Scattering Gaussian distribution

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. [Pg.193]

In Sect. 7.4.6, we discussed various stochastic simulation techniques that include the kinetics of recombination and free-ion yield in multiple ion-pair spurs. No further details will be presented here, but the results will be compared with available experiments. In so doing, we should remember that in the more comprehensive Monte Carlo simulations of Bartczak and Hummel (1986,1987, 1993,1997) Hummel and Bartczak, (1988) the recombination reaction is taken to be fully diffusion-controlled and that the diffusive free path distribution is frequently assumed to be rectangular, consistent with the diffusion coefficient, instead of a more realistic distribution. While the latter assumption can be justified on the basis of the central limit theorem, which guarantees a gaussian distribution for a large number of scatterings, the first assumption is only valid for low-mobility liquids. [Pg.300]

The form factor term, P(q), contains information on the distribution of segments within a single dendrimer. Models can be used to fit the scattering from various types of particles, common ones being a Zimm function which describes scattering from a collection of units with a Gaussian distribution (equation (3a)), a... [Pg.259]

If subsequent analyses of the bulk sample deviate by more than a predetermined amount, the whole batch of results is rejected. Results are thus only accepted if they fall between specified values of s above and below the mean, where Is includes 68%, 2s includes 95% (the normally accepted value), and 3s includes 99.7% of results. The scatter of results usually assumes a symmetrical normal or Gaussian distribution about the mean, as shown in Figs 12.1 and 12.2. [Pg.201]

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... [Pg.262]

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... [Pg.21]

The structure factor S(K) is independent on the scattering power of individual atoms and depends only on the structure of the investigated sample. The experimental RRDF provides information about the probability of finding an atom in a spherical shell at a distance r fi om an arbitrary atom. Successive peaks correspond to nearest-, second- and next-neighbour atomic distribution. Assuming three-dimensional Gaussian distribution of the interatomic distances with standard deviation a. The d(r) function can be finally expressed as follows [11,12] ... [Pg.563]

The broadening values obtained for methane diffusion can be fitted to a molecular jump model with a Gaussian distribution of jump lengths (25,62). Thus, the scattering law becomes a Lorentzian,... [Pg.367]

If X and Sx are determined from a set of normal process runs and a subsequently measured value of X falls more than 2sx away from X, the chances are that something has changed in the process—there is less than a 10% chance that normal scatter can account for the deviation. If the deviation is greater than 3sx, there is less than a 1% chance that normal scatter is the cause. The exact percentages depend on how the measured values are distributed about the mean—whether they follow a Gaussian distribution, for example—and how many points are in the data set used to calculate the mean and standard deviation. [Pg.31]

The probability figures for the Gaussian distribution calculated as areas in Feature 6-2 refer to the probable error for a single measurement. Thus, it is 95.4% probable that a single result from a population will lie within 2cr of the mean jx. If a series of replicate results, each containing N measurements, are taken randomly from a population of results, the mean of each set will show less and less scatter as N increases. The standard deviation of each mean is known as the standard error of the mean and is given the symbol Sy . The standard error is inversely proportional to the square root of the number of data points N used to calculate the mean, as given by Equation 6-6. [Pg.117]

Equation (8.159) is strictly valid for a Gaussian distribution of electric fields. The electric field autocorrelation function is related to the dynamic structure factor S q, t) [compare it with the static scattering function S q) in Eq. (3.121)] ... [Pg.348]

Standard deviation a measure of the scatter in a set of data that fits a Gaussian distribution see also Pooled standard deviation. [Pg.386]

For a Gaussian distribution of the scattered light intensity profile, the intensity ACF, g(2)(xcorr, 0), is related to the electric field ACF, xcorr, 9), via the Siegert relation ... [Pg.444]

Further modifications have to be introduced in order to describe correctly the observed intensity decay for both signs of the scattering angle. In Eq. (10),/(D ) represents a Gaussian distribution of Dy around the mean value Dyo, which was obtained from the parabolic fit. Nl/r is a sign-dependent scaling factor, and in addition a roughness a is introduced. [Pg.431]

If the distribution of particle displacements [rk t) — rt(0) or rk t) f (0)] are described by a Gaussian distribution, then the incoherent and coherent intermediate scattering functions can be written, respectively, in terms of only as ... [Pg.112]

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