Mean values

In ion chromatography data processing the arithmetic mean is used exclusively. The arithmetic mean value % of a random test with n individual measurements %i, %2..is calculated according to Eq. (223)  

Only data resulting from comparable measurements should be used to calculate the mean value. In general, the mean value should be based on at least three individual measurements. Significantly smaller or larger measured values must not be omitted, unless they are proved to be outliers. This strict rule can be softened if the measurements are carried out in one and the same laboratory. In this case, one deviating measured value can be replaced by three others. Mean value calculation is not allowed if the timely sorted measured values exhibit upward or downward trends. 

When a sample is repeatedly analyzed in the laboratory using the same analytical method, results collected will deviate from each other to some extent The deviations, representing a scatter of individual values around a mean value, are denoted as statistical or random errors, a measure of which is the precision. Deviations from the true content of a sample are caused by systematic errors. An analytical method only provides true values if it is free of systematic errors. Random errors make an analytical result less precise while systematic errors give incorrect values. Hence, the precision and accuracy of an analytical method have to be discussed separately. Statements regarding the accuracy are feasible only if the true value is known. 

For an assessment of empirical distributions, the data resulting from an experiment are characterized by numerical indicators. These include mean values and mean variations. A frequency distribution should be defined without these two quantities. Often, only mean values are stated in the result, which is not very expressive. For this reason, results should be supplemented by a statement of the statistical error. 

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To simulate noise of different levels The most unbiased noise was taken as white Gaussian distributed one. Its variance a was chosen as its main parameter, because its mean value equaled zero. The ratio of ct to the maximum level of intensity on the projections... 

The prior knowledge is assumed to be the discrete structure of the image, the statistical independence of the noise values, their stationarity and zero mean value. For this case, the image reconstruction problem can be represented as an adaptive stochastic estimation process [9] with the structure shown in Fig. 1. 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

The supersatiiration e = 8 i(ctj) is tlie mean value of hip, which reflects the presence of other subcritical clusters in the system. 

Note that the relations (23) are valid also if (22) is questionable. Brown [19] refined the approximation (23) by introducing the gn factor, describing the deviation of the mean values for Lj and fi om integers. Validity of the approximation (23) has been checked by means of explicit ab initio calculations, for example, in [20,21]. 

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. 

For a fluid, with no underlying regular structure, the mecin squared displacement gradually increases with time (Figure 6.9). For a solid, however, the mean squared displacement typically oscillates about a mean value. Flowever, if there is diffusion within a solid then tliis can be detected from the mean squared displacement and may be restricted to fewer than three dimensions. For example. Figure 6.10 shows the mean squared displacement calculated for Li+ ions in Li3N at 400 K [Wolf et al. 1984]. This material contains layers of LiiN mobility of the Li" " ions is much greater within these planes than perpendicular to them. 

Sometimes the quantities z and y will fluctuate about non-zero mean values (z) and (y) Under such circumstances it is typical to consider just the fluctuating part and to defint the correlation function as ... 

Equation (7.74) does not require the mean values (z) and (y) to be determined before th( correlation coefficient can be calculated and so values can be accumulated as the simulatioi proceeds. 

After a number, Ntj-iai, of iterations, the mean value of the potential energy would 1 calculated using ... 

Strictly, the mean observed values, (y), which appear in Equation (12.39) should corresponc to the mean of the values for each cross-vahdation group as appropriate rather than the overall mean value of the dependent variables, though often the mean of the entire date set will be used instead. 

Using the mean value theorem for definite integrals... 

Calculation can also explain why in some thiazole dyes vinyiene shift of the first two homologs is larger than the shift between higher members of the series, and also why wavelengths of absorption of nonsymmetrical dyes as calculated by the mean value rule differ from experimental data (6671. This deviation is caused by an interannular no-bond SS-interaction in the monomethine ion. 

The absolute energy of a molecule in molecular mechanics has no intrinsic physical meaning values are useful only for com-... 

Table 2.26a lists the height of an ordinate (Y) as a distance z from the mean, and Table 2.26b the area under the normal curve at a distance z from the mean, expressed as fractions of the total area, 1.000. Returning to Fig. 2.10, we note that 68.27% of the area of the normal distribution curve lies within 1 standard deviation of the center or mean value. Therefore, 31.73% lies outside those limits and 15.86% on each side. Ninety-five percent (actually 95.43%) of the area lies within 2 standard deviations, and 99.73% lies within 3 standard deviations of the mean. Often the last two areas are stated slightly different viz. 95% of the area lies within 1.96cr (approximately 2cr) and 99% lies within approximately 2.5cr. The mean falls at exactly the 50% point for symmetric normal distributions. 

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