Gaussian shape distribution

Band Ideally, a Gaussian-shaped distribution of (1) adjacent wavelengths encountered in spectroscopy or (2) the amount of a compound as it exits from a chromatographic or an electrophoretic column. 

In the above ealeulations of the mean, varianee and standard deviation, we make no prior assumption about the shape of the population distribution. Many of the data distributions eneountered in engineering have a bell-shaped form similar to that showed in Figure 1. In sueh eases, the Normal or Gaussian eontinuous distribution ean be used to model the data using the mean and standard deviation properties. 

Fig. 3.31. Distributions (i)/(Ee) dEe of electron energy (E ) for a low-pressure HF-plasma (suffix pi, Maxwellian with temperature = 80000 K) and an electron beam (suffix eb, simplified to Gaussian shape with 40 eV half-width) (ii) rTx (Ej) ofthe Ej dependent electron impact ionization cross-section for X=Ti...

The crater surfaces obtained in the LA-TOF-MS experiment on the TiN-TiAlN-Fe sample were remarkably smooth and clearly demonstrated the Gaussian intensity distribution of the laser beam. Fig. 4.45 shows an SEM image of the crater after 100 laser pulses (fluence 0.35 J cm ). The crater is symmetrical and bell-shaped. There is no significant distortion of the single layers. Fig. 4.45 is an excellent demonstration of the potential of femtosecond laser ablation, if the laser beam had a flat-top, rather than Gaussian, intensity profile. 

Gel filtration chromatography has been extensively used to determine pore-size distributions of polymeric gels (in particle form). These models generally do not consider details of the shape of the pores, but rather they may consider a distribution of effective average pore sizes. Rodbard [326,327] reviews the various models for pore-size distributions. These include the uniform-pore models of Porath, Squire, and Ostrowski discussed earlier, the Gaussian pore distribution and its approximation developed by Ackers and Henn [3,155,156], the log-normal distribution, and the logistic distribution. 

Although there was some skewing towards low molecular weights particularly for the narrowest distribution, these curves were generally well fit by a Gaussian shape. Furthermore, the same was found for the copolymer fractions shown in Figure 16. Results are summarized in Table V. 

Increasing the number of repeated measurements to infinity, while decreasing more and more the width of classes (bars), normally leads to a symmetrical bell-shaped distribution of the measured values, which is called Gaussian or normal distribution. 

For a stationary spray without scanning, a Gaussian shaped mass distribution typically develops with an annular-jet or discrete-jet atomizer. The radial mass distribution in the spray can be formulated in terms of mass flux)632]... 

Kittel and Abrahams 12S) have predicted an approximately Lorentzian magnetic resonance line shape for a system of spins which are randomly distributed over a small fraction of a large number of possible sites. This effect has been observed in electron spin resonance (124)- Kittel and Abrahams estimate that appreciable deviations from Gaussian shape will occur when the fraction of sites occupied, f, is less than 0.1, in the case of spins of / = H iu a simple cubic lattice with the magnetic field directed... 

If the observed data follow a bell-shaped Gaussian (normal) distribution, the confidence interval Cl of mean is defined by... 

Gaussian distribution Theoretical bell-shaped distribution of measurements when all error is random. The center of the curve is the mean, p, and the width is characterized by the standard deviation, a. A nortnalized Gaussian distribution, also called the normal error curve, has an area of unity and is given by... 

Figure 1.1a shows the Gaussian function. The Lorentzian shape is similar to the Gaussian, but falls off more slowly. The Doppler shift of radiation from an emitting molecule is proportional to its velocity component in the direction of observation. The one-dimensional distribution of speeds in a gas is a Gaussian function. (See any physical-chemistry text.) Hence when Doppler broadening is dominant, we get a Gaussian-shaped line. 

All three components showed a very similar patter in their pH-dependent mobility. These patterns were also similar to the one found in previous CE studies (Hosse and Wilkinson, 2001 Schmitt-Kopplin et al., 1998). The SRFA were comparable to the SRNOM in terms of peak shape changes at lower pH. The two humic acids showed Gaussian mobility distributions and highly symmetric peaks over all the pH range. At pH < 3, the humic acids still showed very sharp and nicely shaped signals... 

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