Relative errors

The analytical chemist often wishes to express the counting error relative to the amount present of the element sought in a simple case we have... 

On the other hand, Davies5 , studying the reaction of adipic add with 1,5-pentanediol in diphenyl oxide or diethylaniline found an order increasing slowly from two with conversion. From this result he concluded that Flory s1,252-254> and Hinshelwood s240,241 interpretations are erroneous. Two remarks must be made about the works of Davies5 experimental errors relative to titrations are rather high and kinetic laws are established for conversions below 50%. Under such conditions the accuracy of experimental determinations of orders is rather poor. 

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

They include simple statistics (e.g., sums, means, standard deviations, coefficient of variation), error analysis terms (e.g., average error, relative error, standard error of estimate), linear regression analysis, and correlation coefficients. 

With the triples correction added, the error relative to experiment is still as large as 15 kJ/mol. More importantly, we are now above experiment and it is reasonable to assume that the inclusion of higher-order excitations (in particular quadruples) would increase this discrepancy even further, perhaps by a few kJ/mol (judging from the differences between the doubles and triples corrections). Extending the coupled-cluster expansion to infinite order, we would eventually reach the exact solution to the nonrelativistic clamped-nuclei electronic Schrodinger equation, with an error of a little more than 15 kJ/mol. Clearly, for agreement with experiment, we must also take into account the effects of nuclear motion and relativity. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

Tab. 13.1 Maximum and average errors (relative to experiment) for CCSD(T) using an spdjg-type basis set in the determination of molecular properties for a number of small molecules containing first-row atoms and hydrogen. Data taken from Ref. [12].

The next step in the development of a compact cc basis set series for Y is the optimization of spd HF sets. These optimizations were carried out for the Ad 5s state only, which is consistent with earlier treatments of the 1st row transition metals. The s, p, and d HF functions were optimized separately and for each series a (MslSpl d)l[2s2p d ExtET set was used as a base set in the optimizations, i.e., [lp d[ m 2s d np, and [2s2p nd. Figure 2 displays the errors relative to the approximate HF limit for each series. For the ns series, the... 

Figure 2. Hartree-Fock errors relative to a (I4sl2pI0d) base set plotted against the number of functions, m, in (ms, 12p, lOd), (I4s,mp, lOd), and(I4s, I2p,md) expansions for the 5s 4d state of yttrium.

In Figures 8 and 9 are shown the data for the dependence of the characteristic film buildup time t on Apg and U. In accord with the model, t is found to be independent of U, with only a very weak dependence on Apg indicated. This latter result could in part be a function of experimental inaccuracy. The data reduction for t introduces no assumptions beyond that needed to draw the exponential flux decline curves such as those shown in Figures 2 and 3. However, an error analysis shows that the maximum errors relative to the exponential curve fits occur at the earlier times of the experiment. This is seen in the typical error curve plotted in Figure 10. The error analysis indicates that during the early fouling stage the relatively crude experimental procedure used is not sufficiently accurate or possibly that the assumed flux decline behavior is not exponential at the early times. In any case, it follows that the accuracy of the determination of 6f is greater than that for t. 

Level of Theory C H (A) Percent Error Ionization (eV) Percent Error Relative Time... 

As shown in Table 2, the inexpensive MMCC(2,3)/PT approach is capable of providing the results which are practically as good as the excellent MMCC(2,3)/CI results. In the case of the 2 S+ and 1 states, which have a strong double excitation character, causing the EOMCCSD approach to fail, the MMCC(2,3)/PT corrections to CCSD/EOMCCSD energies produce the results of the EOMCCSDT quality, reducing the 0.560 and 0.924 eV errors in the EOMCCSD results to 0.102 and 0.090 eV, respectively. For these two states, the errors relative to full Cl obtained with the noniterative MMCC(2,3)/PT approach are 2-3 times smaller than the errors obtained with the much more expensive and iterative CC3 method. For states such as 2 n, which have a partially biexcited character, and for states dominated by single excitations (3 1 11), the MMCC(2,3)/PT results are as... 

Fig. 3. Relation between the minimum number of units in a sample require for sampling errore (relative standard deviations in percentage) of 0.1 and 1 % (y-axis) and the overall composition of a sample (x-axis), for mixtures having two types of particles with a relative difference in composition ranging from 100 to 10% Reprinted wiA permission from W. E. Harris, B. Kratochvil, Analytical Chemistry, 46 (1974), p. 314. Copyright 1974 American Chemical Society...

Variance origin Total error Relative error ... 

Many analytical methods (Examples 2 and 4) cannot be directly or indirectly linked to Si-units with any of the available primary methods of measurement. These can only be used in exceptional cases for a limited range of rather simple problems. Hence, many methods of analysis rely on other methods for the assessment of their accuracy. The instrumentation and measurement process used must be transparent and a full account of sources of error, relative and absolute, must be made. Unfortunately, as Examples 2 and 3 show, with the growing complexity of analytical methods this transparency becomes a major headache for both metrologists and analytical chemists. The relationship between signals measured and the derived concentration becomes a complex calculation [6] and the result of a measurement in many methods can only be traceable to the instrument, its electronics and integrated software [33],... 

Table 3.4 Percent phenol conversion X, asymptotic values of concentration computed with the detailed and the reduced kinetic models for phenol, Ca.co, and trimethylolphenol, Cpi0o> and relevant percent errors relative to the detailed model used as reference (indicated by the superscript °)...

Random errors Relative standard deviation Robust variance Samples and populations Standard deviation of the mean (standard error of the mean)... 

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