Maximum probable error, averages

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... 

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. 

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

Arylcarbonyl Compounds as Initiators for Unsaturated Polyester/ Styrene Copolymerization Systems. Gel Time Determination and Reaction Curves. Ten blanks (gel time of 20 grams Vestopal A without initiator) gave a mean deviation from the average of 3.3% and a maximum deviation of 7.2%. Five measurements with l-phenyl-2-propanone as initiator gave a mean deviation from the average of 3.3% and a maximum deviation of 7.7%. An experimental error of 10% was therefore assumed and proved correct by spot checks. The exceptions (not used in the discussions) are most probably caused by the insolubility of the initiators in the reaction medium. 

When characterizing particulate matter of unknown composition, it is necessary to assume a value for the index of refraction to infer the diameter from a measured intensity ratio. This causes inherent uncertainties in any reported size distribution unless all particles are of a known and uniform composition. In the case of automobile exhaust particles, the composition is certainly unknown and would probably include some combination of carbon particles and lead halides condensed on nuclei. Figure 1 is indicative of expected variations from such a spread of particle compositions. In the range of low a, the characteristic curve for nonabsorbing particles oscillates around an average value which is approximately the n = 1.57 — 0.56i data. Thus the intensity ratio curve for the absorbing soot is a convenient one to assume as the calibration standard for automobile exhaust particulates. Here this assumption results in a maximum error of approximately 30% when measuring particles of unknown composition. 

Table II Summary of the B-sheet propensity data Tm is the midpoint of the thermal denaturation transition. Also shown are the Pb values for the probability of occurrence of each amino acid in B-sheet in proteins of known structure (12). A AAG value is reported for each mutant that is calculated relative to AG Ala GBi1X=AGB1X AGbia- This treatment assumes that within the transition region, A// is independent of temperature. Accordingly, AAG values are reported at a temperature that is within the transition region for all the mutants (60 C). For the standard, BIA, AG = -0.24 kcal/mol at 60 C. Experiments were performed in triplicate, and the results were averaged. The maximum error in the Tm is 0.4 C and in AG to be less than 5%.

Numerical evaluation of the Kassel integral permitted a comparison between theoretical and experimental fall-off behaviour . With an average molecular diameter of 5.5 A the calculated rate coefficient-azoethane pressure curve showed the best agreement with experiment at an effective number of oscillators of 18, somewhat less than half of the maximum 2N— 6. Because of the complexity of the reaction the experimental curve is probably in error, rendering comparison unreliable. Similar calculations for azomethane using the earlier uninhibited kinetic data showed best agreement with experiments at a molecular diameter of 4.7 A and an effective number of oscillators of 12, one half of the total normal modes of vibrations. 

Fortunately, however, the states that are sampled are the most probable states of the system which make the maximum contribution to the thermodynamic quantities. Thus, we obtain the average values (entropy, specific heat) of the thermodynamic quantities with virtually no error. However, to achieve that kind of accuracy, our system still needs to sample a large number of representative states. The sampling of such a large number of states takes a longer time (due to slow relaxation dynamics) at low temperatures, and the utmost care is needed. 

The thick sohd Hne is what would occur if 6 perfectly controlled the error in the reaction probabilities. The solid line with squares is the average of the error, and the dashed line with circles is the maximum error from the systems studied all as a function of S. We see the remarkable result that, even in the worst case of maximum error, 6 reliably controls the error in the reaction probability. Thus, even if we were studying a system in resonance (i.e, a small effective absorption rate F) requiring a larger TabCi would not change. This kind of control is an important aspect of any numerical method, i.e. that one be able to determine a priori how accurate the calculation is and consequently how much computational effort is required. 

In this relation qi is the (average) probability that a particular monomer in the replicated sequence is the correct one. The probability of a perfect replication of a sequence of length Ni therefore is q. This number by itself is less than unity and therefore there must be another factor,, outweighing replication errors. iV, is a measure for the competitive advantage of sequence i. According to Eq. (7.139) the maximum possible length of a sequence satisfying this criterion is... 

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