Expanded uncertainty. See

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. 

For a measurement result to be metrologically traceable, the measurement uncertainty at each level of the calibration hierarchy must be known. Therefore, a calibration standard must have a known uncertainty concerning the quantity value. For a CRM this is included in the certificate. The uncertainty is usually in the form of a confidence interval (expanded uncertainty see chapter 6), which is a range about the certified value that contains the value of the measurand witha particular degree of confidence (usually 95%). There should be sufficient information to convert this confidence interval to a standard uncertainty. Usually the coverage factor ( see chapter 6) is 2, corresponding to infinite degrees of freedom in the calculation of measurement uncertainty, and so the confidence interval can be divided by 2 to obtain uc, the combined standard uncertainty. Suppose this CRM is used to calibrate... 

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 .

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. 

To demonstrate the influence of the CRM uncertainty ox on the determination, the calibration curve obtained in Round No. 73 by the ordinary least squares technique is shown in Fig. 2 by regression solid line 5. The limits of the expanded uncertainty corridor formed by to.95,3 SXo values are shown in Fig. 2 by dotted lines 4 and 6, where t0 95 3=3.18 (see tables of the Student distri-... 

The expanded uncertainty provides the range within which the value of the measurand is believed to be for a particular level of confidence (see Section 6.2). 

Systematic measurement errors may reside implicitly in the customary reporting of pH and ten erature. Chemical hydrogen exchange rates can vary by a factor of ten for each pH unit (see Section 1.2.2) [42]. When pH is recorded to the nearest tenth unit, the expanded uncertainty of the exchange rate coefficient, is no less than 10%. This uncertainty can be reduced substantially. Modern, commercially available pH probes that are calibrated with two-point bracketing have an expanded uncertainty of t/ j(pH) 0.02. When pH is determined at this accuracy, the contribution of f/ j(pH) to the expanded uncertainty of This uncertainty contribution is halved when the... 

Figure 10.3 Typical scenarios arising when measurements of the concentration of analyte are used to assess compliance with an upper specification limit. The vertical lines show the expanded uncertainty (7 on each result (see main text) and the associated curve indicates the inferred probability density function for the value of the measured concentration, emphasizing that there is a larger probability of the value of the measured concentration lying near the centre of the expanded uncertainty interval than near the ends. For a full discussion see Eurachem-CITAC Guide (2007) www.eurachem.ul.pt...

Will critical assessment of the analytical results be possible, with evaluation of an uncertainty budget aiming to determine an expanded uncertainty of the analytical result [29], [30] (see Section 1.6.2)... 

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... 

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. 

The assumption that the water is adsorbed in uniform layers on all the clay surfaces for a wide range of mixtures has been criticized (2, 20). The argument is that the individual clay particles in the clay-water mixture do not expand beyond a certain distance regardless of the quantity of water which is added. The clay layers group themselves into tactoids resulting in two populations of water those molecules which are found between the tactoids and those directly perturbed by the clay layers. If true, this would invalidate the procedure used to calculate the thermodynamic properties of the adsorbed water. However, other workers have reported complete delamination of certain smectites (21., 22). It is not clear under what conditions tactoids will form, or not, and this uncertainty is underlined in (21) (see remarks by Nadeau and Fripiat, pages 146-147). 

Kamlet parameter that depends on expl compn (see article on Velocity of Detonation in this Vol). Their correlation is shown in Fig 17 Kamlet Finger (Ref 23) propose a somewhat simpler empirically-fitted correlation, namely, y/ZE1 = 0.887 >asp0° 4. It should be emphasized that this correlation, as well as the one in Fig. 17, are based on isentrope expansion calcns with all their inherent uncertainties as to the equation of state to be used for the expand-ign detonation products... 

Uncertainty is commonly reported as the standard deviation, a, or its expanded versions, notably 2a or 3a (see Section 10.3.2). The standard deviation or its multiple can be calculated according to Eq. (10.6) by replicate analyses. Although in many instances the reported values are based only on the uncertainty of counting, determined by propagating the uncertainty of the gross count and the background count (see Section 10.3.1), other contributors to the uncertainty can be ignored only if their impact is minor. 

Cjood precision, which has been defined as the close agreement of measured values, is indicated b> a low standard deviation. It is an essential requirement. Good precision and low or /cri> bias equate to good or high accuracy. See Annex C (to be discussed later) for more on precision, bias, and accuracy. The term imcenainty is also frequently used as a surrogate for accuracy in an inverse sense, i.e.. low or acceptable uncertainty is equivalent to high accuracy. Section 8 contains an expanded discussion on these concepts. 

Hguro 3 Cause and effect diagram of a titration in which the standard solution is made up from pure solid. (A) Initial diagram with elements from the measurement model (see eqn [1]). (B) Diagram with expanded influence factors. 7=temperature, s =repeatability, cal = manufacturer s calibration (see also eqn [2]). The mass measurements done by difference will contribute only linearity of calibration to the uncertainty labeled cal . 

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