Population, statistical

RRKM theory assumes a microcanonical ensemble of A vibrational/rotational states within the energy interval E E + dE, so that each of these states is populated statistically with an equal probability [4]. This assumption of a microcanonical distribution means that the unimolecular rate constant for A only depends on energy, and not on the maimer in which A is energized. If N(0) is the number of A molecules excited at / =... 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

Brenner S E, C Chothia and T ] P Hubbard 1997. Population Statistics of Protein Structures Lessons from Structural Classifications. Current Opinion in Structural Biology 7 369-376. 

These equations apply when an entire population is available for measurement. The most common situation in practical problems is one in which the number of measurements is smaller than the entire population. A group of selected measurements smaller than the population is called a sample. Sample statistics are slightly different from population statistics but, for large samples, the equations of sample statistics approach those of population statistics. 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

Bias The systematic or persistent distortion of an estimate from the true value. From sampling theory, bias is a characteristic of the sample estimator of the sufficient statistics for the distribution of interest. Therefore, bias is not a function of the data, but of the method for estimating the population statistics. For example, the method for calculating the sample mean of a normal distribution is an unbiased estimator of the true but unknown population mean. Statistical bias is not a Bayesian concept, because Bayes theorem does not relay on the long-term frequency expections of sample estimators. 

Sampling error In surveys, investigators frequently take measurements (or samples) on the parameters of interest, from which inferences to the true but unknown population are inferred. The inability of the sample statistics to represent the true population statistics is called sample error. There are many reasons why the sample may be inaccurate, from the design of the experiment to the inability of the measuring device. In some cases, the sources of error may be separated (see Variance components). 

Some typical CDFs are now discussed. The data stem from a sample of 48,312 organic crystal structures extracted from the CSD, each diagram being built on sets of 50,000-200,000 distance data. These very high numbers are indispensable small-population statistics is the most underhand enemy of the physical and social sciences. 

Brenner, S.E., Chothia, C., Hubbard, T.J.P. (1997) Population statistics of protein structures lessons from structural classifications. Curr. Opin. Struct. Biol. 7, 369-376. 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

Uncertainty associated with randomly drawing samples from a population. Statistics, such as the mean, estimated from a random sample are subject to random fluctuations when estimated repeatedly for independent sets of samples from the same population. 

In applications dealing with individualized cells, only those fully in view can be analyzed. The largest objects have, proportionally, a higher risk in contact with the image frame and to be discarded, which will ultimately introduce a bias in the population statistics. Some authors suggest a measuring frame be inserted into the visible field [41] or a correction factor, function of the size of the feature relative to the size of the field of view, be applied to the measured size or shape parameter [62]. Border-killing is also used to remove artifacts such as the halo of Fig. 9 a. 

Meijer BC, Wilkinson MHF (1998) Optimized population statistics derived from morphometry In Wilkinson MHF, Schut F (eds) Digital Image Analysis of Microbes. Wiley, New York, p 225... 

When the states are populated statistically, the ratio of the concentrations is just the ratio of the density of states, which may be written down using eqn.(3)... 

For this purpose, we will organize our analysis on the basis of the nations which, during World War Two, came under German rule either in whole or in part, and we will examine the fluctuations exhibited by the Jewish population statistics there. The sequence of the nations corresponds on the whole to that used in Benz s work, where only these countries are dealt with. In comparison, Sarming incorporates more extensive demographic observations, taking into account non-European nations as well, for which reason no strictly defined sequence of nations under German rule can be maintained in his work. 

For each nation or group of nations we shall first give a brief tabular overview of the Jewish population statistics as given in each work. Only where the data given in the two books are at considerable odds will reference to the soundness of the data and their calculation be made in order to determine which author s arguments are better. The reliability of the sources cited by the authors will also be touched on only in cases of dispute. 

Richard Korherr was the leading statistician of the Third Reich. In early 1943, on Himmler s instructions, he drew up a report on the trends which European Jewish population statistics had exhibited since the NS had come to power. Himmler wanted to submit this report to Hitler. After several discussions and some correspondence with Himmler, Korherr revised and shortened his first report.82 These two reports as well as the correspondence that goes with them are counted among the allegedly central pieces of evidence proving the Holocaust, on whose basis G. Wellers, for example, believes he can set the number of victims of the Holocaust at approximately 2 million by late March 1943 alone.83... 

United Nations (1996). Country Population Statistics and Projections 1950—2050. Report. Food and Agricultural Organization of the United Nations, Rome, Italy. 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

Population statistics, e.g. birth defect registers and cancer registers. These are insensitive unless a drug-induced event is highly remarkable or very frequent. 

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