The log-normal distribution

The log-normal distribution is defined as the distribution of a variable, the natural log of which is normally distributed. Thus, for the distribution of the natural log of particle diameters we get  

To obtain the distribution of the particle diameter itself rather than that of its logarithm, we note that F (ln x), the fraction of particles with the logarithm of their diameter less than In x, is the same as F(x), the fraction of particles with diameter less than x. Thus  

In order to write F(x) in terms of x rather than In x, we change the variable of integration  

One should be most careful when converting between cumulative and density distributions for the log-normal distribution. 

The log-normal distribution is skewed with a long tail at large particle sizes. It fits the volume distributions of many powders very well. Because it is skewed, the mode, the median and the mean particle sizes are all different. 

According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. With the log-normal law, it is ratios of equal amounts that are equally likely. In order to maintain a symmetrical bell-shaped curve, it is therefore necessary to plot the relative frequency against size in a geometric progression. 

The equation of the log-normal distribution is obtained by replacing x with z = nx, in equation (2.58). 

Xg is the geometric mean of the distribution (i.e. the arithmetic mean of the logarithms) and Cg is the geometric standard deviation is the standard 

Since the particle size is plotted on a logarithmic scale, the presentation of data on a log-probability graph is particularly useful when the range of sizes is large. The geometric standard deviation can be read from the graph, as with the arithmetic distribution, and is given by  

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The only two distributions we shall consider are the Gaussian distribution ( normal law ) and the log-normal distribution. 

To obtain the expression for the log-normal distribution it is only necessary to substitute for I and a in Equation (1.52) the logarithms of these quantities. One thus obtains... 

Other important probability distributions include tlie Binomial Distribution, the Polynomial Distribution, tlie Normal Distribution, and the Log-Normal Distribution. 

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. 

Let us now give some examples of the log-normal distribution. A representation of several types of these distributions is given on the next page as 5.5.11. 

Illustration Drop size distributions produced by chaotic flows. Affinely deformed drops generate long filaments with a stretching distribution based on the log-normal distribution. The amount of stretching (A) determines the radius of the filament locally as... 

Various theoretical distribution functions have been proposed, such as normal or Gaussian distribution and the log-normal distribution. The simplest case is... 

The frequency distribution of the TMRP measurements shows for both levels of exposure a bi-modal histogram - in contrary to the log-normal distribution of the control values (Figure 7 ) The bimodal distribution could be due to a reaction at the cellular level indicating a critical number of hits, i.e. peak A is representing the cell population with a TMRP unaffected by low level alpha exposure, while peak B represents cells reacting to alpha hits. 

Mugele and Evans14231 proposed the upper-limit distribution function based on their analyses of various distribution functions and comparisons with experimental data. This distribution function is a modified form of the log-normal distribution function, and for droplet volume distribution it is expressed as ... 

The ratio of two normal random variables with zero mean is distributed as a Cauchy variable. Isotopic ratios such as 206Pb/204Pb and 207Pb/204Pb therefore should not be described as normal variables since ratios of ratios (e.g., 207Pb/206Pb) should be distributed with a consistent distribution. A consistent distribution for isotopic ratios is the log-normal distribution. 

The PSD s measured were expressed in terms of the mean diameter and standard deviation of the log normal distribution calculated in two different ways. First they were calculated algebra-... 

It has been reported (33) that MWD of HDPE can be described by the log-normal distribution function (34) ... 

Having selected an appropriate data set, we must select a type of distribution and fit the distribution to the data, or else use an empirical or other nonparametric distribution. There appears to be some mechanistic basis for the log-normal distribution, for environmental concentrations (Ott 1990, 1995). However, in a given situation there may not be very strong theoretical support for a specific type of distribution, log-normal or otherwise. Alternative distributions may need to be considered based on the quality of fit of the distribution to data. Therefore, it is desirable to have quantitative indices that can be used to compare or rank distributions based on agreement with data. The fit of the log-normal distribution (or whatever distributions we may choose) should be evaluated in particular situations, using graphical as well as statistical procedures. 

If we are to use a log-normal distribution (or any other parametric distribution), values have to be assigned to the parameters, based on data or some rational argument. For the log-normal distribution, given the characterization of /< and a as log-scale mean and standard deviation, an obvious approach is to transform values in some suitable dataset to logarithms and use the sample mean (of the logarithms) to estimate fi, and sample standard deviation to estimate o. However, as for distributions of many types, there is more than 1 reasonable approach for estimating lognormal parameters. Below, a brief account is provided of estimation procedures and criteria for evaluation of estimation procedures. 

For certain distributions, the set of values for which the pdf is positive (the support) is unbounded. For example, the pdf of the log-normal distribution is positive for all positive real numbers. Ordinarily, there will be values too extreme to be reasonable, and so it is common to place bounds on the support. However, selecting precise values for the bounds may be a difficult decision. 

Ideally, one would like to describe various size distributions by some relatively simple mathematical function. Because there is no single theoretical basis for a particular function to describe atmospheric aerosols, various empirical matches have been carried out to the experimentally observed size distributions some of these are discussed in detail elsewhere (e.g., see Hinds, 1982). Out of the various mathematical distribution functions for fitting aerosol data, the log-normal distribution (Aitchison and Brown, 1957 Patel et al., 1976) has emerged as the mathematical function that most frequently provides a sufficiently good fit, and hence we briefly discuss its application to the size distribution of atmospheric aerosols. 

Most readers will be familiar with the bell-shaped normal distribution plotted in Fig. 9.12. When applied to the size distribution of particles, for example, such a distribution is fully characterized by the arithmetic mean D and the standard deviation a, where a is defined such that 68% of the particles have sizes in the range D a In the log-normal distribution, the logarithm of the diameter D is assumed to have a normal distribution. (Either logarithms to the base 10 or loga-... 

Although the log-normal distribution in Eq. (B) was given in terms of the distribution of the numbers of particles as a function of size, it can also be applied to... 

An advantage of applying the log-normal distribution to atmospheric aerosols is that the value of the geometric standard deviation, crg, is the same for a given sample for all types of distributions—count, mass, surface, and volume. It is only the value of the geometric mean diameter that changes, depending on the... 

Figure 12. Reconstructed fluorescence spectra of 7-(dimethylamino)-coumarin-4-acetate ion 0.1 and 1 ps after excitation. The solid line represents the best fit of the log normal distribution function to the data. From Ref. 33 with permission, from J. Phys. Chem. 93, 7040 (1988). Copyright 1988, American Chemical Society.

Most systems of fine particles have the log-normal type of particle size distribution. That is, with the logarithm of the particle size, the particle size distribution follows the normal or Gaussian distribution in semilog scales. Therefore, the density function for the log-normal distribution can be expressed by... 

Here, Joi and parameters defining the log-normal distribution. Joi is the median diameter, and cumulative-distribution curve has the value of 0.841 to the median diameter. In Joi and arithmetic mean and the standard deviation of In d, respectively, for the log-normal distribution (Problem 1.3). Note that, for the log-normal distribution, the particle number fraction in a size range of b to b + db is expressed by /N(b) db alternatively, the particle number fraction in a parametric range of Info to Info + d(lnb) is expressed by /N(lnb)d(lnb). 

Solution The data on numbers of particles in each particle range given in Table El.3 can be converted to relative frequencies per unit of particle size as given in Table El. 4. The histogram for the relative frequency per unit of particle size for the data is plotted in Fig. El.2 the histogram yields a total area of bars equal to unity. Superimposed on the histogram is the density function for the normal distribution based on Eqs. (1.24) and (1.30). For this distribution, the values for do and ad are evaluated as 0.342 and 0.181, respectively. Also included in the figure is the density function for the log-normal distribution based on Eq. (1.32a). For this distribution, the values for In doi and od are evaluated as —1.209 and 0.531, respectively. 

A graphical comparison of the three distributions with the given data shown in the figure reveals that the log-normal distribution best approximates the data. 

A number of distribution functions have been identified experimentally for a variety of systems ( 3) and, in particular, the Log-Normal distribution is extensively used for the calculation of the integral and for the evaluation of the moments of the particle size distribution (8—12). The problem with this approach is that, in general, the shape of the particle size distribution is unknown and thus, the average particle diameters obtained are conditional upon... 

Data fit is also critical, and statistical tests for goodness of fit should be applied. The log-normal distribution is used in Europe (Aldenberg and Jaworska 2000) and the log-logistic in the United States, while Australia uses a Burr-type distribution (ANZECC/ARMCANZ 2000 Shao 2000), and Canada is also considering the use of a range of statistical distributions to get the best fit. Not unexpectedly, the type of statistical model chosen will have some effect on the resulting HC5 (see Table 4.7). 

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