Error correlated

Krug et al. have shown that a plot of AG against AH [see Eq. (7-75)] is not subject to the error correlation problem of the AH vs. AS plot provided AG is evaluated at the harmonic mean temperature of the experimental range. (The harmonic mean is the reciprocal of the mean of the reciprocals, i.e., = <1/... 

Earlier analyses making use of AH vs. AS plots generated many p values in the experimentally accessible range, and at least some of these are probably artifacts resulting from the error correlation in this type of plot. Exner s treatment yields p values that may be positive or negative and that are often experimentally inaccessible. Some authors have associated isokinetic relationships and p values with specific chemical phenomena, particularly solvation effects and solvent structure, but skepticism seems justified in view of the treatments of Exner and Krug et al. At the present time an isokinetic relationship should not be claimed solely on the basis of a plot of AH vs. A5, but should be examined by the Exner or Krug methods. 

Results are shown in Table 3. In all runs methanol has been detected as a sole product and its amount within the experiment error correlates with that of CH4 consumed. It is of interest that methane reaction with a-oxygen ... 

Error correlations are often ignored both mathematically and graphically in the treatment of °Th/U data. Nonetheless, error correlations can be as important a parameter in error propagation and evaluation as the errors themselves, and must be considered in all cases. Though sometimes approached with trepidation, the significance of error correlations is easily understood at an intuitive level and demonstrated visually (Fig. 1). Basic examples of procedures for calculating error correlations are given in Appendix I. 

The important, intuitively understandable properties of error correlations are that ... 

The correlation coefficient ranges between -1 and +1. An error correlation of+1 indicates that the errors in both x and y for any particular measurement of an x, y pair will always be in the same direction (either both greater than, or both less than the true values). An error correlation of -1 indicates that the x error will always be in the opposite direction than the y error. An error correlation of zero indicates that the x- andy -errors are independent of one another. 

Figure 1. Results of Monte Carlo simulations for 1500 pairs of x, y points with a mean of 100, Ganssian errors of 10 (1 o), and four different x-y error correlations (p). Elhpses show 95% confidence hmits for the joint x-y distribution. Note that the ellipses extend farther than the 2o range of either the x or j errors themselves—a non-intuitive characteristic of joint distributions that arises because an x- (or y-) value deviating less than expected permits ay- (orx-) value that deviates more than expected.

Th/23Su 1(7 error 234U/238U la error Error correlation... 

Ludwig and Titterington 1994), from which one can calculate the expressions for the corresponding (first order) errors and error correlations,... 

These drawbacks of the Rosholt isochron largely disappear when used for igneous systems, where Th, being the most-abundant actinide, is measured with approximately equal or better precision than the other U-Th isotopes, thus minimizing error correlations. In fact, for data where all analyses have relatively high Osmond diagrams can... 

For TIMS or ICP-MS analyses, error correlations in the measured isotope ratios are almost always negligible, as the ion beam is always much larger that of and the Th ion-beam is almost always much larger than that of °Th (if the latter is not the case, recourse to isochrons is uimecessaiy). 

ThAJ age is calculated. Mathematically, however, the choice of ratios is unimportant, provided that there is a common-denominator isotope to all three axes, and the errors and error correlations for the data points are correctly calculated. Note that, unlike paired x-y isochrons, not only can the errors of the detrital-corrected Th/ U and be... 

Not including error correlations where significant (say >0.3) in isotope-ratio data tables ... 

Using error crosses as isochron data-point symbols where significant x-y error correlations exist ... 

The error correlation between two quantities can be determined empirically, from a number (N) of replicate analyses of pairs of the two quantities (say x and y), and evaluating the expression for the linear correlation (Rickmers and Todd 1967)... 

Fig. 11.6 A scattergram and regression line analysis correlating zone diameters and MICs. The breakpoints of susceptible (MIC <2.0 mg/L, zone diameter >21 mm) and resistant (MIC >8.0 mg/L, zone < 15mm) are shown by the dotted lines. For a complete correlation between MICs and zone diameter, all susceptible, intermediate and resistant isolates should fall in boxes A, B and C, respectively. Errors (correlations outside these boxes) occur (courtesy of Dr Z. Hashm i).

The time measure was plotted as a reciprocal so that a higher number was faster and to prevent skewing of the data by a single very slow performance. Thus, both a quicker performance and fewer errors correlated to more time exploring a novel object. 

This may have been the situation 20 years ago. Today it is pessimistic to a totally unwarranted degree. When based on carefully collected data and careful analysis, it frequently appears that the derived thermal parameters make good sense, in that the ellipsoids correlate well in orientation and magnitude with the chemical structure. Furthermore, there are cases in which x-ray and neutron data yield virtually identical thermal parameters. Some degree of caution in quantitative assessment of accuracy is nevertheless advisable. The reason is that the thermal parameters are sensitive to any systematic errors correlated with scattering angle, and such errors are quite difficult to assess. 

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