An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient

After a measurement, a measured value and a measurement uncertainty are produced as a measurement result. By a repeated measurement, another measurement result is produced. Between the individual results of the two measurements, it is shown that there may be a significant correlation. A correlation coefficient can be determined when a GUM-compliant uncertainty budget for a measurement is available. Utilizing the correlations between the N individual results, an equation is derived to combine the N individual uncertainties of N measurements. Using the newly derived equation including the correlation coefficient, three measurement uncertainties of three measurement results are combined as an example. The combined uncertainty is compared with the uncertainty of a measurement which treats the three individual measurements as one process. Papers published in this section do not necessarily reflect the opinion of the editors, the editorial board, or the publisher.     

Uncertainty assessment of methods for chemical yield determination in measurement of radioactive strontium

Chemical yield determination can be a limiting source to the combined standard uncertainty in measurements of radionuclides. Therefore, the combined standard uncertainties in yield determination in the measurement of radioactive strontium using two methods were evaluated. The two methods compared were the measurement of stable strontium by atomic absorption spectrometry (AAS) and the gamma spectrometric measurement of ⁸⁵Sr. The evaluation showed that using gamma spectrometry for yield determination can reduce the combined standard uncertainty by a factor of three compared to AAS. The expanded uncertainties of AAS and gamma spectrometry were calculated to be 10% and 3.3% (k = 2), respectively.     

Probabilistic uncertainty estimator for gamma-spectroscopy measurements

To properly interpret the quality of a gamma-spectroscopy measurement, an uncertainty estimate must be made. The uncertainty in the efficiency calibration is the dominant component to the total propagated measurement uncertainty for many types of measurements. Any deviations between the as-calibrated geometry and the as-measured geometry contribute to the total uncertainty. A mathematical technique has been developed to evaluate the variations between calibration and measurement conditions. A sensitivity analysis mode identifies those variables with the largest contribution to the uncertainty. The uncertainty mode uses probabilistic techniques for the combined variables to compute average efficiency and uncertainty, and then to propagate those values with the gamma-spectroscopic analysis into the final result for that sample.     

Comparison of the uncertainties calculated for the results of radiochemical determinations using the law of propagation of uncertainty and a Monte Carlo simulation

Quantitative determinations of many radioactive analytes in environmental samples are based on a process in which several independent measurements of different properties are taken. The final results that are calculated using the data have to be evaluated for accuracy and precision. The estimate of the standard deviation, s, also called the combined standard uncertainty (CSU) associated with the result of this combined measurement can be used to evaluate the precision of the result. The CSU can be calculated by applying the law of propagation of uncertainty, which is based on the Taylor series expansion of the equation used to calculate the analytical result. The estimate of s can also be obtained from a Monte Carlo simulation. The data used in this simulation includes the values resulting from the individual measurements, the estimate of the variance of each value, including the type of distribution, and the equation used to calculate the analytical result. A comparison is made between these two methods of estimating the uncertainty of the calculated result.     

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.     

Evaluation of the uncertainty of environmental measurements of radioactivity

Results obtained by measurement of radioactivity have traditionally been associated with an expression of their uncertainty, based on the so-called counting statistics. This is calculated together with the actual result on the assumption that the number of counts observed has a Poisson distribution with equal mean and variance. Most of the nuclear scientific community has, therefore, assumed that it already complied with the latest ISO 17025 requirements. Counting statistics, however, express only the variability observed among repeated measurements of the same sample under the same counting conditions, which is equivalent to the term repeatability used in quantitative analysis. Many other sources of uncertainty need to be taken into account before a statement of the uncertainty of the actual result can be made. As the first link in the traceability chain calibration is always an important uncertainty component in any kind of measurement. For radioactivity measurements in particular we find that counting geometry assumes the greatest importance, because it is often not possible to measure a standard and a control sample under exactly the same conditions. In the case of large samples we have additional uncertainty components associated with sample heterogeneity and its influence on self-absorption and counting efficiency. In this paper we prepared an uncertainty budget for existing data for ¹³⁷Cs in Danish soil, which is shown to account adequately for all sources of uncertainty. This revised version was published online in July 2006 with corrections to the Cover Date.     

The application of isotope dilution thermal ionization mass spectrometry to traceability of chemical measurement and interlaboratory comparisons

In this work, the method of isotope dilution thermal ionization mass spectrometry able to trace to SI was developed to accurately measure trace amount of Cd, Pb, Zn and Cu in sediment, rice, wine, and human serum samples for interlaboratory comparisons. The research focuses on how to apply the primary method correctly, uncertainty evaluation of measurement results, and how to achieve the meaning of traceability to SI by using ID-TIMS. As a result, the measurement results of Cu and Zn in the human serum 1 and 2 with 0.94, 0.83 and 0.49% combined uncertainty, respectively, were accepted by EC-JRC-IRMM as the certified values of the serum samples. The measurement results of Cd and Pb in CCQM-K13 and CCQM-K24 with 3.96, 1.62 and 1.03% combined uncertainty, respectively, are within the degrees of the equivalence. These comparisons at the highest level of measurement are proof that traceability of chemical measurement can be achieved as the traceability chain of ID-TIMS established in this work was used.     

Software support for the Nordtest method of measurement uncertainty evaluation

Software support for the Nordtest method of measurement uncertainty evaluation is described. According to the Nordtest approach, the combined measurement uncertainty is broken down into two main components??the within-laboratory reproducibility (intermediate precision) s _Rw and the uncertainty due to possible laboratory bias u(bias). Both of these can be conveniently estimated from validation and quality control data, thus significantly reducing the need for performing dedicated experiments for estimating detailed uncertainty contributions and thereby making uncertainty estimation easier for routine laboratories. An additional merit of this uncertainty estimation approach is that it reduces the danger of underestimating the uncertainty, which continues to be a problem at routine laboratories. The described software tool??MUkit (measurement uncertainty kit)??fully reflects the versatility of the Nordtest approach: it enables estimating the uncertainty components from different types of data, and the data can be imported using a variety of means such as different laboratory data systems and a dedicated web service as well as manual input. Prior to the development of the MUkit software, a laboratory survey was carried out to identify the needs of laboratories related to uncertainty estimation and other quality assurance procedures, as well as their needs for a practical tool for the calculation of measurement uncertainty.     

Evaluation of measurement uncertainty components associated with results of radiochemical neutron activation analysis for determination of uranium traces

Being aware of the importance to consider every step in the evaluation of the combined measurement uncertainty of the result, the purpose of this work was to evaluate the contribution of the radial thermal neutron flux gradient to the uncertainty budget for trace level uranium determination in biological materials by a radiochemical neutron activation analysis (RNAA). Determination of uranium via the short-lived nuclide ²³⁹U was based on solvent extraction with TBP and measurement of the chemical yield from the gamma-ray spectrum of the isolated fraction via ²³⁵U. It has been shown previously, that radial neutron flux gradient, could have a relevant effect on the final result obtained by RNAA. In the present work, radial neutron flux gradient within the irradiation assembly generally accepted in our lab (standards tapped beside the sample), varied between 93 and 108% around the mean value and contributes approximately 20% to the combined measurement uncertainty of the result.     

Application of the combination of isotope ratio monitoring with isotope dilution mass spectrometry to the determination of glucose in serum

Isotope ratio monitoring combined with n((13)C)/n((12)C) isotope dilution mass spectrometry (IRM/IDMS) provides results of low uncertainty of the order of 0.1% if it is applied to the analysis of simple mixtures as found in organic chemistry, even if only low (13)C spike additives to the sample are used. If the method is applied to the analysis of systems that require large-scale sample preparation prior to the measurement, such as the determination of glucose in serum, the results obtained exhibit a higher uncertainty that is comparable to that of the conventional gas chromatography/isotope dilution mass spectrometry (GC/IDMS) method. The reason for this observation is that the small contribution that the IRM/IDMS method makes to the uncertainty budget of the result is superimposed on a large contribution due to the sample preparation. It appears therefore that the IRM/IDMS method has no advantage over the conventional GC/IDMS method. However, if a series of measurements is carried out, and if a suitable experimental design is chosen, the IRM/IDMS method can provide valuable additional information. The influence of sample preparation on each individual result can be quantified as its deviation from the average value of all results of the series. From these data conclusions can be drawn for an improvement in sample preparation.     

New tools: Expert systems for uncertainty budgets

An important part of quality assurance in any analytical laboratory is the production of comprehensive results integrating uncertainty measurements. Many testing laboratories face the problem that the expenditure required to evaluate small uncertainties (high precision and high accuracy) is often uneconomic. In most cases an uncertainty of high reliability has to be calculated from only a few data (one calibration, few replications, etc.). This problem can be solved by an expert system. To achieve this the analytical procedure has to be structured into a dialouge and divided into parts. The uncertainty has to be calculated for each part of the procedure. Addition of the individual uncertainties results in the combined and expanded uncertainty. During the dialouge the system should advise the analyst how to get an efficient and effective calculation of uncertainty. All calculations, mathematical and statistical procedures have to be surveyable but running the system should not be too time consuming for economic reasons. Within the scope of the EURECA-project initiated by the Eidgenössische Materialprüfungs- und Forschungsanstalt (EMPA), St. Gallen, Switzerland, expert system software is being developed in cooperation with other research institutes and manufacturers of analytical instruments. Using this software it will be possible to calculate the uncertainty for analytical procedures such as titration, atomic emission spectrometry (ICP-OES), atomic absorption spectrometry (AAS) and gas- and liquid chromatography (GC, HPLC).     

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     

Approaching target uncertainty in proficiency testing schemes: experience in the field of water measurement

Good correspondence between evaluations of the proficiency testing data for water measurements according to different scoring schemes was found as a result of implementation of the measurement quality according to the target uncertainty defined in the Drinking Water Directive. The accuracy of the evaluation is determined by the uncertainty of the assigned value, and this uncertainty should be in correspondence with the target uncertainty. Modification of the zeta-score by the thorough application of the target uncertainty is presented. It is convenient for generalised presentation of the proficiency testing data. The allowed bias is included as a linear term on the measured value scale, as it is treated in uncertainty analysis. The importance of implementation of the target measurement uncertainty in other fields of the routine measurements is indicated.     

Quantifying the measurement uncertainty of results from environmental analytical methods

The Eurachem-CITAC Guide Quantifying Uncertainty in Analytical Measurement was put into practice in a public laboratory devoted to environmental analytical measurements. In doing so due regard was given to the provisions of ISO 17025 and an attempt was made to base the entire estimation of measurement uncertainty on available data from the literature or from previously performed validation studies. Most environmental analytical procedures laid down in national or international standards are the result of cooperative efforts and put into effect as part of a compromise between all parties involved, public and private, that also encompasses environmental standards and statutory limits. Central to many procedures is the focus on the measurement of environmental effects rather than on individual chemical species. In this situation it is particularly important to understand the measurement process well enough to produce a realistic uncertainty statement. Environmental analytical methods will be examined as far as necessary, but reference will also be made to analytical methods in general and to physical measurement methods where appropriate. This paper describes ways and means of quantifying uncertainty for frequently practised methods of environmental analysis. It will be shown that operationally defined measurands are no obstacle to the estimation process as described in the Eurachem/CITAC Guide if it is accepted that the dominating component of uncertainty comes from the actual practice of the method as a reproducibility standard deviation.     

Quantifying the measurement uncertainty of results from environmental analytical methods

The Eurachem–CITAC Guide Quantifying Uncertainty in Analytical Measurement was put into practice in a public laboratory devoted to environmental analytical measurements. In doing so due regard was given to the provisions of ISO 17025 and an attempt was made to base the entire estimation of measurement uncertainty on available data from the literature or from previously performed validation studies. Most environmental analytical procedures laid down in national or international standards are the result of cooperative efforts and put into effect as part of a compromise between all parties involved, public and private, that also encompasses environmental standards and statutory limits. Central to many procedures is the focus on the measurement of environmental effects rather than on individual chemical species. In this situation it is particularly important to understand the measurement process well enough to produce a realistic uncertainty statement. Environmental analytical methods will be examined as far as necessary, but reference will also be made to analytical methods in general and to physical measurement methods where appropriate. This paper describes ways and means of quantifying uncertainty for frequently practised methods of environmental analysis. It will be shown that operationally defined measurands are no obstacle to the estimation process as described in the Eurachem/CITAC Guide if it is accepted that the dominating component of uncertainty comes from the actual practice of the method as a reproducibility standard deviation.     

Evaluation of the combined measurement uncertainty in isotope dilution by MC-ICP-MS

The combination of metrological weighing, the measurement of isotope amount ratios by a multicollector inductively coupled plasma mass spectrometer (MC-ICP-MS) and the use of high-purity reference materials are the cornerstones to achieve improved results for the amount content of lead in wine by the reversed isotope dilution technique. Isotope dilution mass spectrometry (IDMS) and reversed IDMS have the potential to be a so-called primary method, with which close comparability and well-stated combined measurement uncertainties can be obtained.This work describes the detailed uncertainty budget determination using the ISO-GUM approach. The traces of lead in wine were separated from the matrix by ion exchange chromatography after HNO(3)/H(2)O(2) microwave digestion. The thallium isotope amount ratio ( n((205)Tl)/ n((203)Tl)) was used to correct for mass discrimination using an exponential model approach. The corrected lead isotope amount ratio n((206)Pb)/ n((208)Pb) for the isotopic standard SRM 981 measured in our laboratory was compared with ratio values considered to be the least uncertain. The result has been compared in a so-called pilot study "lead in wine" organised by the CCQM (Comité Consultatif pour la Quantité de Matière, BIPM, Paris; the highest measurement authority for analytical chemical measurements).The result for the lead amount content k(Pb) and the corresponding expanded uncertainty U given by our laboratory was:k(Pb)=1.329 x 10-10mol g-1 (amount content of lead in wine)U[k(Pb)]=1.0 x 10-12mol g-1 (expanded uncertainty U=kxuc, k=2)The uncertainty of the main influence parameter of the combined measurement uncertainty was determined to be the isotope amount ratio R(206,B) of the blend between the enriched spike and the sample.     

Recognizing heterogeneous distribution of platinum group elements (PGE) in geological materials by means of the Re-Os isotope system

The identification of uncertainties caused by sample inhomogeneity, as distinct from those caused by sample preparation and measurement, is a challenging task. Use of chemometric methods to separate and estimate these contributions to the combined standard uncertainty of a measurement (uc) of an analytical result requires complex experiments. The difficulty of platinum group element measurement makes this task even more complex. But unless it can be demonstrated that sample inhomogeneity is the major contributor to the high variability of an analytical result one should be careful not to mistakenly attribute this to a nugget effect. In this contribution we are able to demonstrate in two special cases that irreproducible results (up to 90% RSD) for analysis of Os and Re in the pg g(-1) to ng g(-1) range are truly caused by a nugget effect and not by inadequacies of the analytical method.     

Uncertainty of standard addition experiments: a novel approach to include the uncertainty associated with the standard in the model equation

A new model equation for determining the measurement result in standard addition experiments was derived and successfully applied to the quantitative determination of rhodium in automotive catalysts. Existing equations for standard addition experiments with gravimetric preparation were changed in order to integrate the novel idea of including the uncertainty associated with the standard into the model equation. Using this novel equation combined with the ordinary least squares algorithm for the regression line also yielded a new formula for the associated measurement uncertainty. This uncertainty accounts for the first time for the uncertainty associated with the standard. The derivation for the model equation and the resulting associated measurement uncertainty is shown for gravimetric standard addition experiments both with and without an internal standard.     

Uncertainty sources in UV-Vis spectrophotometric measurement

An overview is given of the most important uncertainty sources that affect analytical UV-Vis spectrophotometric measurements. Altogether, eight uncertainty sources are discussed that are expected to have influence in chemical analysis. It is demonstrated that the well-known intrinsic (or “physical”) sources of uncertainty that originate from the instrument itself (repeatability of spectrophotometer reading, spectrophotometer drift, stray light, etc.) often have significantly lower contributions to the combined uncertainty of the result than the “chemical” sources of uncertainty that originate from the object under study (interference from the constituents of the matrix, decomposition of the photometric complex, etc.). Although selectivity of a photometric procedure is often considered more a validation topic than an uncertainty topic, it is very often important to include it also in the uncertainty budget.Usually the most difficult part of uncertainty estimation of a chemical measurement result is to evaluate the magnitude of the actual uncertainty components, especially the chemical ones. For most of the uncertainty sources discussed in this paper, approaches for their evaluation are given. A generic uncertainty budget for absorbance is presented. Electronic Supplementary Material Supplementary material is available for this article at     

Optimization of the determination of chemical oxygen demand in wastewaters

Chemical oxygen demand (COD) is one of the most relevant chemical parameters for the management of wastewater treatment facilities including the control of the quality of an effluent. The adequacy of decisions based on COD values relies on the quality of the measurements. Cost effective management of the minor sources of uncertainty can be applied to the analytical procedure without affecting measurement quality. This work presents a detailed assessment of the determination of COD values in wastewaters, according to ISO6060:1989 standard, which can support reduction of both measurement uncertainty and cost of analysis. This assessment includes the definition of the measurement traceability chain and the validation of the measurement procedure supported on sound and objective criteria. Detailed models of the measurement performance, including uncertainty, developed from the Differential Approach, were successfully validated by proficiency tests. The assumption of the measurement function linearity of the uncertainty propagation law was tested through the comparison with the numerical Kragten method. The gathered information supported the definition of strategies for measurement uncertainty or cost reduction. The developed models are available as electronic supplementary material, in an MS-Excel file, to be updated with the user's data.