Statistics populations

It is clear that the nature of the electromagnetic phenomena is the same for optics and radio wave, the only experimental differences being that radiowave photons are far below the spectral density of noise of actual detectors and that the temperature of the source is such that each mode is statistically populated by many photons in the radio wave domain whereas the probability of presence of photons is very small in the optical domain. 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

The rotational population distributions were Boltzmann in nature, characterized by 7Ji = 640 35 K. This seems substantially lower than yet somewhat larger than the temperature associated with the translational degree of freedom. The lambda doublet species were statistically populated. The population ratio of i =l/t =0 was roughly 0.09, consistent with a vibrational temperature Ty— 1120 35K. The same rotational and spin-orbit distributions were obtained for molecules desorbed in t = 1 as for f = 0 levels. Finally, there was no dependence in the J-state distributions on desorption angle. 

In some textbooks, a confidence interval is described as the interval within which there is a certain probability of finding the true value of the estimated quantity. Does the term true used in this sense indicate the statistical population value (e.g., p if one is estimating a mean) or the bias-free value (e.g., 6.21% iron in a mineral) Could these two interpretations of true value be a source of misunderstanding in conversations between a statistician and a geologist ... 

Therefore a particular method was chosen (4). We worked on a statistical population of crystals in order to minimize the dispersion and on simultaneous measurement of all faces in order to compare their growth rate under the same conditions of supersaturation and temperature. Therefore classical (R,o ) isotherms were obtained. Experimentally we grew at the same time and in the same solution a single crystal and twin. Whereas growth rate measurements of the forms hOL are relatively simple (thanks to the fact that the b axis is a binary axis) (Figure lb), the kinetic measurements of the p 110 and p llO forms are more difficult. 

If the assessment endpoint is a distribution, or a statistic from a distribution (e.g., 95th percentile), it is essential to be clear how the distribution is interpreted (Suter 1998, p 129). If it is a frequency distribution, to what statistical population does the distribution refer For example, does the distribution represent a population of individuals, an assemblage of species, a number of locations treated with pesticides, or a series of time periods The answer to this question has substantial implications for the structure of the assessment model and the types of data required. 

If the number of measurements is very large (in the case of a statistical population), x becomes the real mean, p, which will be identical to the exact value x0 in the absence... 

The square root of the variance is called the standard deviation, designated either as s when the number of measurements n is small or as a when a significantly large statistical population is available. The standard deviation s, an indication of the dispersion of a group of measurements, is thus expressed in the same units as x. 

The derivation of equations like (29) and (30) for ktt relies on the use of equilibrium statistical mechanics to calculate statistical populations amongst various vibrational distributions. Electron transfer is assumed to be a slow process on the timescale for vibrational equilibration. In that limit, electron transfer occurs occasionally from a vibrational distribution of the reactants to a vibrational distribution of the products and the population of the initial distribution is rapidly reestablished. However, for vibrational levels near the intersection region, this assumption may not be valid and, in general, it may not be valid for any case where electronic coupling is significant. 

The quantum mechanical result predicts a complex temperature dependence for kobs arising from the statistical population factors in equation (28). However, in the limit that the classical approximation works reasonably well, and assuming that the temperature dependence of vet is negligible, kobs is predicted to vary with T as shown in equation (43). 

If an infinite number of identical, quantitative measurements could be made on a biosystem, this series of numerical values would constitute a statistical population. The average of all of these numbers would be the true value of the measurement. It is obviously not possible to achieve this in practice. The alternative is to obtain a relatively small sample of data, which is a subset of the infinite population data. The significance and precision of these data are then determined by statistical analysis. 

Connor (1997) summarized a collection of Values from CD literature. Treated as statistical populations, the complex stabilities appear to be reasonably described as normal distribution in log Kii, with the mean log value equal to 2.11, 2.69, and 2.55 for, p- andy-CDs, respectively. 

The planning team specifies the parameter of interest, which is the statistical parameter that characterizes the population. The parameter of interest provides a reasonable estimate of the true contaminant concentration, and it may be the mean, median or percentile of a statistical population. 

We need to make a decision related to the disposition of soil that has been excavated from the subsurface at a site with lead contamination history. Excavated soil suspected of containing lead has been stockpiled. We may use this soil as backfill (i.e. place it back into the ground), if the mean lead concentration in it is below the action level of 100 milligram per kilogram (mg/kg). To decide whether the soil is acceptable as backfill, we will sample the soil and analyze it for lead. The mean concentration of lead in soil will represent the statistical population parameter. 

While in the photodissociation of H2O through the AlB state the two possible A-doublets of OH(2n) are populated in a highly nonstatistical way, the two spin-orbit states, 0H( n ) and 0H( n3/2), are perfectly statistically populated. Unlike for the A-doublets there is a priori no geometrical reason to expect a difference in the spin-orbit states other than that given by the 2j + 1 statistical weighting factor. Since j = N + 1/2 for 2n3/2 and j = N — 1/2 for 2n1/2, the statistical weighting factor is (N + l)/N. Therefore, the population ratio 2n3/2 / 2n1/2 multiplied by N/(N-1-1) must be 1 for a statistical distribution as it is indeed measured in the bulk, in the beam, as well as in the dissociation of single rotational states of H2O (Andresen and Schinke 1987). The reason for the statistical... 

We can confirm this point of view by looking at paramagnetic shifts and rs (for water) in the complexes of supposedly different water coordination. We find that in the Ln(III) series of complexes, LnY, anomalies often occur at around Tm(III) especially. We conclude that the exact structures of Ln(III) complexes like those of Na(I), K(I) and Ca(II) can not be represented by single simple pictures and we must refer to statistical populations of complexes of different structure at equilibrium. The complexity of the structural and dynamic features of this A-subgroup chemistry has been used to functional advantage both by man in cements and plasters and by biology in shells and bones and in messengers. 

The reaction H 4- C102 has only been studied by laser-induced fluorescence detection yielding (Fv (OH)> 0.37 and (FR (OH)> 0.26, leaving 0.37 of the reaction energy to be distributed between translation and internal excitation of CIO. Again, a statistical population of spin doublets was found and a slight preference for the Il+ X-doublet but not as marked a preference as for H + NOz. 

There remain residuals Yi f, which we may subject to statistical analysis. To reduce these to the same statistical population we define a weighted residual 5 ... 

Random sample A sample selected from a statistical population such that each individual has an equal probability of being selected (USEPA, 1992a, 1997c). 

N to O along the amide chain in the temperature range between 15 and 295 K. The authors estimate that any such tautomerization would account for no more than 1% of the total statistical population of the crystal. 

Initially, it was observed that successive analyses were essentially identical and showed little variation with time. The concentration levels of zinc, lead, and "copper (contaminated with silver) remained relatively consistent for six days, declined over a period of about a day and then assumed a new low level for one more week. The results of the observations made at the Scripps pier are summarized in Table I. Only those determinations made by standard addition are included in the table values. Thus, the data represent approximately 50 individual measurements. A study of Table I reveals that the measurements in die period 8-10 to 8-15 belong to a different statistical population than the measurements made during 8-17 to 8-24. Within each population each... 

One can therefore expect that monomeric reactions, which result in a non-statistical population among the vibrational states, will be strongly affected by the formation of complexes. In these cases one may observe a thermal-ization of the vibrational energy distribution. This is an example where even a single spectator can simulate solvent effects by providing low-frequency vibrations which behave like phonons in the condensed phase. 

The headspace sampling technique can yield useful results if sufficient numbers of samples can be collected to use statistical populations to suggest anomalous areas. One should always exercise caution, however, with respect to characterisation of gas composition, since evaporation during the collection stage always occurs, resulting in the relative depletion of the lighter gases. 

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