The Lognormal Distribution

A measured aerosol size distribution can be reported as a table of the distribution values for dozens of diameters. For many applications carrying around hundreds or thousands of aerosol distribution values is awkward. In these cases it is often convenient to use a relatively simple mathematical function to describe the atmospheric aerosol distribution. These functions are semiempirical in nature and have been chosen because they match well observed shapes of ambient distributions. Of the various mathematical functions that have been proposed, the lognormal distribution (Aitchison and Brown 1957) often provides a good fit and is regularly used in atmospheric applications. A series of other distributions are discussed in the next section. 

The normal distribution for a quantity u defined from — oc u oo is given by 

The normal distribution has the characteristic bell shape, with a maximum at u. The standard deviation, 7 , quantifies the width of the distribution, and 68% of the area below the curve is in the range u ct . 

A quantity u is lognormally distributed if its logarithm is normally distributed. Either the natural (In u) or the base 10 logarithm (log u) can be used, but since the former is more common, we will express our results in terms of In Dp. An aerosol population is therefore log-normally distributed if u = In Dp satisfies (8.31), or 

FIGURE 8.7 Aerosol distribution functions, nn(Dp), n°N (log Dp) and n (ln Dp) for a lognormally distributed aerosol distribution Dpg = 0.8 pm and ag = 1.5 versus log Dp. Even if all three functions describe the same aerosol population, they differ from each other because they use a different independent variable. The aerosol number is the area below the n (log Dp) curve. 

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The varianee for any set of data ean be ealeulated without referenee to the prior distribution as diseussed in Appendix I. It follows that the varianee equation is also independent of a prior distribution. Here it is assumed that in all the eases the output funetion is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output funetion is robustly Normal in all eases does not strietly apply, partieularly when variables are in eertain eombination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Misehke (1996) and Siddal (1983) for guidanee on using the varianee equation. 

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

The Reactor Safety Study extensively used the lognormal distribution (equation 2.5-6) to represent the variability in failure rates. If plotted on logarithmic graph paper, the lopnormal distribution is normally distributed. 

The justification for the use of the lognormal is the modified Central Limit Theorem (Section 2.5.2.5). However, if the lognormal distribution is used for estimating the very low failure frequencies associated with the tails of the distribution, this approach is conservative because the low-frequency tails of the lognormal distribution generally extend farther from the median than the actual structural resistance or response data can extend. 

Figure 2. Distribution of the surface radium concentration data from the National Airborne Radiometric Reconnaissance survey for 394 1° by 2° quadrangles covering most of the contiguous 48 states. The distribution parameters are calculated from the data and the lognormal distribution based on the geometric mean., and standard deviation from the data is shown as a solid curve.

Ahrens, L.H. (1954). The lognormal distribution of the elements 1 (A fundamental law of geochemistry and its subsidiary). Geochimica et Cosmochimica Acta 5 49-75. 

The Statistical Model. The residue levels of the individual specimens in a particular subpopulation (e.g., a given Census Division and age, sex, race category) are assumed to follow a lognormal distribution. Previous studies on NHATS data have found the lognormal distribution to be appropriate and goodness of fit tests performed on the collected data verified that the assumption is still reasonable. The lognormal model assumes only non-negative values and allows the variances of the different subpopulation distributions to increase with the mean levels. This distribution is commonly used to model pollutant levels in the environment. 

For oral to inhalation route-to-route extrapolation (28 substances), the Predicted Inhalation No-Adverse-Effect Level (NAEL) was often higher than the observed NOAEL (for inhalation) implicating that the substance was considered less toxic after extrapolation when compared with the experimental observations. Based on the 95th percentile of the lognormal distribution of the ratios between the predicted NAEL and the observed NOAEL, UFs ranging from 75 to 201 for the different extrapolation methodologies were found. 

KEMl (2003) suggested that, if necessary, extrapolation can be performed from subchronic to chronic exposure and that such an extrapolation should be based on the distribution of NOAEL ratios reported by Vermeire et al. (2001). If the 95th percentile is chosen, i.e., covering 95% of the substances compared, the corresponding assessment factor is 16. Extrapolation from subacute to chronic exposure should preferably not be performed, but if it is necessary a similar approach is suggested for this extrapolation, an assessment factor of 39 corresponds to the 95th percentile based on the lognormal distribution of NOAEL ratios from subacute and chronic exposure studies in Vermeire et al. (2001). 

Aitchison, J., and J. A. C. Brown, The Lognormal Distribution, Cambridge Univ. Press, London, 1957. 

We discuss here two major processes, the absorption growth process where particle nuclei grow up with absorption of metal atoms, and the coalescence growth process where particles grow up by collision of particle nuclei or clusters (2). The combination of these processes leads to two well-known distributions, the normal-like distribution and the lognormal distribution, depending on the growth condition. These distributions are frequently found for the size distribution of small particles. The... 

Fig. 9.4.2 Size histogram of In small particles produced by a gas flow-cold trap method in acetone. The pressure of He and Ar mixed gas was 1.3 kPa. The ordinate represents the number of samples at a given size interval. Tolal number of samples, 250. Broken line, calculated curve from the lognormal distribution with a = 2.13 and d = 20 nm. (From Ref. 4.)...

The value clearly varies from sample to sample and is a slightly biased (likely to underestimate the true value) estimate of the 95th percentile of the lognormal distribution, B, i.e. 

The Effect of Imperfect Sampling. Let St be the random variable which estimates the true value of the lognormally distributed random variable (x). The value of (St) is determined by sampling (x) and analyzing with a process which has inherent uncertainty associated with it. The uncertainty is described by the coefficient of variation of the analysis, CV. If CV < 0.3, then %) can be modelled adequately as a lognormally distributed random varible characterized by GM and GSV as defined below (1). 

As an example, 80 batches with four observations per batch were each simulated for the following random variation forms normal, exponential, and lognormal, (see Fig. 5 A-C). x and R charts were constructed for each set as if the true random variation were normal. The charts appear in Figs. 2-4. The results appear in Table 2. This table shows that roughly the same number of points falls outside the x control limits, regardless of the form of the random variation. However, the lognormal distribution has many more R values outside the control limits than the other four distributions. The operator of the process would mistakenly think this process was frequently out of control. The R chart shows greater susceptibility to nonnormality in the random error structure. 

Since atmospheric aerosols comprise particles with a wide range of sizes, it is often convenient to use mathematical models to describe the atmospheric aerosol distribution (Seinfeld and Pandis, 1998). A series of mathematical models have been proposed, of which the lognormal distribution has been the most used in atmospheric applications (Seinfeld and Pandis, 1998 Horvath, 2000). Useful discussions of the various aerosol size distribution models are provided by Seinfeld and Pandis (1998) and Jaenicke (1998). In general, atmospheric aerosols size distributions are shown graphically in terms of the volume (or mass) distributions, surface area distributions, or number distributions as a function of particle size (Jaenicke, 1998). 

Figure 4.5. Modeling of volume-averaging effect with the lognormal distribution function on the real parts of the linear (a) and cubic (b) susceptibilities.

The theory was tested with the aid of an ample data array on low-frequency magnetic spectra of solid Co-Cu nanoparticle systems. In doing so, we combined it with the two most popular volume distribution functions. When the linear and cubic dynamic susceptibilities are taken into account simultaneously, the fitting procedure yields a unique set of magnetic and statistical parameters and enables us to conclude the best appropriate form of the model distribution function (histogram). For the case under study it is the lognormal distribution. 

When constructing input distributions for an uncertainty analysis, it is often useful to present the range of values in terms of a standard probability distribution. It is important that the selected distribution be matched to the range and moments of any available data. In some cases, it is appropriate to simply use the raw data or a custom distribution. Other more commonly used standard probability distributions include the normal distribution, the lognormal distribution, the uniform distribution, the log-uniform distribution and the triangular distribution. For the case-study presented below, we use lognormal distributions. 

Figure A2.2 Probability plot showing the distribution of water sample concentration data in oceans and fresh water and the lognormal distribution used to represent these concentration data...

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