Uncertainty interval

Users expect certified values to be correct - with a probability of 95 % - within the stated uncertainty intervals. They assume, perhaps naively, that all statements of uncertainty are the same. In practice the stated uncertainties may have quite different meanings because they have been based on quite diflferent principles. This issue is discussed in more detail below. But for most users neither the differences nor the consequences of the differences are always evident, or understood. 

The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (confidence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value p and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. 

The uncertainty interval includes the estimated mean y and is given by... 

The uncertainty of OLS calibration is characterized by the following special standard deviations and uncertainty intervals ... 

The following uncertainty intervals resulting from Eqs. (6.19) to (6.25) are of practical interest ... 

Estimates of the variance and uncertainty intervals in robust calibration can be taken from the literature (Huber [1981] Rousseeuw and Leroy [1987]). 

The critical value yc represents the smallest measurement value that can be distinguished from the blank yBL with a given level of significance P = 1 — a. In the most general case, the critical value is estimated from the average blank and its uncertainty interval U(yBL)... 

A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the significant deviations have to be summarized by means of the law of error propagation see Sect. 4.2. 

Results of ultra trace analyses are sometimes characterized by relatively high uncertainties up to more than 100%. In such cases it is not allowed that the lower uncertainty limit falls below zero. Results like, e.g., (0.07 0.10) must be replaced by such as (0.07 + 0.10/ — 0.07) or (0.07/q o°), respectively. That means, the total uncertainty interval (confidence interval, prediction interval is 0...0.17). In general, when the confidence interval includes a negative content (concentration), the result has to be given in the form... 

Other means like the median (see Eq. (4.22)) or the geometrical mean (see Eq. (4.18)) etc. have to be reported in a similar way together with the belonging uncertainty interval, e.g.,... 

Geometrical means have unsymmetrical uncertainty intervals which are characterized by a dispersion factor v (see Sect. 4.1.2, Eq. (4.20)) and a covering factor k (see Sect. 4.2). Corresponding results should be given in the form... 

That means the uncertainty interval covers the range x/(U(xgeom))... x-U(xgeom) = x/(kv).. .x-kv. Negative values of concentration at the lower limit of uncertainty do not appear in this case. 

Formally, an analytical result x,- can be calculated from y, by means of the corresponding calibration function. When this result (from repeated measurements) should be reported, it must be taken into account that the relative uncertainty amounts minimally 100% (see Sect. 7.5, item (1) p. 201) and, therefore, it holds that (x x)- That means, that the uncertainty interval of analytical results calculated from measured values nearby the critical value covers a range of about 0... 2x. As additional information, the limit of quantification, xLq, should be given. 

Comparison of test values with a conventional true value ( reference value ) of a (certified) reference material (RM, CRM). In method development and validation of analytical procedures, the comparison of experimental results with standards of diverse kind (laboratory standards, certified reference materials, primary standards) plays an essential role. The decision as to whether an experimental result hits the reference value depends not only from the result itself but also from its uncertainty interval. 

A B Measurement Calibration 95.1 3.2 95.1 5.1 101.7 T The result is false because the > uncertainty interval does not J include the certified value... 

C Preparation 95.1 6.9 The uncertainty interval includes the certified value therefore, it cannot be considered to be false... 

Fig. 8.4. Found result xexp in relation to the reference value RV a illustrates the location of R V without the uncertainty intervals A and B on the one hand and within the interval C on the other b represents in addition the uncertainty interval of the reference value...

Another assessment will be found if the uncertainty of the reference value is considered likewise and, therefore, the f-test according to Eqs. (8.7) or (8.8) is applied. The corresponding f-values (tA = 5.96, tB = 4.00, tc = 3.02) are larger in each case than the critical f-value 2.05 (on the basis a = 0.05, nexp = 10, and nBM = 20). That means that the comparison will become sharper if the relatively small uncertainty interval of RV is included into the test and, therefore, the result C is assessed to be false, too. 

In the least-squares analyses of equation 1, the individual enthalpies of formation were weighted inversely as the squares of the uncertainty intervals. 

Local sensitivity analysis is of limited value when the chemical system is non-linear. In this case global methods, which vary the parameters over the range of their possible values, are preferable. Two global uncertainty methods have been used in this work, a screening method, the so-called Morris One-At-A-Time (MOAT) analysis and a Monte Carlo analysis with Latin Hypercube Sampling (Saltelli et al., 2000 Zador et al., submitted, 20041). The analyses were performed by varying rate parameters, branching ratios and constrained concentrations within their uncertainty interval,... 

Assuming that only random errors affect the laboratory determinations of a given reaction enthalpy, the overall uncertainty interval associated with the mean value (ATH) of a set of n experiments is usually taken as twice the standard deviation of the mean (erm) ... 

Here, the uncertainty intervals are standard deviations multiplied by Student s t factor for 95% probability and 5 degrees of freedom (t = 2.571) [48]. 

Based on these estimates and literature values for the fugacity, Oldani and Bor obtained equation 14.31, from which they derived the reaction enthalpy and entropy at the mean temperature of the experimental temperature range (T = 272 K) Ar//2°72 = 58.6 2.6 kJ mol-1 and ATS 72 = 304 10 J K-1 mol-1. The uncertainty intervals are standard deviations multiplied by Student s t factor for 95% probability and 18 degrees of freedom (t = 2.101) [48]. 

A significant contribution to the uncertainty interval assigned to the O-H bond dissociation enthalpy in benzoic acid comes from the estimate of the activation enthalpy for the radical recombination. The experimental determination of this quantity is not easy because diffusion-controlled recombination rate constants are very high (109 mol-1 dm3 s 1 or larger) [180]. Therefore, most thermochemical data derived from kinetic experiments in solution rely on some similar assumptions. 

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