Uncertainties compared with errors

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

Determine the uncertainty for the gravimetric analysis described in Example 8.1. (a) How does your result compare with the expected accuracy of 0.1-0.2% for precipitation gravimetry (b) What sources of error might account for any discrepancy between the most probable measurement error and the expected accuracy ... 

One of the limitations of the portable field survey instruments in the measurement of americium is that their quantitative accuracy depends on how well the lateral and vertical distribution of americium in the soil compares with the calibration parameters used. These methods can provide a rapid assessment of americium levels on or below surfaces in a particular environment however, laboratory-based analyses of samples procured from these environmental surfaces must be performed in order to ensure accurate quantification of americium (and other radionuclides). This is due, in part, to the strong self absorption of the 59.5 keV gamma-ray by environmental media, such as soil. Consequently, the uncertainty in the depth distribution of americium and the density of the environmental media may contribute to a >30% error in the field survey measurements. Currently, refinements in calibration strategies are being developed to improve both the precision and accuracy (10%) of gamma-ray spectroscopy measurements of americium within contaminated soils (Fong and Alvarez 1997). 

Overall, there are always considerable uncertainties associated with an economic evaluation. In addition to the errors associated with the estimation of capital and operating costs, the project life or interest rates are not known with any certainty. The important thing is that different projects, and options within projects, are compared on the basis of consistent assumptions. Thus, even though the evaluation will be uncertain in an absolute sense, it will still be meaningful in a relative sense for choosing between options. 

First of all, we should make a clear distinction between accuracy and precision. Accuracy is a measure of how close a given value is to the true value, whereas precision is a measure of the uncertainty in the value or how reproducible the value is. For example, if we were to measure the width of a standard piece of paper using a ruler, we might find that it is 21.5 cm, give or take 0.1 cm. The give or take (i.e., the uncertainty) value of 0.1 cm is the precision of the measurement, which is determined by how close we are able to reproduce the measurement with the ruler. However, it is possible that when the ruler is compared with a standard unit of measure it is found to be in error by, say, 0.2 cm. Thus the accuracy of the ruler is limited, which contributes to the uncertainty of the measurement, although we may not know what this limitation is unless we can compare our instrument to one we know to be true. 

When the uncertainties in the values are included, it is found that the statistically determined mass scattering efficiences are not significantly different than those calculated theoretically. For example, representing the uncertainty as twice the standard error results in a statistically inferred sulfate mass scattering efficiency of 5.0 i 1.2 as compared with a theoretical value of 3.2 + 1.9. 

The theoretical error of the sum of all nonrecoil contributions is about 1 Hz, at least an order of magnitude smaller than the uncertainty introduced by the proton anomalous magnetic moment k, and we did not write it explicitly in (12.23). In relative units this theoretical error is about 2 x 10 °, to be compared with the estimate of the same error 1.2 x 10 made in [67]. Reduction of the theoretical error by three orders of magnitude emphasizes the progress achieved in calculations of nonrecoil corrections during the last years. 

Values of K0 estimated in this way for several nonpolar molecules in type-A zeolite and in chabazite are compared with experimental data in Table I. For most of the hydrocarbons in both zeolites the predicted and experimental values agree to within about 35%. The accuracy with which the experimental values of K0 are known is not high since these values are calculated from the intercepts of plots of In K vs. l/T. A variation in K0 of 35% corresponds only to an error of about 0.25 kcal/mole in the value of qo, and this is of the same order as the experimental uncertainty. 

Rate constants are a function of temperature and should be compared at the same temperature whenever possible. Arrhenius equations for die reactions in Table 1 are provided and these equations can often be used as substitutes to calculate the temperature dependence for similar reactions. In practice, the error introduced by temperature uncertainties in the normal range of 25-80 C is not particularly important in the comparison of two indirecdy measured rate constants. Because competitive rate studies are so simple to conduct, synthetic chemists can (and sometimes do ) measure the rates of reactions that are required to plan a new synthetic method or total synthesis. Such quick and dirty experiments are often conducted at only one temperature. A temperature in the middle of die normal range (-50 °C) is particularly useful because the rate constants that are obtained can be directly compared (with the above provisos) to rate constants in the normal range. 

The surface distribution for mean annual h results from two properties of atmospheric flow conservation of h following the large-scale flow and the maintenance of the vertical profile of h by convective processes. These features of the climate system allow one to quantify the expected errors for assuming that mean annual h is invariant with longitude and altitude for the present-day distribution. Forest et al. (1999) examined the distribution and calculated the expected error from assuming zonal invariance to be 4.5 kJ/kg for the mean annual climate. This error translates to an altitude error of 460 m and is compared with an equivalent error of 540 m from the mean annual temperature approach. Moreover, the uncertainty of the terrestrial lapse rate, y(, increases the expected error in elevation as elevations increase, particularly when small lapse rates are assumed. 

Using a relative rate method, rate constants for the gas-phase reactions of O3 with 1- and 3-methylcyclopentene, 1-, 3- and 4-methylcyclohexene, 1-methylcycloheptene, cw-cyclooctene, 1- and 3-methylcyclooctene, cycloocta-1,3- and 1,5-diene, and cyclo-octa-l,3,5,7-tetraene have been measured at 296 2 K and atmospheric pressure. The rate constants obtained (in units of 10-18 cm3 molecule-1 s-1) are as follows 1-methylcyclopentene, 832 24 3-methylcyclopentene, 334 12 1-methylcyclohex-ene, 146 10 3-methylcyclohexene, 55.3 2.6 4-methylcyclohexene, 73.1 3.6 1-methylcycloheptene, 930 24 d.s-cyclooclcnc, 386 23 1-methylcyclooctene, 1420 100 3-methylcyclooctene, 139 9 d.v.d.v-cycloocta-1,3-diene, 20.0 1.4 cycloocta- 1,5-diene, 152 10 and cycloocta-l,3,5,7-tetraene, 2.60 0.19 the indicated errors are two least-squares standard deviations and do not include the uncertainties in the rate constants for the reference alkenes (propene, but-l-ene, d.s-but-2-ene, trans-but-2-ene, 2-methylbut-2-ene, and terpinolene). These rate data were compared with the few available literature data, and the effects of methyl substitution have been discussed.50... 

Frish and Kraulinya and, most recently, by Czajkowski, Skardis, and Krause [71] and Czajkowski, Krause and Skardis [96]. Frish and Bochkova [97, 98] studied excitation transfer from the 6 aPr and 6 aP0 mercury atoms excited by collisions with electrons in a discharge, to various states in sodium. Kraulinya [99] optically excited the Hg(6 aPJ state and followed the excitation transfer to sodium by monitoring the intensities of the collisionally sensitized sodium lines. Her results which are quoted within 30% — 50% are summarized in Table 4.5 and are compared with the cross sections determined by Czajkowski, Skardis and Krause [71], The considerable discrepancies between the two sets of results are apparently due to errors arising from the trapping of mercury resonance radiation [100, 28] which must have particularly affected Kraulinya s results, and from the uncertainty in the determination of the mercury and sodium vapor densities in the binary mixture. 

How does the order of magnitude of the error introduced into the experimental result by the assumption of the perfect-gas law in Eq. (VI-8) compare with the uncertainties inherent in the measurements in this experiment What is the magnitude of the uncertainty introduced by lack of knowledge of the specific heat of the sample Does your A//value pertain to the initial or the final temperature ... 

If more accurate wavelength measurements have been made with a spectrometer, repeat the calculations and compare the uncertainties with those of the simpler diffraction experiment described above. Using your best frequency results, solve the three force constant equations for k, and k i. You will find two possible sets of these constants since the solution of the equations yields a quadratic expression. Choose between these two by recognizing that the interaction constant k i is usually much smaller than k or k Note if the frequencies used are appreciably in error, the solution may yield imaginary roots.) Typically the force constants for single, double, and triple CC bonds are about 500,1000, and 1500 N m respectively. What does your value of A, imply about the CC bond type in benzene ... 

A commonly recommended and accepted way of checking accuracy is the analysis of CRMs along with routinely analyzed samples. The RM(s) applied for this purpose should obviously be as similar as possible in both the type of matrix and content of determined element(s). In this case, it can be assumed that the possible systematic errors resulting from matrix effects, spectral interferences, etc. are the same or at least similar in the CRM and in the analyzed samples. The result of measurement of f/m can be then compared with the certified value Aj-ef Cref, where f/m and (/ref expanded uncertainties of the measurement result and certified value, respectively. Good agreement of the result of measurement and the certified value confirms correctness of analysis. Taking into account numerical values, the following condition should be fulfilled [60] ... 

The shifts of the xenon 3d./, binding energies are, with one exception, well within experimental error of the values determined by Karlsson, Siegbahn, and Bartlett. For XeF, we have found a value of 7.64 0.04 eV compared with their value of 7.88 0.18, the difference being only sightly more than the combined uncertainties. 

Equation (7-4) indicates that the solubility product includes an activity-coefficient term, a term which has been assumed to be unity up to this time. The introduction to this chapter pointed out that errors arising from neglect of the effects of the activity coefficient are usually small when compared with several uncertainties or side reactions. The activity coefficient in Equation (7-4) depends on the kind and concentration of all electrolytes in solution, not merely those involved directly with the precipitate. The correction to solubility calculations that must be made to account for the activity-coefficient effect is known as the diverse ion effect. The appropriate background is discussed in Chapter 2, and Problems 2-1,2-2, and 2-3 are examples of the calculations. For 1 1 electrolytes in solution, activity coefficients can usually be assumed to be unity when concentrations are much less than 0.1 M. Common ion and diverse ion effects can be significant at the same time, for example, when a large excess of common ion is added in a precipitation. The diverse ion effect is one of the reasons that the haphazard addition of a large excess of precipitant should be avoided. 

The CCSD(T)//MP4(SDQ)/aug-cc-pVTZ method, with the extrapolation to the complete basis set limit using the aug-cc-pVTZ and aug-cc-pVQZ basis sets, yielded a standard deviation of 0.58kcal/mol when compared to a select set of experimental values of gas-phase deprotonation reactions compiled in the NIST online database, a data set with uncertainty of values extremely useful in determining accurate pKa calculations and will be referenced throughout the review. 

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