EURACHEM / CITAC Guide Quantifying Uncertainty in Analytical Measurement Second Edition QUAM: 2000. P1 EURACHEM/ CITAC Guide Quantifying Uncertainty in Analytical Measurement Second Edition Editors S L R Ellison ( LGC, UK) M Rosslein ( EMPA, Switzerland) A Williams ( UK)

Composition of the Working Group EURACHEM members A Williams Chairman UK S Ellison Secretary LGC, Teddington, UK M Berglund Institute for Reference Materials and Measurements, Belgium W Haesselbarth Bundesanstalt fur Materialforschung und Prufung, Germany K Hedegaard EUROM II R Kaarls Netherlands Measurement Institute, The Netherlands M Månsson SP Swedish National Testing and Research Institute, Sweden M Rösslein EMPA St. Gallen, Switzerland R Stephany National Institute of Public Health and the Environment, The Netherlands A van der Veen Netherlands Measurement Institute, The Netherlands W Wegscheider University of Mining and Metallurgy, Leoben, Austria H van de Wiel National Institute of Public Health and the Environment, The Netherlands R Wood Food Standards Agency, UK CITAC Members Pan Xiu Rong Director, NRCCRM,. China M Salit National Institute of Science and Technology USA A Squirrell NATA, Australia K Yasuda Hitachi Ltd, Japan AOAC Representatives R Johnson Agricultural Analytical Services, Texas State Chemist, USA Jung Keun Lee U. S. F. D. A. Washington D Mowrey Eli Lilly & Co., Greenfield, USA IAEA Representatives P De Regge IAEA Vienna A Fajgelj IAEA Vienna EA Representative D Galsworthy, UKAS, UK Acknowledgements This document has been produced primarily by a joint EURACHEM/ CITAC Working Group with the composition shown ( right). The editors are grateful to all these individuals and organisations and to others who have contributed comments, advice and assistance.

Production of this Guide was in part supported under contract with the UK Department of Trade and Industry as part of the National Measurement System Valid Analytical Measurement ( VAM) Programme. Quantifying Uncertainty Contents QUAM: 2000. P1 Page i CONTENTS FOREWORD TO THE SECOND EDITION 1 1. SCOPE AND FIELD OF APPLICATION 3 2. UNCERTAINTY 4 2.1. DEFINITION OF UNCERTAINTY 4 2.2. UNCERTAINTY SOURCES 4 2.3. UNCERTAINTY COMPONENTS 4 2.4. ERROR AND UNCERTAINTY 5 3. ANALYTICAL MEASUREMENT AND UNCERTAINTY 7 3.1. METHOD VALIDATION 7 3.2. CONDUCT OF EXPERIMENTAL STUDIES OF METHOD PERFORMANCE 8 3.3. TRACEABILITY 9 4. THE PROCESS OF MEASUREMENT UNCERTAINTY ESTIMATION 11 5. STEP 1. SPECIFICATION OF THE MEASURAND 13 6. STEP 2. IDENTIFYING UNCERTAINTY SOURCES 14 7. STEP 3. QUANTIFYING UNCERTAINTY 16 7.1. INTRODUCTION 16 7.2. UNCERTAINTY EVALUATION PROCEDURE 16 7.3. RELEVANCE OF PRIOR STUDIES 17 7.4. EVALUATING UNCERTAINTY BY QUANTIFICATION OF INDIVIDUAL COMPONENTS 17 7.5. CLOSELY MATCHED CERTIFIED REFERENCE MATERIALS 17 7.6. USING PRIOR COLLABORATIVE METHOD DEVELOPMENT AND VALIDATION STUDY DATA 17 7.7. USING IN HOUSE DEVELOPMENT AND VALIDATION STUDIES 18 7.8. EVALUATION OF UNCERTAINTY FOR EMPIRICAL METHODS 20 7.9. EVALUATION OF UNCERTAINTY FOR AD HOC METHODS 20 7.10. QUANTIFICATION OF INDIVIDUAL COMPONENTS 21 7.11. EXPERIMENTAL ESTIMATION OF INDIVIDUAL UNCERTAINTY CONTRIBUTIONS 21 7.12. ESTIMATION BASED ON OTHER RESULTS OR DATA 22 7.13. MODELLING FROM THEORETICAL PRINCIPLES 22 7.14. ESTIMATION BASED ON JUDGEMENT 22 7.15. SIGNIFICANCE OF BIAS 24 8. STEP 4. CALCULATING THE COMBINED UNCERTAINTY 25 8.1. STANDARD UNCERTAINTIES 25 8.2. COMBINED STANDARD UNCERTAINTY 25 8.3. EXPANDED UNCERTAINTY 27 Quantifying Uncertainty Contents QUAM: 2000. P1 Page ii 9. REPORTING UNCERTAINTY 29 9.1. GENERAL 29 9.2. INFORMATION REQUIRED 29 9.3. REPORTING STANDARD UNCERTAINTY 29 9.4. REPORTING EXPANDED UNCERTAINTY 29 9.5. NUMERICAL EXPRESSION OF RESULTS 30 9.6. COMPLIANCE AGAINST LIMITS 30 APPENDIX A. EXAMPLES 32 INTRODUCTION 32 EXAMPLE A1: PREPARATION OF A CALIBRATION STANDARD 34 EXAMPLE A2: STANDARDISING A SODIUM HYDROXIDE SOLUTION 40 EXAMPLE A3: AN ACID/ BASE TITRATION 50 EXAMPLE A4: UNCERTAINTY ESTIMATION FROM IN HOUSE VALIDATION STUDIES. DETERMINATION OF ORGANOPHOSPHORUS PESTICIDES IN BREAD. 59 EXAMPLE A5: DETERMINATION OF CADMIUM RELEASE FROM CERAMIC WARE BY ATOMIC ABSORPTION SPECTROMETRY 70 EXAMPLE A6: THE DETERMINATION OF CRUDE FIBRE IN ANIMAL FEEDING STUFFS 79 EXAMPLE A7: DETERMINATION OF THE AMOUNT OF LEAD IN WATER USING DOUBLE ISOTOPE DILUTION AND INDUCTIVELY COUPLED PLASMA MASS SPECTROMETRY 87 APPENDIX B. DEFINITIONS 95 APPENDIX C. UNCERTAINTIES IN ANALYTICAL PROCESSES 99 APPENDIX D. ANALYSING UNCERTAINTY SOURCES 100 D. 1 INTRODUCTION 100 D. 2 PRINCIPLES OF APPROACH 100 D. 3 CAUSE AND EFFECT ANALYSIS 100 D. 4 EXAMPLE 101 APPENDIX E. USEFUL STATISTICAL PROCEDURES 102 E. 1 DISTRIBUTION FUNCTIONS 102 E. 2 SPREADSHEET METHOD FOR UNCERTAINTY CALCULATION 104 E. 3 UNCERTAINTIES FROM LINEAR LEAST SQUARES CALIBRATION 106 E. 4: DOCUMENTING UNCERTAINTY DEPENDENT ON ANALYTE LEVEL 108 APPENDIX F. MEASUREMENT UNCERTAINTY AT THE LIMIT OF DETECTION/ LIMIT OF DETERMINATION 112 F1. INTRODUCTION 112 F2. OBSERVATIONS AND ESTIMATES 112 F3. INTERPRETED RESULTS AND COMPLIANCE STATEMENTS 113 APPENDIX G. COMMON SOURCES AND VALUES OF UNCERTAINTY 114 APPENDIX H. BIBLIOGRAPHY 120 Quantifying Uncertainty QUAM: 2000. P1 Page 1 Foreword to the Second Edition Many important decisions are based on the results of chemical quantitative analysis; the results are used, for example, to estimate yields, to check materials against specifications or statutory limits, or to estimate monetary value. Whenever decisions are based on analytical results, it is important to have some indication of the quality of the results, that is, the extent to which they can be relied on for the purpose in hand. Users of the results of chemical analysis, particularly in those areas concerned with international trade, are coming under increasing pressure to eliminate the replication of effort frequently expended in obtaining them. Confidence in data obtained outside the user's own organisation is a prerequisite to meeting 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 and are providing data of the required quality. Such measures include: the use of validated methods of analysis; the use of defined internal quality control procedures; participation in proficiency testing schemes; accreditation based on ISO 17025 [ H. 1], and establishing traceability of the results of the measurements In analytical chemistry, there has been great emphasis on the precision of results obtained using a specified method, rather than on their traceability to a defined standard or SI unit. This has led the use of " official methods" to fulfil legislative and trading requirements. However as there is now a formal requirement to establish the confidence of results it is essential that a measurement result is traceable to a defined reference such as a SI unit, reference material or, where applicable, a defined or empirical ( sec. 5.2.) method. Internal quality control procedures, proficiency testing and accreditation can be an aid in establishing evidence of traceability to a given standard.

As a consequence of these requirements, chemists are, for their part, coming under increasing pressure to demonstrate the quality of their results, and in particular to demonstrate their fitness for purpose by giving a measure of the confidence that can be placed on the result. This is expected to include the degree to which a result would be expected to agree with other results, normally irrespective of the analytical methods used. One useful measure of this is measurement uncertainty.

Although the concept of measurement uncertainty has been recognised by chemists for many years, it was the publication in 1993 of the " Guide to the Expression of Uncertainty in Measurement" [ H. 2] by ISO in collaboration with BIPM, IEC, IFCC, IUPAC, IUPAP and OIML, which formally established general rules for evaluating and expressing uncertainty in measurement across a broad spectrum of measurements. This EURACHEM document shows how the concepts in the ISO Guide may be applied in chemical measurement. It first introduces the concept of uncertainty and the distinction between uncertainty and error. This is followed by a description of the steps involved in the evaluation of uncertainty with the processes illustrated by worked examples in Appendix A.

The evaluation of uncertainty requires the analyst to look closely at all the possible sources of uncertainty. However, although a detailed study of this kind may require a considerable effort, it is essential that the effort expended should not be disproportionate. In practice a preliminary study will quickly identify the most significant sources of uncertainty and, as the examples show, the value obtained for the combined uncertainty is almost entirely controlled by the major contributions. A good estimate of uncertainty can be made by concentrating effort on the largest contributions. Further, once evaluated for a given method applied in a particular laboratory ( i. e. a particular measurement procedure), the uncertainty estimate obtained may be reliably applied to subsequent results obtained by the method in the same laboratory, provided that this is justified by the relevant quality control data. No further effort should be necessary unless the procedure itself or the equipment used is changed, in which case the uncertainty estimate would be reviewed as part of the normal re validation.

The first edition of the EURACHEM Guide for " Quantifying Uncertainty in Analytical Measurement" [ H. 3] was published in 1995 based on the ISO Guide. Quantifying Uncertainty Foreword to the Second Edition QUAM: 2000. P1 Page 2 This second edition of the EURACHEM Guide has been prepared in the light of practical experience of uncertainty estimation in chemistry laboratories and the even greater awareness of the need to introduce formal quality assurance procedures by laboratories. The second edition stresses that the procedures introduced by a laboratory to estimate its measurement uncertainty should be integrated with existing quality assurance measures, since these measures frequently provide much of the information required to evaluate the measurement uncertainty. The guide therefore provides explicitly for the use of validation and related data in the construction of uncertainty estimates in full compliance with formal ISO Guide principles. The approach is also consistent with the requirements of ISO 17025: 1999 [ H. 1]

NOTE Worked examples are given in Appendix A. A numbered list of definitions is given at Appendix B. The convention is adopted of printing defined terms in bold face upon their first occurrence in the text, with a reference to Appendix B enclosed in square brackets. The definitions are, in the main, taken from the International vocabulary of basic and general standard terms in Metrology ( VIM) [ H. 4], the Guide [ H. 2] and ISO 3534 ( Statistics  Vocabulary and symbols) [ H. 5]. Appendix C shows, in general terms, the overall structure of a chemical analysis leading to a measurement result. Appendix D describes a general procedure which can be used to identify uncertainty components and plan further experiments as required; Appendix E describes some statistical operations used in uncertainty estimation in analytical chemistry. Appendix F discusses measurement uncertainty near detection limits. Appendix G lists many common uncertainty sources and methods of estimating the value of the uncertainties. A bibliography is provided at Appendix H. Quantifying Uncertainty Scope and Field of Application QUAM: 2000. P1 Page 3 1. Scope and Field of Application 1.1. This Guide gives detailed guidance for the evaluation and expression of uncertainty in quantitative chemical analysis, based on the approach taken in the ISO " Guide to the Expression of Uncertainty in Measurement" [ H. 2]. It is applicable at all levels of accuracy and in all fields  from routine analysis to basic research and to empirical and rational methods ( see section 5.3.). Some common areas in which chemical measurements are 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.

· Measurements associated with the development and certification of reference materials.

· Research and development.

1.2. Note that additional guidance will be required in some cases. In particular, reference material value assignment using consensus methods ( including multiple measurement methods) is not covered, and the use of uncertainty estimates in compliance statements and the expression and use of uncertainty at low levels may require additional guidance. Uncertainties associated with sampling operations are not explicitly treated.

1.3. Since formal quality assurance measures have been introduced by laboratories in a number of sectors this second EURACHEM Guide is now able to illustrate how data from the following procedures may be used for the estimation of measurement uncertainty:

· Evaluation of the effect of the identified sources of uncertainty on the analytical result for a single method implemented as a defined measurement procedure [ B. 8] 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 competent laboratories.

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

1.4. It is assumed throughout this Guide that, whether carrying out measurements or assessing the performance of the measurement procedure, effective quality assurance and control measures are in place to ensure that the measurement process is stable and in control. Such measures normally include, for example, appropriately qualified staff, proper maintenance and calibration of equipment and reagents, use of appropriate reference standards, documented measurement procedures and use of appropriate check standards and control charts. Reference [ H. 6] provides further information on analytical QA procedures.

NOTE: This paragraph implies that all analytical methods are assumed in this guide to be implemented via fully documented procedures. Any general reference to analytical methods accordingly implies the presence of such a procedure. Strictly, measurement uncertainty can only be applied to the results of such a procedure and not to a more general method of measurement [ B. 9]. Quantifying Uncertainty Uncertainty QUAM: 2000. P1 Page 4 2. Uncertainty 2.1. Definition of uncertainty 2.1.1. The definition of the term uncertainty ( of measurement) used in this protocol and taken from the current version adopted for the International Vocabulary of Basic and General Terms in Metrology [ H. 4] is:

" A parameter associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand"

Note 1 The parameter may be, for example, a standard deviation [ B. 23] ( or a given multiple of it), or the width of a confidence interval.

NOTE 2 Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results of series of measurements and can be characterised by standard deviations. The other components, which also can be characterised by standard deviations, are evaluated from assumed probability distributions based on experience or other information. The ISO Guide refers to these different cases as Type A and Type B estimations respectively.

2.1.2. In many cases in chemical analysis, the measurand [ B. 6] will be the concentration* of an analyte. However chemical analysis is used to measure other quantities, e. g. colour, pH, etc., and therefore the general term " measurand" will be used.

2.1.3. The definition of uncertainty given above focuses on the range of values that the analyst believes could reasonably be attributed to the measurand.

2.1.4. In general use, the word uncertainty relates to the general concept of doubt. In this guide, the * In this guide, the unqualified term " concentration" applies to any of the particular quantities mass concentration, amount concentration, number concentration or volume concentration unless units are quoted ( e. g. a concentration quoted in mg l 1 is evidently a mass concentration). Note also that many other quantities used to express composition, such as mass fraction, substance content and mole fraction, can be directly related to concentration. word uncertainty, without adjectives, refers either to a parameter associated with the definition above, or to the limited knowledge about a particular value. Uncertainty of measurement does not imply doubt about the validity of a measurement; on the contrary, knowledge of the uncertainty implies increased confidence in the validity of a measurement result.

2.2. Uncertainty sources 2.2.1. In practice the uncertainty on the result may arise from many possible sources, including examples such as incomplete definition, sampling, matrix effects and interferences, environmental conditions, uncertainties of masses and volumetric equipment, reference values, approximations and assumptions incorporated in the measurement method and procedure, and random variation ( a fuller description of uncertainty sources is given in section 6.7.)

2.3. Uncertainty components 2.3.1. In estimating the overall uncertainty, it may be necessary to take each source of uncertainty and treat it separately to obtain the contribution from that source. Each of the separate contributions to uncertainty is referred to as an uncertainty component. When expressed as a standard deviation, an uncertainty component is known as a standard uncertainty [ B. 13]. If there is correlation between any components then this has to be taken into account by determining the covariance. However, it is often possible to evaluate the combined effect of several components. This may reduce the overall effort involved and, where components whose contribution is evaluated together are correlated, there may be no additional need to take account of the correlation.

2.3.2. For a measurement result y, the total uncertainty, termed combined standard uncertainty [ B. 14] and denoted by uc( y), is an estimated standard deviation equal to the positive square root of the total variance obtained by combining all the uncertainty components, however evaluated, using the law of propagation of uncertainty ( see section 8.). Quantifying Uncertainty Uncertainty QUAM: 2000. P1 Page 5 2.3.3. For most purposes in analytical chemistry, an expanded uncertainty [ B. 15] U, should be used. The expanded uncertainty provides an interval within which the value of the measurand is believed to lie with a higher level of confidence. U is obtained by multiplying uc( y), the combined standard uncertainty, by a coverage factor [ B. 16] k. The choice of the factor k is based on the level of confidence desired. For an approximate level of confidence of 95%, k is 2.

NOTE The coverage factor k should always be stated so that the combined standard uncertainty of the measured quantity can be recovered for use in calculating the combined standard uncertainty of other measurement results that may depend on that quantity.

2.4. Error and uncertainty 2.4.1. It is important to distinguish between error and uncertainty. Error [ B. 19] is defined as the difference between an individual result and the true value [ B. 3] of the measurand. As such, error is a single value. In principle, the value of a known error can be applied as a correction to the result.

NOTE Error is an idealised concept and errors cannot be known exactly.

2.4.2. Uncertainty, on the other hand, takes the form of a range, and, if estimated for an analytical procedure and defined sample type, may apply to all determinations so described. In general, the value of the uncertainty cannot be used to correct a measurement result.

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

2.4.4. The uncertainty of the result of a measurement should never be interpreted as representing the error itself, nor the error remaining after correction.

2.4.5. An error is regarded as having two components, namely, a random component and a systematic component.

2.4.6. Random error [ B. 20] typically arises from unpredictable variations of influence quantities. These random effects give rise to variations in repeated observations of the measurand. The random error of an analytical result cannot be compensated for, but it can usually be reduced by increasing the number of observations.

NOTE 1 The experimental standard deviation of the arithmetic mean [ B. 22] or average of a series of observations is not the random error of the mean, although it is so referred to in some publications on uncertainty. It is instead a measure of the uncertainty of the mean due to some random effects. The exact value of the random error in the mean arising from these effects cannot be known.

2.4.7. Systematic error [ B. 21] is defined as a component of error which, in the course of a number 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 by increasing the number of analyses under constant measurement conditions.

2.4.8. Constant systematic errors, such as failing to make an allowance for a reagent blank in an assay, or inaccuracies in a multi point instrument calibration, are constant for a given level of the measurement value but may vary with the level of the measurement value.

2.4.9. Effects which change systematically in magnitude during a series of analyses, caused, for example by inadequate control of experimental conditions, give rise to systematic errors that are not constant.

EXAMPLES:

1. A gradual increase in the temperature of a set of samples during a chemical analysis can lead to progressive changes in the result.

2. Sensors and probes that exhibit ageing effects over the time scale of an experiment can also introduce non constant systematic errors.

2.4.10. The result of a measurement should be corrected for all recognised significant systematic effects.

NOTE Measuring instruments and systems are often adjusted or calibrated using measurement standards and reference materials to correct for systematic effects. The uncertainties associated with these standards and materials and the uncertainty in the correction must still be taken into account.

2.4.11. A further type of error is a spurious error, or blunder. Errors of this type invalidate a measurement and typically arise through human Quantifying Uncertainty Uncertainty QUAM: 2000. P1 Page 6 failure or instrument malfunction. Transposing digits in a number while recording data, an air bubble lodged in a spectrophotometer flowthrough cell, or accidental cross contamination of test items are common examples of this type of error.

2.4.12. Measurements for which errors such as these have been detected should be rejected and no attempt should be made to incorporate the errors into any statistical analysis. However, errors such as digit transposition can be corrected ( exactly), particularly if they occur in the leading digits. 2.4.13. Spurious errors are not always obvious and, where a sufficient number of replicate measurements is available, it is usually appropriate to apply an outlier test to check for the presence of suspect members in the data set. Any positive result obtained from such a test should be considered with care and, where possible, referred back to the originator for confirmation. It is generally not wise to reject a value on purely statistical grounds.

2.4.14. Uncertainties estimated using this guide are not intended to allow for the possibility of spurious errors/ blunders. Quantifying Uncertainty Analytical Measurement and Uncertainty QUAM: 2000. P1 Page 7 3. Analytical Measurement and Uncertainty 3.1. Method validation 3.1.1. In practice, the fitness for purpose of analytical methods applied for routine testing is most commonly assessed through method validation studies [ H. 7]. Such studies produce data on overall performance and on individual influence factors which can be applied to the estimation of uncertainty associated with the results of the method in normal use.

3.1.2. Method validation studies rely on the determination of overall method performance parameters. These are obtained during method development and interlaboratory study or following in house validation protocols. Individual sources of error or uncertainty are typically investigated only when significant compared to the overall precision measures in use. The emphasis is primarily on identifying and removing ( rather than correcting for) significant effects. This leads to a situation in which the majority of potentially significant influence factors have been identified, checked for significance compared to overall precision, and shown to be negligible. Under these circumstances, the data available to analysts consists primarily of overall performance figures, together with evidence of insignificance of most effects and some measurements of any remaining significant effects.

3.1.3. Validation studies for quantitative analytical methods typically determine some or all of the following parameters:

Precision. The principal precision measures include repeatability standard deviation sr, reproducibility standard deviation s R, ( ISO 3534 1) and intermediate precision, sometimes denoted s Zi, with i denoting the number of factors varied ( ISO 5725 3: 1994). The repeatability sr indicates the variability observed within a laboratory, over a short time, using a single operator, item of equipment etc. sr may be estimated within a laboratory or by inter laboratory study. Interlaboratory reproducibility standard deviation s R for a particular method may only be estimated directly by interlaboratory study; it shows the variability obtained when different laboratories analyse the same sample. Intermediate precision relates to the variation in results observed when one or more factors, such as time, equipment and operator, are varied within a laboratory; different figures are obtained depending on which factors are held constant. Intermediate precision estimates are most commonly determined within laboratories but may also be determined by interlaboratory study. The observed precision of an analytical procedure is an essential component of overall uncertainty, whether determined by combination of individual variances or by study of the complete method in operation.

Bias. The bias of an analytical method is usually determined by study of relevant reference materials or by spiking studies. The determination of overall bias with respect to appropriate reference values is important in establishing traceability [ B. 12] to recognised standards ( see section 3.2). Bias may be expressed as analytical recovery ( value observed divided by value expected). Bias should be shown to be negligible or corrected for, but in either case the uncertainty associated with the determination of the bias remains an essential component of overall uncertainty.

Linearity. Linearity is an important property of methods used to make measurements at a range of concentrations. The linearity of the response to pure standards and to realistic samples may be determined. Linearity is not generally quantified, but is checked for by inspection or using significance tests for non linearity. Significant non linearity is usually corrected for by use of non linear calibration functions or eliminated by choice of more restricted operating range. Any remaining deviations from linearity are normally sufficiently accounted for by overall precision estimates covering several concentrations, or within any uncertainties associated with calibration ( Appendix E. 3).

Detection limit. During method validation, the detection limit is normally determined only to establish the lower end of the practical operating range of a method. Though uncertainties near the detection limit may require careful consideration and special treatment ( Appendix F), the detection limit, however determined, is not of direct relevance to uncertainty estimation. Quantifying Uncertainty Analytical Measurement and Uncertainty QUAM: 2000. P1 Page 8 Robustness or ruggedness. Many method development or validation protocols require that sensitivity to particular parameters be investigated directly. This is usually done by a preliminary ` ruggedness test', in which the effect of one or more parameter changes is observed. If significant ( compared to the precision of the ruggedness test) a more detailed study is carried out to measure the size of the effect, and a permitted operating interval chosen accordingly. Ruggedness test data can therefore provide information on the effect of important parameters.

Selectivity/ specificity. Though loosely defined, both terms relate to the degree to which a method responds uniquely to the required analyte. Typical selectivity studies investigate the effects of likely interferents, usually by adding the potential interferent to both blank and fortified samples and observing the response. The results are normally used to demonstrate that the practical effects are not significant. However, since the studies measure changes in response directly, it is possible to use the data to estimate the uncertainty associated with potential interferences, given knowledge of the range of interferent concentrations.

3.2. Conduct of experimental studies of method performance 3.2.1. The detailed design and execution of method validation and method performance studies is covered extensively elsewhere [ H. 7] and will not be repeated here. However, the main principles as they affect the relevance of a study applied to uncertainty estimation are pertinent and are considered below.

3.2.2. Representativeness is essential. That is, studies should, as far as possible, be conducted to provide a realistic survey of the number and range of effects operating during normal use of the method, as well as covering the concentration ranges and sample types within the scope of the method. Where a factor has been representatively varied during the course of a precision experiment, for example, the effects of that factor appear directly in the observed variance and need no additional study unless further method optimisation is desirable.

3.2.3. In this context, representative variation means that an influence parameter must take a distribution of values appropriate to the uncertainty in the parameter in question. For continuous parameters, this may be a permitted range or stated uncertainty; for discontinuous factors such as sample matrix, this range corresponds to the variety of types permitted or encountered in normal use of the method. Note that representativeness extends not only to the range of values, but to their distribution.

3.2.4. In selecting factors for variation, it is important to ensure that the larger effects are varied where possible. For example, where day to day variation ( perhaps arising from recalibration effects) is substantial compared to repeatability, two determinations on each of five days will provide a better estimate of intermediate precision than five determinations on each of two days. Ten single determinations on separate days will be better still, subject to sufficient control, though this will provide no additional information on within day repeatability.

3.2.5. It is generally simpler to treat data obtained from random selection than from systematic variation. For example, experiments performed at random times over a sufficient period will usually include representative ambient temperature effects, while experiments performed systematically at 24 hour intervals may be subject to bias due to regular ambient temperature variation during the working day. The former experiment needs only evaluate the overall standard deviation; in the latter, systematic variation of ambient temperature is required, followed by adjustment to allow for the actual distribution of temperatures. Random variation is, however, less efficient. A small number of systematic studies can quickly establish the size of an effect, whereas it will typically take well over 30 determinations to establish an uncertainty contribution to better than about 20% relative accuracy. Where possible, therefore, it is often preferable to investigate small numbers of major effects systematically.

3.2.6. Where factors are known or suspected to interact, it is important to ensure that the effect of interaction is accounted for. This may be achieved either by ensuring random selection from different levels of interacting parameters, or by careful systematic design to obtain both variance and covariance information.

3.2.7. In carrying out studies of overall bias, it is important that the reference materials and values are relevant to the materials under routine test.

3.2.8. Any study undertaken to investigate and test for the significance of an effect should have sufficient power to detect such effects before they become practically significant. Quantifying Uncertainty Analytical Measurement and Uncertainty QUAM: 2000. P1 Page 9 3.3. Traceability 3.3.1. It is important to be able to compare results from different laboratories, or from the same laboratory at different times, with confidence. This is achieved by ensuring that all laboratories are using the same measurement scale, or the same ` reference points'. In many cases this is achieved by establishing a chain of calibrations leading to primary national or international standards, ideally ( for long term consistency) the Systeme Internationale ( SI) units of measurement. A familiar example is the case of analytical balances; each balance is calibrated using reference masses which are themselves checked ( ultimately) against national standards and so on to the primary reference kilogram. This unbroken chain of comparisons leading to a known reference value provides ` traceability' to a common reference point, ensuring that different operators are using the same units of measurement. In routine measurement, the consistency of measurements between one laboratory ( or time) and another is greatly aided by establishing traceability for all relevant intermediate measurements used to obtain or control a measurement result. Traceability is therefore an important concept in all branches of measurement.

3.3.2. Traceability is formally defined [ H. 4] as:

" The property of the result of a measurement or the value of a standard whereby it can be related to stated references, usually national or international standards, through an unbroken chain of comparisons all having stated uncertainties."

The reference to uncertainty arises because the agreement between laboratories is limited, in part, by uncertainties incurred in each laboratory's traceability chain. Traceability is accordingly intimately linked to uncertainty. Traceability provides the means of placing all related measurements on a consistent measurement scale, while uncertainty characterises the ` strength' of the links in the chain and the agreement to be expected between laboratories making similar measurements.

3.3.3. In general, the uncertainty on a result which is traceable to a particular reference, will be the uncertainty on that reference together with the uncertainty on making the measurement relative to that reference. 3.3.4. Traceability of the result of the complete analytical procedure should be established by a combination of the following procedures:

1. Use of traceable standards to calibrate the measuring equipment 2. By using, or by comparison to the results of, a primary method 3. By using a pure substance RM.

4. By using an appropriate matrix Certified Reference Material ( CRM)

5. By using an accepted, closely defined procedure.

Each procedure is discussed in turn below.

3.3.5. Calibration of measuring equipment In all cases, the calibration of the measuring equipment used must be traceable to appropriate standards. The quantification stage of the analytical procedure is often calibrated using a pure substance reference material, whose value is traceable to the SI. This practice provides traceability of the results to SI for this part of the procedure. However, it is also necessary to establish traceability for the results of operations prior to the quantification stage, such as extraction and sample clean up, using additional procedures.

3.3.6. Measurements using Primary Methods A primary method is currently described as follows:

" A primary method of measurement is a method having the highest metrological qualities, whose operation is completely described and understood in terms of SI units and whose results are accepted without reference to a standard of the same quantity."

The result of a primary method is normally traceable directly to the SI, and is of the smallest achievable uncertainty with respect to this reference. Primary methods are normally implemented only by National Measurement Institutes and are rarely applied to routine testing or calibration. Where applicable, traceability to the results of a primary method is achieved by direct comparison of measurement results between the primary method and test or calibration method.

3.3.7. Measurements using a pure substance Reference Material ( RM).

Traceability can be demonstrated by measurement of a sample composed of, or Quantifying Uncertainty Analytical Measurement and Uncertainty QUAM: 2000. P1 Page 10 containing, a known quantity of a pure substance RM. This may be achieved, for example, by spiking or by standard additions. However, it is always necessary to evaluate the difference in response of the measurement system to the standard used and the sample under test. Unfortunately, for many chemical analyses and in the particular case of spiking or standard additions, both the correction for the difference in response and its uncertainty may be large. Thus, although the traceability of the result to SI units can in principle be established, in practice, in all but the most simple cases, the uncertainty on the result may be unacceptably large or even unquantifiable. If the uncertainty is unquantifiable then traceability has not been established 3.3.8. Measurement on a Certified Reference Material ( CRM)

Traceability may be demonstrated through comparison of measurement results on a certified matrix CRM with the certified value( s). This procedure can reduce the uncertainty compared to the use of a pure substance RM where there is a suitable matrix CRM available. If the value of the CRM is traceable to SI, then these measurements provide traceability to SI units and the evaluation of the uncertainty utilising reference materials is discussed in 7.5. However, even in this case, the uncertainty on the result may be unacceptably large or even unquantifiable, particularly if there is not a good match between the composition of the sample and the reference material.

3.3.9. Measurement using an accepted procedure.

Adequate comparability can often only be achieved through use of a closely defined and generally accepted procedure. The procedure will normally be defined in terms of input parameters; for example a specified set of extraction times, particle sizes etc. The results of applying such a procedure are considered traceable when the values of these input parameters are traceable to stated references in the usual way. The uncertainty on the results arises both from uncertainties in the specified input parameters and from the effects of incomplete specification and variability in execution ( see section 7.8.1.). Where the results of an alternative method or procedure are expected to be comparable to the results of such an accepted procedure, traceability to the accepted values is achieved by comparing the results obtained by accepted and alternative procedures. Quantifying Uncertainty The Uncertainty Estimation Process QUAM: 2000. P1 Page 11 4. The Process of Measurement Uncertainty Estimation 4.1. Uncertainty estimation is simple in principle. The following paragraphs summarise the tasks that need to be performed in order to obtain an estimate of the uncertainty associated with a measurement result. Subsequent chapters provide additional guidance applicable in different circumstances, particularly relating to the use of data from method validation studies and the use of formal uncertainty propagation principles. The steps involved are:

Step 1. Specify measurand Write down a clear statement of what is being measured, including the relationship between the measurand and the input quantities ( e. g. measured quantities, constants, calibration standard values etc.) upon which it depends. Where possible, include corrections for known systematic effects. The specification information should be given in the relevant Standard Operating Procedure ( SOP) or other method description.

Step 2. Identify uncertainty sources List the possible sources of uncertainty. This will include sources that contribute to the uncertainty on the parameters in the relationship specified in Step 1, but may include other sources and must include sources arising from chemical assumptions. A general procedure for forming a structured list is suggested at Appendix D. Step 3. Quantify uncertainty components Measure or estimate the size of the uncertainty component associated with each potential source of uncertainty identified. It is often possible to estimate or determine a single contribution to uncertainty associated with a number of separate sources. It is also important to consider whether available data accounts sufficiently for all sources of uncertainty, and plan additional experiments and studies carefully to ensure that all sources of uncertainty are adequately accounted for.

Step 4. Calculate combined uncertainty The information obtained in step 3 will consist of a number of quantified contributions to overall uncertainty, whether associated with individual sources or with the combined effects of several sources. The contributions have to be expressed as standard deviations, and combined according to the appropriate rules, to give a combined standard uncertainty. The appropriate coverage factor should be applied to give an expanded uncertainty.

Figure 1 shows the process schematically.

4.2. The following chapters provide guidance on the execution of all the steps listed above and shows how the procedure may be simplified depending on the information that is available about the combined effect of a number of sources. Quantifying Uncertainty The Uncertainty Estimation Process QUAM: 2000. P1 Page 12 Figure 1: The Uncertainty Estimation Process Specify Measurand Identify Uncertainty Sources Simplify by grouping sources covered by existing data Quantify remaining components Quantify grouped components Convert components to standard deviations Calculate combined standard uncertainty END Calculate Expanded uncertainty Review and if necessary re evaluate large components START Step 1 Step 2 Step 3 Step 4 Quantifying Uncertainty Step 1. Specification of the Measurand QUAM: 2000. P1 Page 13 5. Step 1. Specification of the Measurand 5.1. In the context of uncertainty estimation, " specification of the measurand" requires both a clear and unambiguous statement of what is being measured, and a quantitative expression relating the value of the measurand to the parameters on which it depends. These parameters may be other measurands, quantities which are not directly measured, or constants. It should also be clear whether a sampling step is included within the procedure or not. If it is, estimation of uncertainties associated with the sampling procedure need to be considered. All of this information should be in the Standard Operating Procedure ( SOP).

5.2. In analytical measurement, it is particularly important to distinguish between measurements intended to produce results which are independent of the method used, and those which are not so intended. The latter are often referred to as empirical methods. The following examples may clarify the point further.

EXAMPLES:

1. Methods for the determination of the amount of nickel present in an alloy are normally expected to yield the same result, in the same units, usually expressed as a mass or mole fraction. In principle, any systematic effect due to method bias or matrix would need to be corrected for, though it is more usual to ensure that any such effect is small. Results would not normally need to quote the particular method used, except for information. The method is not empirical.

2. Determinations of " extractable fat" may differ substantially, depending on the extraction conditions specified. Since " extractable fat" is entirely dependent on choice of conditions, the method used is empirical. It is not meaningful to consider correction for bias intrinsic to the method, since the measurand is defined by the method used. Results are generally reported with reference to the method, uncorrected for any bias intrinsic to the method. The method is considered empirical.

3. In circumstances where variations in the substrate, or matrix, have large and unpredictable effects, a procedure is often developed with the sole aim of achieving comparability between laboratories measuring the same material. The procedure may then be adopted as a local, national or international standard method on which trading or other decisions are taken, with no intent to obtain an absolute measure of the true amount of analyte present. Corrections for method bias or matrix effect are ignored by convention ( whether or not they have been minimised in method development). Results are normally reported uncorrected for matrix or method bias. The method is considered to be empirical.

5.3. The distinction between empirical and nonempirical ( sometimes called rational) methods is important because it affects the estimation of uncertainty. In examples 2 and 3 above, because of the conventions employed, uncertainties associated with some quite large effects are not relevant in normal use. Due consideration should accordingly be given to whether the results are expected to be dependent upon, or independent of, the method in use and only those effects relevant to the result as reported should be included in the uncertainty estimate. Quantifying Uncertainty QUAM: 2000. P1 Page 14 6. Step 2. Identifying Uncertainty Sources 6.1. A comprehensive list of relevant sources of uncertainty should be assembled. At this stage, it is not necessary to be concerned about the quantification of individual components; the aim is to be completely clear about what should be considered. In Step 3, the best way of treating each source will be considered.

6.2. In forming the required list of uncertainty sources it is usually convenient to start with the basic expression used to calculate the measurand from intermediate values. All the parameters in this expression may have an uncertainty associated with their value and are therefore potential uncertainty sources. In addition there may be other parameters that do not appear explicitly in the expression used to calculate the value of the measurand, but which nevertheless affect the measurement results, 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 ( Uncertainties in Analytical Processes).

6.3. The cause and effect diagram described in Appendix D is a very convenient way of listing the uncertainty sources, showing how they relate to each other and indicating their influence on the uncertainty of the result. It also helps to avoid double counting of sources. Although the list of uncertainty sources can be prepared in other ways, the cause and effect diagram is used in the following chapters and in all of the examples in Appendix A. Additional information is given in Appendix D ( Analysing uncertainty sources).

6.4. Once the list of uncertainty sources is assembled, their effects on the result can, in principle, be represented by a formal measurement model, in which each effect is associated with a parameter or variable in an equation. The equation then forms a complete model of the measurement process in terms of all the individual factors affecting the result. This function may be very complicated and it may not be possible to write it down explicitly. Where possible, however, this should be done, as the form of the expression will generally determine the method of combining individual uncertainty contributions. 6.5. It may additionally be useful to consider a measurement procedure as a series of discrete operations ( sometimes termed unit operations), each of which may be assessed separately to obtain estimates of uncertainty associated with them. This is a particularly useful approach where similar measurement procedures share common unit operations. The separate uncertainties for each operation then form contributions to the overall uncertainty.

6.6. In practice, it is more usual in analytical measurement to consider uncertainties associated with elements of overall method performance, such as observable precision and bias measured with respect to appropriate reference materials. These contributions generally form the dominant contributions to the uncertainty estimate, and are best modelled as separate effects on the result. It is then necessary to evaluate other possible contributions only to check their significance, quantifying only those that are significant. Further guidance on this approach, which applies particularly to the use of method validation data, is given in section 7.2.1.

6.7. Typical sources of uncertainty are · Sampling Where in house or field sampling form part of the specified procedure, effects such as random variations between different samples and any potential for bias in the sampling procedure form components of uncertainty affecting the final result.

· Storage Conditions Where test items are stored for any period prior to analysis, the storage conditions may affect the results. The duration of storage as well as conditions during storage should therefore be considered as uncertainty sources.

· Instrument effects Instrument effects may include, for example, the limits of accuracy on the calibration of an analytical balance; a temperature controller that may maintain a mean temperature which Quantifying Uncertainty Step 2. Identifying Uncertainty Sources QUAM: 2000. P1 Page 15 differs ( within specification) from its indicated set point; an auto analyser that could be subject to carry over effects.

· Reagent purity The concentration of a volumetric solution will not be known exactly even if the parent material has been assayed, since some uncertainty related to the assaying procedure remains. Many organic dyestuffs, for instance, are not 100% pure and can contain isomers and inorganic salts. The purity of such substances is usually stated by manufacturers as being not less than a specified level. Any assumptions about the degree of purity will introduce an element of uncertainty.

· Assumed stoichiometry Where an analytical process is assumed to follow a particular reaction stoichiometry, it may be necessary to allow for departures from the expected stoichiometry, or for incomplete reaction or side reactions.

· Measurement conditions For example, volumetric glassware may be used at an ambient temperature different from that at which it was calibrated. Gross temperature effects should be corrected for, but any uncertainty in the temperature of liquid and glass should be considered. Similarly, humidity may be important where materials are sensitive to possible changes in humidity.

· Sample effects The recovery of an analyte from a complex matrix, or an instrument response, may be affected by composition of the matrix. Analyte speciation may further compound this effect. The stability of a sample/ analyte may change during analysis because of a changing thermal regime or photolytic effect.

When a ` spike' is used to estimate recovery, the recovery of the analyte from the sample may differ from the recovery of the spike, introducing an uncertainty which needs to be evaluated.

· Computational effects Selection of the calibration model, e. g. using a straight line calibration on a curved response, leads to poorer fit and higher uncertainty.

Truncation and round off can lead to inaccuracies in the final result. Since these are rarely predictable, an uncertainty allowance may be necessary.

· Blank Correction There will be an uncertainty on both the value and the appropriateness of the blank correction. This is particularly important in trace analysis.

· Operator effects Possibility of reading a meter or scale consistently high or low.

Possibility of making a slightly different interpretation of the method.

· Random effects Random effects contribute to the uncertainty in all determinations. This entry should be included in the list as a matter of course.

NOTE: These sources are not necessarily independent. Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 16 7. Step 3. Quantifying Uncertainty 7.1. Introduction 7.1.1. Having identified the uncertainty sources as explained in Step 2 ( Chapter 6), the next step is to quantify the uncertainty arising from these sources. This can be done by · evaluating the uncertainty arising from each individual source and then combining them as described in Chapter 8. Examples A1 to A3 illustrate the use of this procedure.

or · by determining directly the combined contribution to the uncertainty on the result from some or all of these sources using method performance data. Examples A4 to A6 represent applications of this procedure.

In practice, a combination of these is usually necessary and convenient.

7.1.2. Whichever of these approaches is used, most of the information needed to evaluate the uncertainty is likely to be already available from the results of validation studies, from QA/ QC data and from other experimental work that has been carried out to check the performance of the method. However, data may not be available to evaluate the uncertainty from all of the sources and it may be necessary to carry out further work as described in sections 7.10. to 7.14.

7.2. Uncertainty evaluation procedure 7.2.1. The procedure used for estimating the overall uncertainty depends on the data available about the method performance. The stages involved in developing the procedure are · Reconcile the information requirements with the available data First, the list of uncertainty sources should be examined to see which sources of uncertainty are accounted for by the available data, whether by explicit study of the particular contribution or by implicit variation within the course of whole method experiments. These sources should be checked against the list prepared in Step 2 and any remaining sources should be listed to provide an auditable record of which contributions to the uncertainty have been included. · Plan to obtain the further data required For sources of uncertainty not adequately covered by existing data, either seek additional information from the literature or standing data ( certificates, equipment specifications etc.), or plan experiments to obtain the required additional data. Additional experiments may take the form of specific studies of a single contribution to uncertainty, or the usual method performance studies conducted to ensure representative variation of important factors.

7.2.2. It is important to recognise that not all of the components will make a significant contribution to the combined uncertainty; indeed, in practice it is likely that only a small number will. Unless there is a large number of them, components that are less than one third of the largest need not be evaluated in detail. A preliminary estimate of the contribution of each component or combination of components to the uncertainty should be made and those that are not significant eliminated.

7.2.3. The following sections provide guidance on the procedures to be adopted, depending on the data available and on the additional information required. Section 7.3. presents requirements for the use of prior experimental study data, including validation data. Section 7.4. briefly discusses evaluation of uncertainty solely from individual sources of uncertainty. This may be necessary for all, or for very few of the sources identified, depending on the data available, and is consequently also considered in later sections. Sections 7.5. to 7.9. describe the evaluation of uncertainty in a range of circumstances. Section 7.5. applies when using closely matched reference materials. Section 7.6. covers the use of collaborative study data and 7.7. the use of inhouse validation data. 7.8. describes special considerations for empirical methods and 7.9. covers ad hoc methods. Methods for quantifying individual components of uncertainty, including experimental studies, documentary and other data, modelling, and professional judgement are covered in more detail in sections 7.10. to 7.14. Section 7.15. covers the treatment of known bias in uncertainty estimation. Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 17 7.3. Relevance of prior studies 7.3.1. When uncertainty estimates are based at least partly on prior studies of method performance, it is necessary to demonstrate the validity of applying prior study results. Typically, this will consist of:

· Demonstration that a comparable precision to that obtained previously can be achieved.

· Demonstration that the use of the bias data obtained previously is justified, typically through determination of bias on relevant reference materials ( see, for example, ISO Guide 33 [ H. 8]), by appropriate spiking studies, or by satisfactory performance on relevant proficiency schemes or other laboratory intercomparisons.

· Continued performance within statistical control as shown by regular QC sample results and the implementation of effective analytical quality assurance procedures.

7.3.2. Where the conditions above are met, and the method is operated within its scope and field of application, it is normally acceptable to apply the data from prior studies ( including validation studies) directly to uncertainty estimates in the laboratory in question.

7.4. Evaluating uncertainty by quantification of individual components 7.4.1. In some cases, particularly when little or no method performance data is available, the most suitable procedure may be to evaluate each uncertainty component separately.

7.4.2. The general procedure used in combining individual components is to prepare a detailed quantitative model of the experimental procedure ( cf. sections 5. and 6., especially 6.4.), assess the standard uncertainties associated with the individual input parameters, and combine them using the law of propagation of uncertainties as described in Section 8.

7.4.3. In the interests of clarity, detailed guidance on the assessment of individual contributions by experimental and other means is deferred to sections 7.10. to 7.14. Examples A1 to A3 in Appendix A provide detailed illustrations of the procedure. Extensive guidance on the application of this procedure is also given in the ISO Guide [ H. 2]. 7.5. Closely matched certified reference materials 7.5.1. Measurements on certified reference materials are normally carried out as part of method validation or re validation, effectively constituting a calibration of the whole measurement procedure against a traceable reference. Because this procedure provides information on the combined effect of many of the potential sources of uncertainty, it provides very good data for the assessment of uncertainty. Further details are given in section 7.7.4.

NOTE: ISO Guide 33 [ H. 8] gives a useful account of the use of reference materials in checking method performance.

7.6. Uncertainty estimation using prior collaborative method development and validation study data 7.6.1. A collaborative study carried out to validate a published method, for example according to the AOAC/ IUPAC protocol [ H. 9] or ISO 5725 standard [ H. 10], is a valuable source of data to support an uncertainty estimate. The data typically include estimates of reproducibility standard deviation, sR, for several levels of response, a linear estimate of the dependence of sR on level of response, and may include an estimate of bias based on CRM studies. How this data can be utilised depends on the factors taken into account when the study was carried out. During the ` reconciliation' stage indicated above ( section 7.2.), it is necessary to identify any sources of uncertainty that are not covered by the collaborative study data. The sources which may need particular consideration are:

· Sampling. Collaborative studies rarely include a sampling step. If the method used in house involves sub sampling, or the measurand ( see Specification) is estimating a bulk property from a small sample, then the effects of sampling should be investigated and their effects included.

· Pre treatment. In most studies, samples are homogenised, and may additionally be stabilised, before distribution. It may be necessary to investigate and add the effects of the particular pre treatment procedures applied in house.

· Method bias. Method bias is often examined prior to or during interlaboratory study, where possible by comparison with reference Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 18 methods or materials. Where the bias itself, the uncertainty in the reference values used, and the precision associated with the bias check, are all small compared to s R, no additional allowance need be made for bias uncertainty. Otherwise, it will be necessary to make additional allowances.

· Variation in conditions. Laboratories participating in a study may tend towards the means of allowed ranges of experimental conditions, resulting in an underestimate of the range of results possible within the method definition. Where such effects have been investigated and shown to be insignificant across their full permitted range, however, no further allowance is required.

· Changes in sample matrix. The uncertainty arising from matrix compositions or levels of interferents outside the range covered by the study will need to be considered.

7.6.2. Each significant source of uncertainty not covered by the collaborative study data should be evaluated in the form of a standard uncertainty and combined with the reproducibility standard deviation s R in the usual way ( section 8.)

7.6.3. For methods operating within their defined scope, when the reconciliation stage shows that all the identified sources have been included in the validation study or when the contributions from any remaining sources such as those discussed in section 7.6.1. have been shown to be negligible, then the reproducibility standard deviation s R, adjusted for concentration if necessary, may be used as the combined standard uncertainty.

7.6.4. The use of this procedure is shown in example A6 ( Appendix A)

7.7. Uncertainty estimation using inhouse development and validation studies 7.7.1. In house development and validation studies consist chiefly of the determination of the method performance parameters indicated in section 3.1.3. Uncertainty estimation from these parameters utilises:

· The best available estimate of overall precision.

· The best available estimate( s) of overall bias and its uncertainty. · Quantification of any uncertainties associated with effects incompletely accounted for in the above overall performance studies.

Precision study 7.7.2. The precision should be estimated as far as possible over an extended time period, and chosen to allow natural variation of all factors affecting the result. This can be obtained from · The standard deviation of results for a typical sample analysed several times over a period of time, using different analysts and equipment where possible ( the results of measurements on QC check samples can provide this information).

· The standard deviation obtained from replicate analyses performed on each of several samples.

NOTE: Replicates should be performed at materially different times to obtain estimates of intermediate precision; within batch replication provides estimates of repeatability only.

· From formal multi factor experimental designs, analysed by ANOVA to provide separate variance estimates for each factor.

7.7.3. Note that precision frequently varies significantly with the level of response. For example, the observed standard deviation often increases significantly and systematically with analyte concentration. In such cases, the uncertainty estimate should be adjusted to allow for the precision applicable to the particular result. Appendix E. 4 gives additional guidance on handling level dependent contributions to uncertainty.

Bias study 7.7.4. Overall bias is best estimated by repeated analysis of a relevant CRM, using the complete measurement procedure. Where this is done, and the bias found to be insignificant, the uncertainty associated with the bias is simply the combination of the standard uncertainty on the CRM value with the standard deviation associated with the bias.

NOTE: Bias estimated in this way combines bias in laboratory performance with any bias intrinsic to the method in use. Special considerations may apply where the method in use is empirical; see section 7.8.1.

· When the reference material is only approximately representative of the test Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 19 materials, additional factors should be considered, including ( as appropriate) differences in composition and homogeneity; reference materials are frequently more homogeneous that test samples. Estimates based on professional judgement should be used, if necessary, to assign these uncertainties ( see section 7.14.).

· Any effects following from different concentrations of analyte; for example, it is not uncommon to find that extraction losses differ between high and low levels of analyte.

7.7.5. Bias for a method under study can also be determined by comparison of the results with those of a reference method. If the results show that the bias is not statistically significant, the standard uncertainty is that for the reference method ( if applicable; see section 7.8.1.), combined with the standard uncertainty associated with the measured difference between methods. The latter contribution to uncertainty is given by the standard deviation term used in the significance test applied to decide whether the difference is statistically significant, as explained in the example below.

EXAMPLE A method ( method 1) for determining the concentration of Selenium is compared with a reference method ( method 2). The results ( in mg 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 a pooled standard deviation sc 205 . 2 2 5 5 ) 1 5 ( 75 . 2 ) 1 5 ( 47 . 1 2 2 = - + - ´ + - ´ = c s and a corresponding value of t:

46 . 0 4 . 1 64 . 0 5 1 5 1 205 . 2 ) 76 . 4 40 . 5 ( = = ÷ ø ö ç è æ + - = t tcrit is 2.3 for 8 degrees of freedom, so there is no significant difference between the means of the results given by the two methods. However, the difference ( 0.64) is compared with a standard deviation term of 1.4 above. This value of 1.4 is the standard deviation associated with the difference, and accordingly represents the relevant contribution to uncertainty associated with the measured bias. 7.7.6. Overall bias can also be estimated by the addition of analyte to a previously studied material. The same considerations apply as for the study of reference materials ( above). In addition, the differential behaviour of added material and material native to the sample should be considered and due allowance made. Such an allowance can be made on the basis of:

· Studies of the distribution of the bias observed for a range of matrices and levels of added analyte.

· Comparison of result observed in a reference material with the recovery of added analyte in the same reference material.

· Judgement on the basis of specific materials with known extreme behaviour. For example, oyster tissue, a common marine tissue reference, is well known for a tendency to coprecipitate some elements with calcium salts on digestion, and may provide an estimate of ` worst case' recovery on which an uncertainty estimate can be based ( e. g. By treating the worst case as an extreme of a rectangular or triangular distribution).

· Judgement on the basis of prior experience.

7.7.7. Bias may also be estimated by comparison of the particular method with a value determined by the method of standard additions, in which known quantities of the analyte are added to the test material, and the correct analyte concentration inferred by extrapolation. The uncertainty associated with the bias is then normally dominated by the uncertainties associated with the extrapolation, combined ( where appropriate) with any significant contributions from the preparation and addition of stock solution.

NOTE: To be directly relevant, the additions should be made to the original sample, rather than a prepared extract.

7.7.8. It is a general requirement of the ISO Guide that corrections should be applied for all recognised and significant systematic effects. Where a correction is applied to allow for a significant overall bias, the uncertainty associated with the bias is estimated as paragraph 7.7.5. described in the case of insignificant bias 7.7.9. Where the bias is significant, but is nonetheless neglected for practical purposes, additional action is necessary ( see section 7.15.). Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 20 Additional factors 7.7.10. The effects of any remaining factors should be estimated separately, either by experimental variation or by prediction from established theory. The uncertainty associated with such factors should be estimated, recorded and combined with other contributions in the normal way.

7.7.11. Where the effect of these remaining factors is demonstrated to be negligible compared to the precision of the study ( i. e. statistically insignificant), it is recommended that an uncertainty contribution equal to the standard deviation associated with the relevant significance test be associated with that factor.

EXAMPLE The effect of a permitted 1 hour extraction time variation is investigated by a t test on five determinations each on the same sample, for the normal extraction time and a time reduced by 1 hour. The means and standard deviations ( in mg l 1) were: Standard time: mean 1.8, standard deviation 0.21; alternate time: mean 1.7, standard deviation 0.17. A t test uses the pooled variance of 037 . 0 ) 1 5 ( ) 1 5 ( 17 . 0 ) 1 5 ( 21 . 0 ) 1 5 ( 2 2 = - + - ´ - + ´ - to obtain 82 . 0 5 1 5 1 037 . 0 ) 7 . 1 8 . 1 ( = ÷ ø ö ç è æ + ´ - = t This is not significant compared to tcrit = 2.3. But note that the difference ( 0.1) is compared with a calculated standard deviation term of ) 5 / 1 5 / 1 ( 037 . 0 + ´ = 0.12. This value is the contribution to uncertainty associated with the effect of permitted variation in extraction time.

7.7.12. Where an effect is detected and is statistically significant, but remains sufficiently small to neglect in practice, the provisions of section 7.15. apply.

7.8. Evaluation of uncertainty for empirical methods 7.8.1. An ` empirical method' is a method agreed upon for the purposes of comparative measurement within a particular field of application where the measurand characteristically depends upon the method in use. The method accordingly defines the measurand. Examples include methods for leachable metals in ceramics and dietary fibre in foodstuffs ( see also section 5.2. and example A5)

7.8.2. Where such a method is in use within its defined field of application, the bias associated with the method is defined as zero. In such circumstances, bias estimation need relate only to the laboratory performance and should not additionally account for bias intrinsic to the method. This has the following implications.

7.8.3. Reference material investigations, whether to demonstrate negligible bias or to measure bias, should be conducted using reference materials certified using the particular method, or for which a value obtained with the particular method is available for comparison.

7.8.4. Where reference materials so characterised are unavailable, overall control of bias is associated with the control of method parameters affecting the result; typically such factors as times, temperatures, masses, volumes etc. The uncertainty associated with these input factors must accordingly be assessed and either shown to be negligible or quantified ( see example A6).

7.8.5. Empirical methods are normally subjected to collaborative studies and hence the uncertainty can be evaluated as described in section 7.6.

7.9. Evaluation of uncertainty for adhoc methods 7.9.1. Ad hoc methods are methods established to carry out exploratory studies in the short term, or for a short run of test materials. Such methods are typically based on standard or well established methods within the laboratory, but are adapted substantially ( for example to study a different analyte) and will not generally justify formal validation studies for the particular material in question.

7.9.2. Since limited effort will be available to establish the relevant uncertainty contributions, it is necessary to rely largely on the known performance of related systems or blocks within these systems. Uncertainty estimation should accordingly be based on known performance on a related system or systems. This performance information should be supported by any study necessary to establish the relevance of the information. The following recommendations assume that such a related system is available and has been examined sufficiently to obtain a reliable uncertainty estimate, or that the method consists of blocks from other methods and that Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 21 the uncertainty in these blocks has been established previously.

7.9.3. As a minimum, it is essential that an estimate of overall bias and an indication of precision be available for the method in question. Bias will ideally be measured against a reference material, but will in practice more commonly be assessed from spike recovery. The considerations of section 7.7.4. then apply, except that spike recoveries should be compared with those observed on the related system to establish the relevance of the prior studies to the ad hoc method in question. The overall bias observed for the ad hoc method, on the materials under test, should be comparable to that observed for the related system, within the requirements of the study.

7.9.4. A minimum precision experiment consists of a duplicate analysis. It is, however, recommended that as many replicates as practical are performed. The precision should be compared with that for the related system; the standard deviation for the ad hoc method should be comparable.

NOTE: It recommended that the comparison be based on inspection. Statistical significance tests ( e. g. an F test) will generally be unreliable with small numbers of replicates and will tend to lead to the conclusion that there is ` no significant difference' simply because of the low power of the test.

7.9.5. Where the above conditions are met unequivocally, the uncertainty estimate for the related system may be applied directly to results obtained by the ad hoc method, making any adjustments appropriate for concentration dependence and other known factors.

7.10. Quantification of individual components 7.10.1. It is nearly always necessary to consider some sources of uncertainty individually. In some cases, this is only necessary for a small number of sources; in others, particularly when little or no method performance data is available, every source may need separate study ( see examples 1,2 and 3 in Appendix A for illustrations). There are several general methods for establishing individual uncertainty components:

§ Experimental variation of input variables § From standing data such as measurement and calibration certificates § By modelling from theoretical principles § Using judgement based on experience or informed by modelling of assumptions These different methods are discussed briefly below.

7.11. Experimental estimation of individual uncertainty contributions 7.11.1. It is often possible and practical to obtain estimates of uncertainty contributions from experimental studies specific to individual parameters.

7.11.2. The standard uncertainty arising from random effects is often measured from repeatability experiments and is quantified in terms of the standard deviation of the measured values. In practice, no more than about fifteen replicates need normally be considered, unless a high precision is required.

7.11.3. Other typical experiments include:

· Study of the effect of a variation of a single parameter on the result. This is particularly appropriate in the case of continuous, controllable parameters, independent of other effects, such as time or temperature. The rate of change of the result with the change in the parameter can be obtained from the experimental data. This is then combined directly with the uncertainty in the parameter to obtain the relevant uncertainty contribution.

NOTE: The change in parameter should be sufficient to change the result substantially compared to the precision available in the study ( e. g. by five times the standard deviation of replicate measurements)

· Robustness studies, systematically examining the significance of moderate changes in parameters. This is particularly appropriate for rapid identification of significant effects, and commonly used for method optimisation. The method can be applied in the case of discrete effects, such as change of matrix, or small equipment configuration changes, which have unpredictable effects on the result. Where a factor is found to be significant, it is normally necessary to investigate further. Where insignificant, the associated uncertainty is ( at least for initial estimation) that obtained from the robustness study. Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 22 · Systematic multifactor experimental designs intended to estimate factor effects and interactions. Such studies are particularly useful where a categorical variable is involved. A categorical variable is one in which the value of the variable is unrelated to the size of the effect; laboratory numbers in a study, analyst names, or sample types are examples of categorical variables. For example, the effect of changes in matrix type ( within a stated method scope) could be estimated from recovery studies carried out in a replicated multiple matrix study. An analysis of variance would then provide within and between matrix components of variance for observed analytical recovery. The betweenmatrix component of variance would provide a standard uncertainty associated with matrix variation.

7.12. Estimation based on other results or data 7.12.1. It is often possible to estimate some of the standard uncertainties using whatever relevant information is available about the uncertainty on the quantity concerned. The following paragraphs suggest some sources of information.

7.12.2. Proficiency testing ( PT) schemes. A laboratory's results from participation in PT schemes can be used as a check on the evaluated uncertainty, since the uncertainty should be compatible with the spread of results obtained by that laboratory over a number of proficiency test rounds. Further, in the special case where · the compositions of samples used in the scheme cover the full range analysed routinely · the assigned values in each round are traceable to appropriate reference values, and · the uncertainty on the assigned value is small compared to the observed spread of results then the dispersion of the differences between the reported values and the assigned values obtained in repeated rounds provides a basis for a good estimate of the uncertainty arising from those parts of the measurement procedure within the scope of the scheme. For example, for a scheme operating with similar materials and analyte levels, the standard deviation of differences would give the standard uncertainty. Of course, systematic deviation from traceable assigned values and any other sources of uncertainty ( such as those noted in section 7.6.1.) must also be taken into account.

7.12.3. Quality Assurance ( QA) data. As noted previously it is necessary to ensure that the quality criteria set out in standard operating procedures are achieved, and that measurements on QA samples show that the criteria continue to be met. Where reference materials are used in QA checks, section 7.5. shows how the data can be used to evaluate uncertainty. Where any other stable material is used, the QA data provides an estimate of intermediate precision ( Section 7.7.2.). QA data also forms a continuing check on the value quoted for the uncertainty. Clearly, the combined uncertainty arising from random effects cannot be less than the standard deviation of the QA measurements.

7.12.4. Suppliers' information. For many sources of uncertainty, calibration certificates or suppliers catalogues provide information. For example, the tolerance of volumetric glassware may be obtained from the manufacturer's catalogue or a calibration certificate relating to a particular item in advance of its use.

7.13. Modelling from theoretical principles 7.13.1. In many cases, well established physical theory provides good models for effects on the result. For example, temperature effects on volumes and densities are well understood. In such cases, uncertainties can be calculated or estimated from the form of the relationship using the uncertainty propagation methods described in section 8.

7.13.2. In other circumstances, it may be necessary to use approximate theoretical models combined with experimental data. For example, where an analytical measurement depends on a timed derivatisation reaction, it may be necessary to assess uncertainties associated with timing. This might be done by simple variation of elapsed time. However, it may be better to establish an approximate rate model from brief experimental studies of the derivatisation kinetics near the concentrations of interest, and assess the uncertainty from the predicted rate of change at a given time. Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 23 7.14. Estimation based on judgement 7.14.1. The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of the measurement method and procedure used. The quality and utility of the uncertainty quoted for the result of a measurement therefore ultimately depends on the understanding, critical analysis, and integrity of those who contribute to the assignment of its value.

7.14.2. Most distributions of data can be interpreted in the sense that it is less likely to observe data in the margins of the distribution than in the centre. The quantification of these distributions and their associated standard deviations is done through repeated measurements.

7.14.3. However, other assessments of intervals may be required in cases when repeated measurements cannot be performed or do not provide a meaningful measure of a particular uncertainty component.

7.14.4. There are numerous instances in analytical chemistry when the latter prevails, and judgement is required. For example:

· An assessment of recovery and its associated uncertainty cannot be made for every single sample. Instead, an assessment is made for classes of samples ( e. g. grouped by type of matrix), and the results applied to all samples of similar type. The degree of similarity is itself an unknown, thus this inference ( from type of matrix to a specific sample) is associated with an extra element of uncertainty that has no frequentistic interpretation.

· The model of the measurement as defined by the specification of the analytical procedure is used for converting the measured quantity to the value of the measurand ( analytical result). This model is  like all models in science  subject to uncertainty. It is only assumed that nature behaves according to the specific model, but this can never be known with ultimate certainty.

· The use of reference materials is highly encouraged, but there remains uncertainty regarding not only the true value, but also regarding the relevance of a particular reference material for the analysis of a specific sample. A judgement is required of the extent to which a proclaimed standard substance reasonably resembles the nature of the samples in a particular situation.

· Another source of uncertainty arises when the measurand is insufficiently defined by the procedure. Consider the determination of " permanganate oxidizable substances" that are undoubtedly different whether one analyses ground water or municipal waste water. Not only factors such as oxidation temperature, but also chemical effects such as matrix composition or interference, may have an influence on this specification.

· A common practice in analytical chemistry calls for spiking with a single substance, such as a close structural analogue or isotopomer, from which either the recovery of the respective native substance or even that of a whole class of compounds is judged. Clearly, the associated uncertainty is experimentally assessable provided the analyst is prepared to study the recovery at all concentration levels and ratios of measurands to the spike, and all " relevant" matrices. But frequently this experimentation is avoided and substituted by judgements on · the concentration dependence of recoveries of measurand,

· the concentration dependence of recoveries of spike,

· the dependence of recoveries on ( sub) type of matrix,

· the identity of binding modes of native and spiked substances.

7.14.5. Judgement of this type is not based on immediate experimental results, but rather on a subjective ( personal) probability, an expression which here can be used synonymously with " degree of belief", " intuitive probability" and " credibility" [ H. 11]. It is also assumed that a degree of belief is not based on a snap judgement, but on a well considered mature judgement of probability.

7.14.6. Although it is recognised that subjective probabilities vary from one person to another, and even from time to time for a single person, they are not arbitrary as they are influenced by common sense, expert knowledge, and by earlier experiments and observations.

7.14.7. This may appear to be a disadvantage, but need not lead in practice to worse estimates than Quantifying Uncertainty Step 3. Quantifying Uncertainty QUAM: 2000. P1 Page 24 those from repeated measurements. This applies particularly if the true, real life, variability in experimental conditions cannot be simulated and the resulting variability in data thus does not give a realistic picture.

7.14.8. A typical problem of this nature arises if long term variability needs to be assessed when no collaborative study data are available. A scientist who dismisses the option of substituting subjective probability for an actually measured one ( when the latter is not available) is likely to ignore important contributions to combined uncertainty, thus being ultimately less objective than one who relies on subjective probabilities.

7.14.9. For the purpose of estimation of combined uncertainties two features of degree of belief estimations are essential:

· degree of belief is regarded as interval valued which is to say that a lower and an upper bound similar to a classical probability distribution is provided,

· the same computational rules apply in combining ' degree of belief' contributions of uncertainty to a combined uncertainty as for standard deviations derived by other methods. 7.15. Significance of bias 7.15.1. It is a general requirement of the ISO Guide that corrections should be applied for all recognised and significant systematic effects.

7.15.2. In deciding whether a known bias can reasonably be neglected, the following approach is recommended:

i) Estimate the combined uncertainty without considering the relevant bias.

ii) Compare the bias with the combined uncertainty.

iii) Where the bias is not significant compared to the combined uncertainty, the bias may be neglected.

iv) Where the bias is significant compared to the combined uncertainty, additional action is required. Appropriate actions might:

· Eliminate or correct for the bias, making due allowance for the uncertainty of the correction.

· Report the observed bias and its uncertainty in addition to the result.

NOTE: Where a known bias is uncorrected by convention, the method should be considered empirical ( see section 7.8). Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty QUAM: 2000. P1 Page 25 8. Step 4. Calculating the Combined Uncertainty 8.1. Standard uncertainties 8.1.1. Before combination, all uncertainty contributions must be expressed as standard uncertainties, that is, as standard deviations. This may involve conversion from some other measure of dispersion. The following rules give some guidance for converting an uncertainty component to a standard deviation.

8.1.2. Where the uncertainty component was evaluated experimentally from the dispersion of repeated measurements, it can readily be expressed as a standard deviation. For the contribution to uncertainty in single measurements, the standard uncertainty is simply the observed standard deviation; for results subjected to averaging, the standard deviation of the mean [ B. 24] is used.

8.1.3. Where an uncertainty estimate is derived from previous results and data, it may already be expressed as a standard deviation. However where a confidence interval is given with a level of confidence, ( in the form ± a at p%) then divide the value a by the appropriate percentage point of the Normal distribution for the level of confidence given to calculate the standard deviation.

EXAMPLE A specification states that a balance reading is within ± 0.2 mg with 95% confidence. From standard tables of percentage points on the normal distribution, a 95% confidence interval is calculated using a value of 1.96 s . Using this figure gives a standard uncertainty of ( 0.2/ 1.96) » 0.1.

8.1.4. If limits of ± a are given without a confidence level and there is reason to expect that extreme values are likely, it is normally appropriate to assume a rectangular distribution, with a standard deviation of a/ Ö 3 ( see Appendix E).

EXAMPLE A 10 ml Grade A volumetric flask is certified to within ± 0.2 ml. The standard uncertainty is 0.2/ Ö 3 » 0.12 ml.

8.1.5. If limits of ± a are given without a confidence level, but there is reason to expect that extreme values are unlikely, it is normally appropriate to assume a triangular distribution, with a standard deviation of a/ Ö 6 ( see Appendix E).

EXAMPLE A 10 ml Grade A volumetric flask is certified to within ± 0.2 ml, but routine in house checks show that extreme values are rare. The standard uncertainty is 0.2/ Ö 6 » 0.08 ml.

8.1.6. Where an estimate is to be made on the basis of judgement, it may be possible to estimate the component directly as a standard deviation. If this is not possible then an estimate should be made of the maximum deviation which could reasonably occur in practice ( excluding simple mistakes). If a smaller value is considered substantially more likely, this estimate should be treated as descriptive of a triangular distribution. If there are no grounds for believing that a small error is more likely than a large error, the estimate should be treated as characterising a rectangular distribution.

8.1.7. Conversion factors for the most commonly used distribution functions are given in Appendix E. 1.

8.2. Combined standard uncertainty 8.2.1. Following the estimation of individual or groups of components of uncertainty and expressing them as standard uncertainties, the next stage is to calculate the combined standard uncertainty using one of the procedures described below.

8.2.2. The general relationship between the combined standard uncertainty uc( y) of a value y and the uncertainty of the independent parameters x1, x2, ... xn on which it depends is uc( y( x1, x2,...)) =

å = n i i i x u c , 1 2 2 ) ( =

å = n i i x y u , 1 2 ) , ( *

where y( x1, x2,..) is a function of several parameters x1, x2..., ci is a sensitivity coefficient evaluated as ci= ¶ y/ ¶ xi, the partial differential of y with respect to xi and u( y, xi) denotes the uncertainty in y arising from the uncertainty in xi. Each variable's contribution u( y, xi) is just the * The ISO Guide uses the shorter form ui( y) instead of u( y, xi) Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty QUAM: 2000. P1 Page 26 square of the associated uncertainty expressed as a standard deviation multiplied by the square of the relevant sensitivity coefficient. These sensitivity coefficients describe how the value of y varies with changes in the parameters x1, x2 etc.

NOTE: Sensitivity coefficients may also be evaluated directly by experiment; this is particularly valuable where no reliable mathematical description of the relationship exists.

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

å å ¹ = = × + = k i n k i k i k i n i i i j i x x u c c x u c x y u , 1 , , 1 2 2 ... , ) , ( ) ( )) ( (

where u( xi, xk) is the covariance between xi and xk and ci and ck are the sensitivity coefficients as described and evaluated in 8.2.2. The covariance is related to the correlation coefficient rik by u( xi, xk) = u( xi) × u( xk) × rik where  1 £ rik £ 1.

8.2.4. These general procedures apply whether the uncertainties are related to single parameters, grouped parameters or to the method as a whole. However, when an uncertainty contribution is associated with the whole procedure, it is usually expressed as an effect on the final result. In such cases, or when the uncertainty on a parameter is expressed directly in terms of its effect on y, the sensitivity coefficient ¶ y/ ¶ xi is equal to 1.0.

EXAMPLE A result of 22 mg l 1 shows a measured standard deviation of 4.1 mg l 1. The standard uncertainty u( y) associated with precision under these conditions is 4.1 mg l 1. The implicit model for the measurement, neglecting other factors for clarity, is y = ( Calculated result) + e where e represents the effect of random variation under the conditions of measurement. ¶ y/ ¶ e is accordingly 1.0 8.2.5. Except for the case above, when the sensitivity coefficient is equal to one, and for the special cases given in Rule 1 and Rule 2 below, the general procedure, requiring the generation of partial differentials or the numerical equivalent must be employed. Appendix E gives details of a numerical method, suggested by Kragten [ H. 12], which makes effective use of spreadsheet software to provide a combined standard uncertainty from input standard uncertainties and a known measurement model. It is recommended that this method, or another appropriate computerbased method, be used for all but the simplest cases.

8.2.6. In some cases, the expressions for combining uncertainties reduce to much simpler forms. Two simple rules for combining standard uncertainties are given here.

Rule 1 For models involving only a sum or difference of quantities, e. g. y=( p+ q+ r+...), the combined standard uncertainty uc( y) is given by ..... ) ( ) ( ..)) , ( ( 2 2 + + = q u p u q p y u c Rule 2 For models involving only a product or quotient, e. g. y=( p ´ q ´ r ´ . ..) or y= p / ( q ´ r ´ . ..), the combined standard uncertainty uc( y) is given by ..... ) ( ) ( ) ( 2 2 + ÷ ÷ ø ö ç ç è æ + ÷ ÷ ø ö ç ç è æ = q q u p p u y y u c where ( u( p)/ p) etc. are the uncertainties in the parameters, expressed as relative standard deviations.

NOTE Subtraction is treated in the same manner as addition, and division in the same way as multiplication.

8.2.7. For the purposes of combining uncertainty components, it is most convenient to break the original mathematical model down to expressions which consist solely of operations covered by one of the rules above. For example, the expression ) ( ) ( r q p o + + should be broken down to the two elements ( o+ p) and ( q+ r). The interim uncertainties for each of these can then be calculated using rule 1 above; these interim uncertainties can then be combined using rule 2 to give the combined standard uncertainty.

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

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

y = 5.02  6.45 + 9.04 = 7.61 26 . 0 22 . 0 05 . 0 13 . 0 ) ( 2 2 2 = + + = y u Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty QUAM: 2000. P1 Page 27 EXAMPLE 2 y = ( op/ qr). The values are o= 2.46, p= 4.32, q= 6.38 and r= 2.99, with standard uncertainties of 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 2 2 2 2 99 . 2 07 . 0 38 . 6 11 . 0 32 . 4 13 . 0 46 . 2 02 . 0 56 . 0 ) (

÷ ø ö ç è æ + ÷ ø ö ç è æ + ÷ ø ö ç è æ + ÷ ø ö ç è æ ´ = y u Þ u( y) = 0.56 ´ 0.043 = 0.024 8.2.9. There are many instances in which the magnitudes of components of uncertainty vary with the level of analyte. For example, uncertainties in recovery may be smaller for high levels of material, or spectroscopic signals may vary randomly on a scale approximately proportional to intensity ( constant coefficient of variation). In such cases, it is important to take account of the changes in the combined standard uncertainty with level of analyte. Approaches include:

· Restricting the specified procedure or uncertainty estimate to a small range of analyte concentrations.

· Providing an uncertainty estimate in the form of a relative standard deviation.

· Explicitly calculating the dependence and recalculating the uncertainty for a given result.

Appendix E4 gives additional information on these approaches.

8.3. Expanded uncertainty 8.3.1. The final stage is to multiply the combined standard uncertainty by the chosen coverage factor in order to obtain an expanded uncertainty. The expanded uncertainty is required to provide an interval which may be expected to encompass a large fraction of the distribution of values which could reasonably be attributed to the measurand.

8.3.2. In choosing a value for the coverage factor k, a number of issues should be considered. These include:

· The level of confidence required · Any knowledge of the underlying distributions · Any knowledge of the number of values used to estimate random effects ( see 8.3.3 below).

8.3.3. For most purposes it is recommended that k is set to 2. However, this value of k may be insufficient where the combined uncertainty is based on statistical observations with relatively few degrees of freedom ( less than about six). The choice of k then depends on the effective number of degrees of freedom.

8.3.4. Where the combined standard uncertainty is dominated by a single contribution with fewer than six degrees of freedom, it is recommended that k be set equal to the two tailed value of Student's t for the number of degrees of freedom associated with that contribution, and for the level of confidence required ( normally 95%). Table 1 ( page 28) gives a short list of values for t.

EXAMPLE:

A combined standard uncertainty for a weighing operation is formed from contributions ucal= 0.01 mg arising from calibration uncertainty and sobs= 0.08 mg based on the standard deviation of five repeated observations. The combined standard uncertainty uc is equal to 2 2 08 . 0 01 . 0 + = 0.081 mg. This is clearly dominated by the repeatability contribution sobs, which is based on five observations, giving 5 1= 4 degrees of freedom. k is accordingly based on Student's t. The two tailed value of t for four degrees of freedom and 95% confidence is, from tables, 2.8; k is accordingly set to 2.8 and the expanded uncertainty U= 2.8 ´ 0.081= 0.23 mg.

8.3.5. The Guide [ H. 2] gives additional guidance on choosing k where a small number of measurements is used to estimate large random effects, and should be referred to when estimating degrees of freedom where several contributions are significant.

8.3.6. Where the distributions concerned are normal, a coverage factor of 2 ( or chosen according to paragraphs 8.3.3. 8.3.5. using a level of confidence of 95%) gives an interval containing approximately 95% of the distribution of values. It is not recommended that this interval is taken to imply a 95% confidence interval without a knowledge of the distribution concerned. Quantifying Uncertainty Step 4. Calculating the Combined Uncertainty QUAM: 2000. P1 Page 28 Table 1: Student's t for 95% confidence ( 2 tailed)

Degrees of freedom n t 1 12.7 2 4.3 3 3.2 4 2.8 5 2.6 6 2.5 Quantifying Uncertainty Reporting Uncertainty QUAM: 2000. P1 Page 29 9. Reporting Uncertainty 9.1. General 9.1.1. The information necessary to report the result of a measurement depends on its intended use. The guiding principles are:

· present sufficient information to allow the result to be re evaluated if new information or data become available · it is preferable to err on the side of providing too much information rather than too little.

9.1.2. When the details of a measurement, including how the uncertainty was determined, depend on references to published documentation, it is imperative that the documentation to hand is kept up to date and consistent with the methods in use.

9.2. Information required 9.2.1. A complete report of a measurement result should include or refer to documentation containing,

· a description of the methods used to calculate the measurement result and its uncertainty from the experimental observations and input data · the values and sources of all corrections and constants used in both the calculation and the uncertainty analysis · a list of all the components of uncertainty with full documentation on how each was evaluated 9.2.2. The data and analysis should be presented in such a way that its important steps can be readily followed and the calculation of the result repeated if necessary.

9.2.3. Where a detailed report including intermediate input values is required, the report should · give the value of each input value, its standard uncertainty and a description of how each was obtained · give the relationship between the result and the input values and any partial derivatives, covariances or correlation coefficients used to account for correlation effects · state the estimated number of degrees of freedom for the standard uncertainty of each input value ( methods for estimating degrees of freedom are given in the ISO Guide [ H. 2]).

NOTE: Where the functional relationship is extremely complex or does not exist explicitly ( for example, it may only exist as a computer program), the relationship may be described in general terms or by citation of appropriate references. In such cases, it must be clear how the result and its uncertainty were obtained.

9.2.4. When reporting the results of routine analysis, it may be sufficient to state only the value of the expanded uncertainty and the value of k.

9.3. Reporting standard uncertainty 9.3.1. When uncertainty is expressed as the combined standard uncertainty uc ( that is, as a single standard deviation), the following form is recommended:

"( Result): x ( units) [ with a] standard uncertainty of uc ( units) [ where standard uncertainty is as defined in the International Vocabulary of Basic and General terms in Metrology, 2nd ed., ISO 1993 and corresponds to one standard deviation.]"

NOTE The use of the symbol ± is not recommended when using standard uncertainty as the symbol is commonly associated with intervals corresponding to high levels of confidence.

Terms in parentheses [] may be omitted or abbreviated as appropriate.

EXAMPLE:

Total nitrogen: 3.52 % w/ w Standard uncertainty: 0.07 % w/ w *

* Standard uncertainty corresponds to one standard deviation.

9.4. Reporting expanded uncertainty 9.4.1. Unless otherwise required, the result x should be stated together with the expanded uncertainty U calculated using a coverage factor Quantifying Uncertainty Reporting Uncertainty QUAM: 2000. P1 Page 30 k= 2 ( or as described in section 8.3.3.). The following form is recommended:

"( Result): ( x ± U) ( units)

[ where] the reported uncertainty is [ an expanded uncertainty as defined in the International Vocabulary of Basic and General terms in metrology, 2nd ed., ISO 1993,] calculated using a coverage factor of 2, [ which gives a level of confidence of approximately 95%]"

Terms in parentheses [] may be omitted or abbreviated as appropriate. The coverage factor should, of course, be adjusted to show the value actually used.

EXAMPLE:

Total nitrogen: ( 3.52 ± 0.14) % w/ w *

* The reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2 which gives a level of confidence of approximately 95%.

9.5. Numerical expression of results 9.5.1. The numerical values of the result and its uncertainty should not be given with an excessive number of digits. Whether expanded uncertainty U or a standard uncertainty u is given, it is seldom necessary to give more than two significant digits for the uncertainty. Results should be rounded to be consistent with the uncertainty given.

9.6. Compliance against limits 9.6.1. Regulatory compliance often requires that a measurand, such as the concentration of a toxic substance, be shown to be within particular limits. Measurement uncertainty clearly has implications for interpretation of analytical results in this context. In particular:

· The uncertainty in the analytical result may need to be taken into account when assessing compliance. · The limits may have been set with some allowance for measurement uncertainties.

Consideration should be given to both factors in any assessment. The following paragraphs give examples of common practice.

9.6.2. Assuming that limits were set with no allowance for uncertainty, four situations are apparent for the case of compliance with an upper limit ( see Figure 2):

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

ii) The result exceeds the limiting value by less than the expanded uncertainty.

iii) The result is below the limiting value by less than the expanded uncertainty iv) The result is less than the limiting value minus the expanded uncertainty.

Case i) is normally interpreted as demonstrating clear non compliance. Case iv) is normally interpreted as demonstrating compliance. Cases ii) and iii) will normally require individual consideration in the light of any agreements with the user of the data. Analogous arguments apply in the case of compliance with a lower limit.

9.6.3. Where it is known or believed that limits have been set with some allowance for uncertainty, a judgement of compliance can reasonably be made only with knowledge of that allowance. An exception arises where compliance is set against a stated method operating in defined circumstances. Implicit in such a requirement is the assumption that the uncertainty, or at least reproducibility, of the stated method is small enough to ignore for practical purposes. In such a case, provided that appropriate quality control is in place, compliance is normally reported only on the value of the particular result. This will normally be stated in any standard taking this approach. Quantifying Uncertainty Reporting Uncertainty QUAM: 2000. P1 Page 31 Figure 2: Uncertainty and compliance limits Upper Control Limit ( 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 QUAM: 2000. P1 Page 32 Appendix A. Examples Introduction General introduction These examples illustrate how the techniques for evaluating uncertainty, described in sections 5 7, can be applied to some typical chemical analyses. They all follow the procedure shown in the flow diagram ( Figure 1 on page 12). The uncertainty sources are identified and set out in a cause and effect diagram ( see appendix D). This helps to avoid double counting of sources and also assists in the grouping together of components whose combined effect can be evaluated. Examples 1 6 illustrate the use of the spreadsheet method of Appendix E. 2 for calculating the combined uncertainties from the calculated contributions u( y, xi).*

Each of examples 1 6 has an introductory summary. This gives an outline of the analytical method, a table of the uncertainty sources and their respective contributions, a graphical comparison of the different contributions, and the combined uncertainty.

Examples 1 3 illustrate the evaluation of the uncertainty by the quantification of the uncertainty arising from each source separately. Each gives a detailed analysis of the uncertainty associated with the measurement of volumes using volumetric glassware and masses from difference weighings. The detail is for illustrative purposes, and should not be taken as a general recommendation as to the level of detail required or the approach taken. For many analyses, the uncertainty associated with these operations will not be significant and such a detailed evaluation will not be necessary. It would be sufficient to use typical values for these operations with due allowance being made for the actual values of the masses and volumes involved.

Example A1 Example A1 deals with the very simple case of the preparation of a calibration standard of cadmium in HNO3 for AAS. Its purpose is to * Section 8.2.2. explains the theory behind the calculated contributions u( y, xi). show how to evaluate the components of uncertainty arising from the basic operations of volume measurement and weighing and how these components are combined to determine the overall uncertainty.

Example A2 This deals with the preparation of a standardised solution of sodium hydroxide ( NaOH) which is standardised against the titrimetric standard potassium hydrogen phthalate ( KHP). It includes the evaluation of uncertainty on simple volume measurements and weighings, as described in example A1, but also examines the uncertainty associated with the titrimetric determination.

Example A3 Example A3 expands on example A2 by including the titration of an HCl against the prepared NaOH solution.

Example A4 This illustrates the use of in house validation data, as described in section 7.7., and shows how the data can be used to evaluate the uncertainty arising from combined effect of a number of sources. It also shows how to evaluate the uncertainty associated with method bias.

Example A5 This shows how to evaluate the uncertainty on results obtained using a standard or " empirical" method to measure the amount of heavy metals leached from ceramic ware using a defined procedure, as described in section 7.2. 7.8. Its purpose is to show how, in the absence of collaborative trial data or ruggedness testing results, it is necessary to consider the uncertainty arising from the range of the parameters ( e. g. temperature, etching time and acid strength) allowed in the method definition. This process is considerably simplified when collaborative study data is available, as is shown in the next example.

Example A6 The sixth example is based on an uncertainty estimate for a crude ( dietary) fibre determination. Quantifying Uncertainty Appendix A. Examples QUAM: 2000. P1 Page 33 Since the analyte is defined only in terms of the standard method, the method is empirical. In this case, collaborative study data, in house QA checks and literature study data were available, permitting the approach described in section 7.6. The in house studies verify that the method is performing as expected on the basis of the collaborative study. The example shows how the use of collaborative study data backed up by inhouse method performance checks can substantially reduce the number of different contributions required to form an uncertainty estimate under these circumstances. Example A7 This gives a detailed description of the evaluation of uncertainty on the measurement of the lead content of a water sample using IDMS. In addition to identifying the possible sources of uncertainty and quantifying them by statistical means the examples shows how it is also necessary to include the evaluation of components based on judgement as described in section 7.14. Use of judgement is a special case of Type B evaluation as described in the ISO Guide [ H. 2]. Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 34 Example A1: Preparation of a Calibration Standard Summary Goal A calibration standard is prepared from a high purity metal ( cadmium) with a concentration of ca. 1000 mg l 1.

Measurement procedure The surface of the high purity metal is cleaned to remove any metal oxide contamination. Afterwards the metal is weighed and then dissolved in nitric acid in a volumetric flask. The stages in the procedure are show in the following flow chart.

Clean metal surface Clean metal surface Weigh metal Weigh metal Dissolve and dilute Dissolve and dilute RESULT RESULT Figure A1. 1: Preparation of cadmium standard Measurand V P m c Cd × × = 1000 [ mg l 1] where cCd : concentration of the calibration standard [ mg l 1]

1000 : conversion factor from [ ml] to [ l]

m : mass 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]

Identification of the uncertainty sources:

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

Purity V m Repeatability Calibration Temperature c( Cd)

m( tare) m( gross)

Repeatability Repeatability Calibration Linearity Sensitivity Calibration Linearity Sensitivity Readability Readability Quantification of the uncertainty components The values and their uncertainties are shown in the Table below.

Combined Standard Uncertainty The combined standard uncertainty for the preparation of a 1002.7 mg l 1 Cd calibration standard is 0.9 mg l 1 The different contributions are shown diagrammatically in Figure A1.2.

Table A1.1: Values and uncertainties Description Value Standard uncertainty Relative standard uncertainty u( x)/ x P Purity of the metal 0.9999 0.000058 0.000058 m Mass 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 the calibration standard 1002.7 mg l 1 0.9 mg l 1 0.0009 Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 35 Figure A1.2: Uncertainty contributions in cadmium standard preparation u( y, xi) ( mg l 1) 0 0.2 0.4 0.6 0.8 1 c( Cd) Purity m V The values of u( y, xi)=( ¶ y/ ¶ xi). u( xi) are taken from Table A1.3 Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 36 Example A1: Preparation of a calibration standard. Detailed discussion A1.1 Introduction This first introductory example discusses the preparation of a calibration standard for atomic absorption spectroscopy ( AAS) from the corresponding high purity metal ( in this example » 1000 mg l 1 Cd in dilute HNO3). Even though the example does not represent an entire analytical measurement, the use of calibration standards is part of nearly every determination, because modern routine analytical measurements are relative measurements, which need a reference standard to provide traceability to the SI.

A1.2 Step 1: Specification The goal of this first step is to write down a clear statement of what is being measured. This specification includes a description of the preparation of the calibration standard and the mathematical relationship between the measurand and the parameters upon which it depends.

Procedure The specific information on how to prepare a calibration standard is normally given in a Standard Operating Procedure ( SOP). The preparation consists of the following stages Figure A1.3: Preparation of cadmium standard Clean metal surface Clean metal surface Weigh metal Weigh metal Dissolve and dilute Dissolve and dilute RESULT RESULT The separate stages are:

i) The surface of the high purity metal is treated with an acid mixture to remove any metaloxide contamination. The cleaning method is provided by the manufacturer of the metal and needs to be carried out to obtain the purity quoted on the certificate.

ii) The volumetric flask ( 100 ml) is weighed without and with the purified metal inside. The balance used has a resolution of 0.01 mg.

iii) 1 ml of nitric acid ( 65% m/ m) and 3 ml of ionfree water are added to the flask to dissolve the cadmium ( approximately 100 mg, weighed accurately). Afterwards the flask is filled with ion free water up to the mark and mixed by inverting the flask at least thirty times.

Calculation:

The measurand in this example is the concentration of the calibration standard solution, which depends upon the weighing of the high purity metal ( Cd), its purity and the volume of the liquid in which it is dissolved. The concentration is given by V P m c Cd × × = 1000 mg l 1 where cCd : concentration of the calibration standard [ mg l 1]

1000 : conversion factor from [ ml] to [ l]

m : mass 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 analysing uncertainty sources The aim of this second step is to list all the uncertainty sources for each of the parameters which affect the value of the measurand.

Purity The purity of the metal ( Cd) is quoted in the supplier's certificate as 99.99 ± 0.01%. P is Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 37 therefore 0.9999 ± 0.0001. These values depend on the effectiveness of the surface cleaning of the high purity metal. If the manufacturer's procedure is strictly followed, no additional uncertainty due to the contamination of the surface with metaloxide needs to be added to the value given in the certificate. There is no information available that 100% of the metal dissolves. Therefore one has to check with a repeated preparation experiment that this contribution can be neglected.

Mass m The second stage of the preparation involves weighing the high purity metal. A 100 ml quantity of a 1000 mg l 1 cadmium solution is to be prepared.

The relevant mass of cadmium is determined by a tared weighing, giving m= 0.10028 g The manufacturer's literature identifies three uncertainty sources for the tared weighing: the repeatability; the readability ( digital resolution) of the balance scale; and the contribution due to the uncertainty in the calibration function of the scale. This calibration function has two potential uncertainty sources, identified as the sensitivity of the balance and its linearity. The sensitivity can be neglected because the mass by difference is done on the same balance over a very narrow range.

NOTE: Buoyancy correction is not considered because all weighing results are quoted on the conventional basis for weighing in air [ H. 19]. The remaining uncertainties are too small to consider. Note 1 in Appendix G refers.

Volume V The volume of the solution contained in the volumetric flask is subject to three major sources of uncertainty:

· The uncertainty in the certified internal volume of the flask.

· Variation in filling the flask to the mark.

· The flask and solution temperatures differing from the temperature at which the volume of the flask was calibrated.

The different effects and their influences are shown as a cause and effect diagram in Figure A1.4 ( see Appendix D for description). Figure A1.4: Uncertainties in Cd Standard preparation Purity V m Repeatability Calibration Temperature c( Cd)

m( tare) m( gross)

Repeatability Repeatability Calibration Linearity Sensitivity Calibration Linearity Sensitivity Readability Readability A1.4 Step 3: Quantifying the uncertainty components In step 3 the size of each identified potential source of uncertainty is either directly measured, estimated using previous experimental results or derived from theoretical analysis.

Purity The purity of the cadmium is given on the certificate as 0.9999 ± 0.0001. Because there is no additional information about the uncertainty value, a rectangular distribution is assumed. To obtain the standard uncertainty u( P) the value of 0.0001 has to be divided by 3 ( see Appendix E1.1)

000058 . 0 3 0001 . 0 ) ( = = P u Mass m The uncertainty associated with the mass of the cadmium is estimated, using the data from the calibration certificate and the manufacturer's recommendations on uncertainty estimation, as 0.05 mg. This estimate takes into account the three contributions identified earlier ( Section A1.3).

NOTE: Detailed calculations for uncertainties in mass can be very intricate, and it is important to refer to manufacturer's literature where mass uncertainties are dominant. In this example, the calculations are omitted for clarity.

Volume V The volume has three major influences; calibration, repeatability and temperature effects. Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 38 i) Calibration: The manufacturer quotes a volume for the flask of 100 ml ± 0.1 ml measured at a temperature of 20 ° C. The value of the uncertainty is given without a confidence level or distribution information, so an assumption is necessary. Here, the standard uncertainty is calculated assuming a triangular distribution.

ml 04 . 0 6 ml 1 . 0 = NOTE: A triangular distribution was chosen, because in an effective production process, the nominal value is more likely than extremes. The resulting distribution is better represented by a triangular distribution than a rectangular one.

ii) Repeatability: The uncertainty due to variations in filling can be estimated from a repeatability experiment on a typical example of the flask used. A series of ten fill and weigh experiments on a typical 100 ml flask gave a standard deviation of 0.02 ml. This can be used directly as a standard uncertainty.

iii) Temperature: According to the manufacturer the flask has been calibrated at a temperature of 20 ° C, whereas the laboratory temperature varies between the limits of ± 4 ° C. The uncertainty from this effect can be calculated from the estimate of the temperature range and the coefficient of the volume expansion. The volume expansion of the liquid is considerably larger than that of the flask, so only the former needs to be considered. The coefficient of volume expansion for water is 2.1 ´ 10   4 ° C   1, which leads to a volume variation of ml 084 . 0 ) 10 1 . 2 4 100 ( 4 ± = ´ ´ ´ ± - The standard uncertainty is calculated using the assumption of a rectangular distribution for the temperature variation i. e.

ml 05 . 0 3 ml 084 . 0 = The three contributions are combined to give the standard uncertainty u( V) of the volume V ml 07 . 0 05 . 0 02 . 0 04 . 0 ) ( 2 2 2 = + + = V u A1.5 Step 4: Calculating the combined standard uncertainty cCd is given by ] l mg [ 1000 1

V P m c Cd × × = The intermediate values, their standard uncertainties and their relative standard uncertainties are summarised overleaf ( Table A1.2)

Using those values, the concentration of the calibration standard is 1 l mg 7 . 1002 0 . 100 9999 . 0 28 . 100 1000 - = ´ ´ = Cd c For this simple multiplicative expression, the uncertainties associated with each component are combined as follows.

2 2 2 ) ( ) ( ) ( ) (

÷ ø ö ç è æ + ÷ ø ö ç è æ + ÷ ø ö ç è æ = V V u m m u P P u c c u Cd Cd c 0009 . 0 0007 . 0 0005 . 0 000058 . 0 2 2 2 = + + = 1 1 l mg 9 . 0 0009 . 0 l mg 7 . 1002 0009 . 0 ) (

- - = ´ = ´ = Cd Cd c c c u It is preferable to derive the combined standard uncertainty ( uc( cCd)) using the spreadsheet method given in Appendix E, since this can be utilised even for complex expressions. The completed spreadsheet is shown in Table A1.3.

The contributions of the different parameters are shown in Figure A1.5. The contribution of the uncertainty on the volume of the flask is the largest and that from the weighing procedure is similar. The uncertainty on the purity of the cadmium has virtually no influence on the overall uncertainty. Table A1.2: Values and Uncertainties Description Value x u( x) u( x)/ x Purity of the metal P 0.9999 0.000058 0.000058 Mass of the metal m ( mg) 100.28 0.05 mg 0.0005 Volume of the flask V ( ml) 100.0 0.07 ml 0.0007 Quantifying Uncertainty Example A1: Preparation of a Calibration Standard QUAM: 2000. P1 Page 39 The expanded uncertainty U( cCd) is obtained by multiplying the combined standard uncertainty with a coverage factor of 2, giving 1 1 l mg 8 . 1 l mg 9 . 0 2 ) ( - - = ´ = Cd c U Table A1.3: Spreadsheet calculation of uncertainty 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 u( y, xi) 0.05816 0.49995  0.70140 11 u( y) 2, u( y, xi) 2 0.74529 0.00338 0.24995 0.49196 12 13 u( c( Cd)) 0.9 The values of the parameters are entered in the second row from C2 to E2. Their standard uncertainties are in the row below ( C3 E3). The spreadsheet copies the values from C2 E2 into the second column from B5 to B7. The result ( c( Cd)) using these values is given in B9. The C5 shows the value of P from C2 plus its uncertainty given in C3. The result of the calculation 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) are the differences of the row ( C9 E9) minus the value given in B9. In row 11 ( C11 E11) the values of row 10 ( C10 E10) are squared and summed to give the value shown in B11. B13 gives the combined standard uncertainty, which is the square root of B11.