Dispersion of a distribution

I. A measure, symbolized by square root of the variance. Hence, it is used to describe the distribution about a mean value. For a normal distribution curve centered on some mean value, fjt, multiples of the standard deviation provides information on what percentage of the values lie within na of that mean. Thus, 68.3% of the values lie within one standard deviation of the mean, 95.5% within 2 cr, and 99.7% within 3 cr. 2. The corresponding statistic, 5, used to estimate the true standard deviation cr = (2(Xi - x) )/(n - 1). See Statistics (A Primer)... 

Often the ratio (or Q) - M /M (dispersity ) is used to describe the broadness of a molar mass distribution. In principle, each ratio of mean values can be used to characterize the dispersity of a distribution, e.g. which results from eq. (4.1.3). 

Another parameter is called the standard deviation, which is designated as O. The square of the standard deviation is used frequently and is called the popular variance, O". Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean [L. If the spread is broad, then the standard deviation will be larger than if it were more constrained. 

The horizontal dispersion of a plume has been modeled by the use of expanding cells well mixed vertically, with the chemistry calculated for each cell (31). The resulting simulation of transformation of NO to NO2 in a power plant plume by infusion of atmospheric ozone is a peaked distribution of NO2 that resembles a plume of the primary pollutants, SO2 and NO. The ozone distribution shows depletion across the plume, with maximum depletion in the center at 20 min travel time from the source, but relatively uniform ozone concentrations back to initial levels at travel distances 1 h from the source. 

If one does not use the short gradient pulse (SGP) approximation, the term A has to be substituted with (A 8/3). In the case of a mono-disperse system, the plot of ln(E) versus y2g282A is a straight line having the absolute value of the slope equal to the self-diffusion coefficient. For polydisperse sample, the signal intensity decay can be interpreted in terms of a distribution of diffusing species ... 

Bias is a measure of trueness . It tells us how close the mean of a set of measurement results is to an assumed true value. Precision, on the other hand, is a measure of the spread or dispersion of a set of results. Precision applies to a set of replicate measurements and tells us how the individual members of that set are distributed about the calculated mean value, regardless of where this mean value lies with respect to the true value. 

An aqueous dispersion of a disperse dye contains an equilibrium distribution of solid dye particles of various sizes. Dyeing takes place from a saturated solution, which is maintained in this state by the presence of undissolved particles of dye. As dyeing proceeds, the smallest insoluble particles dissolve at a rate appropriate to maintain this saturated solution. Only the smallest moieties present, single molecules and dimers, are capable of becoming absorbed by cellulose acetate or polyester fibres. A recent study of three representative Cl Disperse dyes, namely the nitrodiphenylamine Yellow 42 (3.49), the monoazo Red 118 (3.50) and the anthraquinone Violet 26 (3.51), demonstrated that aggregation of dye molecules dissolved in aqueous surfactant solutions does not proceed beyond dimerisation. The proportion present as dimers reached a maximum at a surfactant dye molar ratio of 2 5 for all three dyes, implying the formation of mixed dye-surfactant micelles [52]. 

The chemistry and physics of the vehicle and its components, including factors such as polarity, molecular weight or molecular weight distribution, and viscosity. The dispersibility of a system which contains several different components such as resins depends on the solubility of each individual component in the medium and their compatibility with each other. 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

One major task of statistics is to describe the distribution of a set of data. The most important characteristics of a distribution are the location, the dispersion, the skewness and the kurtosis. These are discussed in the following slides. 

The most important measure for the dispersion of a data distribution is the variance. The variance of a population (with all possible data being known) is the mean of the squares of the deviations of the individual values from the population mean. 

A statistical measure of the dispersion of the distribution of the random variable, typically obtained by taking the expected value of the square of the difference between the random variable and its mean. The variance is the square of the standard deviation. See Statistics (A Primer)... 

It has already been stated that the retention of a solute depends on the magnitude of the distribution coefficient of the solute between the mobile and stationary phases. Furthermore, according to Vant Hoff s Law, the distribution coefficient will vary according to the exponent of the reciprocal of the absolute temperature. In addition, the dispersion of a solute band in a column will be shown to depend on the dlffusivity of the solute In both phases, the viscosity of the mobile phase and also on the distribution coefficient of the solute, all of which vary with temperature. It follows that, for consistent results, the column must be carefully thermostated. The column and its contents have a significant heat capacity and, consequently, it is of little use trytng to thermostat the column in an air bath for satisfactory temperature control, the thermostating medium... 

Detonation Wave, Two-Dimensional. Under this term is known a wave generated by the lateral dispersion of a detonating substance, in other words, the two dimensional motion of the detonation products. Two- dimensional deton waves may be either stationary or unsteady. Various numerical methods have been applied to the solution of a stationary wave and of the distribution of the fluid properties behind a steadily expanding cylindrical detonation wave as described in Refs 56a, 60, 63a, 74, 93a 93b... 

The mean summarises only one aspect of a distribution. We also need some measure of spread or dispersion, the tendency for observations to depart from the central tendency. The standard measure of dispersion is the variance ... 

Distribution Dispersal of a xenobiotic and its derivatives throughout an organism or environmental matrix, including tissue binding and localization. 

It is useful to find a quantity that could serve us as a measure of these density fluctuations. Its simplest characteristic is the dispersion of a number of particles N in some volume V i.e., (N2) — (N)2. The distinctive feature of the classical ideal gas is a simple relation between the dispersion and macroscopic density (TV2) - (TV)2 = (IV) = nV. Moreover, all other fluctuation characteristics of the ideal gas, related to the quantity (Nm, could also be expressed through (TV) or density n. Therefore, in the model of ideal gas the density n is the only parameter characterizing the fluctuation spectrum. Such the particle distribution is called the Poisson distribution. It could be easily generalized for the many-component system, e.g., a mixture of two ideal gases. Each component is characterized here by its density, nA and nB density fluctuations of different components are statistically independent, (IVAIVB) = (Na)(Nb). 

At last, note logical inconsistency of the method presented. Non-uniform concentration distribution, corresponding to the Poisson fluctuation spectrum (2.1.42), is introduced through initial condition imposed on Z(r,t) - see (2.1.71), (2.1.72). However, equation (2.1.42) disagrees with the starting kinetic equation (2.1.40) the solution of the latter in the absence of reaction, Fi = 0, is Ci(r, t — oo) = nj(0). Consequently, we can find dispersion of a number of particles within an arbitrary volume ... 

Shape factors of a different sort are involved in the Taylor dispersion problem. With parabolic flow at mean speed U through a cylindrical tube of radius R, Taylor found that the longitudinal dispersion of a solute from the interaction of the flow distribution and transverse diffusion was R2U2/48D. The number 48 depends on both the geometry of the cross-section and the flow profile. If, however, we insist that the flow should be laminar, then the geometry of the cross-section determines the flow and hence the numerical constant in the Taylor dispersion coefficient. 

Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneous action of molecular diffusion and variation of the velocity of the solvent. A new basis for his analysis is presented here which removes the restrictions imposed on some of the parameters at the expense of describing the distribution of solute in terms of its moments in the direction of flow. It is shown that the rate of growth of the variance is proportional to the sum of the molecular diffusion coefficient, D, and the Taylor diffusion coefficient ko2U2/D, where U is the mean velocity and a is a dimension characteristic of the cross-section of the tube. An expression for k is given in the most general case, and it is shown that a finite distribution of solute tends to become normally distributed. 

A second important result is that the damping function of a distribution having the hole within the range richest in BA is largely similar to that of the inhomogeneous natural copolymer (Figure 9). Since the conversion is practically 100%, the only explanation is that the parts richest in BA are so finely dispersed that they are below the resolving... 

Colloids A system in which finely divided droplets, particles, or bubbles are distributed in another phase. As it is usually used, dispersion implies a distribution without dissolution. An emulsion is an example of a colloidal dispersion see also Colloidal. 

---