Approximate uncertainty calculation

These simple rules for approximate uncertainty calculation have been summarized in the following overview table... 

Combining these rules for approximate uncertainty calculation, it can be ensured that a calculation result expresses the accuracy on which the input data are based. This prevents, for example, that the density of a mortar sample is stated as 2.29583 g/cm when the value has been calculated from a measured quantity of 5.51 g and a measured volume of 2.4 cm. ... 

For most calculations, an approximate uncertainty calculation is adequate. On the other hand, for a problem where the very uncertainty on the result is significant, a complete mathematical uncertainty calculation should be performed following the procedure (f). 

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%. 

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. 

AjH is a weighted average of results from three gas-phase equilibrium reactions studied mass spectrometrically by Hildenbrand (1 ) and Potter et al. ( 2). Analyses of these data are summarized below. The overall range of values is less than 4 kcal mol", suggesting the approximate uncertainty in A H . From the adopted A H we calculate 98.2 2 kcal mol... 

For a determined concentration of 200 pg/ml, the expanded uncertainty would be f/(y) = 200 x 0.292 = 58. This result denotes that, for a result of 200 pg/ml, the concentration of ochratoxin A would be expressed as ochratoxin A (200 58) pg/ml, where the stated uncertainty is an expanded uncertainty calculated using a coverage factor of 2 that corresponds approximately to the 95% confidence interval. ... 

For a measurement procedure an uncertainty budget is estimated based on the standard deviations for each contribution and the measure of the combined uncertainty calculated by taking the square root of the sum of the squared estimates of the standard deviations of all uncertainty contributions. Then the overall uncertainty is obtained by multiplying the combined uncertainty with the factor k = 2. In the case of normal distribution the factor k = 2 means that the limits of the overall uncertainty have a confidence level of approximately 95% [21,22]. 

Jo) parameters from the 35 and 90 GHz experiments were -60 20 MHz and 10.3 0.1 MHz. It was shown that the spectra are relatively insensitive to Jo at these frequencies, but at 326 GHz a more accurate value of the exchange of-82 3 MHz was obtained and this was shown to result in an estimated maximum electron transfer rate between semiquinones of 10 s". The theoretical treatment of the EPR spectra was further validated by the use of the point dipole approximation to calculate the distance between the centres of the quinone rings, which gave a value of 17.2 A that is the same, within experimental uncertainties, as that determined by X-ray crystallography. 

Solution. Calculation with approximate uncertainty determination... 

Thus, the calculations performed with approximate uncertainty assessment have lead to the following result... 

The uncertainties in choice of potential function and in how to approximate the surface distortion contribution combine to make the calculated surface energies of ionic crystals rather uncertain. Some results are given in Table VII-2, but comparison between the various references cited will yield major discrepancies. Experimental verification is difficult (see Section VII-5). Qualitatively, one expects the surface energy of a solid to be distinctly higher than the surface tension of the liquid and, for example, the value of 212 ergs/cm for (100)... 

Due to difficulties and uncertainties in the experimental separation of the porous media [93], and the inevitability of approximations in the analytical treatment [87,89], the nature of the chain movement in a random environment is still far from being well understood, and theoretical predictions are controversial [87,89]. Thus, on the ground of replica calculations within a variational approach, one predicts three regimes [87] in which the chain gyration radius Rg scales with the number of repeatable units N as rI (X for low, R x N for medium, and R x for high... 

The departure of the Eh values from a smooth trend is somewhat over half as large as that of the dissociation energy values in the last row of Table VI. Until the London energy calculations are refined to eliminate the dipole-dipole approximation and other uncertainties, it is not possible to say whether that effect accounts for the entire anomaly or not. In any event a substantial portion of the anomaly may be ascribed to the correlation of the motion of the unshared electron pairs in. the valence shell. 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

Both of these are equations are approximate and are useful only for giving estimates of the value of the of the polyblend or copolymer. To calculate values of more accurately requires additional information such as the coefficients of thermal expansion of both components in both their liquid and glassy states. Given the uncertainty in the numerical value of T, which as we have seen depends on the method by which has been determined, there is little point in developing such arithmetical refinements. 

The European Pharmacopoeia Commission has adopted the policy that the value assigned to a reference substance as a result of an inter-laboratory trial should have an uncertainty not greater than a predetermined value. The following formula (Equation 5.1) may be used to calculate the estimated approximate imcertainty ... 

The uncertainty in the smoothed values depends upon the uncertainties in the original observed values and upon the magnitude of deviations between observed and calculated values. To approximate the contribution of these two effects at the temperature T, the uncertainties ux(T) for the low temperature range are calculated from ... 

In this equation, u(T) represents the uncertainty of the observed data in the vicinity of T and is approximated by fitting a polynomial of order 1-3 to the estimated uncertainties as a function of temperature (other symbols appear in the glossary). Uncertainties in the smoothed data for the high temperature range are calculated using ... 

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