Expanded uncertainty

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

Mechanical Properties and Structural Performance. As a result of the manufacturing process, some cellular plastics have an elongated cell shape and thus exhibit anisotropy in mechanical, thermal, and expansion properties (35,36). Efforts are underway to develop manufacturing techniques that reduce such anisotropy and its effects. In general, higher strengths occur for the paraHel-to-rise direction than in the perpendicular-to-rise orientation. Properties of these materials show variabiUty due to specimen form and position in the bulk material and to uncertainty in the axes with respect to direction of foam rise. Expanded and molded bead products exhibit Httie anisotropy. 

Minimills and other EAF plants ate expanding iato flat-roUed steel products which, by some estimates, requite 50—75% low residual scrap or alternative raw material. Up to 16 million t of new capacity are expected to be added ia the United States between 1994 and 2000 (18). Developments ia other parts of the world also impact scrap use and supply. Possible scrap deficiencies of several million tons have been projected for EAFs ia East Asia and ia parts of Europe. This puts additional strains on the total scrap supply, particularly low residual scrap (19,20). The question of adequate supply of low residual scrap is always a controversial one. Some analysts see serious global shortages ia the first decade of the twenty-first century others are convinced that the scrap iadustry has the capabiUty to produce scrap ia the quantities and quaUty to meet foreseeable demand. This uncertainty ia combination with high scrap prices has led to iacreased use of scrap alternatives where the latter is price competitive with premium scrap. Use of pig iroa has iacreased ia EAF plants and mote capacity is being iastaHed for DRI and HBI outside the United States. 

Analysts must recognize that the end use as well as the uncertainty determines the value of measurements. While the operators may pay the most attention to one set of measurements in making their decisions, another set may be the proper focus for model development and parameter estimation. The predilec tion is to focus on those measurements that the operators Believe in or that the designers/con-trollers originally believed in. While these may not be misleading, they are usually not optimal, and analysts must consciously expand their vision to include others. 

Pauwels (1999) argues that the certified values of CRMs should be presented in the form of an expanded combined uncertainty according to the ISO Guide on the expression of uncertainty in measurement, so that coverage factor should always be clearly mentioned in order to allow an easy recalculation of the combined standard uncertainty. This is needed for uncertainty propagation when the CRM is used for calibration and the ISO Guide should be revised accordingly. The use of the expanded uncertainty has been pohcy in certification by NIST since 1993 (Taylor and Kuyatt 1994). 

If users are to benefit from the implementation and/or verification of traceability in analytical chemistry the unbroken pathway of references must be kept short. The uncertainty of the references (CRMs) used may significantly widen the uncertainty a user must attach to the result of his measiuement when addressing accuracy and traceability through comparison with a CRM. These comparisons should be only considered in a first or second level step as to keep the uncertainties of the results within limits fit for the purpose. The producers of CRMs must keep their uncertainties sufficiently small to allow introduction of the CRM at different points in the analytical pathway, without limiting the usefiilness of results through unduly expanded uncertainties. 

The lattice energy of a molecular compound corresponds to the energy of sublimation at 0 K. This energy cannot be measured directly, but it is equal to the enthalpy of sublimation at a temperature T plus the thermal energy needed to warm the sample from 0 K to this temperature, minus RT. RT is the amount of energy required to expand one mole of a gas at a temperature T to an infinitely small pressure. These amounts of energy, in principle, can be measured and therefore the lattice energy can be determined experimentally in this case. However, the measurement is not simple and is subject to various uncertainties. 

Third, as the size and complexity of the biomolecular systems at hand further expand, there are more uncertainties in the molecular model itself. For example, the resolution of the X-ray structure may not be sufficiently high for identifying the locations of critical water molecules, ions and other components in the system the oxidation states and/or titration states of key reactive groups might be unclear. In those cases, it is important to couple QM/MM to other molecular simulation techniques to establish and to validate the microscopic models before elaborate calculations on the reactive mechanisms are investigated. In this context, pKa and various spectroscopic calculations [113,114] can be very relevant. 

The principles of quality assurance are commonly related to product and process control in manufacturing. Today the field of application greatly expanded to include environmental protection and quality control within analytical chemistry itself, i.e., the quality assurance of analytical measurements. In any field, features of quality cannot be reproduced with any absolute degree of precision but only within certain limits of tolerance. These depend on the uncertainties of both the process under control and the test procedure and additionally from the expense of testing and controlling that may be economically justifiable. 

Figure 1.1 Results from an international intercomparison, IMEP 9, wl expanded uncertainty and ue the combined standard uncertainty [1] (see Chapter 6, Section 6.3). Reproduced by permission of EC-JRC-IRMM (Philip Taylor) from IMEP 9, Trace elements in water III, Cd, certified range 81.0-85.4nmoll-1 .

To allow the uncertainty to be evaluated effectively, a model equation describing the method of analysis is required. The starting point is the equation used to calculate the final result. Intially, we will need to consider the uncertainties associated with the parameters that appear in this equation. It may be necessary to add terms to this equation (i.e. expand the model) to include other parameters that may influence the final result and therefore contribute to the measurement uncertainty. 

Example the concentration of a reference solution is 1000 3 mg I 1. where the reported uncertainty is an expanded uncertainty, calculated using a coverage factor of k = 2, which gives a level of confidence of approximately 95%. 

A coverage factor (usually denoted by the letter k) is used to increase (expand) a standard uncertainty to give the required level of confidence (usually 95%). Expanded uncertainties are discussed in more detail in Section 6.3.6. To convert an expanded uncertainty back to a standard uncertainty, simply divide by the stated coverage factor. In this example, k = 2, so the standard uncertainty is 1.5 mg l-1. 

Expanded uncertainty Stated range (values equally likely across range)... 

Consider the previous example of calculating the concentration of a standard solution. The combined standard uncertainty of 2.69 mg l-1 would be multiplied by a coverage factor of 2 to give an expanded uncertainty of 5.38 mg l-1. We can now report the result as follows concentration of solution = (1004 5) mg 1 1, where the reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2, which gives a level of confidence of approximately 95%. Note that the coverage factor is applied only to the final combined uncertainty. 

It is important to include a statement as to what the value quoted after the represents (i.e. a standard or an expanded uncertainty) so that users of the result interpret the quoted uncertainty correctly. 

In Figure 6.15, only cases (a) and (e) are easy to interpret. Looking at case (a), we see that the measured value is less than the reference value. The upper extreme of the expanded uncertainty is also less than the reference value. We can therefore safely conclude that the concentration of compound X is less than the reference value in case (a) so this particular batch of material can be accepted. In case (e), the measured value exceeds the reference value, as does the lowest extreme of the expanded uncertainty. There is therefore no doubt that the concentration of X exceeds the reference value and the batch of material must therefore be rejected. 

In case (b), although the measured value is less than the reference value, if the expanded uncertainty is taken into account it is possible that the actual concentration of X could exceed the reference value. In case (c), the measurement result equals the reference value. Although the measured value does not exceed the reference value, the expanded uncertainty means that the true value of the concentration of X could exceed the reference value. Finally, in case (d) the measurement value exceeds the reference value but if the expanded uncertainty is taken into account, it is possible that the true value of the concentration of X could be below the reference value. 

There are a number of other scoring systems but these are not as widely used as the z-score system. En numbers take into account the expanded uncertainty in the assigned value (Um ) and the expanded uncertainty in the participant s result... 

En numbers are used when the assigned value has been produced by a reference laboratory, which has provided an estimate of the expanded uncertainty. This scoring method also requires a valid estimate of the expanded uncertainty for each participant s result. A score of En < 1 is considered satisfactory. The acceptability criterion is different from that used for z-, z - or zeta-scores as En numbers are calculated using expanded uncertainties. However, the En number is equal to zeta/2 if a coverage factor of 2 is used to calculate the expanded uncertainties (see Chapter 6, Section 6.3.6). En numbers are not normally used by proficiency testing scheme providers but are often used in calibration studies. 

Expanded uncertainty A quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand. 

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