Error, probable

Gani" derived a group contribution method that shows promise. Initial evaluations show average errors of 5 to 10 percent (10-30 K) on a wide variety of compounds, but larger errors can occur. It is recommended that severaf compounds of Known melting point in the same or a similar family be predic ted in order to estimate the probable error. 

Order-of-magnitude estimate (ratio estimate). Rule-of-thumb method based on cost data for previous similar types of plant probable error within 10 to 50 percent. 

Study estimate (factored estimate). Better than order-of-magnitude requires knowledge of major items of equipment used for feasibihty surveys probable error up to 30 percent. 

Preliminaiy e.stimate (budget-authoiization e.stimate). Requires more detailed information than study estimate probable error up to 20 percent. 

Definitive e.stimate (project-control e.stimate). Based on considerable data prior to preparation of completed drawings and specifications probable error within 10 percent. 

Detailed e.stimate (firm or contractor s e.stimate). Requires completed drawings, specifications, and site suiweys probable error within 5 percent. 

Finally, force field methods are zero-dimensional . It is not possible to asses the probable error of a given result within the method. The quality of the result can only be judged by comparison with other calculations on similar types of molecules, for which relevant experimental data exist. 

A. Standard series method (Section 17.4). The test solution contained in a Nessler tube is diluted to a definite volume, thoroughly mixed, and its colour compared with a series of standards similarly prepared. The concentration of the unknown is then, of course, equal to that of the known solution whose colour it matches exactly. The accuracy of the method will depend upon the concentrations of the standard series the probable error is of the order of + 3 per cent, but may be as high as + 8 per cent. 

The roughly estimated intensity values for GuKa reflections reported by Hassel and Luzanski are sufficient to permit the evaluation of w and u with a probable error of about 0.005 and of v and z to about 0.010. 

All of the observed reflections could be indexed on the basis of a cubic unit cell with Oo = 11.82 A the estimated probable error is 0.01 A. The only systematic absences were hhl with l odd this is characteristic of the space group 0 -PmP>n, which also was reported by von Stackelberg from his single-crystal work on sulfur dioxide hydrate. For 46 H20 and 6 Cl2 in the unit cell the calculated density is 1.26 densities reported by various observers range from 1.23 to 1.29. 

The agreement between doaic4. and d0b . given in Table IV of the previous research is such as to indicate a probable error somewhat greater than 0.1% in the spec-trometric observations reported. 

The reliability factor B was 0276 after the first refinement and 0-211 after the fourth refinement. The parameters from the third and fourth refinements differed very little from one another. The final values are given in Table 1. As large systematic errors were introduced in the refinement process by the unavoidable use of very poor atomic form factors, the probable errors in the parameters as obtained in the refinement were considered to be of questionable significance. For this reason they are not given in the table. The average error was, however, estimated to be 0-001 for the positional parameters and 5% for the compositional parameters. The scattering power of the two atoms of type A was given by the least-squares refinement as only 0-8 times that of aluminum (the fraction... 

Because of the difficulty of obtaining satisfactory photometer records of electron diffraction photographs of gas molecules, we have adapted and extended the visual method to the calculation of radial distribution curves, by making use of the values of (4t sin d/2)/X obtained by the measurement of ring diameters (as in the usual visual method) in conjunction with visually estimated intensities of the rings, as described below. Various tests of the method indicate that the important interatomic distances can be determined in this way to within 1 or 2% (probable error). 

The quantitative comparison of measured. To values and 5 values calculated for models A for mesitylene and hexamethylbenzene is given in Tables XII and XIII. With omission of the innermost maximum and minimum, the unreliable third minimum, and the very weak fifth maximum for each substance, the average values sa/s0 = 1.000 and 1.002, respectively, are found. These correspond to the interatomic distances C — = 1.54 0.01 A. and Car-Car = 1.39 A. in both substances, the 0 —Qa distance being equal to the single-bond distance in aliphatic compounds and the Car—distance to that in benzene to within the probable error of the determination. 

Saunders states. Assuming that the valence electrons at the top of the band have the average hybrid character 3d34s4p2, the interaction energy of one of these valence electrons and an atomic electron, assumed to be approximately a 3d electron, is found to be —2707 cm-, or —0.334 ev, with probable error about 10%. 

Probable errors in assigning the integral distribution curve, as indicated by scatter of the points in Fig. 57, are magnified in the process of taking the slope for the deduction of the differential distribution. Only the approximate location of the maximum and breadth of the latter are experimentally significant. 

One measure of the quality of an estimate of an average Is the confidence limits (or maximum probable error) for the estimate. For averages of Independent samples, the maximum probable error Is... 

Equation 4 also can be evaluated using the nomograph. For example, to determine the maximum probable error that will occur with 95% probability based on n 4 tests when o 20 ppm, first find the point where the diagonal and the line through n 4 and o 20 intersect then extend the line through this point and P 95% to find E 19.6 ppm. 

Student5 (1908) The probable error of a mean. Biometrika 6 1 Thompson M (1995) Uncertainty in an uncertain world. Analyst 120 117N... 

One then proceeds to calculate a value of the rate constant for each pair of points separated by a time A [i.e., a value is calculated from the points corresponding to (0 and A), and (A + tj, 2ti and (A + 2tfj, etc.]. The arithmetic mean of these values is a good representative value of the rate constant. In this technique each data point is used once and only once, and the probable errors of the quantities that are averaged are all of the same order of magnitude. For the first-order case it is apparent from equation 3.1.8 that the average value of the rate constant is given by... 

We keep learning more about the history of noise calculations. It seems that the topic of the noise of a spectrum in the constant-detector-noise case was addressed more than 50 years ago [1], Not only that, but it was done while taking into account the noise of the reference readings. The calculation of the optimum absorbance value was performed using several different criteria for optimum . One of these criteria, which Cole called the Probable Error Method, gives the same results that we obtained for the optimum transmittance value of 32.99%T [2], Cole s approach, however, had several limitations. The main one, from our point of view, is the fact that he directed his equations to represent the absorbance noise as soon as possible in his derivation. Thus his derivation, as well as virtually all the ones since then, bypassed consideration of the behavior of noise of transmittance spectra. This, coupled with the fact that the only place we have found that presented an expression for transmittance noise had a typographical error as we reported in our previous column [3], means that as far as we know, the correct expression for the behavior of transmittance noise has still never been previously reported in the literature. On the other hand, we do have to draw back a bit and admit that the correct expression for the optimum transmittance has been reported. 

The volume fraction of each phase was taken from the fractional area in the transmission electron micrographs. Combined with the values shown in Table 1, the compositions within each phase were calculated and are shown in Table 2. Overall, the results suggest variably 0-20% actual molecular mixing. Noting the probable errors in estimating the experimental Tg s, the and W2 values are probably correct to within +0.05. Thus mixing plays an important role in interpenetration and influences the reinforcement within each phase. 

Probable errors are 0.2 Hz or less for all values reported here except the 13C-F couplings for which the probable errors are 0.4 Hz. b) Values reported by Frankiss, Ref. 63, for 95 % solutions in cyclohexane. 

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