Gaussian distribution equation

The particle size distribution of ball-milled metals and minerals, and atomized metals, follows approximately the Gaussian or normal distribution, in most cases when the logarithn of die diameter is used rather dran the simple diameter. The normal Gaussian distribution equation is... 

The reason for calling equation 8.3-1 a "Gaussian diffusion model" is because it has the form of the normal/Gaussian distribution (equation 2.5-2). Concentration averages for long time intervals may be calculated by averaging the concentrations at grid elements over which the plume passes. 

A critical difference between the transient and CW measurements is that while the CW probe source uniformly illuminates the sample, both the transient pump and probe beams have Gaussian distributions. Equation (7.7) can be rewritten for the transient case as ... 

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... 

The features of an ideal chromatogram are the same as those obtained from a normal distribution of random errors (Gaussian distribution equation (1.2), cf. 21.3). In keeping with the classical notations, fi would correspond to the retention time of the eluting peak, a to the standard deviation of the peak (a2 represents the variance) and y represents the signal, as a function of time, from the detector located at the end of the column (see Fig. 1.3). 

On the chromatogram, a represents the width of the peak at 60.6% of its height and /R the retention time. tR and a have to be measured using the same units (time, distance, or eluted volume if the flow is constant). If a is expressed in units of volume (using the flow), then 4Gaussian distribution, equation (1.8) is preferred. Equation (1.9) is less used because of the distortion of most peaks at the base. 

Many real angular intensity distribution functions have been found to closely approximate a Gaussian and for the remainder of this paper a Gaussian distribution will be used. It should be noted, however, that the angular deconvolution can be used on patterns from specimens with any form of disorientation function. For a Gaussian distribution, equation (2) becomes... 

The approximation = nX was proposed by Craig and derived by Nichols, who pointed out that it is a reasonable approximation if n is large compared with r,n . This equation is analogous to that for the gaussian distribution [Equation (26-1)], where the population mean is replaced by nX, x is replaced by r, and the standard deviation a by y/nXY. 

The lower two curves of Fig. 12 are seen to be symmetrical, and similar in shape to each other. Such curves can be represented by a Gaussian-distribution equation, provided that (a) NB > 50, and (b) Z < Vir/Nr. 

J. Normal distribution Generally, the peak profiles in the chromatograms are assumed to be symmetrical, and are treated by the well-known normal or Gaussian distribution equation ... 

This is in complete agreement with Gaussian distribution. Equation (11). It means, binomial distribution turns into normal distribution in the hmit N oo and aity fixed p. 

The kind of questions that have not heen addressed thus far can he summarized hy asking whether the experimental values for x and s (or SE ) correspond to expectations, i.e., what are the confidence limits (Section 8.2.4) within which the hest measured value x can be regarded as being an acceptable estimate of the true value p, (note that only random error is considered here). To introduce some important concepts, particularly what we mean hy acceptable , we consider first the question of whether or not a normal (Gaussian) distribution (Equation [8.3]) provides an adequate model for a set of replicate measurements as we shall see (Section 8.2.5), modifications to the Gaussian theory are required to account for real-world complications, especially the limited number of replicate measurements that can reahstically be made by the analytical chemist... 

The four generic types of models are Gaussian, numerical, statistical, and physical. The Gaussian models use the Gaussian distribution equation and are widely used to estimate the impact of nonreactive pollutants. Numerical models are more appropriate than Gaussian models for area sources in urban locations that involve reactive pollutants, but numerical models require extremely detailed source and pollutant information and are not widely used. Statistical models are used when scientific information about the chemical and physical processes of a source are incomplete or vague so the use of either Gaussian or numerical models is impractical. Finally, physical models... 

Standard Deviation of Counting Data In contrast to Equation al-14 (Appendix 1) for a Gaussian distribution. Equation 32-9 (or a Poisson distribution contains no corresponding standard deviation term. It can be shown that the breadth of curves such as those in Figure 32-1 depend only on the total number of counts for any given period. That is,... 

The most common mathematical approach has been the box model (also known as top-hat or slab model), which estimates overall features of the cloud such as mean radius, mean height, and mean cloud temperature without calculating detailed features of the cloud in any spatial dimension. Some models of this class impose a Gaussian distribution equating to the average condition. 

Provided a CD Cotton effect is approximated by the Gaussian distribution. Equation [3] is obtained ... 

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