An uncertainty evaluation for multiple measurements by GUM

An approach for uncertainty evaluation is proposed to determine the overall uncertainty by combining the uncertainties of the individual results from multiple measurements. It is accomplished by the separate combinations of the individual random and systematic components of the uncertainties of the individual results. The approach is useful when the individual results are not statistically different. It is recognized that, owing to the correlation, the uncertainty resulting from systematic effects is not reduced by multiple measurements. On the contrary, the uncertainty resulting from random effects can be reduced. Received: 3 May 2002 Accepted: 16 July 2002     

Correlation between repeated measurements: bane and boon of the GUM approach to the uncertainty of measurement

With measurement uncertainty estimation accounting for all relevant uncertainty contributions, the results of measurements using the same procedure on different objects or samples may no longer be considered as being independent, and correlations have to be taken into account. For this purpose, a simple approximation for the estimation of covariances is derived and applied to the estimation of uncertainty for some basic combinations of two measurement results. This covariance estimate is also applied to the estimation of uncertainty for the mean value of the results of replicate measurements on the same object or sample.

An uncertainty evaluation for multiple measurements by GUM,II

The former approach is extended to determine the overall uncertainty by combining the uncertainties of the individual results of n measurements where the difference in the individual results is statistically significant. The difference, defined as an unrecognized uncertainty arising from a random effect, is considered as an additional uncertainty. Now, it is possible to determine the overall uncertainty by combining the uncertainties of the individual results whether the individual results are statistically different or not.     

Application of consistency checking to evaluation of uncertainty in multiple replicate measurements

Use of repeated measurements in quantitative chemical analysis is common but leads to the problem of how to combine the measurement values and produce a result with an uncertainty following the GUM. There is often confusion between repeated indications or observations of an input quantity, for whose uncertainty the GUM prescribes a type A evaluation, and complete measurements repeated on multiple sub-samples, as considered here. A solution for combining repeated measurement results and their individual uncertainties based on simple interval logic is proposed here. The individual measurement values and their uncertainties are compared with the calculated average value to see if this implies that another, possibly unknown, source of uncertainty is present. The model of the individual results is modified for this possible between-replicate effect so that the repeated measurements are consistent. Lack of consistency is a strong indication that the measurement is not fully under control and needs further development or investigation. This is not always possible, however and the method given here is proposed to ensure that the values of the repeated measurements agree with each other. A simple numerical example is given showing how the method can be implemented in practice.     

An uncertainty budget for trace analysis by isotope-dilution ICP-MS with proper consideration of correlation

Isotope-dilution mass spectrometry (IDMS) is considered to be a method without significant correction factors. It is also believed that this method is well understood. But unfortunately a large number of different uncertainty budgets have been published that consider different correction factors. These differences lead to conflicting combined uncertainties especially in trace analysis. It is described how the known correction factors must be considered in the uncertainty budget of values determined by IDMS combined with ICP-MS (ICP-IDMS). The corrections applied are dead time, background, interference, mass discrimination, blank correction and air buoyancy.IDMS measurements consist always of a series of isotope abundance ratio measurements and can be done according to different measurement protocols. Because the measurement protocols of IDMS are often rather sophisticated, correlations of influence quantities are difficult to identify. Therefore the measurement protocol has to be carefully considered in the specification of the measurand and a strategy is presented to properly account for these correlations. This will be exemplified for the estimation of mass fractions of platinum group elements (PGEs) and Re in the geological reference material UB-N (from CRPG-CNRS, Nancy in France) with ICP-IDMS. The PGEs with more than one isotope and the element Re are measured with on-line cation-exchange chromatography coupled to a quadrupole ICP-MS. All contents are below 10 µg kg^–1. Only osmium is separated from the matrix by direct sparging of OsO₄ into the plasma. This leads to transient signals for all PGEs and Re. It is possible to estimate the combined uncertainties and keep them favourably small despite the low contents, the transient signals and the sophisticated correction model.     

Model-based measurement uncertainty evaluation,with applications in testing

Basic concepts and implementations of the model-based approach to uncertainty evaluation are discussed. One implementation is that of the law of propagation of uncertainty with the use of the central limit theorem as recommended in the Guide to the expression of uncertainty in measurement (GUM). Another is the propagation of distributions, the subject of a supplemental guide to the GUM, which is at an advanced stage of development. It falls in the category of other analytical and numerical methods indicated in the GUM. Two testing applications are used to illustrate the principles: tennis-ball rebound and tensile strength.

From GUM to alternative methods for measurement uncertainty evaluation

Since the advent of the Guide to the expression of Uncertainty in Measurement (GUM) in 1995 laying the principles of uncertainty evaluation numerous projects have been carried out to develop alternative practical methods that are easier to implement namely when it is impossible to model the measurement process for technical or economical aspects. In this paper, the author presents the recent evolution of measurement uncertainty evaluation methods. The evaluation of measurement uncertainty can be presented according to two axes based on intralaboratory and interlaboratory approaches. The intralaboratory approach includes “the modelling approach” (application of the procedure described in section 8 of the GUM, known as GUM uncertainty framework) and “the single laboratory validation approach”. The interlaboratory approaches are based on collaborative studies and they are respectively named “interlaboratory validation approach” and “proficiency testing approach”.     

Determination of uncertainty for hardness measurement: proposal of the standard,available software

After the next revision of the ISO-standards for hardness testing (ISO 6506–6508) an estimation of the uncertainty (ISO/IEC 17025, ISO 5725) is mandatory. The two proposed approaches of part one (ISO 6506–6508) of the ISO Standards for hardness are exemplified. The calculation approach implemented in an Excel-based file, which is available via , is announced.Presented at the Metro Trade Workshop on Traceability and Measurement Uncertainty in Testing, 30–31 January 2003, Berlin, Germany     

Uranium assay determination using Davies and Gray titration: an overview and implementation of GUM for uncertainty evaluation

The International Organization for Standardization (ISO) Guide to the expression of Uncertainty in Measurement (GUM) was developed to meet the demand for a standardized way of evaluating and expressing uncertainties. The Davies and Gray (D&G) titrimetry method is routinely used in nuclear safeguards for uranium accountability measurement and a statement of the uncertainty that can reasonably be attributed to the measured assay value is therefore of importance. A mathematical model for an uncertainty evaluation of D&G measurements in compliance with ISO GUM is presented. This is illustrated by a numerical example and the utilization of the uncertainty budget is explored.     

Uncertainty evaluation for the quantification of low masses of benzo[a]pyrene: Comparison between the Law of Propagation of Uncertainty and the Monte Carlo method

A proper evaluation of the uncertainty associated to the quantification of micropollutants in the environment, like Polycyclic Aromatic Hydrocarbons (PAHs), is crucial for the reliability of the measurement results. The present work describes a comparison between the uncertainty evaluation carried out according to the GUM uncertainty framework and the Monte Carlo (MC) method. This comparison was carried out starting from real data sets obtained from the quantification of benzo[a]pyrene (BaP), spiked on filters commonly used for airborne particulate matter sampling. BaP was chosen as target analyte as it is listed in the current European legislation as marker of the carcinogenic risk for the whole class of PAHs.     

Dynamic correlation for MCSCF wave functions: An effective potential method

A method is suggested which allows the inclusion of dynamic correlation into CASSCF calculations. An effective Coulomb hole potential is added to the Hamiltonian. The potential has a simple form, which allows its implementation into existing LCAO programs using Gaussian integral packages. The parameters appearing in the potential are determined by fitting to empirical valence correlation energies for first row atoms. Calculations of ionization energies and electron affinities show considerable improvement compared to the MCSCF values. Test calculations on three molecules give the following results, H₂ r _e=0.745 (0.741) Å, D _e=4.62 (4.75) eV; N₂ r _e=1.099 (1.098) Å, D _e= 10.42 (9.91) eV; O₂ r _e=1.198 (1.207) Å, D _e=4.73 (5.21) eV. Experimental values within parenthesis. On leave from: Institute of Organic Chemistry, Polish Academy of Sciences, PL-01-224 Warszawa 42, ul. Kasprzaka 44, Poland.     

Reflections regarding uncertainty of measurement, on the results of a Nordic fatigue test interlaboratory comparison

This paper presents the experiences of calculation and reporting uncertainty of measurement in fatigue testing. Six Nordic laboratories performed fatigue tests on steel specimens. The laboratories also reported their results concerning uncertainty of measurement and how they calculated it. The results show large differences in the way the uncertainties of measurement were calculated and reported. No laboratory included the most significant uncertainty source, bending stress (due to misalignment of the testing machine, incorrect specimens and/or incorrectly mounted specimens), when calculating the uncertainty of measurement. Several laboratories did not calculate the uncertainty of measurement in accordance with the Guide to the Expression of Uncertainty in Measurement (GUM) [1].     

Uncertainty in spectrophotometric analysis--"error propagation break up", a novel statistical method for uncertainty management

In this work, is given the Combined Standard Uncertainty (CSU) calculation procedure, which can be applied in spectrophotometric measurements. For the assessment of the computations, different approaches are discussed, such as the contribution to the Combined Standard Uncertainty of the reproducibility, the repeatability, the total bias, the calibration curve, and the type of the measurand. Results of inter-laboratory measurements confirmed the assumptions. For the minimization of the errors propagation a controlled experimental procedure was applied by this laboratory, called “errors propagation break-up” (ERBs). The uncertainty of sample concentration from a reference curve dominates the Combined Standard Uncertainty. The contribution of the method and the laboratory bias (total bias) to the CSU is insignificant under controlled conditions of a measurement. This work develops a simple methodology that can be utilized to evaluate the uncertainty and errors control on routine methods used both by academic researchers or the industrial sector.     

Interlaboratory study of a liquid chromatography method for erythromycin: determination of uncertainty

Erythromycin is a mixture of macrolide antibiotics produced by Saccharopolyspora erythreas during fermentation. A new method for the analysis of erythromycin by liquid chromatography has previously been developed. It makes use of an Astec C18 polymeric column. After validation in one laboratory, the method was now validated in an interlaboratory study. Validation studies are commonly used to test the fitness of the analytical method prior to its use for routine quality testing. The data derived in the interlaboratory study can be used to make an uncertainty statement as well. The relationship between validation and uncertainty statement is not clear for many analysts and there is a need to show how the existing data, derived during validation, can be used in practice. Eight laboratories participated in this interlaboratory study. The set-up allowed the determination of the repeatability variance, s(2)r and the between-laboratory variance, s(2)L. Combination of s(2)r and s(2)L results in the reproducibility variance s(2)R. It has been shown how these data can be used in future by a single laboratory that wants to make an uncertainty statement concerning the same analysis.     

Validation of the uncertainty evaluation for the determination of metals in solid samples by atomic spectrometry

 Every analytical result should be expressed with some indication of its quality. The uncertainty as defined by Eurachem ("parameter associated with the result of a measurement that characterises the dispersion of the values that could reasonably be attributed to the, . . ., quantity subjected to measurement") is a good tool to accomplish this goal in quantitative analysis. Eurachem has produced a guide to the estimation of the uncertainty attached to an analytical result. Indeed, the estimation of the total uncertainty by using uncertainty propagation laws is components-dependent. The estimation of some of those components is based on subjective criteria. The identification of the uncertainty sources and of their importance, for the same method, can vary from analyst to analyst. It is important to develop tools which will support each choice and approximation. In this work, the comparison of an estimated uncertainty with an experimentally assessed one, through a variance test, is performed. This approach is applied to the determination by atomic absorption of manganese in digested samples of lettuce leaves. The total uncertainty estimation is calculated assuming 100% digestion efficiency with negligible uncertainty. This assumption was tested. Received: 3 November 1997 · Accepted: 2 January 1998     

Setting up a decision rule from estimated uncertainty: emission limit value for PCDD and PCDF incineration plants in Wallonia, Belgium

Complex analytical procedures are often required to prove the non-compliance with a specific legislation. In the case of a small overlap of the limit, integration of the method uncertainty in the decision-making process is essential. The decision rule proposed in Wallonia, Belgium, for the non-compliance of waste incineration plants with the EU limit value for PCDD and PCDF emissions is presented. The method uncertainty was estimated annually over 6 years from duplicate measurements using two top-down approaches. Depending on the congener, the standard uncertainty varies from 30 to 85%, with a good correlation between calculations. The analytical contribution was estimated using a bottom-up evaluation. The impact of the sampling step was deduced from the whole estimation and represents more than 80% of the total uncertainty budget. No optimisation is foreseen at this time because of practical field constraints. Based on the average fraction of each congener, the uncertainty associated with the measurement result has been established and shows a high stability over the years. Using this value, a guard band has been calculated and will be proposed to the regulatory body. Presented at the Measurement Uncertainty Symposium, Berlin, Germany, April 2008.     

Accuracy and long-term reproducibility of lead isotopic measurements by multiple-collector inductively coupled plasma mass spectrometry using an external method for correction of mass discrimination

The precision of isotopic measurements of Pb by thermal ionization mass spectrometry (TIMS) is limited by the fact that this element does not possess an invariant isotope ratio that can be used for the correction of mass fractionation by internal normalization. Multiple-collector inductively coupled plasma mass spectrometry (MC-ICPMS) can overcome this limitation, because with plasma ionization, elements with overlapping mass ranges are thought to display identical mass discrimination. With respect to Pb, this can be exploited by the addition of Tl to the sample solutions; the mass discrimination factor obtained for Tl can then be used for the correction of the measured Pb isotope ratios. In this article we present the results of a detailed study that investigates the accuracy and precision of such an external correction technique for mass discrimination based upon the results of multiple analyses of a mixed standard solution of NIST SRM-981 Pb and SRM-997 Tl. Our data indicate that normalization of the Pb isotope ratios to the certified isotopic composition of SRM-997 Tl produces Pb isotopic results that are significantly lower than recently published reference values by TIMS. This systematic offset can be eliminated by renormalization of the Pb data to a different Tl isotopic composition to obtain an empirically determined mass discrimination factor for Pb that generates accurate results. It is furthermore shown that a linear law is least suited for the correction of mass discrimination, whereas a power or exponential law function provide significantly more accurate and precise results. In detail, it appears that a power law may provide the most appropriate correction procedure, because the corrected Pb isotope ratios display less residual correlations with mass discrimination compared to the exponentially corrected data. Using an exponential or power law correction our results, obtained over a period of over seven months, display a precision (2σ) of better than 60 parts per million (ppm) for ²⁰⁸Pb/²⁰⁶Pb and ²⁰⁷Pb/²⁰⁶Pb and of better than 350 ppm for ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb. This represents a significant improvement compared to conventional TIMS techniques and demonstrates the potential of MC-ICPMS for routine, high-precision measurements of Pb isotopic compositions.     

Use of uncertainty information for estimation of certified values. Comparison of four approaches using data from a recent certification exercise

Four approaches for estimation of reference values and their respective uncertainties of characterisation were compared using data from the recently finalised certified reference materials ERM-EC680k and ERM-EC681k, elements in plastics. Reference values and uncertainties of characterisation were estimated as mean of laboratory means and their respective standard deviations, using equal weights and the weighting procedure of Mandel–Paule. In addition, two approaches taking into consideration uncertainty information reported by the participants, namely the consistency check and simulation procedure proposed by Cox for CCQM Key comparisons and an approach suggested by Pauwels et al. (Accred Qual Assur 3:180–184, 2000) were used. No difference between the equally-weighted and Mandel–Paule consensus means was observed and the reference value from the Cox approach was in all cases within ±2 u _char of each consensus mean. Uncertainties varied between the three approaches. Uncertainties derived from equally-weighted mean of means approach are on average 14% above uncertainties using the Mandel–Paule consensus mean, 36% above the uncertainties estimated by Pauwels et al., and 54% above the uncertainties from the Cox approach. Robustness of the uncertainty estimation against incorrect estimation of uncertainties was assessed. Assumption of a 50% uncertainty of the individual uncertainties resulted in an uncertainty of 30% of the uncertainty of characterisation. Differences between the four approaches are negligible for this dataset when combined with the uncertainty contribution from heterogeneity and stability as prescribed in ISO Guide 35. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.