydoes any analytical laboratory have a temperaturecontroller to keep the room temperature thatlevel. A mean room temperature of 25°C or evenhigher is more common at the bench during thesummertime. Therefore the final result has to becalculated using the actual volumes and not thecalibrated volumes at 20°C. A volume iscorrected for the temperature effect according to

)]C20(1[ o−−=′ TVV α

where′V :actual volume at the mean temperature T

V :volume calibrated at 20°Cα :expansion coefficient of an aqueous

solution [°C–1]T :actual mean temperature in the laboratory

[°C]

Table A3.3: Acid-base Titration – spreadsheet calculation of uncertainty

A B C D E F G H1 m(KHP) P(KHP) V(T2) V(T1) F(KHP) V(HCl)2 value 0.3888 1.0 14.89 18.64 204.2212 153 Uncertainty 0.00013 0.00029 0.015 0.016 0.0038 0.01145 m(KHP) 0.3888 0.38893 0.3888 0.3888 0.3888 0.3888 0.38886 P(KHP) 1.0 1.0 1.00029 1.0 1.0 1.0 1.07 V(T2) 14.89 14.89 14.89 14.905 14.89 14.89 14.898 V(T1) 18.64 18.64 18.64 18.64 18.656 18.64 18.649 F(KHP) 204.2212 204.2212 204.2212 204.2212 204.2212 204.2250 204.2212

10 V(HCl) 15 15 15 15 15 15 15.0111112 c(HCl) 0.101387 0.101421 0.101417 0.101489 0.101300 0.101385 0.10131313 0.000034 0.000029 0.000102 -0.000087 -0.0000019 -0.00007414 2.55E-8 1.1E-9 8.64E-10 1.043E-8 7.56E-9 3.56E-12 5.52E-91516 u(c(HCl)) 0.00016



The equation of the measurand has to be rewrittenin the following way:

−−⋅−−−−

×

⋅⋅=

′⋅′′

⋅⋅⋅

=

)]C20(1[)]C20(1[)]C20(1[

1000

1000

1

2

1

2

oo

o

TVTV

TV

F

Pm

VV

V

F

Pmc

HClT

T

KHP

KHPKHP

HClT

T

KHP

KHPKHPHCl

ααα

This expression can be reduced making theassumption that the mean temperature T and theexpansion coefficient of an aqueous solution αare the same for all three volumes

−−⋅⋅×

⋅⋅=

C)]20(1[

1000

1

2oTVV

V

F

Pmc

HClT

T

KHP

KHPKHPHCl

α

This gives a slightly different result for the HClconcentration at 20°C:

1

4

lmol10149.0

)]2025(101.21[1564.182236.20489.140.13888.01000

−

−

⋅=

−⋅−⋅⋅⋅⋅⋅⋅

=HClc

The figure is still within the range given by thecombined standard uncertainty of the result at amean temperature of 20°C. All the calculationand assumptions within the section have noinfluence on to the evaluation of the combinedstandard uncertainty because still a temperaturevariation of ±4°C at the mean room temperatureof 25°C is assumed.

Visual end-point detection

A bias is introduced if the indicatorphenolphthalein is used for a visual end-pointdetection instead of the automatic titration systemextracting the equivalence-point out of the shapeof the pH curve recorded with a combined pH-electrode. The change of colour from transparentto red/purple occurs between pH 8.2 and 9.8leading to an excess volume, introducing a biascompared to the end-point detection employing apH meter. Investigations have shown that theexcess volume is around 0.05 ml with a standarduncertainty for the visual detection of the end-point of approximately 0.03 ml.

The bias arising from the excess volume has to beconsidered in the calculation of the final result.The actual volume for the visual end-pointdetection is given by

V V VT Ind T Excess1 1; = +

where

VT Ind1; :volume from a visual end-pointdetection

VT1 :volume at the equivalence-point

VExcess :excess volume needed to change thecolour of phenolphthalein

The volume correction quoted above leads to thefollowing changes in the equation of themeasurand

HClExcessIndTKHP

ExcessIndTKHPKHPHCl VVVF

VVPmc

⋅−⋅

−⋅⋅⋅=

)(

)(1000

;1

;2

The standard uncertainties u(VT2) and u(VT1) haveto be recalculated using the standard uncertaintyof the visual end-point detection as theuncertainty component of the repeatability of theend-point detection.

ml034.003.0009.0012.0004.0

)()(

2222

;11

=+++=

−= ExcessIndTT VVuVu

ml033.003.0007.0012.0004.0

)()(

2222

;22

=+++=

−= ExcessIndTT VVuVu

The combined standard uncertainty

1lmol0003.0)( −⋅=HClc cu

is considerable larger than before.

Triple determination to obtain the final result

The two step experiment is performed three timesto obtain the final result and the tripledetermination leads to the opportunity tocalculate the overall repeatability of theexperiment directly from the standard deviationin the final result. Therefore all the run to runvariations are combined to one single component,which represents the overall experimentalrepeatability as shown in the in the cause andeffect diagram (Figure A3.7).



The uncertainty components are quantified in thefollowing way:

m(KHP)

Linearity: mg087.0315.0 =

mg12.087.02)( 2 =⋅=⇒ KHPmu

P(KHP)

Purity: 00029.030005.0 =

V(T2)

calibration: ml012.0603.0 =

temperature: ml007.034101.215 4 =⋅⋅⋅ −

( ) ml014.0007.0012.0 222 =+=⇒ TVu

Repeatability

The quality log of the triple determination showsa mean long term standard deviation of theexperiment of 0.0006 mol⋅l–1. It is notrecommended to use the actual standard deviationobtained from the three determinations becausethis value has itself an uncertainty of 52%. Thestandard deviation of 0.0006 mol⋅l–1 is divided bythe square root of 3 to obtain the standarduncertainty of the triple determination. (Threeindependent measurements)

1lmol00017.030003.0 −⋅==Rep

V(HCl)


temperature: ml007.034101.215 4 =⋅⋅⋅ −

( ) ml01.0007.0008.0 22 =+=⇒ HClVu

F(KHP)

( ) 1molg0038.0 −⋅=KHPFu

V(T1)


temperature:ml009.034101.219 4 =⋅⋅⋅ −

( ) ml015.0009.0012.0 221 =+=⇒ TVu

All the values of the uncertainty components aresummarised in Table A3.4. The combinedstandard uncertainty is 0.00023 mol⋅l–1, which isnot a significant reduction due to the tripledetermination. The comparison of the uncertaintycontributions in the histogram, shown in FigureA3.8, highlights some of the reasons for thatresult.

Figure A3.8: Replicated Acid-base Titrationvalues and uncertainties

0 0.0005 0.001 0.0015 0.002 0.0025

c(HCl)

V (HCl)

F(KHP)

V (T1)

Rep

V (T2)

P(KHP)

m(KHP)


Figure A3.7: Acid-base Titration values and uncertainties (repeatability grouped)

c(HCl)

P(KHP)V(T2)

V(HCl)

Calibration

Temperature

Calibration

Temperature

End-point

Bias m(KHP)

V(T1) F(KHP)

Calibration

Temperature

End-point

BiasRepeatability

m(tare)

calibration

linearity

sensitivity

m(gross)

calibration

linearity

sensitivity

same balance

m(KHP)

V(T2)

end-point-V(T2)

V(1)

end-point-V(T1)

V(HCl)



Table A3.4: Replicated Acid-base Titration values and uncertainties

Description approximate Value Standard

uncertainty

Relative standarduncertainty




Rep Repeatability of the determination 0.10140 mol l-1 0.00017 mol l-1 0.0017






Example A4: Uncertainty estimation from in-house validation studies.Determination of organophosphorus pesticides in bread.

Summary

Goal

The amount of an organophosorus pesticidesresidue in bread is determined employing anextraction and a GC-procedure.


The numerous stages needed to determine theamount of organophosorus pesticides residue areshown in Figure A4.1

Measurand:

610⋅⋅⋅⋅

⋅⋅= hom

sampleref

oprefopop F

mRI

VcIP mg kg-1

wherePop :Level of pesticide in the sample [mg kg-1]Iop :Peak intensity of the sample extractcref :Mass concentration of the referencestandard [g ml-1]Vop :Final volume of the extract [ml]106 :Conversion factor from [g/g] to [mg kg-1]Iref :Peak intensity of the reference standardRec :Recoverymsample:Weight of the investigated sub-sample [g]Fhom :Correction factor for sample

inhomogeneity


The relevant uncertainty sources are shown in thecause and effect diagram in Figure A4.2.

Quantification of the uncertainty components:

Based on in-house validation data, the three majorcontributions are listed in Table A4.1 and showndiagramatically in Figure A4.3.

Figure A4.1: Organophosphorus pesticidesanalysis

RESULT

Homogenise

Extraction

Clean-up

'Bulk up'

GCDetermination

GCCalibration

Preparation ofcalibrationstandard

Table A4.1: Uncertainties in pesticide analysis



Remark

Repeatability(1) 1.0 0.27 0.27 Duplicate tests of differenttypes of samples

Bias (Rec) (2) 0.9 0.043 0.048 Spiked samples

Other sources (3)

(Homogeneity)

1.0 0.2 0.2 Estimations founded on modelassumptions

u(Pop)/Pop - - - - 0.34 Relative standard uncertainty



Figure A4.2: Uncertainty sources in pesticide analysis

P(op)

I(op) c(ref) V(op)

m(sample)I(ref)

V(op)

Calibration(2)

Temperature(2)

dilution

dilutionCalibration(2)

V(ref)

V(ref)Calibration(2)Temperature(2)

m(ref)

Calibration(lin)(2)

m(ref)

Purity(ref)I(op)

Calibration(3)

Recovery(2)

m(gross)

I(ref)

Calibration(3)Calibration(2)

Linearity

m(tare)

m(sample)

Calibration(2)

Linearity

F(hom)(3)

Repeatability(1)

Figure A4.3: Uncertainties in pesticide analysis

0 0.1 0.2 0.3 0.4

u(rel,Pop)

Homogeneity

Bias (Rec)

Repeatability




Example A4: Determination of organophosphorus pesticides in bread: Detaileddiscussion.

A4.1 Introduction

This example illustrates the way in which in-house validation data can be used to quantify themeasurement uncertainty. The aim of themeasurement is to determine the amount of anorganophosphorus pesticides residue in bread.The validation scheme and experiments establishtraceability by measurements on spiked samples.It is assumed the uncertainty due to any differencein response of the measurement to the spike andthe analyte in the sample is small compared withthe total uncertainty on the result.


The specification of the measurand for moreextensive analytical methods is best done by acomprehensive description of the different stagesof the analytical method and by providing theequation of the measurand.

Procedure

The measurement procedure is illustratedschematically in Figure A4.4. The separate stagesare:

i) Homogenisation: The complete sample isdivided into small (approx. 2 cm) fragments, arandom selection is made of about 15 of these,and the sub-sample homogenised. Where extremeinhomogeneity is suspected proportional samplingis used before blending.

ii) Weighing of sub-sampling for analysis givesmass msample

iii) Extraction: Quantitative extraction of theanalyte with organic solvent, decanting anddrying through a sodium sulphate columns, andconcentration of the extract using a Kedurna-Danish apparatus.

iv) Liquid-liquid extraction:Acetonitrile/hexane liquid partition, washing theacetonitrile extract with hexane, drying thehexane layer through sodium sulphate column.

v) Concentration of the washed extract by gasblown-down of extract to near dryness.

vi) Dilution to standard volume Vop (approx.2 ml) in a graduated 10 tube.

vii) Measurement: Injection and GC measurementof 5 µl of sample extract to give the peak intensityIop .

viii) Preparation of an approximately 5 µg ml-1

standard (actual mass concentration cref ).

ix) GC calibration using the standard preparedbefore and injection and GC measurement of 5 µlof the standard to give a reference peak intensityIref .

Figure A4.4: Organophosphorus pesticidesanalysis

RESULT

Homogenise

Extraction

Clean-up

'Bulk up'

GCDetermination

GCCalibration

Preparation ofcalibration

standard



Calculation

The mass concentration cop in the final sample isgiven by

1mlg −⋅=ref

oprefop I

Icc

and the estimated Pop of the level of pesticide inthe bulk sample (in mg kg-1) is given by

16 kgmg10 −⋅⋅

⋅=

sample

opopop mRec

VcP

which leads to the comprehensive equation of themeasurand

1-6 kgmg10⋅⋅⋅

⋅⋅=

sampleref

oprefopop mRecI

VcIP

wherePop :Level of pesticide in the sample [mg kg-1]Iop :Peak intensity of the sample extractcref :Mass concentration of the reference

standard [g ml-1]Vop :Final volume of the extract [ml]106 :Conversion factor from [g/g] to [mg kg-1]Iref :Peak intensity of the reference standardRec :Recoverymsample:Weight of the investigated sub-sample [g]

Scope

The analytical method is applicable to a smallrange of chemically similar pesticides at levelsbetween 0.01 and 2 mg kg-1 with different kind ofbreads as matrix.


The identification of all relevant uncertaintysources for such a complex analytical procedureis best done by drafting a cause and effectdiagram. All parameters of the equations of themeasurand represent the main branches of thediagram. Afterwards any further factor is added tothe diagram considering each step in theanalytical procedure. (A 4.2) until thecontributory factors become sufficiently remote.This leads to the following diagram:

The sample inhomogeneity is not a parameter inthe original equation of the measurand, but itappears to be a significant effect in the analyticalprocedure. Therefore a new main branchrepresenting the sample inhomogeneity is addedto the cause and effect diagram.

Finally the uncertainty branch due to theinhomogeneity of the sample has to be included inthe calculation of the measurand. To show theeffect of uncertainties arising from that source

Figure A4.5: Cause and effect diagram with added main branch for sample inhomogeneity

P(op)

I(op) c(ref) V(op)

m(sample)I(ref)

Precision

Calibration

Temperature

dilution

PrecisionCalibration

V(ref)

PrecisionCalibrationTemperature

PrecisionCalibration(lin)

m(ref)

Purity(ref)

Precision

Calibration

Recovery

m(gross)Precision

Calibration Precision

Calibration

Linearity

Sensitivity

m(tare)

Precision

Calibration

Linearity

Sensitivity

F(hom)



clearly, it is useful to write

]kgmg[106⋅⋅⋅

⋅⋅⋅=

sampleref

oprefophomop mRecI

VcIFP

where from Fhom is a correction factor assumed tobe unity in the original calculation. This makes itclear that the uncertainties in the correction factormust be included in the estimation of the overalluncertainty. The final expression also shows howthe uncertainty will apply.

NOTE: Correction factors: This approach is quitegeneral, and may be very valuable inhighlighting hidden assumptions. In principle,every measurement has associated with it suchcorrection factors, which are normallyassumed to be unity. For example, theuncertainty in cop can be expressed as astandard uncertainty for cop, or as the standarduncertainty which represents the uncertainty ina correction factor. In the latter case, the valueis identically the uncertainty for cop expressedas a relative standard deviation.


The quantification of the different uncertaintycomponents utilises data from three major stepsfrom the in-house development and validationstudies:

The best feasible estimation of the overall run torun variation of the analytical process.

The best possible estimation of the overall bias(Rec) and its uncertainty.

Quantification of any uncertainties associatedwith effects incompletely accounted for theoverall performance studies.

Some rearrangement of influence quantitiesidentified previous the cause and effect diagram(Figure A4.) has to be done to accommodate thesethree major steps.

1. Precision study

The overall run to run variation (precision) of theanalytical procedure was performed with anumber of duplicate tests for typicalorganophosphorus pesticides found in differentbread samples. The results are collected in TableA4.2. The overall estimated standard deviation s= 0.382.

The normalised difference data (the differencedivided by the mean) provides a measure of theoverall run to run variability. To obtain theestimated relative standard uncertainty for singledeterminations, the standard deviation of thenormalised differences is taken and divided by

2 to correct from a standard deviation forpairwise differences to the standard uncertaintyfor the single values. This gives a value for the

Figure A4.6: Cause and effect diagram after rearrangement to accommodate the data of thevalidation study

P(op)

I(op) c(ref) V(op)

m(sample)I(ref)

V(op)

Calibration

Temperature

dilution

dilutionCalibration

V(ref)

V(ref)CalibrationTemperature

m(ref)

Calibration(lin)

m(ref)

Purity(ref)I(op)

Calibration

Recovery

m(gross)

I(ref)

CalibrationCalibration

Linearity m(tare)

m(sample)

Calibration

Linearity

F(hom)

Repeatability



standard uncertainty due to run to run variation ofthe overall analytical process of 27.02382.0 =

NOTE: At first sight it may seem that duplicate testsprovide insufficient degrees of freedom. But itis not the goal to obtain very accurate numbersfor the precision of the analytical process forone specific pesticide in one special kind ofbread. It is more important in this study to testa wide variety of different materials andsample levels for a representative selection oftypical organophosphorus pesticides. This isdone in the most efficient way by duplicatetests on many materials, providing (for therepeatability estimate) approximately onedegree of freedom for each material studied induplicate.

2. Bias study

The bias of the analytical procedure wasinvestigated during the in-house validation studyusing spiked samples. Table A4.3 collects theresults of a long term study of spiked samples ofvarious types.

The relevant line (marked with grey colour) is the"bread" entry line which shows a mean recoveryfor forty-two samples of 90%, with a standarddeviation (s) of 28%. The standard uncertainty

was calculated as the standard deviation of themean 0432.04228.0)( ==Recu . There arethree possible cases arising for the value of therecovery Rec

1) Rec taking into account )(Recu is notsignificantly different from 1 so no correction isapplied.

2) Rec taking into account )(Recu issignificantly different from 1 and a correction isapplied.

3) Rec taking into account )(Recu issignificantly different from 1 but a correction isnot applied.

A significance test is used to determine whetherthe recovery is significantly different from 1 . Thetest statistic t is calculated using the followingequation

( )315.2

0432.09.01

)(

1=

−=

−=

Recu

Rect

This value is compared with the 2-tailed criticalvalue tcrit, for n–1 degrees of freedom at 95%confidence (where n is the number of results usedto estimate Rec ). If t is greater or equal than the

Table A4.2: Results of duplicate pesticide analysis

Residue D1[mg.kg-1]

D2[mg.kg-1]

Mean[mg.kg-1]

DifferenceD1-D2

Difference/mean

Malathion 1.30 1.30 1.30 0.00 0.000Malathion 1.30 0.90 1.10 0.40 0.364Malathion 0.57 0.53 0.55 0.04 0.073Malathion 0.16 0.26 0.21 -0.10 -0.476Malathion 0.65 0.58 0.62 0.07 0.114Pirimiphos Methyl 0.04 0.04 0.04 0.00 0.000Chlorpyrifos Methyl 0.08 0.09 0.085 -0.01 -0.118Pirimiphos Methyl 0.02 0.02 0.02 0.00 0.000Chlorpyrifos Methyl 0.01 0.02 0.015 -0.01 -0.667Pirimiphos Methyl 0.02 0.01 0.015 0.01 0.667Chlorpyrifos Methyl 0.03 0.02 0.025 0.01 0.400Chlorpyrifos Methyl 0.04 0.06 0.05 -0.02 -0.400Pirimiphos Methyl 0.07 0.08 0.75 -0.10 -0.133Chlorpyrifos Methyl 0.01 0.01 0.10 0.00 0.000Pirimiphos Methyl 0.06 0.03 0.045 0.03 0.667



critical value tcrit than Rec is significantlydifferent from 1.

021.231.2 41; ≅≥= crittt

In this example a correction factor (1/ Rec ) isbeing applied and therefore Rec is explicitlyincluded in the calculation of the result.

3. Other sources of uncertainty

The cause and effect diagram in Figure A4.7shows which other sources of uncertainty have tobe examined and eventually considered in thecalculation of the measurement uncertainty.

(1) Considered during the variability investigationof the analytical procedure.

(2) Considered during the bias study of theanalytical procedure.

(3) To be considered during the evaluation of theother sources of uncertainty.

All balances and the important volumetricmeasuring devices are under regular control. Thebias study takes into account the influence of thecalibration of the different volumetric measuringdevices because during the investigation variousvolumetric flasks and pipettes have been used.The extensive variability studies, which lasted formore than half a year, determined also the

influence of the environmental temperature ontothe result.

The purity of the reference standard is given bythe manufacturer as 99.53% ±0.06%. The purityis a potential additional other uncertainty sourcewith a standard uncertainty of

00035.030006.0 = (Rectangular distribution).But its value is much too small to be consideredany further as an essential uncertaintycontribution.

Another feasible influence quantity is thenonlinearity of the signal of the examinedorganophosphorus pesticides within the givenconcentration range. The in-house validationstudy has proven that this is not the case.

The homogeneity of the bread sub-sample is thelast remaining other uncertainty source. Noliterature data were available on the distributionof trace organic components in bread products,despite an extensive literature search (at first sightthis is surprising, but most food analysts attempthomogenisation rather than evaluateinhomogeneity separately). Nor was it practical tomeasure homogeneity directly. Therefore itscontribution has been estimated on the basis ofthe sampling method used.

Table A4.3: Results of pesticide recovery studies

Substrate ResidueType

Conc.[mg.kg–1]

N1) Mean 2) [%] s 2)[%]

Waste Oil PCB 10.0 8 84 9Butter OC 0.65 33 109 12Compound Animal Feed I OC 0.325 100 90 9Animal & Vegetable Fats I OC 0.33 34 102 24Brassicas 1987 OC 0.32 32 104 18Bread OP 0.13 42 90 28Rusks OP 0.13 30 84 27Meat & Bone Feeds OC 0.325 8 95 12Maize Gluten Feeds OC 0.325 9 92 9Rape Feed I OC 0.325 11 89 13Wheat Feed I OC 0.325 25 88 9Soya Feed I OC 0.325 13 85 19Barley Feed I OC 0.325 9 84 22(1) The number of experiments carried out(2) The mean and sample standard deviation s are given as percentage recoveries.



To aid the estimation, a number of feasiblepesticide residue distribution scenarios wereconsidered, and a simple binomial statisticaldistribution used to calculate the standarduncertainty for the total included in the analysedsample (see section A4.6). The scenarios, andtheir calculated relative standard uncertainty inthe amount of pesticide in the final sample were:

§ Residue distributed on the top surface only:0.58.

§ Residue distributed evenly over the surfaceonly: 0.20.

§ Residue distributed evenly through thesample, but reduced in concentration byevaporative loss or decomposition close to thesurface: 0.05-0.10 (depending on the "surfacelayer" thickness).

Scenario (a) is specifically catered for byproportional sampling or completehomogenisation: It would arise in the case ofdecorative additions (whole grains) added to onesurface. Scenario (b) is therefore considered thelikely worst case. Scenario (c) is considered themost probable, but cannot be readilydistinguished from (b). On this basis, the value of0.20 was chosen.

NOTE: For more details on modelling inhomogeneitysee the last section of this example.


During the in-house validation study of theanalytical procedure the repeatability, the biasand all other feasible uncertainty sources hadbeen thoroughly investigated. Their values anduncertainties are collected in Table A4.4.

Only the relative value of the combined standarduncertainty can be calculated because theuncertainty contribution for the entire range of theanalyte is evaluated.

opopc

op

c

PPu

P

u

⋅=⇒

=++=

34.0)(

34.02.0048.027.0 222

The standard spreadsheet for this case (TableA4.5) takes the form shown in Table A4.5.

The values of the parameters are entered in thesecond row from C2 to E2. Their standarduncertainties are in the row below (C3-E3). Thespreadsheet copies the values from C2-E2 into thesecond column from B5 to B7. The result usingthese values is given in B9. The C5 shows thevalue of the repeatability from C2 plus itsuncertainty given in C3. The result of thecalculation using the values C5-C7 is given in C9.The columns D and E follow a similar procedure.The values shown in the row 10 (C10-E10) arethe differences of the row (C9-E9) minus the

Figure A4.7: Evaluation of other sources of uncertainty

P(op)

I(op) c(ref) V(op)

m(sample)I(ref)

V(op)

Calibration(2)

Temperature(2)

dilution

dilutionCalibration(2)

V(ref)

V(ref)Calibration(2)Temperature(2)

m(ref)

Calibration(lin)(2)

m(ref)

Purity(ref)I(op)

Calibration(3)

Recovery(2)

m(gross)

I(ref)

Calibration(3)Calibration(2)

Linearity

m(tare)

m(sample)

Calibration(2)

Linearity

F(hom)(3)

Repeatability(1)



value given in B9. In row 11 (C11-E11) thevalues of row 10 (C10-E10) are squared andsummed to give the value shown in B11. B13gives the combined standard uncertainty, which isthe square root of B11.

The size of the three different contributions canbe compared by employing a histogram (FigureA4.8) showing their relative standarduncertainties.

The repeatability is the largest contribution to themeasurement uncertainty. Since this component is

derived from the overall variability in the method,further experiemts would be needed to showwhere improvements could be made. However theuncertainty could be reduced significantly byhomogenising the whole loaf before taking asample.

The expanded uncertainty U(Pop) is calculated bymultiplying the combined standard uncertaintywith a coverage factor of 2 to give:

opopop PPPU ⋅=⋅⋅= 68.0234.0)(



Relative standarduncertainty u(x)

Remark

Repeatability(1) 1.0 0.27 0.27 Duplicate tests of differenttypes of samples

Bias (Rec) (2) 0.9 0.043 0.048 Spiked samples

Other sources (3)

(Homogeneity)

1.0 0.2 0.2 Estimations founded on modelassumptions

u(Pop)/Pop - - - - 0.34 Relative standard uncertainty


A B C D E

1 Repeatability Bias Homogeneity2 value 1.0 0.9 1.03 uncertainty 0.27 0.043 0.24

5 Repeatability 1.0 1.27 1.0 1.06 Bias 0.9 0.9 0.9043 0.97 Homogeneity 1.0 1.0 1.0 1.28

9 1.1111 1.4111 1.1058 1.33310 0.30 -0.0053 0.22211 0.1394 0.09 0.000028 0.0493812

13 ur(Pop) 0.37



A4.6 Special aspect:

Modelling inhomogeneity for organophosphoruspesticide uncertainty

Assuming that all of the material of interest in asample can be extracted for analysis irrespectiveof its state, the worst case for inhomogeneity isthe situation where some part or parts of a samplecontain all of the substance of interest. A moregeneral, but closely related, case is that in whichtwo levels, say L1 and L2 of the material arepresent in different parts of the whole sample.The effect of such inhomogeneity in the case ofrandom sub-sampling can be estimated usingbinomial statistics. The values required are themean µ and the standard deviation σ of theamount of material in n equal portions selectedrandomly after separation.

These values are given by

( )µ = ⋅ + ⇒n p l p l1 1 2 2

( )µ = ⋅ − +np l l nl1 1 2 2 [1]

( ) ( )σ 21 1 1 2

21= ⋅ − ⋅ −np p l l [2]

where l1 and l2 are the amount of substance inportions from regions in the sample containingtotal fraction L1 and L2 respectively, of the totalamount X, and p1 and p2 are the probabilities ofselecting portions from those regions (n must besmall compared to the total number of portionsfrom which the selection is made).

The figures shown above were calculated asfollows, assuming that a typical sample loaf isapproximately 241212 ×× cm, using a portionsize of 222 ×× cm (total of 432 portions) and

assuming 15 such portions are selected at randomand homogenised.

Scenario (a)

The material is confined to a single large face (thetop) of the sample. L2 is therefore zero as is l2;and L1=1. Each portion including part of the topsurface will contain an amount l1 of the material.For the dimensions given, clearly one in six(2/12) of the portions meet this criterion, p1 istherefore 1/6, or 0.167, and l1 is X/72 (i.e. thereare 72 "top" portions).

This gives

11 5.2167.015 ll ⋅=⋅⋅=µ

21

21

2 08.2)17.01(167.015 ll ⋅=⋅−⋅⋅=σ

12

1 44.108.2 ll ⋅=⋅=⇒ σ

58.0==⇒µσ

RSD

NOTE: To calculate the level X in the entire sample, µis multiplied back up by 432/15, giving amean estimate of X of

XX

lX =⋅=⋅⋅=72

725.215432

1

This result is typical of random sampling; theexpectation value of the mean is exactly themean value of the population. For randomsampling, there is thus no contribution tooverall uncertainty other that the run to runvariability, expressed as σ or RSD here.

Scenario (b)

The material is distributed evenly over the wholesurface. Following similar arguments andassuming that all surface portions contain the

Figure A4.8: Uncertainties in pesticide analysis

0 0.1 0.2 0.3 0.4

u(rel,Pop)

Homogeneity

Bias (Rec)

Repeatability




same amount l1 of material, l2 is again zero, and p1

is, using the dimensions above, given by

63.0)241212(

)2088()241212(1 =

⋅⋅⋅⋅−⋅⋅

=p

i.e. p1 is that fraction of sample in the "outer"2 cm. Using the same assumptions then

2721 Xl = .

NOTE: The change in value from scenario (a)

This gives:

11 5.963.015 ll ⋅=⋅⋅=µ

21

21

2 5.3)63.01(63.015 ll ⋅=⋅−⋅⋅=σ

12

1 87.15.3 ll ⋅=⋅=⇒ σ

2.0==⇒µσ

RSD

Scenario (c)

The amount of material near the surface isreduced to zero by evaporative or other loss. Thiscase can be examined most simply by consideringit as the inverse of scenario (b), with p1=0.37 andl1 equal to X/160. This gives

11 6.537.015 ll ⋅=⋅⋅=µ

21

21

2 5.3)37.01(37.015 ll ⋅=⋅−⋅⋅=σ

12

1 87.15.3 ll ⋅=⋅=⇒ σ

33.0==⇒µσ

RSD

However, if the loss extends to a depth less than

the size of the portion removed, as would beexpected, each portion contains some material l1

and l2 would therefore both be non-zero. Takingthe case where all outer portions contain 50%"centre" and 50% "outer" parts of the sample

2962 121 Xlll =⇒⋅=

( )222

221

6.201537.0151537.015

lll

lll

⋅=⋅+⋅⋅=⋅+−⋅⋅=µ

22

221

2 5.3)()37.01(37.015 lll =−⋅−⋅⋅=σ

giving an RSD of 09.06.2087.1 =

In the current model, this corresponds to a depthof 1 cm through which material is lost.Examination of typical bread samples shows crustthickness typically of 1 cm or less, and taking thisto be the depth to which the material of interest islost (crust formation itself inhibits lost below thisdepth), it follows that realistic variants onscenario (c) will give values of µσ not above0.09.

NOTE: In this case, the reduction in uncertainty arisesbecause the inhomogeneity is on a smallerscale than the portion taken forhomogenisation. In general this will lead to areduced contribution to uncertainty, it followsthat no additional modelling need be done forcases where larger numbers of smallinclusions (such as grains incorporated in thebulk of a loaf) contain disproportionateamounts of the material of interest. Providedthat the probability of such an inclusion beingincorporated into the portions taken forhomogenisation is large enough, then thecontribution to uncertainty will not exceed anyalready calculated in the scenarios above.



Example 5 -Empirical method: Determination of cadmium release fromceramic ware by atomic absorption spectrometry

Summary

Goal

The amount of released cadmium from ceramicware is determined using atomic absorptionspectrometry. The employed procedure is theempirical method BS 6748.


The different stages to determine the amount ofreleased cadmium from ceramic are given in theflow chart (Figure A5.1).

Measurand:

20 dmmg −⋅⋅⋅⋅⋅

= temptimeacidV

L fffda

Vcr

The variables are described in Table A5.1.


The relevant uncertainty sources are shown in thecause and effect diagram at Figure A5.2.

Quantification of the uncertainty sources:

The size of the different contributions is givenin Table A5.1 and shown diagrammatically inFigure A5.2

Figure A5.1: Extractable metal procedure

RESULT

Preparation

Conditioning ofsurface area

Fill with 4% v/vacetic acid

Leaching

Homogenise

Preparation ofcalibrationstandards

AASCalibration

AASDetermination

Table A5.1: Uncertainties in extractable cadmium determination



c0 Content of cadmium in the extractionsolution

0.26 mg/l 0.018 mg/l 0.064

VL Volume of the leachate 0.332 l 0.0018 l 0.0056

aV Surface area of the vessel 2.37 dm2 0.06 dm2 0.025

facid Influence of the acid concentration 1.0 0.0008 0.0008

ftime Influence of the duration 1.0 0.001 0.001

ftemp Influence of temperature 1.0 0.06 0.06

r Mass of cadmium leached per unitarea

0.036 mg/dm2 0.0033 mg/dm2 0.09



Figure A5.2: Uncertainty sources in leachable Cadmium determination

Result r

c(0) V(L)

da(V)

Reading

Calibration

Temperature

1 length

2 length

area

Filling

calibrationcurve

f(temperature)f(time)

f(acid)

Figure A5.3: Uncertainties in leachable Cd determination

0 0.02 0.04 0.06 0.08 0.1

r

f(temp)

f(time)

f(acid)

a(V)

V(L)

c(0)




Example A5. Determination of cadmium release from ceramic ware by atomicabsorption spectrometry. Detailed discussion.

A5.1 Introduction

This example demonstrates the uncertaintyevaluation of an empirical method, in this case(BS 6748) limits of metal release form ceramicware, glassware, glass-ceramic ware and vitreousenamel ware. The test is used to determine byatomic absorption spectroscopy (AAS) theamount of lead or cadmium leached from thesurface of ceramic ware by a 4% (v/v) aqueoussolution of acetic acid. The results obtained withthis analytical procedure (empirical method) areonly traceable within the boundaries of itsspecifications. There is no traceability to the SI-units.


The following extract from BS 6748:1986 “Limitsof metal release from ceramic ware, glass ware,glass ceramic ware and vitreous enamel ware“forms the specification for the measurand.

A5.2.1 Reagents

Water, complying with the requirements of BS3978.

Acetic acid CH3COOH, glacial.

Acetic acid solution 4% v/v 40 ml of glacial aceticacid is added to 500 ml of water and made up to1 litre. The solution is freshly prepared prior touse.

Standard metal solutions

1000 ±1 mg Pb in 1 l at 4% (v/v) acetic acid.

500 ±0.5 mg Cd in 1 l of 4% (v/v) acetic acid.

A5.2.2 Apparatus

Atomic absorption spectrophotometer, with adetection limit of at least 0.2 mg/l Pb (in 4% v/vacetic acid) and 0.02 mg/l Cd ( in 4% v/v aceticacid).

Laboratory glassware, volumetric glassware of atleast class B of Borosilicate glass incapable ofreleasing detectable levels of lead or cadmiuminto 4% acetic acid during the test procedure.

A5.2.3 Preparation of samples

Samples are to be washed at 40 ±5°C in anaqueous solution containing 1 mg/l of domesticliquid detergent, rinsed with water (as specified

above), drained and wiped dry with clean filterpaper. Areas of the samples, which do not contactfoodstuffs in normal use, are covered afterwashing and drying with a suitable protectivecoating.

A5.2.4 Procedure

The analytical procedure is illustratedschematically in Figure A5.4. The different stepsare:

The sample is conditioned to 22 ±2°C . Whereappropriate (‘category 1’ articles the surface areaof the article is determined.

The conditioned sample is filled with 4% v/v acidsolution at 22 ±2°C to a level no more than 1 mmfrom the overflow point, measured from the upperrim of the sample, and to no more than 6 mmfrom the extreme edge of a sample with a flat orsloping rim.

The quantity of 4% v/v acetic acid required or

RESULT

Preparation

Conditioning ofsurface area

Fill with 4% v/vacetic acid

Leaching

Homogenise

Preparation ofcalibrationstandards

AASCalibration

AASDetermination

Figure A5.4: Extractable metal procedure



used is recorded to an accuracy of ±2%.

The sample is allowed to stand at 22 ±2°C for24 hours (in darkness if cadmium is determined)with due precaution to prevent evaporation loss.

The extract solution is homogenised, by stirring,or by other means, without loss of solution orabrasion of the surface being tested and a portiontaken for analysis by AAS.

A5.2.5 Analysis

The AAS instrument is set up according to themanufacturer’s instruction using wavelengths of217.0 nm for lead determination and 228.8 nm forcadmium determination with appropriatecorrection for background absorption effects.

Provided that absorbance values of the dilutestandard metal solutions an of the 4% v/v aceticacid solution indicate minimal drift, the resultmay be calculated from a manually preparedcalibration curve (below), or by using thecalibration bracketing technique.

A5.2.6 Calculation of results from a manuallyprepared calibration curve

The lead or cadmium content, c0 expressed inmg/l at the extraction solution, is given by theequation:

1-

1

000 lmg

)(d

B

BAc ⋅

−=

wherec0 :content of lead or cadmium of the

extraction solution [mg/l]A0 :absorbance of the lead or cadmium in the

extraction solutionB1 :slope of the manually prepared calibration

curveB0 :intercept of the manually prepared

calibration curve [mg/l]d :the factor by which the sample was diluted

NOTE: The calibration curve should be chosen tohave absorbance values within the range ofthat of the sample extract or diluted sampleextract.

A5.2.7 Test report

The test report is to include, inter alia:

§ the nature of the article under test.

§ the surface area or volume, as appropriate, ofthe article.

§ the amount of lead and/or cadmium in thetotal quantity of the extracting solutionexpressed as milligrams of Pb or Cd persquare decimetre of surface area for category1 articles or milligrams of Pb or Cd per litreof the volume for category 2 and 3 articles.

NOTE: This extract from BS 6748:1986 is reproducedwith the permission of BSI. Complete copiescan be obtained by post from BSI customerservices, 389 Chiswick Leigh Road, LondonW4 4AL England J 0181 996 7000.

A5.3 Step 2: Identity and analysinguncertainty sources

Step 1 describes an ’empirical method’. If such amethod is used within its defined field ofapplication, the bias of the method is defined aszero. Therefore bias estimation relates to thelaboratory performance and not to the biasintrinsic to the method. Because no referencematerial certified for this standardised method isavailable overall control of bias is related with thecontrol of method parameters influencing theresult. Such influence quantities are time,temperature, mass and volumes, etc.

The concentration of lead or cadmium in theacetic acid is determined by atomic absorptionspectrometry. For vessels that cannot be filledcompletely the empirical method call for theresult to be expressed as mass r of Pb or Cdleached per unit area. r is given by

2-

1

000 dmmg)(

dBa

BAVd

a

Vcr

V

L

V

L ⋅⋅

−⋅=⋅

⋅=

wherer :mass of Cd or Pb leached per unit area

[mg dm-2]VL :the volume of the leachate [l]aV :the surface area of the vessel [dm2]c0 :content of lead or cadmium in the

extraction solution [mg l-1]A0 :absorbance of the metal in the sample

extractB0 :intercept of the calibration curveB1 :slope of the calibration curved :factor by which the sample was diluted

The first part of the above equation of themeasurand is used to draft the basic cause andeffect diagram (Figure A5.5).



Figure A5.5:Initial cause and effect diagram

Result r

c(0) V(L)

da(V)

Reading

Calibration

Temperature

1 length

2 length

area

Filling

calibrationcurve

There is no reference material certified for thisempirical method and which would help to assessthe laboratory performance. Hence all the feasibleinfluence quantities such as temperature, time ofthe leaching process and acid concentration haveto be considered. To accommodate the additionalinfluence quantities the equation is expanded bythe respective correction factors leading to

temptimeacidV

L fffda

Vcr ⋅⋅⋅⋅

⋅= 0

These additional factors are also included in therevised cause and effect diagram (Figure A5.6).

NOTE: The latitude in temperature permitted by thestandard is a case of an uncertainty arising as aresult of incomplete specification of the

measurand. Taking the effect of temperatureinto account allows estimation of the range ofresults which could be reported whilstcomplying with the empirical method as wellas is practically possible. Note particularlythat variations in the result caused by differentoperating temperatures within the rangecannot reasonably descried as bias as theyrepresent results obtained in accordance withthe specification.

A5.4 Step 3: Quantifying uncertainty sources

The aim of this step is to quantify the uncertaintyarising from each of the previously identifiedsources. This can be done be either usingexperimental data or well based assumptions.

Dilution factor d

For the current example no dilution of theleaching solution is necessary, therefore nouncertainty contribution have to be accounted for.

VL

Filling: The empirical method requires the vesselto be filled ‘to within 1 mm from the brim’. For atypical drinking or kitchen utensil, 1 mm willrepresent about 1% of the height of the vessel.The vessel will therefore be 99.5 ±0.5% filled (i.e. VL will be approximately 0.995 ±0.005 of thevessel’s volume).

Temperature: The temperature of the acetic acidhas to be 22 ±2ºC. This temperature range leadsto an uncertainty in the determined volume, dueto a considerable larger volume expansion of the

Figure A5.6:Cause and effect diagram with added hidden assumptions (correction factors)

Result r

c(0) V(L)

da(V)

Reading

Calibration

Temperature

1 length

2 length

area

Filling

calibrationcurve

f(temperature)f(time)

f(acid)



liquid compared with the vessel. The standarduncertainty assuming a rectangular temperaturedistribution is

ml08.03

2322101.2 4=

⋅⋅⋅ −

Reading: The volume VL used is to be recorded towithin 2%, in practice, use of a measuringcylinder allows an inaccuracy of about 1% (i.e.0.01·VL). The standard uncertainty is calculatedassuming a triangular distribution.

Calibration: The volume is calibrated accordingto the manufacturer’s specification within therange of ±2.5 ml for a 500 ml measuring cylinder.The standard uncertainty is obtained assuming atriangular distribution.

For this example a volume of 332 ml is used andthe four uncertainty components are combinedaccordingly

( )

ml83.165.2

633201.0

)08.0(6

332005.0

22

22

=

+

⋅+

+

⋅

=LVu

c0

The amount of leached cadmium is calculatedusing a manually prepared calibration curve. Forthis purpose five calibration standards, with aconcentration 0.1 mg/l, 0.3 mg/l, 0.5 mg/l,0.7 mg/l and 0.9 mg/l, were prepared from a 500±0.5 mg/l cadmium reference standard. The linearleast square fitting procedure used assumes thatthe uncertainties of the values of the abscissa areconsiderable smaller than the uncertainty on thevalues of the ordinate. Therefore the usualuncertainty calculation procedures for c0 onlyreflect the uncertainty in the absorbance and notthe uncertainty of the calibration standards. In thiscase the uncertainty of the calibration standards issufficiently small for this procedure to be used.

The five calibration standards were measuredthree times each providing the following results.

Concentration[mg/l]

1 2 3

0.1 0.028 0.029 0.029

0.3 0.084 0.083 0.081

0.5 0.135 0.131 0.133

0.7 0.180 0.181 0.183

0.9 0.215 0.230 0.216

The calibration curve is given by

01 BBcA ij +⋅=

whereAj :jth measurement of the absorbance of the ith

calibration standardci :concentration of the ith calibration standardB1 :slopeB0 :interceptand the results of the linear least square fit are

value uncertaintyB1 0.2410 0.0050B0 0.0087 0.0029

with a multiple R-squared of 0.9944 and theresidual standard error is 0.005486. R is thecorrelation coefficient for the linear least squarefit.

The actual leach solution was measured twiceleading to a concentration c0 of 0.26 mg/l. Thecalculation of the uncertainty u(c0) of the linearleast square fitting procedure is describedthoroughly in Appendix E3. Therefore only ashort description of the different calculation stepsis given here.



u(c0) is given by

10

2

20

10

lmg018.0)(2.1

)5.026.0(151

21

241.0005486.0

)(11)(

−⋅=⇒

−++=

−++=

cu

S

cc

npB

scu

xx

with the residual standard deviation s given by

005486.02

)]([1

210

=−

⋅+−=

∑=

n

cBBA

s

n

jjj

and

2.1)(1

2 =−= ∑=

n

jjxx ccS

whereB1 :slopep :number of measurement to determine c0

n :number of measurement for the calibrationc0 :determined cadmium concentration of theleached solutionc :mean value of the different calibrationstandards (n number of measurements)i :index for the number of calibrationstandardsj :index for the number of measurements toobtain the calibration curve

Area aV

Length measurement: The total surface area of thesample vessel was calculated, from measureddimensions, to be 2.37 dm2. Since the item isapproximately cylindrical but not perfectlyregular, measurements are estimated to be within2 mm at 95% confidence. Typical dimensions arebetween 1.0 dm and 2.0 dm leading to anestimated dimensional measurement uncertaintyof 1 mm (after dividing the 95% figure by 1.96).Area measurements typically require two lengthmeasurements height and width respectively (i.e.1.45 dm and 1.64 dm)

Area: Since the item has not a perfect geometricshape, there is also an uncertainty in any areacalculation; in this example, this is estimated tocontribute an additional 5% at 95% confidence.

The uncertainty contribution of the lengthmeasurement and area itself are combined in theusual way.

2

222

dm06.0)(

96.138.205.0

01.001.0)(

=⇒

⋅

++=

V

V

au

au

ftemp

A number of studies of the effect of temperatureon metal release from ceramic ware have beenundertaken(1-5). In general the temperature effect issubstantial, and a near-exponential increase inmetal release with temperature is observed until

Figure A5.7:Linear least square fit and uncertainty interval for duplicate determinations

Concentration of Cadmium [mg/l]

Abs

orpt

ion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.05

0.10

0.15

0.20

0.25



limiting values are reached. Only one study hasgiven an indication of effects in the range of an20-25°C; from the graphical informationpresented the change in metal release withtemperature near 25°C is approximately linear,with a gradient of approximately 5%/°C. For the±2°C range allowed by the empirical method thisleads to a factor ftemp of 1 ±0.1. Converting this toa standard uncertainty gives, assuming arectangular distribution:

u(ftemp)= 06.031.0 =

ftime

For a relative slow process such as leaching, theamount leached will be approximatelyproportional to time for small changes in the time.Krinitz and Franco1 found a mean change inconcentration over the last 6 hours of leachingwas approximately 1.8 mg/l in 86, that is, about0.3%/h. For a time of 24 ±0.5h c0 will thereforeneed correction by a factor ftime of 1 ±(0.5·0.003)=1± 0.0015. This is a rectangular distributionleading to the standard uncertainty

001.030015.0)( ≅=timefu .

facid

One study of the effect of acid concentration onlead release showed that changing concentrationfrom 4 to 5% v/v increased the lead released froma particular ceramic batch from 92.9 to 101.9mg/l, i.e. a change in facid of

097.09.92)9.929.101( =− or close to 0.1.Another study, using a hot leach method, showeda comparable change (50% change in leadextracted on a change of from 2 to 6% v/v)3.

Assuming this effect as approximately linear withacid concentration gives an estimated change infacid of approximately 0.1 per % v/v change in acidconcentration. In a separate experiment theconcentration and its standard uncertainty havebeen established using titration with astandardised NaOH titre. (3.996% v/vu = 0.008% v/v). Taking the uncertainty of0.008% v/v on the acid concentration suggests anuncertainty for facid of 0.008·0.1 = 0.0008. As theuncertainty on the acid concentration is alreadyexpressed as a standard uncertainty, this value canbe used directly as the uncertainty associated withfacid.

NOTE: In principle, the uncertainty value would needcorrecting for the assumption that the singlestudy above is sufficiently representative of allceramics. The present value does, however,give a reasonable estimate of the magnitude ofthe uncertainty.


The amount of leached cadmium per unit area isgiven by

2-0 dmmgtemptimeacidV

L fffa

Vcr ⋅⋅⋅

⋅=

The intermediate values and their standarduncertainties are collected in Table A5.2.Employing those values

2dmmg036.00.10.10.137.2

332.026.0 −⋅=⋅⋅⋅⋅

=r

In order to calculate the combined standarduncertainty of a multiplicative expression (asabove) the standard uncertainties of each

Table A5.2: Intermediate values and uncertainties for leachable Cadmium analysis

Description Value Standard

uncertainty

Relative standarduncertainty

c0 Content of cadmium in the extractionsolution

0.26 mg/l 0.018 mg/l 0.064

VL Volume of the leachate 0.332 l 0.0018 l 0.0056

aV Surface area of the vessel 2.37 dm2 0.06 dm2 0.025

facid Influence of the acid concentration 1.0 0.0008 0.0008

ftime Influence of the duration 1.0 0.001 0.001

ftemp Influence of temperature 1.0 0.06 0.06



component is used in the following way.

( )( ) ( ) ( )

( ) ( ) ( )

09.006.0001.00008.0

025.00056.0064.0222

222

222

222

0

0

=+++

++=

+

+

+

+

+

=

temp

temp

time

time

acid

acid

V

V

L

L

c

f

fu

f

fu

f

fu

a

au

V

Vu

c

cu

r

ru

2dmmg0033.009.0)( −⋅=⋅=⇒ rrcu

The simpler spreadsheet approach to calculate thecombined standard uncertainty is shown below. Acomprehensive introduction to the method isgiven in Appendix E.

The values of the parameters are entered in thesecond row from C2 to H2. Their standarduncertainties are in the row below (C3-H3). Thespreadsheet copies the values from C2-H2 intothe second column from B5 to B10. The result (r)using these values is given in B12. The C5 showsthe value of c0 from C2 plus its uncertainty givenin C3. The result of the calculation using thevalues C5-C10 is given in C12. The columns D

and H follow a similar procedure. The valuesshown in the row 13 (C13-H13) are thedifferences of the row (C13-H13) minus the valuegiven in B12. In row 14 (C14-H14) the values ofrow 13 (C13-H13) are squared and summed togive the value shown in B14. B16 gives thecombined standard uncertainty, which is thesquare root of B11.

The contributions of the different parameters andinfluence quantities to the measurementuncertainty are illustrated in Figure A5.8,comparing their relative standard uncertainties.

The expanded uncertainty U(r) is obtained byapplying a coverage factor of 2

2dmmg007.020033.0)( −⋅=⋅=rU

Thus the amount of released cadmium measuredaccording to BS 6748:1986

0.036 ±0.007 mg dm-2

where the stated uncertainty is calculated using acoverage factor of 2.

Table A5.3: Spreadsheet calculation of uncertainty for leachable Cadmium analysis

A B C D E F G H

1 c0 VL aV facid ftime ftemp

2 value 0.26 0.322 2.37 1.0 1.0 1.03 uncertainty 0.018 0.0018 0.06 0.0008 0.001 0.0645 c0 0.26 0.278 0.26 0.26 0.26 0.26 0.266 VL 0.332 0.332 0.3338 0.332 0.332 0.332 0.3327 aV 2.37 2.37 2.37 2.43 2.37 2.37 2.378 facid 1.0 1.0 1.0 1.0 1.0008 1.0 1.09 ftime 1.0 1.0 1.0 1.0 1.0 1.001 1.010 ftemp 1.0 1.0 1.0 1.0 1.0 1.0 1.061112 0.036422 0.038943 0.036619 0.035523 0.036451 0.036458 0.03860713 0.002521 0.000197 -0.000899 0.000029 0.000036 0.00218514 1.199 E-5 6.36 E-6 3.90 E-8 8.09 E-7 8.49 E-10 1.33 E-9 4.78 E-61516 uc(r) 0.0034



A5.6 References for Example 5

1. B. Krinitz, V. Franco, J.AOAC 56 869-875 (1973)

2. B. Krinitz, J. AOAC 61, 1124-1129 (1978)

3. J.H. Gould, S. W. Butler, K. W. Boyer, E. A. Stelle, J. AOAC 66, 610-619 (1983)

4. T. D. Seht, S. Sircar, M. Z. Hasan, Bull. Environ Contam. Toxicol. 10, 51-56 (1973)

5. J.H. Gould, S. W. Butler, E. A. Steele, J. AOAC 66, 1112-1116 (1983)

Figure A5.8: Uncertainties in leachable Cd determination

0 0.02 0.04 0.06 0.08 0.1

r

f(temp)

f(time)

f(acid)

a(V)

V(L)

c(0)




Example A6.The Determination of Crude Fibre in Animal Feeding Stuffs

Summary

Goal

The determination of crude fibre (“Dietary fibre”)by a regulatory standard method.


The measurement procedure is a standardisedprocedure involving the general steps outlined inFigure A6.1. These are repeated for a blanksample to obtain a blank correction.

Measurand

The fibre content as a percentage of the sample byweight, Cfibre, is given by:

Cfibre = ( )

acb 100×−

Where:a is the mass (g) of the sample.

(Approximately 1 g)b is the loss of mass (g) after ashing during

the determination;c is the loss of mass (g) after ashing during

the blank test.

Identification of uncertainty sources

A full cause and effect diagram is provided in thedetailed discussion as Figure A6.9.

Quantification of uncertainty components

Laboratory experiments showed that the methodwas performing in house in a manner that fullyjustified adoption of collaborative studyreproducibility data. No other contributions weresignificant in general. At low levels it wasnecessary to add an allowance for the specific

drying procedure used. Typical resultinguncertainty estimates are tabulated below (asstandard uncertainties) (Table A6.1).

Figure A6.1: Fibre determination.

Grind and weigh sample

Acid digestion

Alkaline digestion

Dry & weighresidue

Ash & weighresidue

RESULT

Ash & weighresidue

Table A6.1: Combined standard uncertainties

Fibre content

(%w/w)

Standard uncertainty

u(Cfibre) (%w/w)

Standard uncertainty as

CV(%)

2·5· 0 29 0 115 0 312 2. . .+ = 12

5 0·4 8

10 0·6 6



Example A6.The Determination of Crude Fibre in Animal Feeding Stuffs:Detailed discussion

A6.1 Introduction

Crude fibre is defined in the method scope as theamount of fat-free organic substances which areinsoluble in acid and alkaline media; theprocedure is standardised and its results useddirectly. Changes in the procedure change themeasurand; this is accordingly an example of anempirical method.

This is a statutory method for which collaborativetrial data (repeatability and reproducibility) wereavailable. The precision experiments describedwere planned as part of the in-house evaluation ofthe method performance. There is no suitablereference material (i.e. certified by the samemethod) available for this method.


The specification of the measurand for moreextensive analytical methods is best done by acomprehensive description of the different stagesof the analytical method and by providing theequation of the measurand

Procedure

The procedure, a complex digestion, filtration,drying, ashing and weighing procedure which isalso repeated for a blank crucible, is summarisedin Figure A6.2. The aim is to digest mostcomponents, leaving behind all the undigestedmaterial. The organic material is ashed, leavingan inorganic residue. The difference between thedry organic/inorganic residue weight and theashed residue weight is the “fibre content”. Themain stages are:i) Grind the sample to pass through a 1mm

sieveii) Weigh 1g of the sample into a weighed

crucibleiii) Add a set of acid digestion reagents at stated

concentrations and volumes. Boil for astated, standardised time, filter and wash theresidue.

iv) Add standard alkali digestion reagents and

boil for the required time, filter, wash andrinse with acetone.

v) Dry to constant weight at a standardisedtemperature (“constant weight” is notdefined within the published method; nor areother drying conditions such as aircirculation or dispersion of the residue).

vi) Record the dry residue weight.vii) Ash at a stated temperature to “constant

weight” (in practice realised by ashing for aset time decided after in house studies).

viii) Weigh the ashed residue and calculate thefibre content by difference, after subtractingthe residue weight found for the blankcrucible.

Measurand

The fibre content as a percentage of the sample byweight, Cfibre, is given by:

Cfibre = ( )

a

cb 100×−

Where:a is the mass (g) of the sample.Approximately 1 g of sample is taken foranalysis;b is the loss of mass (g) after ashing duringthe determination;c is the loss of mass (g) after ashing duringthe blank test.


A range of sources of uncertainty were identified.these are shown in the cause and effect diagramfor the method (see Figure A6.9). This diagramwas simplified to remove duplication followingthe procedures in Appendix D; this, together withremoval of insignificant components, leads to thesimplified cause and effect diagram in FigureA6.10.



Since prior collaborative and in-house study datawere available for the method, the use of thesedata is closely related to the evaluation ofdifferent contributions to uncertainty and isaccordingly discussed further below.


Collaborative trial results

The method has been the subject of a

collaborative trial. Five different feeding stuffsrepresenting typical fibre and fat concentrationswere analysed in the trial. Participants in the trialcarried out all stages of the method, includinggrinding of the samples. The repeatability andreproducibility estimates obtained from the trialare presented in Table A6.2.

As part of the in-house evaluation of the method,experiments were planned to evaluate therepeatability (within batch precision) for feeding

Figure A6.2: Flow diagram illustrating the stages in the regulatory method for thedetermination of fibre in animal feeding stuffs

Grind sample to passthrough 1 mm sieve

Weigh 1 g of sample into crucible

Add filter aid, anti-foaming agentfollowed by 150 ml boiling H2SO4

Add anti-foaming agent followedby 150 ml boiling KOH

Boil vigorously for 30 mins

Filter and wash with 3x30 ml boiling water

Boil vigorously for 30 mins

Filter and wash with 3x30 ml boiling water

Apply vacuum, wash with 3x25 ml acetone

Dry to constant weight at 130 °C

Ash to constant weight at 475-500 °C

Calculate the % crude fibre content

Weigh crucible for blank test



stuffs with fibre concentrations similar to those ofthe samples analysed in the collaborative trial.The results are summarised in Table A6.2. Eachestimate of in-house repeatability is based on 5replicates.

The estimates of repeatability obtained in-housewere comparable to those obtained from thecollaborative trial. This indicates that the methodprecision in this particular laboratory is similar tothat of the laboratories which took part in thecollaborative trial. It is therefore acceptable touse the reproducibility standard deviation fromthe collaborative trial in the uncertainty budgetfor the method. To complete the uncertaintybudget we need to consider whether there are anyother effects not covered by the collaborative trialwhich need to be addressed. The collaborativetrial covered different sample matrices and thepre-treatment of samples, as the participants weresupplied with samples which required grindingprior to analysis. The uncertainties associatedwith matrix effects and sample pre-treatment donot therefore require any additional consideration.Other parameters which affect the result relate tothe extraction and drying conditions used in themethod. These were investigated separately toensure the laboratory bias was under control (i.e.,small compared to the reproducibility standarddeviation). The parameters considered arediscussed below.

Loss of mass on ashing

As there is no appropriate reference material forthis method, in-house bias has to be assessed by

considering the uncertainties associated withindividual stages of the method. Several factorswill contribute to the uncertainty associated withthe loss of mass after ashing:§ acid concentration;§ alkali concentration;§ acid digestion time;§ alkali digestion time;§ drying temperature and time;§ ashing temperature and time.

Reagent concentrations and digestion times

The effects of acid concentration, alkaliconcentration, acid digestion time and alkalidigestion time have been studied in previouslypublished papers. In these studies, the effect ofchanges in the parameter on the result of theanalysis was evaluated. For each parameter thesensitivity coefficient (i.e., the rate of change inthe final result with changes in the parameter) andthe uncertainty in the parameter were calculated.

The uncertainties given in Table A6.3 are smallcompared to the reproducibility figures presentedin Table A6.2. For example, the reproducibilitystandard deviation for a sample containing2·3 %(w/w) fibre is 0·293 %(w/w). Theuncertainty associated with variations in the aciddigestion time is estimated as 0·021 %(w/w) (i.e.,2·3 × 0·009). We can therefore safely neglect theuncertainties associated with variations in thesemethod parameters.

Table A6.2: Summary of results from collaborative trial of the method and in-houserepeatability check

Fibre content (% w/w)

Collaborative trial results

Sample Mean Reproducibilitystandard

deviation (sR)

Repeatabilitystandard

deviation (sr)

In-houserepeatability

standarddeviation

A 2· 3 0·293 0·198 0·193

B 12·1 0·563 0·358 0·312

C 5·4 0·390 0·264 0·259

D 3·4 0·347 0·232 0·213

E 10·1 0·575 0·391 0·327



Drying temperature and time

No prior data were available. The method statesthat the sample should be dried at 130 °C to“constant weight”. In this case the sample isdried for 3 hours at 130 °C and then weighed. Itis then dried for a further hour and re-weighed.Constant weight is defined in this laboratory as achange of less than 2 mg between successiveweighings. In an in-house study, replicatesamples of four feeding stuffs were dried at 110,130 and 150 °C and weighed after 3 and 4 hoursdrying time. In the majority of cases the weightchange between 3 and 4 hours was less than 2 mg.This was therefore taken as the worst caseestimate of the uncertainty in the weight change

on drying. ±2 mg is a rectangular distributionwhich is converted to a standard uncertainty bydividing by √3. The uncertainty in the weightrecorded after drying to constant weight istherefore 0·00115 g. The method specifies asample weight of 1 g. For a 1 g sample, theuncertainty in drying to constant weightcorresponds to a standard uncertainty of0·115 %(w/w) in the fibre content. This source ofuncertainty is independent of the fibre content ofthe sample. There will therefore be a fixedcontribution of 0·115 %(w/w) to the uncertaintybudget for each sample, regardless of theconcentration of fibre in the sample. At all fibreconcentrations, this uncertainty is smaller than thereproducibility standard deviation, and for all butthe lowest fibre concentrations is less than 1/3 ofthe sR value. Again this source of uncertainty canusually be neglected. However for low fibreconcentrations, this uncertainty is more than 1/3of the sR value so an additional term should beincluded in the uncertainty budget (see TableA6.4).

Ashing temperature and time

The method requires the sample to be ashed at475 to 500 °C for at least 30 mins. A publishedstudy on the effect of ashing conditions involveddetermining fibre content at a number of differentashing temperature/time combinations, rangingfrom 450 °C for 30 minutes to 650 °C for 3 hours.No significant difference was observed betweenthe fibre contents obtained under the differentconditions. The effect on the final result of smallvariations in ashing temperature and time cantherefore be assumed to be negligible.

Loss of mass after blank ashing

No experimental data were available for thisparameter. However, as discussed above, theeffects of variations in this parameter are likely tobe small.

NOTES ON Table A6.3

(1) The sensitivity coefficients were estimated byplotting the normalised change in fibre contentagainst reagent strength or digestion time.Linear regression was then used to calculatethe rate of change of the result of the analysiswith changes in the parameter.

(2) The standard uncertainties in theconcentrations of the acid and alkali solutionswere calculated from estimates of theprecision and accuracy of the volumetricglassware used in their preparation,temperature effects etc. (see workshops 2 and5-7 for further examples of calculatinguncertainties for the concentrations ofsolutions).

(3) The method specifies a digestion time of 30minutes. The digestion time is controlled towithin ±5 minutes. This is a rectangulardistribution which is converted to a standarduncertainty by dividing by 3 .

(4) The uncertainty in the final result, as a relativestandard deviation, is calculated bymultiplying the sensitivity coefficient by theuncertainty in the parameter.

Table A6.3: Uncertainties associated with method parameters

Parameter Sensitivitycoefficient(1)

Uncertainty inparameter

Uncertainty in finalresult as RSD(4)

acid concentration 0·23 (mol/l)-1 0·0013 mol/l(2) 0·00030

alkali concentration 0·21 (mol/l)-1 0·0023 mol/l 0·00048

acid digestion time 0·0031 min-1 2·89 mins(3) 0·0090

alkali digestion time 0·0025 min-1 2·89 mins 0·0072




This is an example of an empirical method forwhich collaborative trial data were available. Thein-house repeatability was evaluated and found tobe comparable to that predicted by thecollaborative trial. It is therefore appropriate touse the sR values from the collaborative trial. Thediscussion presented in Step 3leads to theconclusion that, with the exception of the effectof drying conditions at low fibre concentrations,the other sources of uncertainty identified are allsmall in comparison to sR. The performance ofthe laboratory producing the uncertainty estimateis therefore comparable to that of the laboratorieswhich took part in the trial. In cases such as thisthe uncertainty estimate can be based on thereproducibility standard deviation, sR, obtained

from the collaborative trial. For samples with afibre content of 2.5 %(w/w), an additional termhas been included to take account of theuncertainty associated with the drying conditions.


Typical standard uncertainties for a range of fibreconcentrations are given in the table below:

Expanded uncertainty

Typical expanded uncertainties are given in thetable below. These were calculated using acoverage factor of 2 which gives a level ofconfidence of approximately 95%.

Table A6.4: Combined standard uncertainties

Fibre content

(%w/w)


u(Cfibre) (%w/w)

Standard uncertainty as

CV(%)

2·5· 0 29 0 115 0 312 2. . .+ = 12

5 0·4 8

10 0·6 6

Table A6.5: Expanded uncertainties

Fibre content

(%w/w)

Expanded uncertainty

U(Cfibre) (%w/w)

Expanded uncertainty as

CV (%)

2·5 0·62 25

5 0·8 16

10 0·12 12

Fig

ure

A6.

9: C

ause

and

eff

ect

diag

ram

for

the

det

erm

inat

ion

of f

ibre

in a

nim

al f

eedi

ng s

tuff

s

Cru

de F

ibre

(%)

Los

s of

mas

s af

ter

ashi

ng (

b)

Los

s of

mas

s af

ter

blan

k as

hing

(c)

Mas

s sa

mpl

e (a

)P

reci

sion

sam

ple

wei

ght p

reci

sion

wei

ghin

g pr

ecis

ion

wei

ght o

f cru

cibl

ebe

fore

ash

ing

dryi

ng te

mp

wei

ght o

f sam

ple

and

cruc

ible

afte

r ash

ing

ashi

ng te

mp

ashi

ng ti

me

wei

ghin

g of

cru

cibl

e

bala

nce

linea

rity

bala

nce

calib

ratio

n

extra

ctio

n pr

ecis

ion

wei

ght o

fcr

ucib

le a

fter

ashi

ng

dryi

ng ti

me

bala

nce

calib

ratio

n

bala

nce

linea

rity

ashi

ng te

mp

ashi

ng ti

me

bala

nce

calib

ratio

nba

lanc

elin

earit

yw

eigh

ing

of c

ruci

ble

wei

ghin

g of

cru

cibl

e

acid

dig

est

alka

li di

gest

alka

li

alka

li co

ncdi

gest

con

ditio

ns

boili

ngra

te

extra

ctio

n tim

eac

id v

ol

acid

con

c

dige

st c

ondi

tions

boili

ng ra

te

extra

ctio

n tim

e

bala

nce

linea

rity

bala

nce

calib

ratio

n

dryi

ng te

mp

dryi

ng ti

me

wei

ghin

g of

cru

cibl

e

ashi

ng p

reci

sion

bala

nce

calib

ratio

n

bala

nce

linea

rity

wei

ght

of

sam

ple

and

cruc

ible

bef

ore

ashi

ng

acid

dig

est*

alka

li di

gest

*

*The

bra

nche

s fe

edin

g in

to th

ese

“aci

ddi

gest

” an

d “a

lkal

i dig

est”

bra

nche

sha

ve b

een

omitt

ed fo

r cla

rity.

The

sam

e fa

ctor

s af

fect

them

as

for t

hesa

mpl

e (i.

e., d

iges

t con

ditio

ns, a

cid

conc

etc

.).



Fig

ure

A6.

10:

Sim

plif

ied

caus

e an

d ef

fect

dia

gram

Cru

de F

ibre

(%)

Los

s of

mas

s af

ter

ashi

ng (

b)

Los

s of

mas

s af

ter

blan

k as

hing

(c)

Pre

cisi

on

sam

ple

wei

ght p

reci

sion

wei

ghin

g pr

ecis

ion

wei

ght o

f cru

cibl

ebe

fore

ash

ing

dryi

ng te

mp

ashi

ng te

mp

ashi

ng ti

me

extra

ctio

nw

eigh

t of

cruc

ible

afte

ras

hing

dryi

ng ti

me

ashi

ng te

mp

ashi

ng ti

me

acid

dig

est

alka

li di

gest

alka

li vo

l

alka

li co

ncdi

gest

con

ditio

ns

boili

ng ra

te

extra

ctio

n tim

eac

id v

ol

acid

con

c

dige

st c

ondi

tions

boili

ng ra

te

extra

ctio

n tim

e

dryi

ng te

mp

dryi

ng ti

me

ashi

ng p

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sion

wei

ght o

f sam

ple

and

cruc

ible

afte

r ash

ing

wei

ght

of

sam

ple

and

cruc

ible

bef

ore

ashi

ng

acid

dig

est

alka

li di

gest





Example A7 - Determination of the amount of lead in water using DoubleIsotope Dilution and Inductively Coupled Plasma Mass Spectrometry

A7.1 Introduction

This example will illustrate how the uncertaintyconcept can be applied to a measurement of theamount of lead in a water sample using DoubleIsotope Dilution Mass Spectrometry (IDMS) andICP-MS.

General introduction to Double IDMS

IDMS is one of the techniques that is recognisedby CCQM to have the potential to be a primarymethod of measurement, and therefore a welldefined expression which describes how themeasurand is calculated is available. In thesimplest case of isotope dilution using a certifiedspike, which is an enriched isotopic referencematerial, to measure the amount of an elementpresent in a sample, the expression takes thisform:

( )

( )∑∑

⋅

⋅⋅

⋅−⋅

⋅−⋅⋅⋅=

i

i

RK

RK

RKRK

RKRK

m

mcc

yiyi

xixi

x1x1bb

bby1y1

x

yyx

(1)

where cx and cy are the amount of the element inthe sample and the spike respectively. The symbolc is used instead of k for the amount to avoidconfusion with K-factors. mx and my are mass ofsample and spike respectively. Rx, Ry and Rb arethe isotope amount ratios. The indexes x, y and brepresent the sample, the spike and the blendrespectively. One isotope, usually the mostabundant in the sample, is selected and all isotoperatios are expressed relative to it. A particular pairof isotopes, the reference isotope and preferablythe most abundant isotope in the spike, is thenselected as monitor ratio, e.g. n(208Pb)/n(206Pb).Rxi and Ryi are all the possible isotope ratios in thesample and the spike respectively. For thereference isotope this ratio is unity. Kxi, Kyi and Kb

are the correction factors for mass discrimination,for a particular isotope ratio, in sample, spike andblend respectively. The K-factors are calculatedusing a certified isotopic reference materialaccording to eqn. (2).

observed

certified0bias0 where;

R

RKKKK =+= (2)

where K0 is the mass discrimination correctionfactor at time 0, Kbias is a bias factor coming into

effect as soon as the K-factor is applied to correcta ratio measured at a different time. The Kbias canalso include other possible sources of biases likemultiplier dead time correction and backgroundcorrection. Rcertified is the certified isotope amountratio taken from the certificate of an isotopicreference material and Robserved is the measuredisotope ratio of this isotopic reference material. InIDMS experiments, using Inductively CoupledPlasma Mass Spectrometry (ICP-MS), massfractionation will vary with time which leads toeqn. (1) where all isotope amount ratios need tobe individually corrected for mass discrimination.

The availability of certified material enriched in aspecific isotope is often very scarce. To overcomethis ‘double’ IDMS is frequently used. The ideahere is to use a material of natural isotopiccomposition as primary assay standard. Toperform double IDMS we need to make anotherblend, here called blend b’. Blend b is the blendbetween sample and spike from eqn. (1). Thistime, for blend b’, we use the well characterisedprimary assay standard with the amount contentcz. This gives us a similar expression to eqn. (1):

( )

( )∑∑

⋅

⋅⋅

⋅−⋅

⋅−⋅⋅⋅=

i

i

zz

yy

RK

RK

RKRK

RKRK

m

mcc

yiyi

zizi

11bb

bb11

z

yyz ''

'''

(3)

where cz is the element amount content of theprimary assay standard solution and mz the massof the primary assay standard when preparing thenew blend. m’y is the mass of the enriched spikesolution, K’b, R’b, Kz1 and Rz1 are the K-factor andthe ratio for the new blend and the assay standardrespectively. The index z thus represents theassay standard. Equation (1) and (3) are similarand in order to eliminate cy from the expressionswe divide equation (1) with equation (3):

( )

( )( )

( )∑∑∑∑

⋅

⋅⋅

⋅−⋅

⋅−⋅⋅⋅

⋅

⋅⋅

⋅−⋅

⋅−⋅⋅⋅

=

i

i

i

i

RK

RK

RKRK

RKRK

m

mc

RK

RK

RKRK

RKRK

m

mc

c

c

yiyi

zizi

z1z1bb

bby1y1

z

yy

yiyi

xixi

x1x1bb

bby1y1

x

yy

z

x

''

'''

(4)



Simplifying this equation we get:

( )

( )∑∑

⋅

⋅⋅

⋅−⋅⋅−⋅

×

⋅−⋅

⋅−⋅⋅⋅⋅=

i

i

RK

RK

RKRK

RKRK

RKRK

RKRK

m

m

m

mcc

zizi

xixi

bby1y1

z1z1bb

x1x1bb

bby1y1

y

z

x

yzx

''''

'(5)

This is the final equation. For reference, theparameters are summarised in Table A7.1.


Calculation procedure for the amount content cx

For this determination of lead in water, fourblends each of the b’, (assay + spike), and b,(sample + spike), were prepared. This gives atotal of 4 values for cx. One of thesedeterminations will be described in detailfollowing Table A7.2, steps 1 to 4. The reportedvalue for cx will be the average of the fourreplicates. The number of digits displayed for theparameters in the calculations will sometimes bemore than what would be appropriate, but this isto minimise rounding off errors.

Table A7.2: General procedure

Step Description

1 Preparing the primary assaystandard

2 Preparation of blends: b’ and b

3 Measurement of isotope ratios

4 Calculation of the amountcontent of Pb in the sample, cx

5 Estimating the uncertainty in cx

Calculation of the Molar Mass

Due to natural variations in the isotopiccomposition of certain elements, e.g. Pb, themolar mass, M, for the primary assay standard hasto be determined since this will affect the amountcontent cz,. Note that this is not the case when cz

already is expressed in mol·g-1. The molar mass,M(E), for an element E, is numerically equal tothe atomic weight of element E, Ar(E). The atomicweight can be calculated according to the generalexpression:

∑

∑

=

==p

i

p

i

R

MRA

1i

1ii

r )E( (6)

where Ri are all true isotope amount ratios for theelement E and Mi are the tabulated nuclidemasses.

Note that the isotope amount ratios in eqn. (7)have to be absolute ratios, that is, they have to becorrected for mass discrimination. With the use ofproper indexes this gives equation (8). For thecalculation, nuclide masses, Mi, were taken from

Table A7.1. Summary of IDMS parameters

Param. Description

cz Amount content of the primary assaystandard

mx Mass of sample in blend bmy Mass of enriched spike in blend bm`y Mass of enriched spike in blend b’mz Mass of primary assay standard in

blend b’Rb Measured ratio of blend bR`b Measured ratio of blend b’Rx1 Measured ratio of the enriched

isotope to the reference isotope. Herein the sample

Ry1 As above but in the enriched spikeRz1 As above but in the primary assay

standardKb Mass bias correction of Rb

K`b Mass bias correction of R’b

Ky1 Mass bias correction of Ry1

Kzi Mass bias correction factors for allratios of a particular element,correcting for mass discrimination inthe measured ratios of the primaryassay standard. An element with 3isotopes would give Kz1, Kz2 and Kz3.

Kxi As above but for the sampleRzi All ratios in the primary assay

standard, Rz1, Rz2 etc.Rxi All ratios in the sample



G. Audi and A.H. Wapstra, Nuclear Physics,A565 (1993) while Ratios, Rzi, and K-factors, Kzi,were measured and taken from Table A7.8.

)7(21036.207

)1,Pb(

1

1iziz

1iziziz

−

=

=

⋅=

=

∑

∑

molg

RK

MRKAssayM

p

i

p

i

Measurement of K-factors and isotope amountratios

To correct for mass discrimination, a correctionfactor, K, is used, see equation (2). The K-factorcan be calculated using a reference materialcertified for isotopic composition. In this case theisotopically certified reference material NISTSRM 981 was used to monitor a possible changein the K-factor. The K-factor is measured beforeand after the ratio it will correct. A typical samplesequence could be: 1. (Blank), 2. (NIST SRM981), 3. (Blank), 4. (Blend 1), 5. (Blank), 6.(NIST SRM 981), 7. (Blank), 8. (Sample), etc.

The blank measurements are not only used forblank correction, they are also used formonitoring the number of counts for the blank.No new measurement run was started until theblank counts were stable and back to a normallevel. Note that sample, blends, spike and assaystandard were diluted to an appropriate amountcontent prior to the measurements. The results ofratio measurements, calculated K-factors andmasses are summarised in Table A7.8 togetherwith the calculated amount content of lead in theprimary assay standard, Assay 2.

Preparing the primary assay standard andcalculating the amount content, cz.

Two primary assay standards were produced,each from a different piece of metallic lead with achemical purity of w=99.999 mass percent. Thetwo pieces came from the same batch of highpurity lead. The pieces were dissolved in about10mL of 1:3 w/w HNO3:water with the aid of heatand then further diluted. The values from one ofthe produced standard assays will be displayed.

0.36544g lead, m1, was dissolved and diluted to atotal of d1=196.14 g 0.5M HNO3, this solution isnamed Assay 1. A more diluted solution wasneeded and m2=1.0292g of Assay 1, was diluted toa total mass of d2=99.931g 0.5 M HNO3. Thissolution is named Assay 2. The amount content of

Pb in Assay 2, cz, is then calculated according toeqn. (8)

( )( ) )8(gmol092606.0gmol102606.9

1,Pb1

118

1

1

2

2

−−− ⋅=⋅⋅=

⋅⋅

⋅=AssayMd

wm

d

mc z

Preparation of the blends

The mass fraction of the spike is known to beroughly 20µg Pb per g solution and the massfraction of Pb in the sample is also known to be inthis range. In Table A7.3 are the weighing datafor the two blends used in this example.

Table A7.3

Blend b b’

Solutionsused

Spike Sample

Spike Assay2

Parameter my mx m’y mz

Mass (g) 1.1360 1.0440 1.0654 1.1029

Calculation of the unknown amount content cx

Inserting the measured and calculated data, seeTable A7.8., into equation (5) gives cx=0.0537377µmol·g -1. The results from all four replicates aregiven in Table A7.4.

Table A7.4

cx (µmol·g -1)

Replicate 1 (our example) 0.0537377

Replicate 2 0.0536208



Average 0.05370

Experimental standarddeviation (s)

0.00011

A7.3 Steps 2 and 3: Identifying andQuantifying uncertainty sources

Strategy for the uncertainty calculation

If eqn. (2,6,7) were to be included in the finalIDMS eqn. (5), the sheer number of parameterswould make the equation almost impossible tohandle. To keep it simpler, K-factors and amount



content of the standard assay solution and theirassociated uncertainties can be treated separatelyand then introduced into the IDMS equation (5).In this case it will not affect the final combineduncertainty of cx, and it is advisable to simplifyfor practical reasons.

For calculating the combined standarduncertainty, uc(cx), the values from one of thedeterminations, as described in A7.2, will beused. The combined uncertainty of cx will becalculated using the spreadsheet method. Thismethod is described in Appendix E.

Uncertainty on the K-factors

A cause and effect diagram is constructed belowfor the uncertainty on the K-factors.

K

K0 Kbias

Rcertified

Robserved

i) Uncertainty on K0

K is calculated according to equation (2) andusing the values of Kx1 as an example gives forK0:

99917.01699.21681.2

observed

certified)1(0 ===

R

RK x (10)

To calculate the uncertainty on K0 we first look atthe certificate where the certified ratio, 2.1681,has a stated uncertainty of 0.0008 based on a 95%confidence interval. To convert an uncertaintybased on a 95% confidence interval to standarduncertainty we divide by 2. This gives thecertified ratio a standard uncertainty ofu(Rcertified)=0.0004. The observed amount ratio,Robserved=n(208Pb)/n(206Pb), had a relative standarduncertainty, (RSu), of 0.25%. For the K-factor,the combined uncertainty can be calculated,following Appendix D.5, as:

( )

)11(002505.0

0025.01681.20004.0

99917.0)( 22

)x1(0c

=

+

⋅=Ku

This clearly points out that the uncertaintycontribution from the certified ratios arenegligible. Henceforth, the uncertainties on the

measured ratios, Robserved, will be used for theuncertainties on K0.

Uncertainty on Kbias

This bias factor is introduced to compensate fordrifts in the value of the mass discriminationfactor. As can be seen in the cause and effectdiagram above, and in eqn.(2), there is a biasassociated with every K-factor. The values ofthese biases are in our case not known, and avalue of 0 is applied. An uncertainty is, of course,associated with every bias and this has to be takeninto consideration when calculating the finaluncertainty. In principle a bias would be appliedas in eqn. (12), using an excerpt from eqn. (5) andthe parameters Ky1 and Ry1 to demonstrate thisprinciple.

( )....

.....

...)1()1(.... y1bias0

x ⋅−⋅+

⋅=RyKyK

c (12)

The drawback with this approach is that thiswould increase the number of parameters andwould make the uncertainty calculation lessmanageable. Therefore the type B, biasuncertainties, are included later in the spreadsheetcalculation as an additional contribution in thespreadsheet equation, see eqn. (14). In thisexample the uncertainty from a bias has beenestimated as a fraction of the type A contributionof that particular Robserved. Note that the bias isNOT, in any way, a function of the standarddeviation of Robserved, it just gives a convenientbase for the estimation of the variation in apossible bias.

To explain how the bias uncertainty isimplemented let us look at eqn. (13) which is thegeneral equation used when applying thespreadsheet model. The square of the combineduncertainty is the sum of the uncertaintycontributions from the different parameters.

( )2iii

2c )())(()( ∑ −+= xfxuxfyu (13)

In our case the uncertainties in the biases wereestimated to be 20% of the type A contributionsof Robserved and hence 20% of K0. An example ofhow it is applied in the generic case using eqn.(13) is seen below:

( ) ( )[ ]( ) ( ) ( )[ ] ...)(K%20

)(...)(2

000

2000x

2c

+−+⋅+

−++=

KfuKf

KfKuKfcu

c

c

(14)



In eqn. (14), the first term is the uncertaintyassociated with variability; the second (20%…) isan estimate of the uncertainty associated withsystematic effects which have not been observed.Eqn (14) is used to calculate the experimentalcombined uncertainty by combining theexperimental standard uncertainties and theestimated standard uncertainties of the parametersin eqn (5). These uncertainties are given in TableA7.8 columns 3 and 4.

In Table A7.8 there is a summary of theparameters, their value and their experimentalstandard uncertainties. For the calculation of theexperimental combined uncertainty the conceptdescribed in eqn. (14) was applied. In the nextstep, the calculation of the final combineduncertainty, the number of measurements of everyparameter needs to be taken into account. In thisexample every ratio was measured eight times andevery type A uncertainty has to be divided by √8.Implementing this, in eqn. (14), gives:

( ) ( )[ ]

( ) ( ) ( )[ ] ...)(%208

)(...)(

200c0

200c0

x2

c

+−+⋅+

−++=

KfKuKf

KfKuKfcu

(15)

In the last two columns in Table A7.8 thecontribution of the type A and type Buncertainties to the final uncertainty can be seen.The value at the bottom of these two columns isthe final combined uncertainty for the measurand,cx, calculated using eqn. (15).

Uncertainty of the weighed masses

In this case a dedicated mass metrology labperformed the weighings. The procedure appliedwas a bracketing technique using calibratedweights and a comparator. The bracketingtechnique was repeated at least six times for everysample mass determination. Buoyancy correctionwas applied. Stoichiometry and impuritycorrections were not applied in this case. Theuncertainties from the weighing certificates weretreated as standard uncertainties and are given inTable A7.8.

Uncertainty in the amount content of the StandardAssay Solution, cz

i) Uncertainty in the atomic weight of Pb

First the combined uncertainty of the molar massof the assay solution, Assay 1, will be calculated.

The following parameters are known or have beenmeasured:

Table A7.5

Value StandardUncertainty

Type1

Rz1 2.1429 0.0054 A

Kz1 0.9989 0.0025 A

Kz2 1 0 A

Kz3 0.9993 0.0035 A

Kz4 1.0002 0.0060 A

Rz2 1 0 A

Rz3 0.9147 0.0032 A

Rz4 0.05870 0.00035 A

M1 207.976636 0.000003 B

M2 205.974449 0.000003 B

M3 206.975880 0.000003 B

M4 203.973028 0.000003 B1 Type A (statistical evaluation) or Type B (other)

The equation used to calculate the molar mass isgiven by eqn (16):

(16)

To calculate the combined standard uncertainty ofthe molar mass of Pb in the standard assaysolution the spreadsheet model described inAppendix E was used. There were eightmeasurements of every ratio and K-factor. Thisgave a molar mass of M(Pb, Assay 1)=(207.21036±0.00085) g·mol-1. The uncertaintywas calculated according to the concept outlinedin A7.3.5

ii) Calculation of the combined standarduncertainty in determining cz

To calculate the uncertainty on the amountcontent of Pb in the standard assay solution, cz thedata from A7.2.1 and equation (8) will be used.The uncertainties were taken from the weighingcertificates, see A7.3.3. All parameters used inequation (8) are given with their uncertainties inTable A7.6.

z4z4z3z3z2z2z1z1

4z4z43z3z32z2z21z1z1

)1,Pb(

RKRKRKRK

MRKMRKMRKMRK

AssayM

⋅+⋅+⋅+⋅⋅⋅+⋅⋅+⋅⋅+⋅⋅

=



The amount content, cz, was calculated usingequation (7). Following Appendix D.5 thecombined standard uncertainty in cz, is calculatedaccording to: uc(cz)=0.000028. This givescz=(0.092606±0.000028) µmol·g-1 and aRSuc(cz)=0.03%

To calculate uc(cx), for replicate 1, thespreadsheet model was applied, see Appendix E.The uncertainty budget for replicate 1 will berepresentative for the measurement. Due to thenumber of parameters in equation (5) thespreadsheet will not be displayed. The value ofthe parameters and their uncertainties as well asthe combined uncertainty of cx can be seen inTable A7.8.


In Table A7.7 the average and the experimentalstandard deviation of the four replicates aredisplayed. The numbers are taken from TableA7.4 and Table A7.8.

Table A7.7

Replicate 1 Mean of replicates1-4

cx= 0.05374 cx= 0.05370 µmol·g -1

uc(cx)= 0.00019 s1= 0.00011 µmol·g -1

1 Note, this is the experimental standard uncertainty and notthe standard deviation of the mean.

We can now compare the type A contributionfrom the experimental combined uncertainty,which is 83% of 0.00043 µmol·g -1, see TableA7.8, with the experimental standard deviation ofthe four replicates, which is 0.00011 µmol·g -1, seeTable A7.7. The experimental combineduncertainty, is larger than the obtainedexperimental standard deviation of the fourreplicates. This indicates that the experimentalstandard deviation is fully explained by the typeA contributions and that no further type Acontribution, due to the making of the blendsneeds to be considered. There could however be abias associated with the preparations of theblends. In this example a possible bias in thepreparation of the blends is judged to be coveredby the bias associated with the K-factors. Theamount content of lead in the water sample isthen:

cx=(0.05370±0.00038) µmol·g-1

The result is presented with an expandeduncertainty using a coverage factor of 2.

Table A7.6

Value Uncertainty

Mass of lead piece,m1

(g)0.36544 0.00005

Total mass firstdilution, d1 (g)

196.14 0.03

Aliquot of first dilution,m2 (g)

1.0292 0.0002

Total mass of seconddilution, d2 (g)

99.931 0.01

Purity of the metalliclead piece, w (massfraction)

0.99999 0.000005

Molar mass of Pb in theAssay Material, M(g·mol-1)

207.21036 0.00085



Table A7.8

Experimentalstandard

uncertainty

Estimatedstandard

uncertainty

Contribution to

experimental uc

(%)

Contribution to

final uc (%)

Parameter value

A B A B A B

cz 0.092606 0.000028 0.2 0.7

mx 1.0440 0.0002 0.1 0.3

my 1.1360 0.0002 0.1 0.3

m`y 1.0654 0.0002 0.1 0.3

mz 1.1029 0.0002 0.1 0.3

Rb 0.29360 0.00073 14.3 8.6

R`b 0.5050 0.0013 18.1 10.9

Rx1 2.1402 0.0054 4.3 2.6

Ry1 0.000640 0.000040 0.0 0.0

Rz1 2.1429 0.0054 6.6 3.9

Kb 0.9987 0.0025 14.3 2.9 8.6 13.8

K`b 0.9983 0.0025 18.1 3.6 10.9 17.4

Kx1 0.9992 0.0025 4.3 0.9 2.6 4.1

Ky1 0.9999 0.0025 0.0 0.0 0.0 0.0

Kz1 0.9989 0.0025 6.6 1.3 3.9 6.3

Kx2 1 0.0 0.0 0.0 0.0

Kx3 1.0004 0.0035 1.0 0.2 0.6 1.0

Kx4 1.0010 0.0060 0.0 0.0 0.0 0.0

Kz2 1 0.0 0.0 0.0 0.0

Kz3 0.9993 0.0035 1.0 0.2 0.6 1.0

Kz4 1.0000 0.0060 0.0 0.0 0.0 0.0

Rx2 1 0.0 0.0

Rx3 0.9142 0.0032 1.0 0.6

Rx4 0.05901 0.00035 0.0 0.0

Rz2 1 0.0 0.0

Rz3 0.9147 0.0032 1.0 0.6

Rz4 0.05870 0.00035 0.0 0.0

cx= 0.05374 µmol·g -1 90.6% 9.4% 54.5% 45.5%

uc(cx)= 0.00042 µmol·g -

10.00019 µmol·g -1

Quantifying Uncertainty Appendix B - Definitions


Appendix B - Definitions

General

B.1 Accuracy of measurement

The closeness of the agreement betweenthe result of a measurement and a truevalue of the measurand [G.4, 3.5].

NOTE 1 "Accuracy" is a qualitative concept.

NOTE 2 The term "precision" should not beused for "accuracy".

B.2 Precision

The closeness of agreement betweenindependent test results obtained understipulated conditions. [3534-1, 3.14]

NOTE 1 Precision depends only on thedistribution of random errors and doesnot relate to the true value or thespecified value.

NOTE 2 The measure of precision is usuallyexpressed in terms of imprecision andcomputed as a standard deviation of thetest results. Less precision is reflected bya larger standard deviation.

NOTE 3 "Independent test results" meansresults obtained in a manner notinfluenced by any previous result on thesame or similar test object. Quantitativemeasures of precision depend criticallyon the stipulated conditions.Repeatability and reproducibilityconditions are particular sets of extremestipulated conditions.

B.3 True value

Value consistent with the definition of agiven particular quantity [G.4, 1.19].

NOTE 1 This is a value that would be obtainedby a perfect measurement.

NOTE 2 True values are by natureindeterminate.

NOTE 3 The indefinite article "a" rather thanthe definite article "the" is used inconjunction with "true value" becausethere may be many values consistent

with the definition of a given particularquantity.

B.4 Conventional true value

Value attributed to a particular quantityand accepted, sometimes by convention,as having an uncertainty appropriate for agiven purpose. [G.4, 1.20].

EXAMPLES

a) at a given location, the value assignedto the quantity realised by a referencestandard may be taken as a conventionaltrue value.

b) the CODATA (1986) recommendedvalue for the Avogadro constant, NA:6.0221367×1023 mol-1

NOTE 1 "Conventional true value" issometimes called assigned value, bestestimate of the value, conventional valueor reference value.

NOTE 2 Frequently, a number of results ofmeasurements of a quantity is used toestablish a conventional true value.

B.5 Influence quantity

A quantity that is not the measurand butthat affects the result of the measurement[G.4, 2.7].

EXAMPLES

1. Temperature of a micrometer used tomeasure length;

2. Frequency in the measurement of analternating electric potential difference;

3. Bilirubin concentration in themeasurement of haemoglobinconcentration in human blood plasma.

Measurement

B.6 Measurand

Particular quantity subject tomeasurement. [G.4, 2.6]



NOTE The specification of a measurand mayrequire statements about quantities suchas time, temperature and pressure..

B.7 Measurement

Set of operations having the object ofdetermining a value of a quantity [G.4,2.1].

B.8 Measurement Procedure

Set of operations, described specifically,used in the performance of measurementsaccording to a given method [G.4, 2.5].

NOTE A measurement procedure is usuallyrecorded in a document that is sometimesitself a "measurement procedure" (or ameasurement method) and is usually insufficient detail to enable an operator tocarry out a measurement withoutadditional information.

B.9 Method of measurement

A logical sequence of operations,described generically, used in theperformance of measurements [G.4, 2.4].

NOTE Methods of measurement may bequalified in various ways such as:

- substitution method

- differential method

- null method

B.10 Result of a measurement

Value attributed to a measurand,obtained by measurement [G.4, 3.1].

NOTE 1 When the term "result of ameasurement" is used, it should be madeclear whether it refers to:- The indication.- The uncorrected result.- The corrected result.and whether several values are averaged.

NOTE 2 A complete statement of the result of ameasurement includes information aboutthe uncertainty of measurement.

Uncertainty

B.11 Uncertainty (of measurement)

Parameter associated with the result of ameasurement, that characterises thedispersion of the values that couldreasonably be attributed to themeasurand [G.4, 3.9].

NOTE 1 The parameter may be, for example, astandard deviation (or a given multipleof it), or the width of a confidenceinterval.

NOTE 2 Uncertainty of measurementcomprises, in general, many components.Some of these components may beevaluated from the statistical distributionof the results of a series of measurementsand can be characterised by experimentalstandard deviations. The othercomponents, which can also becharacterised by standard deviations, areevaluated from assumed probabilitydistributions based on experience orother information.

NOTE 3 It is understood that the result of themeasurement is the best estimate of thevalue of the measurand and that allcomponents of uncertainty, includingthose arising from systematic effects,such as components associated withcorrections and reference standards,contribute to the dispersion.

B.12 Traceability

"the property of the result of ameasurement or the value of a standardwhereby it can be related to statedreferences, usually national orinternational standards, through anunbroken chain of comparisons allhaving a stated uncertainties" [VIM G.4]

B.13 Standard uncertainty

u(xi) uncertainty of the result xi of ameasurement expressed as a standarddeviation. [G.2, 2.3.1]

B.15 Combined standard uncertainty

uc(y) standard uncertainty of the result y of ameasurement when the result is obtainedfrom the values of a number of otherquantities, equal to the positive square



root of a sum of terms, the terms beingthe variances or covariances of theseother quantities weighted according tohow the measurement result varies withthese quantities. [G.2, 2.3.4].

B.16 Expanded uncertainty

U Quantity defining an interval about theresult of a measurement that may beexpected to encompass a large fraction ofthe distribution of values that couldreasonably be attributed to themeasurand. [G.2, 2.3.5]

NOTE 1 The fraction may be regarded as thecoverage probability or level ofconfidence of the interval.

NOTE 2 To associate a specific level ofconfidence with the interval defined bythe expanded uncertainty requiresexplicit or implicit assumptionsregarding the probability distributioncharacterised by the measurement resultand its combined standard uncertainty.The level of confidence that may beattributed to this interval can be knownonly to the extent to which suchassumptions can be justified.

NOTE 3 An expanded uncertainty U iscalculated from a combined standarduncertainty uc and a coverage factor kusing

U = k × uc

B.17 Coverage factor

k numerical factor used as a multiplier ofthe combined standard uncertainty inorder to obtain an expanded uncertainty[G.2, 2.3.6].

NOTE A coverage factor is typically in therange 2 to 3.

B.18 Type A evaluation (of uncertainty)

Method of evaluation of uncertainty bythe statistical analysis of series ofobservations [G.2, 2.3.2].

B.19 Type B evaluation (of uncertainty)

Method of evaluation of uncertainty bymeans other than the statistical analysisof series of observations [G.2, 2.3.3]

Error

B.20 Error (of measurement)

The result of a measurement minus a truevalue of the measurand [G.4, 3.10].

NOTE 1 Since a true value cannot bedetermined, in practice a conventionaltrue value is used.

B.21 Random error

Result of a measurement minus the meanthat would result from an infinite numberof measurements of the same measurandcarried out under repeatability conditions[G.4, 3.13].

NOTE 1 Random error is equal to error minussystematic error.

NOTE 2 Because only a finite number ofmeasurements can be made, it is possibleto determine only an estimate of randomerror.

B.22 Systematic error

Mean that would result from an infinitenumber of measurements of the samemeasurand carried out under repeatabilityconditions minus a true value of themeasurand. [G.4, 3.14].

NOTE 1: Systematic error is equal to errorminus random error.

NOTE 2: Like true value, systematic error andits causes cannot be known.

Statistical terms

B.23 Arithmetic mean

x arithmetic mean value of a sample of nresults.

x

x ii= =∑

1

n

n



B.24 Sample Standard Deviation

s an estimate of the population standarddeviation σ from a sample of n results.

1

)(1

2

−

−=

∑=

n

xxs

n

ii

B.25 Standard deviation of the mean

xs The standard deviation of the mean x ofn values taken from a population is givenby

n

ssx =

The terms "standard error" and "standarderror of the mean" have also been used todescribe the same quantity.

B.26 Relative Standard Deviation (RSD)

RSD an estimate of the standard deviation of apopulation from a sample of n resultsdivided by the mean of that sample.Often known as coefficient of variation(CV). Also frequently stated as apercentage.

x

s=RSD

Quantifying Uncertainty Appendix C - Structure of analysis


Appendix C – Uncertainties in Analytical Processes

C.1 In order to identify the possible sources ofuncertainty in an analytical procedure it is helpfulto break down the analysis into a set of genericsteps:1. Sampling

2. Sample preparation

3. Presentation of Certified ReferenceMaterials to the measuring system

4. Calibration of Instrument

5. Analysis (data acquisition)

6. Data processing

7. Presentation of results

8. Interpretation of results

C.2 These steps can be further broken down bycontributions to the uncertainty for each. Thefollowing list, though not necessarilycomprehensive, provides guidance on factorswhich should be considered.

1. Sampling Homogeneity. Effects of specific sampling strategy (e.g.

random, stratified random, proportionaletc.)

Effects of movement of bulk medium(particularly density selection)

Physical state of bulk (solid, liquid, gas) Temperature and pressure effects. Does sampling process affect

composition? e.g. differential adsorptionin sampling system.

2. Sample preparation Homogenisation and/or sub-sampling

effects. Drying. Milling. Dissolution. Extraction. Contamination.

Derivatisation (chemical effects) Dilution errors. Concentration. Control of speciation effects.

3. Presentation of Certified ReferenceMaterials to the measuring system Uncertainty for CRM. CRM match to sample

4. Calibration of instrument Instrument calibration errors using a

Certified Reference Material. Reference material and its uncertainty. Sample match to calibrant Instrument precision

5. Analysis Carry-over in auto analysers. Operator effects, e.g. colour blindness,

parallax, other systematic errors. Interferences from the matrix, reagents or

other analytes. Reagent purity. Instrument parameter settings, e.g.

integration parameters Run-to-run precision

6. Data Processing Averaging. Control of rounding and truncating. Statistics. Processing algorithms (model fitting, e.g.

linear least squares).

7. Presentation of Results Final result. Estimate of uncertainty. Confidence level.

8. Interpretation of Results Against limits/bounds. Regulatory compliance. Fitness for purpose.

Quantifying Uncertainty Appendix D – Analysing Uncertainty Sources


Appendix D: Analysing uncertainty sources

D.1 Introduction

It is commonly necessary to develop and record alist of sources of uncertainty relevant to ananalytical method. It is often useful to structurethis process, both to ensure comprehensivecoverage and to avoid over-counting. Thefollowing procedure, (based on a previouslypublished method [G.11]), provides one possiblemeans of developing a suitable, structuredanalysis of uncertainty contributions.

D.2 Principles of approach

D.2.1 The strategy has two stages:

• Identifying the effects on a result

In practice, the necessary structured analysisis effected using a cause and effect diagram(sometimes known as an Ishikawa or‘fishbone’ diagram) [G.12]

• Simplifying and resolving duplication

The initial list is refined to simplifypresentation and ensure that effects are notunnecessarily duplicated.

D.3 Cause and effect analysis

D.3.1 The principles of constructing a cause andeffect diagram are described fully elsewhere. Theprocedure employed is as follows:

1. Write the complete equation for the result. Theparameters in the equation form the mainbranches of the diagram. It is almost alwaysnecessary to add a main branch representing anominal correction for overall bias, usually asrecovery, and this is accordinglyrecommended at this stage if appropriate.

2. Consider each step of the method and add anyfurther factors to the diagram, workingoutwards from the main effects. Examplesinclude environmental and matrix effects.

3. For each branch, add contributory factors untileffects become sufficiently remote, that is,until effects on the result are negligible

4. Resolve duplications and re-arrange to clarifycontributions and group related causes. It is

convenient to group precision terms at thisstage on a separate precision branch.

D.3.2 The final stage of the cause and effectanalysis requires further elucidation.Duplications arise naturally in detailingcontributions separately for every inputparameter. For example, a run-to-run variabilityelement is always present, at least nominally, forany influence factor; these effects contribute toany overall variance observed for the method asa whole and should not be added in separately ifalready so accounted for. Similarly, it is commonto find the same instrument used to weighmaterials, leading to over-counting of itscalibration uncertainties. These considerationslead to the following additional rules forrefinement of the diagram (though they applyequally well to any structured list of effects):

• Cancelling effects: remove both. Forexample, in a weight by difference, twoweights are determined, both subject to thebalance ‘zero bias’. The zero bias will cancelout of the weight by difference, and can beremoved from the branches corresponding tothe separate weighings.

• Similar effect, same time: combine into asingle input. For example, run-to-runvariation on many inputs can be combinedinto an overall run-to-run precision ‘branch’.Some caution is required; specifically,variability in operations carried outindividually for every determination can becombined, whereas variability in operationscarried out on complete batches (such asinstrument calibration) will only beobservable in between-batch measures ofprecision.

• Different instances: re-label. It is common tofind similarly named effects which actuallyrefer to different instances of similarmeasurements. These must be clearlydistinguished before proceeding.

D.3.3 This form of analysis does not lead touniquely structured lists. In the present example,temperature may be seen as either a direct effecton the density to be measured, or as an effect on

Quantifying Uncertainty Appendix D – Analysing Uncertainty Sources


the measured mass of material contained in adensity bottle; either could form the initialstructure. In practice this does not affect theutility of the method. Provided that all significanteffects appear once, somewhere in the list, theoverall methodology remains effective.

D.3.4 Once the cause-and-effect analysis iscomplete, it may be appropriate to return to theoriginal equation for the result and add any newterms (such as temperature) to the equation.

D.4 Example

D.4.1 The procedure is illustrated by reference toa simplified direct density measurement.Consider the case of direct determination of thedensity d(EtOH) of ethanol by weighing a knownvolume V in a suitable volumetric vessel of tareweight Mtare and gross weight including ethanolMgross. The density is calculated from

d(EtOH)=(Mgross - Mtare)/V

For clarity, only three effects will be considered:Equipment calibration, Temperature, and theprecision of each determination. Figures D1-D3illustrate the process graphically.

D.4.2 A cause and effect diagram consists of ahierarchical structure culminating in a singleoutcome. For the present purpose, this outcomeis a particular analytical result (‘d(EtOH)’ inFigure D1). The ‘branches’ leading to theoutcome are the contributory effects, whichinclude both the results of particular intermediatemeasurements and other factors, such asenvironmental or matrix effects. Each branchmay in turn have further contributory effects.These ‘effects’ comprise all factors affecting theresult, whether variable or constant; uncertaintiesin any of these effects will clearly contribute touncertainty in the result.

D.4.3 Figure D1 shows a possible diagramobtained directly from application of steps 1-3.The main branches are the parameters in theequation, and effects on each are represented bysubsidiary branches. Note that there are two‘temperature’ effects, three ‘precision’ effectsand three ‘calibration’ effects.

D.4.4 Figure D2 shows precision andtemperature effects each grouped togetherfollowing the second rule (same effect/time);temperature may be treated as a single effect ondensity, while the individual variations in each

determination contribute to variation observed inreplication of the entire method.

D.4.5 The calibration bias on the two weighingscancels, and can be removed (Figure D3)following the first refinement rule (cancellation).

D.4.6 Finally, the remaining ‘calibration’branches would need to be distinguished as two(different) contributions owing to possible non-linearity of balance response, together with thecalibration uncertainty associated with thevolumetric determination.

Figure D1: Initial list

d(EtOH)d(EtOH)

M(gross) M(tare)

Volume

TemperatureTemperature

Calibration


Calibration

Lin*. Bias

Lin*. Bias

Precision

Precision

*Lin. = Linearity

Figure D2: Combination of similar effects

d(EtOH)

M(gross) M(tare)

Volume

Temperature

Temperature

Calibration


Calibration

Lin. Bias

Lin. Bias

Precision

Precision

Precision

Temperature

Figure D3: Cancellation

d(EtOH)

M(gross) M(tare)

Volume

Calibration

Calibration

Calibration

Lin. Bias

Lin. Bias

Precision

Temperature

Same balance: bias cancels

Quantifying Uncertainty Appendix E – Statistical Procedures


Appendix E - Useful Statistical Procedures

E.1 Distribution functions

The following table shows how to calculate a standard uncertainty from the parameters of the two mostimportant distribution functions, and gives an indication of the circumstances in which each should be used .

EXAMPLE

A chemist estimates a contributory factor as not less than 7 or more than 10, but feels that the value could beanywhere in between, with no idea of whether any part of the range is more likely than another. This is adescription of a rectangular distribution function with a range 2a=3 (semi range of a=1.5). Using the functionbelow for a rectangular distribution, an estimate of the standard uncertainty can be calculated. Using the aboverange, a=1.5, results in a standard uncertainty of (1.5/√3) = 0.87.

Rectangular distribution

Form Use when: Uncertainty

2a ( = ±a )

1/2a

x

• A certificate or other specification giveslimits without specifying a level ofconfidence (e.g. 25ml ± 0.05ml)

• An estimate is made in the form of amaximum range (±a) with no knowledgeof the shape of the distribution.

3)(

axu =

Triangular distribution


2a ( = ± a )

1/a

x

• The available information concerning x isless limited than for a rectangulardistribution. Values close to x are morelikely than near the bounds.

• An estimate is made in the form of amaximum range (±a) described by asymmetric distribution.

6)(

axu =



Normal distribution


2σ

x

• An estimate is made from repeatedobservations of a randomly varyingprocess.

• An uncertainty is given in the form of astandard deviation s, a relative standarddeviation xs / , or a coefficient ofvariance CV% without specifying thedistribution.

• An uncertainty is given in the form of a95% (or other) confidence interval Iwithout specifying the distribution.

u(x) = s

u(x) = s

u(x)=x.( s x/ )

u(x)= x⋅100

%CV

u(x) = I/2 (for Iat 95%)



E.2 Spreadsheet method for uncertainty calculation

E.2.1 A standard spreadsheet can be used tosimplify the calculations shown in Section 8. Theprocedure takes advantage of an approximatenumerical method of differentiation, and requiresknowledge only of the calculation used to derivethe final result (including any necessarycorrection factors or influences) and of thenumerical values of the parameters and theiruncertainties. The description here follows that ofKragten [G.9].

E.2.2 In the expression for u(y(x1, x2...xn))

∑∑==

⋅⋅+

⋅

nkini

ikxkxy

ixy

ixuixy

,1,,1

),(

2

)( s∂∂

∂∂

∂∂

provided that either y(x1, x2...xn) is linear in xi oru(xi) is small compared to xi, the partialdifferentials (∂y/∂xi) can be approximated by:-

)()())((

i

iii

i xuxyxuxy

xy −+

≈∂∂

Multiplying by u(xi) to obtain the uncertaintyu(y,xi) in y due to the uncertainty in xi gives

u(y,xi) ≈ y(x1,x2,..(xi+u(xi))..xn)-y(x1,x2,..xi..xn)

Thus u(y,xi) is just the difference between thevalues of y calculated for [xi + u(xi)] and xi

respectively.

E.2.3 The assumption of linearity or small valuesof u(xi)/xi will not be closely met in all cases.Nonetheless, the method does provide acceptableaccuracy for practical purposes when consideredagainst the necessary approximations made inestimating the values of u(xi). Reference G.9discusses the point more fully and suggestsmethods of checking the validity of theassumption.

E.2.4 The basic spreadsheet is set up as follows,assuming that the result y is a function of the fourparameters p, q, r, and s:i) Enter the values of p, q, etc. and the formula

for calculating y in column A of thespreadsheet. Copy column A across thefollowing columns once for every variable in y(see Figure E2.1). It is convenient to place thevalues of the uncertainties u(p), u(q) and so onin row 1 as shown.

ii) Add u(p) to p in cell B3, u(q) to q in cell C4etc., as in Figure E2.2. On recalculating thespreadsheet, cell B8 then becomes

f(p+u(p), q ,r..) (denoted by f (p’, q, r, ..) inFigures E2.2 and E2.3), cell C8 becomesf(p, q+u(q), r,..) etc.

iii) In row 9 enter row 8 minus A8 (for example,cell B9 becomes B8-A8). This gives the valuesof u(y,p) as

u(y,p)=f (p+u(p), q, r ..) - f (p,q,r ..) etc.iv) To obtain the standard uncertainty on y, these

individual contributions are squared, addedtogether and then the square root taken, byentering u(y,p)2 in row 10 (Figure E2.3) andputting the square root of their sum in A10.That is, cell A10 is set to the formula

SQRT(SUM(B10+C10+D10+E10))

which gives the standard uncertainty on y.

E.2.5 The contents of the cells B10, C10 etc.show the contributions of the individualuncertainty components to the uncertainty on yand hence it is easy to see which components aresignificant.

E.2.6 It is straightforward to allow updatedcalculations as individual parameter valueschange or uncertainties are refined. In step i)above, rather than copying column A directly tocolumns B-E, copy the values p to s by reference,that is, cells B3 to E3 all reference A3, B4 to E4reference A4 etc. The horizontal arrows in FigureE2.1 show the referencing for row 3. Note thatcells B8 to E8 should still reference the values incolumns B to E respectively, as shown for columnB by the vertical arrows in Figure E2.1. In step ii)above, add the references to row 1 by reference(as shown by the arrows in Figure E2.1). Forexample, cell B3 becomes A3+B1, cell C4becomes A4+C1 etc. Changes to eitherparameters or uncertainties will then be reflectedimmediately in the overall result at A8 and thecombined standard uncertainty at A10.

E.2.7 If any of the variables are correlated, thenecessary additional term is added to the SUM inA10. For example, if p and q are correlated, witha correlation coefficient r(p,q), then the extra term2×r(p,q) ×u(y,p) ×u(y,q) is added to the calculatedsum before taking the square root. Correlation cantherefore easily be included by adding suitableextra terms to the spreadsheet.



Figure E2.1A B C D E

1 u(p) u(q) u(r) u(s)23 p p p p p4 q q q q q5 r r r r r6 s s s s s78 y=f(p,q,..) y=f(p,q,..) y=f(p,q,..) y=f(p,q,..) y=f(p,q,..)91011


1 u(p) u(q) u(r) u(s)23 p p+u(p) p p p4 q q q+u(q) q q5 r r r r+u(r) r6 s s s s s+u(s)78 y=f(p,q,..) y=f(p’,...) y=f(..q’,..) y=f(..r’,..) y=f(..s’,..)9 u(y,p) u(y,q) u(y,r) u(y,s)1011


1 u(p) u(q) u(r) u(s)23 p p+u(p) p p p4 q q q+u(q) q q5 r r r r+u(r) r6 s s s s s+u(s)78 y=f(p,q,..) y=f(p’,...) y=f(..q’,..) y=f(..r’,..) y=f(..s’,..)9 u(y,p) u(y,q) u(y,r) u(y,s)10 u(y) u(y,p)2 u(y,q)2 u(y,r)2 u(y,s)2

11



E. 3 Uncertainties from Linear Least Squares Calibration

E.3.1 An analytical method or instrument is oftencalibrated by observing the responses, y, todifferent levels of the analyte, x. In most casesthis relationship is taken to be linear viz.:

y = b0 + b1x

The concentration xobs of the analyte from asample which produces an observed response yobs

is then given by:-

xobs = (yobs - c)/b1

It is usual to determine the constants b1 and b0 byleast squares regression on a set of n values (xi,yi).

E.3.2 There are four main sources of uncertaintyto consider in arriving at an uncertainty on theestimated concentration xobs:

• Random variations in measurement of y,affecting both the reference responses yi andthe measured response yobs.

• Random effects resulting in errors in theassigned reference values xi.

• Values of xi and yi may be subject to aconstant unknown offset e.g. arising when thevalues of x are obtained from serial dilutionof a stock solution

• The assumption of linearity may not be valid

Of these, the most significant for normal practiceare random variations in y, and methods ofestimating uncertainty for this source are detailedhere. The remaining sources are also consideredbriefly to give an indication of methods available.

E.3.3 The uncertainty u(xobs, y) in a predictedvalue xobs due to variability in y can be estimatedin several ways:

From calculated variance and covariance:

If the values of b1 and b0, their variances var(b1),var(b0) and their covariance, covar(b1,b0), aredetermined by the method of least squares, thevariance on x, var(x) is given by

21

01012 )var(),(covar2)var()var(

)var(

b

bbbxbxy

x

+⋅⋅+⋅+

=

and the corresponding uncertainty u(xobs, y) is√var(x).

From the RMS error or the variance ofresiduals S.

var(x) is approximately equal to 21

2 / bS , where S2

is the variance of the y values about the fittedline:

Sy y

ni2

2

2=

−

−∑ ( )

and ( )y yi i− is the residual for the ith point. Scan also be calculated from the RMS error using

n

yyi∑ −=

2)(errorRMS

It follows that S is given by

2)errorRMS( 22

−⋅=

n

nS

From the correlation coefficient r

The correlation coefficient r together with therange R(y) of the y values can be used to obtainan approximate estimate of S using

121

)(R2

22 ryS

−⋅=

If using this value of S shows that var(x) is notsignificant compared with the other componentsof the uncertainty, then it is not necessary toobtain a better estimate of it. However if it issignificant then a better estimate will be required.

From the calibration data

Given a set of data (xi,yi), the uncertainty u(xobs, y)in xobs arising from random variability in y valuesis given by

u(xobs,y) =

∑−∑

−++⋅

−⋅

−∑

))()((

)(11222

1

2

21

2

)2(

)(

nxxb

yy

nii

obs

nbiyiy

where ( )y yi i− is the residual for the ith point, nis the number of data points in the calibration, b1

the calculated best fit gradient, and )( yyobs − the



difference between yobs and the mean y of the yi

values.

Other methods

Some software gives the standard deviation s(yc)on a value of y calculated from the fitted line forsome new value of x and this can be used tocalculate var(x) since :-

var(x) = [ s(yc) /b1 ]2

If s(yc) is not given then it can be calculatedfrom:-

−⋅++⋅=

D

xxn

nSys c

222 )(1

1)(

where S2 is the variance of the y values about thefitted line defined above and

( )∑ ∑−⋅=22 )( ii xxnD

In most cases it is sufficient to use an estimate ofD obtained from the range R of the n values of xused in the calibration and then:-

⋅

⋅−++⋅= 2

222 12)(1

1)(Rn

xxn

Sys c

and at the extreme of the calibration range

+⋅=

nSys c

41)( 22

This is a sufficient approximation for most caseswhere the var(x) is not a dominant component ofthe final uncertainty.

E.3.4 The reference values xi may each haveuncertainties which propagate through to the finalresult. In practice, uncertainties in these valuesare usually small compared to uncertainties in thesystem responses yi, and may be ignored. An

approximate estimate of the uncertainty u(xobs, xi)in a predicted value xobs due to uncertainties in xi

is

u(xobs, xi) ≈ u(xi)/n

where n is the number of xi values used in thecalibration. This expression can be used to checkthe significance of u(xobs, xi).

E.3.5 The uncertainty arising from the assumptionof a linear relationship between y and x is notnormally large enough to require an additionalestimate. Providing the residuals show that thereis no significant systematic deviation from thisassumed relationship, the uncertainty arising fromthis assumption (in addition to that covered by theresulting increase in y variance) can be taken tobe negligible. If the residuals show a systematictrend then it may be necessary to include higherterms in the calibration function. Methods ofcalculating var(x) in these cases are given instandard texts. It is also possible to make ajudgement based on the size of the systematictrend.

E.3.6 The values of x and y may be subject to aconstant unknown offset (e.g. arising when thevalues of x are obtained from serial dilution of astock solution which has an uncertainty on itscertified value) If the standard uncertainties on yand x from these effects are u(y,const) andu(x,const), then the uncertainty on theinterpolated value xobs is given by:-

u(xobs)2 =

u(x,const)2 + (u(y,const)/b1)2 + var(x)

E.3.7 The overall uncertainty arising fromcalculation from a linear calibration can then becalculated in the normal way from the fourcomponents above.



Appendix E4: Documenting uncertainty dependent on analyte level

E4.1 Introduction

E4.1.1 It is often observed in chemicalmeasurement that, over a large range ofanalyte levels, dominant contributions to theoverall uncertainty vary approximatelyproportionately to the level of analyte, that isu(x) ∝ x. In such cases it is often sensible toquote uncertainties as relative standarddeviations or, for example, coefficient ofvariation (%CV). However, at low levels,other effects dominate and theproportionality is lost (for example, directproportionality leads to zero estimateduncertainties as the observed levels approachzero). In these circumstances, or where arelatively narrow range of analyte level isinvolved, it is sensible to quote an absolutevalue for the uncertainty. This section setsout a general approach to recordinguncertainty information where variation ofuncertainty with analyte level is an issue.

E4.2 Basis of approach

E4.2.1 To allow for both proportionality ofuncertainty and the possibility of anessentially constant value with level, thefollowing general expression is used:

21

20 )()( sxsxu ⋅+= [1]

where

u(x) is the combined standard uncertaintyin the result x (that is, the uncertaintyexpressed as a standard deviation)

s0 represents a constant contribution to theoverall uncertainty

s1 is a proportionality constant.

The expression is based on the normalmethod of combining of two contributions tooverall uncertainty, assuming onecontribution (s0) is constant and one (x.s1)proportional to the result. Figure E4.1 showsthe form of this expression.

NOTE: The approach above is practical onlywhere it is possible to calculate a largenumber of values. Where experimentalstudy is employed, it will not often bepossible to establish the relevantparabolic relationship. In suchcircumstances, an adequate

approximation can be obtained by simplelinear regression through four or morecombined uncertainties obtained atdifferent analyte concentrations. Thisprocedure is consistent with thatemployed in studies of reproducibilityand repeatability according to ISO5725:1994. The relevant expression isthen u x s x s( ) ' . '≈ +0 1

E4.2.2 The figure can be divided intoapproximate regions (A to C on the figure):

A: The uncertainty is dominated by the terms0, and is approximately constant andclose to s0.

B: Both terms contribute significantly; theresulting uncertainty is significantlyhigher than either s0 or x.s1, and somecurvature is visible.

C: The term x.s1 dominates; the uncertaintyrises approximately linearly withincreasing x and is close to x.s1.

E4.2.3 Note that in many experimental casesthe complete form of the curve will not beapparent. Very often, the whole reportingrange of analyte level permitted by the scopeof the method falls within a single chartregion; the result is a number of special casesdealt with in more detail below.

E4.3 Documenting level-dependentuncertainty data

E4.3.1 In general, uncertainties can bedocumented in the form of a value for eachof s0 and s1. The values can be used toprovide an uncertainty estimate across thescope of the method. This is particularlyvaluable when calculations for wellcharacterised methods are implemented oncomputer systems, where the general form ofthe equation can be implementedindependently of the values of theparameters (one of which may be zero - seebelow). It is accordingly recommended that,except in the special cases outlined below orwhere the dependence is strong but notlinear*, uncertainties are documented in the

* An important example of non-linear dependenceis the effect of instrument noise on absorbance



form of values for a constant termrepresented by s0 and a variable termrepresented by s1.

E4.4. Special cases

E4.4.1. Uncertainty not dependent on levelof analyte (s0 dominant)

Uncertainty will generally be effectivelyindependent of observed analyteconcentration when:

• The result is close to zero (for example,within the stated detection limit for themethod). Region A in Figure E4.1

• The possible range of results (stated inthe method scope or in a statement ofscope for the uncertainty estimate) issmall compared to the observed level.

Under these circumstances, the value of s1

can be recorded as zero. s0 is normally thecalculated standard uncertainty.

E4.4.2. Uncertainty entirely dependent onanalyte (s1 dominant)

Where the result is far from zero (forexample, above a ‘limit of determination’)and there is clear evidence that theuncertainty changes proportionally with thelevel of analyte permitted within the scope ofthe method, the term x.s1 dominates (seeRegion C in Figure E4.1). Under thesecircumstances, and where the method scopedoes not include levels of analyte near zero,s0 may reasonably be recorded as zero and s1

is simply the uncertainty expressed as arelative standard deviation.

measurement at high absorbances near the upperlimit of the instrument capability. This isparticularly pronounced where absorbance iscalculated from transmittance (as in infraredspectroscopy). Under these circumstances,baseline noise causes very large uncertainties inhigh absorbance figures, and the uncertainty risesmuch faster than a simple linear estimate wouldpredict. The usual approach is to reduce theabsorbance, typically by dilution, to bring theabsorbance figures well within the working range;the linear model used here will then normally beadequate. Other examples include the ‘sigmoidal’response of some immunoassay methods.

E4.4.3. Intermediate dependence

In intermediate cases, and in particular wherethe situation corresponds to region B infigure 1, two approaches can be taken:

a) Applying variable dependence

The more general approach is to determine,record and use both s0 and s1. Uncertaintyestimates, when required, can then beproduced on the basis of the reported result.This remains the recommended approachwhere practical.

NOTE: See the note to section E4.2.

b) Applying a fixed approximation

An alternative which may be used in generaltesting and where

• the dependence is not strong (that is,evidence for proportionality is weak)

or• the range of results expected is

moderate

leading in either case to uncertainties whichdo not vary by more than about 15% from anaverage uncertainty estimate, it will often bereasonable to calculate and quote a fixedvalue of uncertainty for general use, basedon the mean value of results expected. Thatis,

eithera mean or typical value for x is used tocalculate a fixed uncertainty estimate, andthis is used in place of individuallycalculated estimates

ora single standard deviation has beenobtained, based on studies of materialscovering the full range of analyte levelspermitted (within the scope of theuncertainty estimate), and there is littleevidence to justify an assumption ofproportionality. This should generally betreated as a case of zero dependence, andthe relevant standard deviation recordedas s0.

E4.5. Determining s0 and s1

E4.5.1. In the special cases in which oneterm dominates, it will normally be sufficientto use the uncertainty as standard deviation



or relative standard deviation respectively asvalues of s0 and s1. Where the dependence isless obvious, however, it may be necessary todetermine s0 and s1 indirectly from a series ofestimates of uncertainty at different analytelevels.

E4.5.1. Given a calculation of combineduncertainty from the various components,some of which depend on analyte level whileothers do not, it will normally be possible toinvestigate the dependence of overalluncertainty on analyte level by simulation.The procedure is as follows:1: Calculate (or obtain experimentally)

uncertainties u(xi) for at least ten levelsxi of analyte, covering the full rangepermitted.

2. Plot u(xi)2 against xi2

3. By linear regression, obtain estimates ofm and c for the line u(x)2 = m.x2 + c

4. Calculate s0 and s1 from s0 = √c, s1 = √m5. Record s0 and s1

E4.6.Reporting

E4.6.1. The approach outlined here permits

estimation of a standard uncertainty for anysingle result. In principle, where uncertaintyinformation is to be reported, it will be in theform of

[result] ± [uncertainty]

where the uncertainty as standard deviationis calculated as above, and if necessaryexpanded (usually by a factor of two) to giveincreased confidence. Where a number ofresults are reported together, however, it maybe possible, and is perfectly acceptable, togive an estimate of uncertainty applicable toall results reported.

E4.6.1. Table E4.2 gives some examples.The uncertainty figures for a list of differentanalytes may usefully be tabulated followingsimilar principles.

NOTE: Where a ‘detection limit’ or ‘reportinglimit’ is used to give results in the form“<x” or “nd”, it will normally benecessary to quote the limits used inaddition to the uncertainties applicable toresults above reporting limits.

Table E4.2: Summarising uncertainty for several samples

Situation Dominant term Reporting example(s)

Uncertainty essentially constantacross all results

s0 or fixed approximation(sections E4.4.1. or E4.4.3.a)

Standard deviation: expandeduncertainty; 95% confidenceinterval

Uncertainty generallyproportional to level

x.s1

(see section E4.4.2.)relative standard deviation;coefficient of variance (%cv)

Mixture of proportionality andlower limiting value foruncertainty

Intermediate case(section E4.4.3.)

quote %cv or rsd together withlower limit as standarddeviation.



Figure E4.1: Variation of uncertainty with observed result

CBA

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6 7 8

Result x

Uncertainty u(X)

s(0)

x.s(1)

u(x)

Uncertaintyapproximately

equal to s0

Uncertaintysignificantlygreater than

either s0 or x.s1

Uncertaintyapproximatelyequal to x. s1

Quantifying Uncertainty Appendix F – Uncertainty Sources


Appendix F - Common sources and values of uncertainty

The following tables summarise typical examples of uncertainty components from amongthose found in the EURACHEM document. The tables give:

• The particular measurand or experimental procedure (determining mass, volumeetc)

• The main components and sources of uncertainty in each case

• A suggested method of determining the uncertainty arising from each source.

• An example of a typical case

The tables are intended only to summarise the examples and to indicate general methodsof estimating uncertainties in analysis. They are not intended to be comprehensive, norshould the values given be used directly without independent justification. The valuesmay, however, help in deciding whether a particular component is significant.

Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Mas

s (a

bsol

ute)

Bal

ance

cal

ibra

tion

unce

rtain

tyLi

mite

d ac

cura

cy in

calib

ratio

nSt

ated

on

calib

ratio

n ce

rtific

ate,

conv

erte

d to

sta

ndar

d de

viat

ion

4-fig

ure

bala

nce

0.5

mg

Line

arity

i) Ex

perim

ent,

with

rang

e of

certi

fied

wei

ghts

ii) M

anuf

actu

rer's

spe

cific

atio

n

ca. 0

.5x

last

sign

ifica

ntdi

git

Dai

ly d

rift

Var

ious

Stan

dard

dev

iatio

n of

long

term

chec

k w

eigh

ings

. Cal

cula

te a

sR

SD if

nec

essa

ry.

ca. 0

.5x

last

sign

ifica

ntdi

git.

Run

to ru

n va

riatio

nV

ario

usSt

anda

rd d

evia

tion

of s

ucce

ssiv

esa

mpl

e or

che

ck w

eigh

ings

ca. 0

.5x

last

sign

ifica

ntdi

git.

TOTA

L sp

ecifi

catio

nun

certa

inty

Com

bina

tion

of a

bove

Com

bine

abo

ve a

s st

anda

rdde

viat

ions

4-fig

ure

bala

nce

0.5

mg

Cal

ibra

tion

wei

ght/s

ampl

e de

nsity

mis

mat

ch

The

mis

mat

ch c

ause

s a

diff

eren

ce in

the

effe

ct o

fat

mos

pher

ic b

uoya

ncy.

To c

orre

ct, c

alcu

late

atm

osph

eric

buoy

ancy

ef

fect

an

d su

btra

ctbu

oyan

cy

effe

ct

on

calib

ratio

nw

eigh

t.

100

g w

ater

10 g

Nic

kel

+0.1

g

< 1

mg



Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Mas

s(b

y di

ffer

ence

)R

un to

run

varia

tion

Var

ious

Stan

dard

dev

iatio

n of

suc

cess

ive

sam

ple

or c

heck

wei

ghin

gs2-

Figu

reB

alan

ce10

g ch

eck

wei

ght:

s=0.

03g

Bal

ance

Lin

earit

yi)

Expe

rimen

t, w

ith ra

nge

ofce

rtifie

d w

eigh

ts

ii) M

anuf

actu

rer's

spe

cific

atio

n

4-fig

ure

bala

nce

QC

sho

ws

s=0.

07m

g fo

rra

nge

ofw

eigh

ts

TOTA

L:C

ombi

natio

n of

abo

veC

ombi

ne a

bove

as

stan

dard

devi

atio

ns4-

figur

eba

lanc

e: R

un-to

-ru

n: s

= 0.

07m

g.Sp

ec =

±0.

1mg

at 9

5%co

nfid

ence

22

96.11.0

07.0

+

=0.

087m

g



Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Vol

ume

(liqu

id)

Cal

ibra

tion

unce

rtain

tyLi

mite

d ac

cura

cy in

calib

ratio

nSt

ated

on

man

ufac

ture

r'ssp

ecifi

catio

n, c

onve

rted

tost

anda

rd d

evia

tion.

For A

STM

cla

ss A

gla

ssw

are

ofvo

lum

e V

, the

lim

it is

appr

oxim

atel

y V

0.6 /

200

10 m

l (G

rade

A)

0.02

/ √3

=0.

01 m

l*

Tem

pera

ture

Tem

pera

ture

var

iatio

nfr

om th

e ca

libra

tion

tem

pera

ture

cau

ses

adi

ffer

ence

in th

e vo

lum

eat

the

stan

dard

tem

pera

ture

.

∆T.

α/2

.√3

give

s th

e re

lativ

est

anda

rd d

evia

tion,

whe

re ∆

T is

the

poss

ible

tem

pera

ture

rang

ean

d α

the

coef

ficie

nt o

f vol

ume

expa

nsio

n of

the

liqui

d. α

isap

prox

imat

ely

2 x1

0-4

K-1

for

wat

er a

nd 1

x 1

0-3

K-1

for

orga

nic

liqui

ds.

100

ml w

ater

0.03

ml f

orop

erat

ing

with

in 3

°Cof

the

stat

edop

erat

ing

tem

pera

ture

Run

to ru

n va

riatio

nV

ario

usSt

anda

rd d

evia

tion

of s

ucce

ssiv

ech

eck

deliv

erie

s (f

ound

by

wei

ghin

g)

25 m

l pip

ette

Rep

licat

efil

l/wei

gh:

s =

0.0

092

ml

*Ass

umin

g re

ctan

gula

r dis

tribu

tion



Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Ref

eren

ce m

ater

ial

conc

entra

tion

Purit

yIm

purit

ies

redu

ce th

eam

ount

of r

efer

ence

mat

eria

l pre

sent

. Rea

ctiv

eim

purit

ies

may

inte

rfer

ew

ith th

e m

easu

rem

ent.

Stat

ed o

n m

anuf

actu

rer’

sce

rtific

ate.

Ref

eren

ce c

ertif

icat

esus

ually

giv

e un

qual

ified

lim

its;

thes

e sh

ould

acc

ordi

ngly

be

treat

ed a

s re

ctan

gula

rdi

strib

utio

ns a

nd d

ivid

ed b

y √3

.N

ote:

whe

re th

e na

ture

of t

heim

purit

ies

is n

ot s

tate

d,ad

ditio

nal a

llow

ance

or c

heck

sm

ay n

eed

to b

e m

ade

to e

stab

lish

limits

for i

nter

fere

nce

etc.

Ref

eren

cepo

tass

ium

hydr

ogen

phth

alat

ece

rtifie

d as

99.

9±0

.1%

0.1/

√3 =

0.06

%

Con

cent

ratio

n(c

ertif

ied)

Cer

tifie

d un

certa

inty

inre

fere

nce

mat

eria

lco

ncen

tratio

n.

Stat

ed o

n m

anuf

actu

rer’

sce

rtific

ate.

Ref

eren

ce c

ertif

icat

esus

ually

giv

e un

qual

ified

lim

its;

thes

e sh

ould

acc

ordi

ngly

be

treat

ed a

s re

ctan

gula

rdi

strib

utio

ns a

nd d

ivid

ed b

y √3

.

Cad

miu

mac

etat

e in

4%

acet

ic a

cid.

Cer

tifie

d as

1000

±2 m

g.l-1

.

2/√3

= 1

.2m

g.l-1

(0.0

012

asR

SD)*

Con

cent

ratio

n (m

ade

up fr

om c

ertif

ied

mat

eria

l)

Com

bina

tion

ofun

certa

intie

s in

refe

renc

eva

lues

and

inte

rmed

iate

step

s

Com

bine

val

ues

for p

rior s

teps

as

RSD

thro

ugho

ut.

Cad

miu

mac

etat

e af

ter

thre

e di

lutio

nsfr

om 1

000

to 0

.5m

g.l-1

0034

.0

0017

.0

0021

.0

0017

.0

0012

.0

2222

=

+++

as R

SD

*Ass

umin

g re

ctan

gula

r dis

tribu

tion



Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Abs

orba

nce

Inst

rum

ent c

alib

ratio

n

Not

e: th

is c

ompo

nent

rela

tes

to a

bsor

banc

ere

adin

g ve

rsus

refe

renc

e ab

sorb

ance

,no

t to

the

calib

ratio

nof

con

cent

ratio

nag

ains

t abs

orba

nce

read

ing

Lim

ited

accu

racy

inca

libra

tion.

Stat

ed o

n ca

libra

tion

certi

ficat

eas

lim

its, c

onve

rted

to s

tand

ard

devi

atio

n

Run

to ru

n va

riatio

nV

ario

usSt

anda

rd d

evia

tion

of re

plic

ate

dete

rmin

atio

ns, o

r QA

perf

orm

ance

.

Mea

n of

7 A

Aab

sorb

ance

read

ings

with

s=1.

63

1.63

/√7

= 0.

62

Sam

plin

gH

omog

enei

tySu

b-sa

mpl

ing

from

inho

mog

eneo

us m

ater

ial

will

not

gen

eral

lyre

pres

ent t

he b

ulk

exac

tly.

Not

e: ra

ndom

sam

plin

gw

ill g

ener

ally

resu

lt in

zero

bia

s. It

may

be

nece

ssar

y to

che

ck th

atsa

mpl

ing

is a

ctua

llyra

ndom

.

i) St

anda

rd d

evia

tion

of s

epar

ate

sub-

sam

ple

resu

lts (i

f th

ein

hom

ogen

eity

is la

rge

rela

tive

toan

alyt

ical

acc

urac

y).

ii) E

stim

ated

sta

ndar

d de

viat

ion.

Sam

plin

g fr

ombr

ead

ofas

sum

ed tw

o-va

lued

inho

mog

enei

ty

For 1

5 po

rtion

sfr

om 7

2co

ntam

inat

edan

d 36

0un

cont

amin

ated

bulk

por

tions

:R

SD =

0.5

8



Det

erm

inat

ion

Unc

erta

inty

Cau

seM

etho

d of

det

erm

inat

ion

Typ

ical

val

ues

Com

pone

nts

Exa

mpl

eV

alue

Extra

ctio

nre

cove

ryM

ean

reco

very

Extra

ctio

n is

rare

lyco

mpl

ete

and

may

add

or

incl

ude

inte

rfer

ents

.

Rec

over

y ca

lcul

ated

as

perc

enta

ge re

cove

ry fr

omco

mpa

rabl

e re

fere

nce

mat

eria

lor

repr

esen

tativ

e sp

ikin

g.U

ncer

tain

ty o

btai

ned

from

stan

dard

dev

iatio

n of

mea

n of

reco

very

exp

erim

ents

.

Not

e: re

cove

ry m

ay a

lso

beca

lcul

ated

dire

ctly

from

prev

ious

ly m

easu

red

parti

tion

coef

ficie

nts.

Rec

over

y of

pest

icid

e fr

ombr

ead;

42

expe

rimen

ts,

mea

n 90

%,

s=28

%

28/√

42=

4.3%

(0.0

48 a

sR

SD)

Run

to ru

n va

riatio

n in

reco

very

Var

ious

Stan

dard

dev

iatio

n of

repl

icat

eex

perim

ents

.R

ecov

ery

ofpe

stic

ides

from

brea

d fr

ompa

ired

repl

icat

eda

ta.

0.31

as

RSD

.Se

e te

xt fo

rca

lcul

atio

n.



Quantifying Uncertainty Appendix G - Bibliography


Appendix G - Bibliography

G.1. ISO/IEC Guide 25:1990 (3rd Edition). General Requirements for the Competence of Calibrationand Testing Laboratories. ISO, Geneva (1990)

G.2. Guide To The Expression Of Uncertainty In Measurement. ISO, Geneva, Switzerland 1993.(ISBN 92-67-10188-9)

G.3. EURACHEM, Quantifying Uncertainty in Analytical Measurement. Laboratory of theGovernment Chemist, London 1995. ISBN 0-948926-08-2

G.4. International Vocabulary of basic and general standard terms in Metrology. ISO, Geneva,Switzerland 1993 (ISBN 92-67-10175-1)

G.5. ISO 3534:1993. Statistics - Vocabulary and Symbols. ISO, Geneva, Switzerland 1993

G.6. EURACHEM, The Fitness for Purpose of Analytical Methods. 1998 (ISBN 0-948926-12-0)

G.7. ISO/IEC Guide 33:1989, “Uses of Certified Reference Materials”. ISO, Geneva 1989

G.8. I.J.Good, "Degree of Belief" 1982, in Encyclopaedia of Statistical Sciences, Vol. 2, Wiley, NewYork .

G.9. J. Kragten, “Calculating standard deviations and confidence intervals with a universally applicablespreadsheet technique”, Analyst, 119, 2161-2166 (1994)

G.10. British Standard BS 6748:1986. Limits of metal release from ceramic ware, glassware, glassceramic ware and vitreous enamel ware.

G.11.S L R Ellison, V Barwick. Accred. Qual. Assur. (1998) 3 101-105.

G.12.ISO 9004-4:1993, Total Quality Management. Part 2. Guidelines for quality improvement, (ISO1993)

Documents

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