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                <h1 class="wi-article-title article-title-main">Log-normal Distributions across the Sciences: Keys and Clues: On the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probability—normal or log-normal: That is the question
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                                    <a class="linked-name" href="javascript:;">Markus Abbt</a>
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                                                <div class="ww-citation-primary">BioScience (2001) 51 (5): 341-352.</div>
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    <p>Eckhard Limpert, Werner A. Stahel, Markus Abbt; Log-normal Distributions across the Sciences: Keys and Clues: On the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probability—normal or log-normal: That is the question. <em>BioScience</em> 2001; 51 (5): 341-352. doi: 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2</p>
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<p><strong>As the need grows for conceptualization,</strong> formalization, and abstraction in biology, so too does mathematics' relevance to the field (<a reveal-id="I0006-3568-51-5-341-FAGERSTROM1" data-open="I0006-3568-51-5-341-FAGERSTROM1" class="link link-ref link-reveal xref-bibr">Fagerström et al. 1996</a>). Mathematics is particularly important for analyzing and characterizing random variation of, for example, size and weight of individuals in populations, their sensitivity to chemicals, and time-to-event cases, such as the amount of time an individual needs to recover from illness. The frequency distribution of such data is a major factor determining the type of statistical analysis that can be validly carried out on any data set. Many widely used statistical methods, such as ANOVA (analysis of variance) and regression analysis, require that the data be normally distributed, but only rarely is the frequency distribution of data tested when these techniques are used.</p><p>The Gaussian (normal) distribution is most often assumed to describe the random variation that occurs in the data from many scientific disciplines; the well-known bell-shaped curve can easily be characterized and described by two values: the arithmetic mean <div class="overline"><em>x</em></div> and the standard deviation s, so that data sets are commonly described by the expression <div class="overline"><em>x</em></div> ± s. A historical example of a normal distribution is that of chest measurements of Scottish soldiers made by Quetelet, Belgian founder of modern social statistics (<a reveal-id="I0006-3568-51-5-341-SWOBODA1" data-open="I0006-3568-51-5-341-SWOBODA1" class="link link-ref link-reveal xref-bibr">Swoboda 1974</a>). In addition, such disparate phenomena as milk production by cows and random deviations from target values in industrial processes fit a normal distribution.</p><p>However, many measurements show a more or less skewed distribution. Skewed distributions are particularly common when mean values are low, variances large, and values cannot be negative, as is the case, for example, with species abundance, lengths of latent periods of infectious diseases, and distribution of mineral resources in the Earth's crust. Such skewed distributions often closely fit the log-normal distribution (<a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown 1957</a>, <a reveal-id="I0006-3568-51-5-341-CROW1" data-open="I0006-3568-51-5-341-CROW1" class="link link-ref link-reveal xref-bibr">Crow and Shimizu 1988</a>, <a reveal-id="I0006-3568-51-5-341-LEE1" data-open="I0006-3568-51-5-341-LEE1" class="link link-ref link-reveal xref-bibr">Lee 1992</a>, <a reveal-id="I0006-3568-51-5-341-JOHNSON1" data-open="I0006-3568-51-5-341-JOHNSON1" class="link link-ref link-reveal xref-bibr">Johnson et al. 1994</a>, <a reveal-id="I0006-3568-51-5-341-SACHS1" data-open="I0006-3568-51-5-341-SACHS1" class="link link-ref link-reveal xref-bibr">Sachs 1997</a>). Examples fitting the normal distribution, which is symmetrical, and the log-normal distribution, which is skewed, are given in <a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="link link-reveal link-table xref-fig">Figure 1</a>. Note that body height fits both distributions.</p><p>Often, biological mechanisms induce log-normal distributions (<a reveal-id="I0006-3568-51-5-341-KOCH1" data-open="I0006-3568-51-5-341-KOCH1" class="link link-ref link-reveal xref-bibr">Koch 1966</a>), as when, for instance, exponential growth is combined with further symmetrical variation: With a mean concentration of, say, 10<sup>6</sup> bacteria, one cell division more—or less—will lead to 2 × 10<sup>6</sup>—or 5 — 10<sup>5</sup>—cells. Thus, the range will be asymmetrical—to be precise, multiplied or divided by 2 around the mean. The skewed size distribution may be why “exceptionally” big fruit are reported in journals year after year in autumn. Such exceptions, however, may well be the rule: Inheritance of fruit and flower size has long been known to fit the log-normal distribution (<a reveal-id="I0006-3568-51-5-341-GROTH1" data-open="I0006-3568-51-5-341-GROTH1" class="link link-ref link-reveal xref-bibr">Groth 1914</a>, <a reveal-id="I0006-3568-51-5-341-POWERS1" data-open="I0006-3568-51-5-341-POWERS1" class="link link-ref link-reveal xref-bibr">Powers 1936</a>, <a reveal-id="I0006-3568-51-5-341-SINNOT1" data-open="I0006-3568-51-5-341-SINNOT1" class="link link-ref link-reveal xref-bibr">Sinnot 1937</a>).</p><p>What is the difference between normal and log-normal variability? Both forms of variability are based on a variety of forces acting independently of one another. A major difference, however, is that the effects can be additive or multiplicative, thus leading to normal or log-normal distributions, respectively.</p><p>Some basic principles of additive and multiplicative effects can easily be demonstrated with the help of two ordinary dice with sides numbered from 1 to 6. Adding the two numbers, which is the principle of most games, leads to values from 2 to 12, with a mean of 7, and a symmetrical frequency distribution. The total range can be described as 7 plus or minus 5 (that is, 7 ± 5) where, in this case, 5 is not the standard deviation. Multiplying the two numbers, however, leads to values between 1 and 36 with a highly skewed distribution. The total variability can be described as 6 multiplied or divided by 6 (or 6 ×/ 6). In this case, the symmetry has moved to the multiplicative level.</p><p>Although these examples are neither normal nor log-normal distributions, they do clearly indicate that additive and multiplicative effects give rise to different distributions. Thus, we cannot describe both types of distribution in the same way. Unfortunately, however, common belief has it that quantitative variability is generally bell shaped and symmetrical. The current practice in science is to use symmetrical bars in graphs to indicate standard deviations or errors, and the sign ± to summarize data, even though the data or the underlying principles may suggest skewed distributions (<a reveal-id="I0006-3568-51-5-341-FACTOR1" data-open="I0006-3568-51-5-341-FACTOR1" class="link link-ref link-reveal xref-bibr">Factor et al. 2000</a>, <a reveal-id="I0006-3568-51-5-341-KEESING1" data-open="I0006-3568-51-5-341-KEESING1" class="link link-ref link-reveal xref-bibr">Keesing 2000</a>, <a reveal-id="I0006-3568-51-5-341-LENAOUR1" data-open="I0006-3568-51-5-341-LENAOUR1" class="link link-ref link-reveal xref-bibr">Le Naour et al. 2000</a>, <a reveal-id="I0006-3568-51-5-341-RHEW1" data-open="I0006-3568-51-5-341-RHEW1" class="link link-ref link-reveal xref-bibr">Rhew et al. 2000</a>). In a number of cases the variability is clearly asymmetrical because subtracting three standard deviations from the mean produces negative values, as in the example 100 ± 50. Moreover, the example of the dice shows that the established way to characterize symmetrical, additive variability with the sign ± (plus or minus) has its equivalent in the handy sign ×/ (times or divided by), which will be discussed further below.</p><p>Log-normal distributions are usually characterized in terms of the log-transformed variable, using as parameters the expected value, or mean, of its distribution, and the standard deviation. This characterization can be advantageous as, by definition, log-normal distributions are symmetrical again at the log level.</p><p>Unfortunately, the widespread aversion to statistics becomes even more pronounced as soon as logarithms are involved. This may be the major reason that log-normal distributions are so little understood in general, which leads to frequent misunderstandings and errors. Plotting the data can help, but graphs are difficult to communicate orally. In short, current ways of handling log-normal distributions are often unwieldy.</p><p>To get an idea of a sample, most people prefer to think in terms of the original rather than the log-transformed data. This conception is indeed feasible and advisable for log-normal data, too, because the familiar properties of the normal distribution have their analogies in the log-normal distribution. To improve comprehension of log-normal distributions, to encourage their proper use, and to show their importance in life, we present a novel physical model for generating log-normal distributions, thus filling a 100-year-old gap. We also demonstrate the evolution and use of parameters allowing characterization of the data at the original scale. Moreover, we compare log-normal distributions from a variety of branches of science to elucidate patterns of variability, thereby reemphasizing the importance of log-normal distributions in life.</p>                        <h2 scrollto-destination=37941474 id="37941474" class="section-title">A physical model demonstrating the genesis of log-normal distributions</h2>
<p>There was reason for <a reveal-id="I0006-3568-51-5-341-GALTON2" data-open="I0006-3568-51-5-341-GALTON2" class="link link-ref link-reveal xref-bibr">Galton (1889)</a> to complain about colleagues who were interested only in averages and ignored random variability. In his thinking, variability was even part of the “charms of statistics.” Consequently, he presented a simple physical model to give a clear visualization of binomial and, finally, normal variability and its derivation.</p><p><a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="link link-reveal link-table xref-fig">Figure 2a</a> shows a further development of this “Galton board,” in which particles fall down a board and are deviated at decision points (the tips of the triangular obstacles) either left or right with equal probability. (Galton used simple nails instead of the isosceles triangles shown here, so his invention resembles a pinball machine or the Japanese game Pachinko.) The normal distribution created by the board reflects the cumulative additive effects of the sequence of decision points.</p><p>A particle leaving the funnel at the top meets the tip of the first obstacle and is deviated to the left or right by a distance <em>c</em> with equal probability. It then meets the corresponding triangle in the second row, and is again deviated in the same manner, and so forth. The deviation of the particle from one row to the next is a realization of a random variable with possible values +c and −c, and with equal probability for both of them. Finally, after passing <em>r</em> rows of triangles, the particle falls into one of the r + 1 receptacles at the bottom. The probabilities of ending up in these receptacles, numbered 0, 1,…,<em>r</em>, follow a binomial law with parameters <em>r</em> and <em>p</em> = 0.5. When many particles have made their way through the obstacles, the height of the particles piled up in the several receptacles will be approximately proportional to the binomial probabilities.</p><p>For a large number of rows, the probabilities approach a normal density function according to the central limit theorem. In its simplest form, this mathematical law states that the sum of many (<em>r</em>) independent, identically distributed random variables is, in the limit as <em>r</em>→∞, normally distributed. Therefore, a Galton board with many rows of obstacles shows normal density as the expected height of particle piles in the receptacles, and its mechanism captures the idea of a sum of <em>r</em> independent random variables.</p><p><a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="link link-reveal link-table xref-fig">Figure 2b</a> shows how Galton's construction was modified to describe the distribution of a product of such variables, which ultimately leads to a log-normal distribution. To this aim, scalene triangles are needed (although they appear to be isosceles in the figure), with the longer side to the right. Let the distance from the left edge of the board to the tip of the first obstacle below the funnel be x<sub>m</sub>. The lower corners of the first triangle are at x<sub>m</sub> · c and x<sub>m</sub>/c (ignoring the gap necessary to allow the particles to pass between the obstacles). Therefore, the particle meets the tip of a triangle in the next row at X = x<sub>m</sub> · c, or X = x<sub>m</sub> /c, with equal probabilities for both values. In the second and following rows, the triangles with the tip at distance x from the left edge have lower corners at x · c and x/c (up to the gap width). Thus, the horizontal position of a particle is multiplied in each row by a random variable with equal probabilities for its two possible values c and 1/c.</p><p>Once again, the probabilities of particles falling into any receptacle follow the same binomial law as in Galton's device, but because the receptacles on the right are wider than those on the left, the height of accumulated particles is a “histogram” skewed to the left. For a large number of rows, the heights approach a log-normal distribution. This follows from the multiplicative version of the central limit theorem, which proves that the product of many independent, identically distributed, positive random variables has approximately a log-normal distribution. Computer implementations of the models shown in <a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="link link-reveal link-table xref-fig">Figure 2</a> also are available at the Web site <a class="link" href="http://stat.ethz.ch/vis/log-normal" target="_blank">http://stat.ethz.ch/vis/log-normal</a> (<a reveal-id="I0006-3568-51-5-341-GUT1" data-open="I0006-3568-51-5-341-GUT1" class="link link-ref link-reveal xref-bibr">Gut et al. 2000</a>).</p><p>J. C. Kapteyn designed the direct predecessor of the log-normal machine (<a reveal-id="I0006-3568-51-5-341-KAPTEYN1" data-open="I0006-3568-51-5-341-KAPTEYN1" class="link link-ref link-reveal xref-bibr">Kapteyn 1903</a>, <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown 1957</a>). For that machine, isosceles triangles were used instead of the skewed shape described here. Because the triangles' width is proportional to their horizontal position, this model also leads to a log-normal distribution. However, the isosceles triangles with increasingly wide sides to the right of the entry point have a hidden logical disadvantage: The median of the particle flow shifts to the left. In contrast, there is no such shift and the median remains below the entry point of the particles in the log-normal board presented here (which was designed by author E. L.). Moreover, the isosceles triangles in the Kapteyn board create additive effects at each decision point, in contrast to the multiplicative, log-normal effects apparent in <a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="link link-reveal link-table xref-fig">Figure 2b</a>.</p><p>Consequently, the log-normal board presented here is a physical representation of the multiplicative central limit theorem in probability theory.</p>                        <h2 scrollto-destination=37941492 id="37941492" class="section-title">Basic properties of log-normal distributions</h2>
<p>The basic properties of log-normal distribution were established long ago (<a reveal-id="I0006-3568-51-5-341-WEBER1" data-open="I0006-3568-51-5-341-WEBER1" class="link link-ref link-reveal xref-bibr">Weber 1834</a>, <a reveal-id="I0006-3568-51-5-341-FECHNER1" data-open="I0006-3568-51-5-341-FECHNER1" class="link link-ref link-reveal xref-bibr">Fechner 1860</a>, <a reveal-id="I0006-3568-51-5-341-FECHNER2" data-open="I0006-3568-51-5-341-FECHNER2" class="link link-ref link-reveal xref-bibr">1897</a>, <a reveal-id="I0006-3568-51-5-341-GALTON1" data-open="I0006-3568-51-5-341-GALTON1" class="link link-ref link-reveal xref-bibr">Galton 1879</a>, <a reveal-id="I0006-3568-51-5-341-MCALISTER1" data-open="I0006-3568-51-5-341-MCALISTER1" class="link link-ref link-reveal xref-bibr">McAlister 1879</a>, <a reveal-id="I0006-3568-51-5-341-GIBRAT1" data-open="I0006-3568-51-5-341-GIBRAT1" class="link link-ref link-reveal xref-bibr">Gibrat 1931</a>, <a reveal-id="I0006-3568-51-5-341-GADDUM1" data-open="I0006-3568-51-5-341-GADDUM1" class="link link-ref link-reveal xref-bibr">Gaddum 1945</a>), and it is not difficult to characterize log-normal distributions mathematically. A random variable <em>X</em> is said to be log-normally distributed if log(<em>X</em>) is normally distributed (see the box on the facing page). Only positive values are possible for the variable, and the distribution is skewed to the left <a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="link link-reveal link-table xref-fig">(Figure 3a)</a>.</p><p>Two parameters are needed to specify a log-normal distribution. Traditionally, the mean µ and the standard deviation σ (or the variance σ<sup>2</sup>) of log(<em>X</em>) are used <a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="link link-reveal link-table xref-fig">(Figure 3b)</a>. How-ever, there are clear advantages to using “back-transformed” values (the values are in terms of x, the measured data): <a id="I0006-3568-51-5-341-EQ1" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ1"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq1.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq1.gif" /></a></div></div></div></p><p>We then use <em>X</em> ∼ Λ(µ*,σ*) as a mathematical expression meaning that <em>X</em> is distributed according to the log-normal law with median µ* and multiplicative standard deviation σ*.</p><p>The median of this log-normal distribution is med(<em>X</em>) = µ* = <em>e</em><sup>µ</sup>, since µ is the median of log(<em>X</em>). Thus, the probability that the value of <em>X</em> is greater than µ* is 0.5, as is the probability that the value is less than µ*. The parameter σ*, which we call multiplicative standard deviation, determines the shape of the distribution. <a reveal-id="I0006-3568-51-5-341-F04" data-open="I0006-3568-51-5-341-F04" class="link link-reveal link-table xref-fig">Figure 4</a> shows density curves for some selected values of σ*. Note that µ* is a scale parameter; hence, if <em>X</em> is expressed in different units (or multiplied by a constant for other reasons), then µ* changes accordingly but σ* remains the same.</p><p>Distributions are commonly characterized by their expected value µ and standard deviation σ. In applications for which the log-normal distribution adequately describes the data, these parameters are usually less easy to interpret than the median µ* (<a reveal-id="I0006-3568-51-5-341-MCALISTER1" data-open="I0006-3568-51-5-341-MCALISTER1" class="link link-ref link-reveal xref-bibr">McAlister 1879</a>) and the shape parameter σ*. It is worth noting that σ* is related to the coefficient of variation by a monotonic, increasing transformation (see the box below, <a reveal-id="I0006-3568-51-5-341-EQ4" data-open="I0006-3568-51-5-341-EQ4" class="link link-ref link-reveal xref-default">eq. 2</a>).</p><p>For normally distributed data, the interval µ ± σ covers a probability of 68.3%, while µ ± 2σ covers 95.5% <a reveal-id="I0006-3568-51-5-341-T01" data-open="I0006-3568-51-5-341-T01" class="link link-reveal link-table xref-fig">(Table 1)</a>. The corresponding statements for log-normal quantities are <a id="I0006-3568-51-5-341-EQ2" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ2"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq2.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq2.gif" /></a></div></div></div></p><p>This characterization shows that the operations of multiplying and dividing, which we denote with the sign ×/ (times/divide), help to determine useful intervals for log-normal distributions <a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="link link-reveal link-table xref-fig">(Figure 3)</a>, in the same way that the operations of adding and subtracting (±, or plus/minus) do for normal distributions. <a reveal-id="I0006-3568-51-5-341-T01" data-open="I0006-3568-51-5-341-T01" class="link link-reveal link-table xref-fig">Table 1</a> summarizes and compares some properties of normal and log-normal distributions.</p><p>The sum of several independent normal variables is itself a normal random variable. For quantities with a log-normal distribution, however, multiplication is the relevant operation for combining them in most applications; for example, the product of concentrations determines the speed of a simple chemical reaction. The product of independent log-normal quantities also follows a log-normal distribution. The median of this product is the product of the medians of its factors. The formula for σ* of the product is given in the box below (eq. 3).</p><p>For a log-normal distribution, the most precise (i.e., asymptotically most efficient) method for estimating the parameters µ* and σ* relies on log transformation. The mean and empirical standard deviation of the logarithms of the data are calculated and then back-transformed, as in equation 1. These estimators are called <div class="overline"><em>x</em></div>* and s*, where <div class="overline"><em>x</em></div>* is the geometric mean of the data (<a reveal-id="I0006-3568-51-5-341-MCALISTER1" data-open="I0006-3568-51-5-341-MCALISTER1" class="link link-ref link-reveal xref-bibr">McAlister 1879</a>; eq. 4 in the box below). More robust but less efficient estimates can be obtained from the median and the quartiles of the data, as described in the box below.</p><p>As noted previously, it is not uncommon for data with a log-normal distribution to be characterized in the literature by the arithmetic mean <div class="overline"><em>x</em></div> and the standard deviation s of a sample, but it is still possible to obtain estimates for µ* and σ* (see the box on page 345). For example, <a reveal-id="I0006-3568-51-5-341-STEHMANN1" data-open="I0006-3568-51-5-341-STEHMANN1" class="link link-ref link-reveal xref-bibr">Stehmann and De Waard (1996)</a> describe their data as log-normal, with the arithmetic mean <div class="overline"><em>x</em></div> and standard deviation s as 4.1 ± 3.7. Taking the log-normal nature of the distribution into account, the probability of the corresponding <div class="overline"><em>x</em></div> ± s interval (0.4 to 7.8) turns out to be 88.4% instead of 68.3%. Moreover, 65% of the population are below the mean and almost exclusively within only one standard deviation. In contrast, the proposed characterization, which uses the geometric mean <div class="overline"><em>x</em></div>* and the multiplicative standard deviation s*, reads 3.0 ×/ 2.2 (1.36 to 6.6). This interval covers approximately 68% of the data and thus is more appropriate than the other interval for the skewed data.</p>                        <h2 scrollto-destination=37941515 id="37941515" class="section-title">Comparing log-normal distributions across the sciences</h2>
<p>Examples of log-normal distributions from various branches of science reveal interesting patterns (<a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup>). In general, values of s* vary between 1.1 and 33, with most in the range of approximately 1.4 to 3. The shapes of such distributions are apparent by comparison with selected instances shown in <a reveal-id="I0006-3568-51-5-341-F04" data-open="I0006-3568-51-5-341-F04" class="link link-reveal link-table xref-fig">Figure 4</a>.</p>                        <h3 scrollto-destination=37941519 id="37941519" class="section-title">Geology and mining</h3>
<p>In the Earth's crust, the concentration of elements and their radioactivity usually follow a log-normal distribution. In geology, values of s* in 27 examples varied from 1.17 to 5.6 (<a reveal-id="I0006-3568-51-5-341-RAZUMOVSKY1" data-open="I0006-3568-51-5-341-RAZUMOVSKY1" class="link link-ref link-reveal xref-bibr">Razumovsky 1940</a>, <a reveal-id="I0006-3568-51-5-341-AHRENS1" data-open="I0006-3568-51-5-341-AHRENS1" class="link link-ref link-reveal xref-bibr">Ahrens 1954</a>, <a reveal-id="I0006-3568-51-5-341-MALANCA1" data-open="I0006-3568-51-5-341-MALANCA1" class="link link-ref link-reveal xref-bibr">Malanca et al. 1996</a>); nine other examples are given in <a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup>. A closer look at extensive data from different reefs (<a reveal-id="I0006-3568-51-5-341-KRIGE1" data-open="I0006-3568-51-5-341-KRIGE1" class="link link-ref link-reveal xref-bibr">Krige 1966</a>) indicates that values of s* for gold and uranium increase in concert with the size of the region considered.</p>                        <h3 scrollto-destination=37941523 id="37941523" class="section-title">Human medicine</h3>
<p>A variety of examples from medicine fit the log-normal distribution. Latent periods (time from infection to first symptoms) of infectious diseases have often been shown to be log-normally distributed (<a reveal-id="I0006-3568-51-5-341-SARTWELL1" data-open="I0006-3568-51-5-341-SARTWELL1" class="link link-ref link-reveal xref-bibr">Sartwell 1950</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL2" data-open="I0006-3568-51-5-341-SARTWELL2" class="link link-ref link-reveal xref-bibr">1952</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL3" data-open="I0006-3568-51-5-341-SARTWELL3" class="link link-ref link-reveal xref-bibr">1966</a>, <a reveal-id="I0006-3568-51-5-341-KONDO1" data-open="I0006-3568-51-5-341-KONDO1" class="link link-ref link-reveal xref-bibr">Kondo 1977</a>); approximately 70% of 86 examples reviewed by <a reveal-id="I0006-3568-51-5-341-KONDO1" data-open="I0006-3568-51-5-341-KONDO1" class="link link-ref link-reveal xref-bibr">Kondo (1977)</a> appear to be log-normal. <a reveal-id="I0006-3568-51-5-341-SARTWELL1" data-open="I0006-3568-51-5-341-SARTWELL1" class="link link-ref link-reveal xref-bibr">Sartwell (1950</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL2" data-open="I0006-3568-51-5-341-SARTWELL2" class="link link-ref link-reveal xref-bibr">1952</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL3" data-open="I0006-3568-51-5-341-SARTWELL3" class="link link-ref link-reveal xref-bibr">1966</a>) documents 37 cases fitting the log-normal distribution. A particularly impressive one is that of 5914 soldiers inoculated on the same day with the same batch of faulty vaccine, 1005 of whom developed serum hepatitis.</p><p>Interestingly, despite considerable differences in the median <div class="overline"><em>x</em></div>* of latency periods of various diseases (ranging from 2.3 hours to several months; <a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup>), the majority of s* values were close to 1.5. It might be worth trying to account for the similarities and dissimilarities in s*. For instance, the small s* value of 1.24 in the example of the Scottish soldiers may be due to limited variability within this rather homogeneous group of people. Survival time after diagnosis of four types of cancer is, compared with latent periods of infectious diseases, much more variable, with s* values between 2.5 and 3.2 (<a reveal-id="I0006-3568-51-5-341-BOAG1" data-open="I0006-3568-51-5-341-BOAG1" class="link link-ref link-reveal xref-bibr">Boag 1949</a>, <a reveal-id="I0006-3568-51-5-341-FEINLEIB1" data-open="I0006-3568-51-5-341-FEINLEIB1" class="link link-ref link-reveal xref-bibr">Feinleib and McMahon 1960</a>). It would be interesting to see whether <div class="overline"><em>x</em></div>* and s* values have changed in accord with the changes in diagnosis and treatment of cancer in the last half century. The age of onset of Alzheimer's disease can be characterized with the geometric mean <div class="overline"><em>x</em></div>* of 60 years and s* of 1.16 (<a reveal-id="I0006-3568-51-5-341-HORNER1" data-open="I0006-3568-51-5-341-HORNER1" class="link link-ref link-reveal xref-bibr">Horner 1987</a>).</p>                        <h3 scrollto-destination=37941527 id="37941527" class="section-title">Environment</h3>
<p>The distribution of particles, chemicals, and organisms in the environment is often log-normal. For example, the amounts of rain falling from seeded and unseeded clouds differed significantly (<a reveal-id="I0006-3568-51-5-341-BIONDINI1" data-open="I0006-3568-51-5-341-BIONDINI1" class="link link-ref link-reveal xref-bibr">Biondini 1976</a>), and again s* values were similar (seeding itself accounts for the greater variation with seeded clouds). The parameters for the content of hydroxymethylfurfurol in honey (see <a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="link link-reveal link-table xref-fig">Figure 1b</a>) show that the distribution of the chemical in 1573 samples can be described adequately with just the two values. <a reveal-id="I0006-3568-51-5-341-OTT1" data-open="I0006-3568-51-5-341-OTT1" class="link link-ref link-reveal xref-bibr">Ott (1978)</a> presented data on the Pollutant Standard Index, a measure of air quality. Data were collected for eight US cities; the extremes of <div class="overline"><em>x</em></div>* and s* were found in Los Angeles, Houston, and Seattle, allowing interesting comparisons.</p>                        <h3 scrollto-destination=37941530 id="37941530" class="section-title">Atmospheric sciences and aerobiology</h3>
<p>Another component of air quality is its content of microorganisms, which was—not surprisingly—much higher and less variable in the air of Marseille than in that of an island (<a reveal-id="I0006-3568-51-5-341-DIGIORGIO1" data-open="I0006-3568-51-5-341-DIGIORGIO1" class="link link-ref link-reveal xref-bibr">Di Giorgio et al. 1996</a>). The atmosphere is a major part of life support systems, and many atmospheric physical and chemical properties follow a log-normal distribution law. Among other examples are size distributions of aerosols and clouds and parameters of turbulent processes. In the context of turbulence, the size of which is distributed log-normally (<a reveal-id="I0006-3568-51-5-341-LIMPERT6" data-open="I0006-3568-51-5-341-LIMPERT6" class="link link-ref link-reveal xref-bibr">Limpert et al. 2000b</a>).</p>                        <h3 scrollto-destination=37941532 id="37941532" class="section-title">Phytomedicine and microbiology</h3>
<p>Examples from microbiology and phytomedicine include the distribution of sensitivity to fungicides in populations and distribution of population size. <a reveal-id="I0006-3568-51-5-341-ROMERO1" data-open="I0006-3568-51-5-341-ROMERO1" class="link link-ref link-reveal xref-bibr">Romero and Sutton (1997)</a> analyzed the sensitivity of the banana leaf spot fungus <em>(Mycosphaerella fijiensis)</em> to the fungicide propiconazole in samples from untreated and treated areas in Costa Rica. The differences in <div class="overline"><em>x</em></div>* and s* among the areas can be explained by treatment history. The s* in untreated areas reflects mostly environmental conditions and stabilizing selection. The increase in s* after treatment reflects the widened spectrum of sensitivity, which results from the additional selection caused by use of the chemical.</p><p>Similar results were obtained for the barley mildew pathogen, <em>Blumeria (Erysiphe) graminis f.</em> sp. <em>hordei,</em> and the fungicide triadimenol (<a reveal-id="I0006-3568-51-5-341-LIMPERT3" data-open="I0006-3568-51-5-341-LIMPERT3" class="link link-ref link-reveal xref-bibr">Limpert and Koller 1990</a>) where, again, s* was higher in the treated region. Mildew in Spain, where triadimenol had not been used, represented the original sensitivity. In contrast, in England the pathogen was often treated and was consequently highly resistant, differing by a resistance factor of close to 450 (<div class="overline"><em>x</em></div>* England / <div class="overline"><em>x</em></div>* Spain). To obtain the same control of the resistant population, then, the concentration of the chemical would have to be increased by this factor.</p><p>The abundance of bacteria on plants varies among plant species, type of bacteria, and environment and has been found to be log-normally distributed (<a reveal-id="I0006-3568-51-5-341-HIRANO1" data-open="I0006-3568-51-5-341-HIRANO1" class="link link-ref link-reveal xref-bibr">Hirano et al. 1982</a>, <a reveal-id="I0006-3568-51-5-341-LOPER1" data-open="I0006-3568-51-5-341-LOPER1" class="link link-ref link-reveal xref-bibr">Loper et al. 1984</a>). In the case of bacterial populations on the leaves of corn (<em>Zea mays</em>), the median population size (<div class="overline"><em>x</em></div>*) increased from July and August to October, but the relative variability expressed (s*) remained nearly constant (<a reveal-id="I0006-3568-51-5-341-HIRANO1" data-open="I0006-3568-51-5-341-HIRANO1" class="link link-ref link-reveal xref-bibr">Hirano et al. 1982</a>). Interestingly, whereas s* for the total number of bacteria varied little (from 1.26 to 2.0), that for the subgroup of ice nucleation bacteria varied considerably (from 3.75 to 8.04).</p>                        <h3 scrollto-destination=37941538 id="37941538" class="section-title">Plant physiology</h3>
<p>Recently, convincing evidence was presented from plant physiology indicating that the log-normal distribution fits well to permeability and to solute mobility in plant cuticles (<a reveal-id="I0006-3568-51-5-341-BAUR1" data-open="I0006-3568-51-5-341-BAUR1" class="link link-ref link-reveal xref-bibr">Baur 1997</a>). For the number of combinations of species, plant parts, and chemical compounds studied, the median s* for water permeability of leaves was 1.18. The corresponding s* of isolated cuticles, 1.30, appears to be considerably higher, presumably because of the preparation of cuticles. Again, s* was considerably higher for mobility of the herbicides Dichlorophenoxyacetic acid (2,4-D) and WL110547 (1-(3-fluoromethylphenyl)-5-U-<sup>14</sup>C-phenoxy-1,2,3,4-tetrazole). One explanation for the differences in s* for water and for the other chemicals may be extrapolated from results from food technology, where, for transport through filters, s* is smaller for simple (e.g., spherical) particles than for more complex particles such as rods (E. J. Windhab [Eidgenös sische Technische Hochschule, Zurich, Switzerland], personal communication, 2000).</p><p>Chemicals called accelerators can reduce the variability of mobility. For the combination of <em>Citrus aurantium</em> cuticles and 2,4-D, diethyladipate (accelerator 1) caused s* to fall from 1.38 to 1.17. For the same combination, tributylphosphate (accelerator 2) caused an even greater decrease, from 1.56 to 1.03. Statistical reasoning suggests that these data, with s* values of 1.17 and 1.03, are normally distributed (<a reveal-id="I0006-3568-51-5-341-BAUR1" data-open="I0006-3568-51-5-341-BAUR1" class="link link-ref link-reveal xref-bibr">Baur 1997</a>). However, because the underlying principles of permeability remain the same, we think these cases represent log-normal distributions. Thus, considering only statistical reasons may lead to misclassification, which may handicap further analysis. One question remains: What are the underlying principles of permeability that cause log-normal variability?</p>                        <h3 scrollto-destination=37941542 id="37941542" class="section-title">Ecology</h3>
<p>In the majority of plant and animal communities, the abundance of species follows a (truncated) log-normal distribution (<a reveal-id="I0006-3568-51-5-341-SUGIHARA1" data-open="I0006-3568-51-5-341-SUGIHARA1" class="link link-ref link-reveal xref-bibr">Sugihara 1980</a>, <a reveal-id="I0006-3568-51-5-341-MAGURRAN1" data-open="I0006-3568-51-5-341-MAGURRAN1" class="link link-ref link-reveal xref-bibr">Magurran 1988</a>). Interestingly, the range of s* for birds, fish, moths, plants, or diatoms was very close to that found within one group or another. Based on the data and conclusions of <a reveal-id="I0006-3568-51-5-341-PRESTON1" data-open="I0006-3568-51-5-341-PRESTON1" class="link link-ref link-reveal xref-bibr">Preston (1948)</a>, we determined the most typical value of s* to be 11.6 .<a reveal-id="n4" data-open="n4" class="link link-ref link-reveal xref-fn">4</a></p>                        <h3 scrollto-destination=37941545 id="37941545" class="section-title">Food technology</h3>
<p>Various applications of the log-normal distribution are related to the characterization of structures in food technology and food process engineering. Such disperse structures may be the size and frequency of particles, droplets, and bubbles that are generated in dispersing processes, or they may be the pores in filtering membranes. The latter are typically formed by particles that are also log-normally distributed in diameter. Such particles can also be generated in dry or wet milling processes, in which log-normal distribution is a powerful approximation. The examples of ice cream and mayonnaise given in <a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup> also point to the presence of log-normal distributions in everyday life.</p>                        <h3 scrollto-destination=37941547 id="37941547" class="section-title">Linguistics</h3>
<p>In linguistics, the number of letters per word and the number of words per sentence fit the log-normal distribution. In English telephone conversations, the variability s* of the length of all words used—as well as of different words—was similar (<a reveal-id="I0006-3568-51-5-341-HERDAN1" data-open="I0006-3568-51-5-341-HERDAN1" class="link link-ref link-reveal xref-bibr">Herdan 1958</a>). Likewise, the number of words per sentence varied little between writers (<a reveal-id="I0006-3568-51-5-341-WILLIAMS1" data-open="I0006-3568-51-5-341-WILLIAMS1" class="link link-ref link-reveal xref-bibr">Williams 1940</a>).</p>                        <h3 scrollto-destination=37941550 id="37941550" class="section-title">Social sciences and economics</h3>
<p>Examples of log-normal distributions in the social sciences and economics include age of marriage, farm size, and income. The age of first marriage in Western civilization follows a three-parameter log-normal distribution; the third parameter corresponds to age at puberty (<a reveal-id="I0006-3568-51-5-341-PRESTON3" data-open="I0006-3568-51-5-341-PRESTON3" class="link link-ref link-reveal xref-bibr">Preston 1981</a>). For farm size in England and Wales, both <div class="overline"><em>x</em></div> * and s* increased over 50 years, the former by 38.6% (<a reveal-id="I0006-3568-51-5-341-ALLANSON1" data-open="I0006-3568-51-5-341-ALLANSON1" class="link link-ref link-reveal xref-bibr">Allanson 1992</a>). For income distributions, <div class="overline"><em>x</em></div>* and s* may facilitate comparisons among societies and generations (<a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown 1957</a>, <a reveal-id="I0006-3568-51-5-341-LIMPERT5" data-open="I0006-3568-51-5-341-LIMPERT5" class="link link-ref link-reveal xref-bibr">Limpert et al. 2000a</a>).</p>                        <h2 scrollto-destination=37941553 id="37941553" class="section-title">Typical s* values</h2>
<p>One question arises from the comparison of log-normal distributions across the sciences: To what extent are s* values typical for a certain attribute? In some cases, values of s* appear to be fairly restricted, as is the case for the range of s* for latent periods of diseases—a fact that <a reveal-id="I0006-3568-51-5-341-SARTWELL1" data-open="I0006-3568-51-5-341-SARTWELL1" class="link link-ref link-reveal xref-bibr">Sartwell recognized (1950</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL2" data-open="I0006-3568-51-5-341-SARTWELL2" class="link link-ref link-reveal xref-bibr">1952</a>, <a reveal-id="I0006-3568-51-5-341-SARTWELL3" data-open="I0006-3568-51-5-341-SARTWELL3" class="link link-ref link-reveal xref-bibr">1966</a>) and <a reveal-id="I0006-3568-51-5-341-LAWRENCE1" data-open="I0006-3568-51-5-341-LAWRENCE1" class="link link-ref link-reveal xref-bibr">Lawrence reemphasized (1988a)</a>. Describing patterns of typical skewness at the established log level, <a reveal-id="I0006-3568-51-5-341-LAWRENCE1" data-open="I0006-3568-51-5-341-LAWRENCE1" class="link link-ref link-reveal xref-bibr">Lawrence (1988a</a>, <a reveal-id="I0006-3568-51-5-341-LAWRENCE2" data-open="I0006-3568-51-5-341-LAWRENCE2" class="link link-ref link-reveal xref-bibr">1988b</a>) can be regarded as the direct predecessor of our comparison of s* values across the sciences. <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown (1957)</a>, using graphical methods such as quantile-quantile plots and Lorenz curves, demonstrated that log-normal distributions describing, for example, national income across countries, or income for groups of occupations within a country, show typical shapes.</p><p>A restricted range of variation for a specific research question makes sense. For infectious diseases of humans, for example, the infection processes of the pathogens are similar, as is the genetic variability of the human population. The same appears to hold for survival time after diagnosis of cancer, although the value of s* is higher; this can be attributed to the additional variation caused by cancer recognition and treatment. Other examples with typical ranges of s* come from linguistics. For bacteria on plant surfaces, the range of variation of total bacteria is smaller than that of a group of bacteria, which can be expected because of competition. Thus, the ranges of variation in these cases appear to be typical and meaningful, a fact that might well stimulate future research.</p>                        <h2 scrollto-destination=37941557 id="37941557" class="section-title">Future challenges</h2>
<p>A number of scientific areas—and everyday life—will present opportunities for new comparisons and for more far-reaching analyses of s* values for the applications considered to date. Moreover, our concept can be extended also to descriptions based on sigmoid curves, such as, for example, dose–response relationships.</p>                        <h3 scrollto-destination=37941559 id="37941559" class="section-title">Further comparisons of s* values</h3>
<p>Permeability and mobility are important not only for plant physiology (<a reveal-id="I0006-3568-51-5-341-BAUR1" data-open="I0006-3568-51-5-341-BAUR1" class="link link-ref link-reveal xref-bibr">Baur 1997</a>) but also for many other fields such as soil sciences and human medicine, as well as for some industrial processes. With the help of <div class="overline"><em>x</em></div>* and s*, the mobility of different chemicals through a variety of natural membranes could easily be assessed, allowing for comparisons of the membranes with one another as well as with those for filter actions of soils or with technical membranes and filters. Such comparisons will undoubtedly yield valuable insights.</p>                        <h3 scrollto-destination=37941562 id="37941562" class="section-title">Farther-reaching analyses</h3>
<p>An adequate description of variability is a prerequisite for studying its patterns and estimating variance components. One component that deserves more attention is the variability arising from unknown reasons and chance, commonly called <em>error variation,</em> or in this case, s*<sub><em>E</em></sub>. Such variability can be estimated if other conditions accounting for variability—the environment and genetics, for example—are kept constant. The field of population genetics and fungicide sensitivity, as well as that of permeability and mobility, can demonstrate the benefits of analyses of variance.</p><p>An important parameter of population genetics is migration. Migration among regions leads to population mixing, thus widening the spectrum of fungicide sensitivity encountered in any one of the regions mentioned in the discussion of phytomedicine and microbiology. Population mixing among regions will increase s* but decrease the difference in <div class="overline"><em>x</em></div>*. Migration of spores appears to be greater, for instance, in regions of Costa Rica than in those of Europe (<a reveal-id="I0006-3568-51-5-341-LIMPERT4" data-open="I0006-3568-51-5-341-LIMPERT4" class="link link-ref link-reveal xref-bibr">Limpert et al. 1996</a>, <a reveal-id="I0006-3568-51-5-341-ROMERO1" data-open="I0006-3568-51-5-341-ROMERO1" class="link link-ref link-reveal xref-bibr">Romero and Sutton 1997</a>, <a reveal-id="I0006-3568-51-5-341-LIMPERT2" data-open="I0006-3568-51-5-341-LIMPERT2" class="link link-ref link-reveal xref-bibr">Limpert 1999</a>).</p><p> Another important aim of pesticide research is to estimate resistance risk. As a first approximation, resistance risk is assumed to correlate with s*. However, s* depends on genetic and other causes of variability. Thus, determining its genetic part, s *<sub><em>G</em></sub>, is a task worth undertaking, because genetic variation has a major impact on future evolution (<a reveal-id="I0006-3568-51-5-341-LIMPERT2" data-open="I0006-3568-51-5-341-LIMPERT2" class="link link-ref link-reveal xref-bibr">Limpert 1999</a>). Several aspects of various branches of science are expected to benefit from improved identification of components of s*. In the study of plant physiology and permeability noted above (<a reveal-id="I0006-3568-51-5-341-BAUR1" data-open="I0006-3568-51-5-341-BAUR1" class="link link-ref link-reveal xref-bibr">Baur 1997</a>), for example, determining the effects of accelerators and their share of variability would be elucidative.</p>                        <h3 scrollto-destination=37941567 id="37941567" class="section-title">Sigmoid curves based on log-normal distributions</h3>
<p>Dose-response relations are essential for understanding the control of pests and pathogens (<a reveal-id="I0006-3568-51-5-341-HORSFALL1" data-open="I0006-3568-51-5-341-HORSFALL1" class="link link-ref link-reveal xref-bibr">Horsfall 1956</a>). Equally important are dose-response curves that demonstrate the effects of other chemicals, such as hormones or minerals. Typically, such curves are sigmoid and show the cumulative action of the chemical. If plotted against the logarithm of the chemical dose, the sigmoid is symmetrical and corresponds to the cumulative curve of the log-normal distribution at logarithmic scale <a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="link link-reveal link-table xref-fig">(Figure 3b)</a>. The steepness of the sigmoid curve is inversely proportional to s*, and the geometric mean value <div class="overline"><em>x</em></div>* equals the “ED<sub>50</sub>,” the chemical dose creating 50% of the maximal effect. Considering the general importance of chemical sensitivity, a wide field of further applications opens up in which progress can be expected and in which researchers may find the proposed characterization <div class="overline"><em>x</em></div>* ×/s* advantageous.</p>                        <h2 scrollto-destination=37941570 id="37941570" class="section-title">Normal or log-normal?</h2>
<p>Considering the patterns of normal and log-normal distributions further, as well as the connections and distinctions between them (<a reveal-id="I0006-3568-51-5-341-T01" data-open="I0006-3568-51-5-341-T01" class="link link-reveal link-table xref-fig">Tables 1</a>, <a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="link link-reveal link-table xref-fig">3</a>), is useful for describing and explaining phenomena relating to frequency distributions in life. Some important aspects are discussed below.</p>                        <h3 scrollto-destination=37941573 id="37941573" class="section-title">The range of log-normal variability</h3>
<p>How far can s* values extend beyond the range described, from 1.1 to 33? Toward the high end of the scale of possible s* values, we found one s* larger than 150 for hail energy of clouds (<a reveal-id="I0006-3568-51-5-341-FEDERER1" data-open="I0006-3568-51-5-341-FEDERER1" class="link link-ref link-reveal xref-bibr">Federer et al. 1986</a>, calculations by W. A. S). Values below 1.2 may even be common, and therefore of great interest in science. However, such log-normal distributions are difficult to distinguish from normal ones—see <a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="link link-reveal link-table xref-fig">Figures 1</a> and <a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="link link-reveal link-table xref-fig">3</a>—and thus until now have usually been taken to be normal.</p><p>Because of the general preference for the normal distribution, we were asked to find examples of data that followed a normal distribution but did not match a log-normal distribution. Interestingly, original measurements did not yield any such examples. As noted earlier, even the classic example of the height of women (<a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="link link-reveal link-table xref-fig">Figure 1a</a>; <a reveal-id="I0006-3568-51-5-341-SNEDECOR1" data-open="I0006-3568-51-5-341-SNEDECOR1" class="link link-ref link-reveal xref-bibr">Snedecor and Cochran 1989</a>) fits both distributions equally well. The distribution can be characterized with 62.54 inches ± 2.38 and 62.48 inches ×/ 1.039, respectively. The examples that we found of normally—but not log-normally—distributed data consisted of differences, sums, means, or other functions of original measurements. These findings raise questions about the role of symmetry in quantitative variation in nature.</p>                        <h3 scrollto-destination=37941577 id="37941577" class="section-title">Why the normal distribution is so popular</h3>
<p>Regardless of statistical considerations, there are a number of reasons why the normal distribution is much better known than the log-normal. A major one appears to be symmetry, one of the basic principles realized in nature as well as in our culture and thinking. Thus, probability distributions based on symmetry may have more inherent appeal than skewed ones. Two other reasons relate to simplicity. First, as <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown (1957</a>, p. 2) stated, “Man has found addition an easier operation than multiplication, and so it is not surprising that an additive law of errors was the first to be formulated.” Second, the established, concise description of a normal sample—<div class="overline"><em>x</em></div>±s—is handy, well-known, and sufficient to represent the underlying distribution, which made it easier, until now, to handle normal distributions than to work with log-normal distributions. Another reason relates to the history of the distributions: The normal distribution has been known and applied more than twice as long as its log-normal sister distribution. Finally, the very notion of “normal” conjures more positive associations for nonstatisticians than does “log-normal.” For all of these reasons, the normal or Gaussian distribution is far more familiar than the log-normal distribution is to most people.</p><p>This preference leads to two practical ways to make data look normal even if they are skewed. First, skewed distributions produce large values that may appear to be outliers. It is common practice to reject such observations and conduct the analysis without them, thereby reducing the skewness but introducing bias. Second, skewed data are often grouped together, and their means—which are more normally distributed—are used for further analyses. Of course, following that procedure means that important features of the data may remain undiscovered.</p>                        <h3 scrollto-destination=37941580 id="37941580" class="section-title">Why the log-normal distribution is usually the better model for original data</h3>
<p>As discussed above, the connection between additive effects and the normal distribution parallels that of multiplicative effects and the log-normal distribution. <a reveal-id="I0006-3568-51-5-341-KAPTEYN1" data-open="I0006-3568-51-5-341-KAPTEYN1" class="link link-ref link-reveal xref-bibr">Kapteyn (1903)</a> noted long ago that if data from one-dimensional measurements in nature fit the normal distribution, two- and three-dimensional results such as surfaces and volumes cannot be symmetric. A number of effects that point to the log-normal distribution as an appropriate model have been described in various papers (e.g., <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown 1957</a>, <a reveal-id="I0006-3568-51-5-341-KOCH1" data-open="I0006-3568-51-5-341-KOCH1" class="link link-ref link-reveal xref-bibr">Koch 1966</a>, <a reveal-id="I0006-3568-51-5-341-KOCH2" data-open="I0006-3568-51-5-341-KOCH2" class="link link-ref link-reveal xref-bibr">1969</a>, <a reveal-id="I0006-3568-51-5-341-CROW1" data-open="I0006-3568-51-5-341-CROW1" class="link link-ref link-reveal xref-bibr">Crow and Shimizu 1988</a>). Interestingly, even in biological systematics, which is the science of classification, the number of, say, species per family was expected to fit log-normality (<a reveal-id="I0006-3568-51-5-341-KOCH1" data-open="I0006-3568-51-5-341-KOCH1" class="link link-ref link-reveal xref-bibr">Koch 1966</a>).</p><p>The most basic indicator of the importance of the log-normal distribution may be even more general, however. Clearly, chemistry and physics are fundamental in life, and the prevailing operation in the laws of these disciplines is multiplication. In chemistry, for instance, the velocity of a simple reaction depends on the product of the concentrations of the molecules involved. Equilibrium conditions likewise are governed by factors that act in a multiplicative way. From this, a major contrast becomes obvious: The reasons governing frequency distributions in nature usually favor the log-normal, whereas people are in favor of the normal.</p><p>For small coefficients of variation, normal and log-normal distributions both fit well. For these cases, it is natural to choose the distribution found appropriate for related cases exhibiting increased variability, which corresponds to the law governing the reasons of variability. This will most often be the log-normal.</p>                        <h2 scrollto-destination=37941585 id="37941585" class="section-title">Conclusion</h2>
<p>This article shows, in a nutshell, the fundamental role of the log-normal distribution and provides insights for gaining a deeper comprehension of that role. Compared with established methods for describing log-normal distributions <a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="link link-reveal link-table xref-fig">(Table 3)</a>, the proposed characterization by <div class="overline"><em>x</em></div>* and s* offers several advantages, some of which have been described before (<a reveal-id="I0006-3568-51-5-341-SARTWELL1" data-open="I0006-3568-51-5-341-SARTWELL1" class="link link-ref link-reveal xref-bibr">Sartwell 1950</a>, <a reveal-id="I0006-3568-51-5-341-AHRENS1" data-open="I0006-3568-51-5-341-AHRENS1" class="link link-ref link-reveal xref-bibr">Ahrens 1954</a>, <a reveal-id="I0006-3568-51-5-341-LIMPERT1" data-open="I0006-3568-51-5-341-LIMPERT1" class="link link-ref link-reveal xref-bibr">Limpert 1993</a>). Both <div class="overline"><em>x</em></div>* and s* describe the data directly at their original scale, they are easy to calculate and imagine, and they even allow mental calculation and estimation. The proposed characterization does not appear to have any major disadvantage.</p><p>On the first page of their book, <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown (1957)</a> stated that, compared with its sister distributions, the normal and the binomial, the log-normal distribution “has remained the Cinderella of distributions, the interest of writers in the learned journals being curiously sporadic and that of the authors of statistical textbooks but faintly aroused.” This is indeed true: Despite abundant, increasing evidence that log-normal distributions are widespread in the physical, biological, and social sciences, and in economics, log-normal knowledge has remained dispersed. The question now is this: Can we begin to bring the wealth of knowledge we have on normal and log-normal distributions to the public? We feel that doing so would lead to a general preference for the log-normal, or multiplicative normal, distribution over the Gaussian distribution when describing original data.</p>                        <h2 scrollto-destination=37941589 id="37941589" class="backacknowledgements-title">Acknowledgments</h2>
<p>This work was supported by a grant from the Swiss Federal Institute of Technology (Zurich) and by COST (Coordination of Science and Technology in Europe) at Brussels and Bern. Patrick Flütsch, Swiss Federal Institute of Technology, constructed the physical models. We are grateful to Roy Snaydon, professor emeritus at the University of Reading, United Kingdom, to Rebecca Chasan, and to four anonymous reviewers for valuable comments on the manuscript. We thank Donna Verdier and Herman Marshall for getting the paper into good shape for publication, and E. L. also thanks Gerhard Wenzel, professor of agronomy and plant breeding at Technical University, Munich, for helpful discussions.</p><p>Because of his fundamental and comprehensive contribution to the understanding of skewed distributions close to 100 years ago, our paper is dedicated to the Dutch astronomer Jacobus Cornelius Kapteyn (<a reveal-id="I0006-3568-51-5-341-VANDERHEIJDEN1" data-open="I0006-3568-51-5-341-VANDERHEIJDEN1" class="link link-ref link-reveal xref-bibr">van der Heijden 2000</a>).</p>                        <h2 scrollto-destination=37941593 id="37941593" class="backreferences-title">References cited</h2>
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</div></div></div></div></div>                        <h2 scrollto-destination=37941594 id="37941594" class="backappendixgroup-title">Appendices</h2>
                        <h4 scrollto-destination=37941597 id="37941597" class="section-title">Definition and properties of the log-normal distribution</h4>
<p>A random variable <em>X</em> is log-normally distributed if log(<em>X</em>) has a normal distribution. Usually, natural logarithms are used, but other bases would lead to the same family of distributions, with rescaled parameters. The probability density function of such a random variable has the form <a id="I0006-3568-51-5-341-EQ3" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ3"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq3.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq3.gif" /></a></div></div></div> A shift parameter can be included to define a three-parameter family. This may be adequate if the data cannot be smaller than a certain bound different from zero (cf. <a reveal-id="I0006-3568-51-5-341-AITCHISON1" data-open="I0006-3568-51-5-341-AITCHISON1" class="link link-ref link-reveal xref-bibr">Aitchison and Brown 1957</a>, page 14). The mean and variance are exp(µ + σ/2) and (exp(σ<sup>2</sup>)−1)exp(2µ+σ<sup>2</sup>), respectively, and therefore, the coefficient of variation is <a id="I0006-3568-51-5-341-EQ4" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ4"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq4.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq4.gif" /></a></div></div></div> which is a function in σ only.
The product of two independent log-normally distributed random variables has the shape parameter <a id="I0006-3568-51-5-341-EQ5" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ5"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq5.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq5.gif" /></a></div></div></div> since the variances at the log-transformed variables add.
Estimation: The asymptotically most efficient (maximum likelihood) estimators are <a id="I0006-3568-51-5-341-EQ6" class="jumplink-placeholder"> </a><div class="formula-wrap"><div class="disp-formula" content-id="I0006-3568-51-5-341-EQ6"><div class="fig-graphic"><a class="jumplink-placeholder" id="i0006-3568-51-5-341-eq6.gif"><img class="content-image" src="" alt="formula" path-from-xml="i0006-3568-51-5-341-eq6.gif" /></a></div></div></div> The quartiles q1 and q2 lead to a more robust estimate (q1/q2)<sup>c</sup> for s*, where 1/c = 1.349 = 2 · Φ<sup>−1</sup> (0.75), Φ<sub>−1</sub> denoting the inverse standard normal distribution function. If the mean <div class="overline"><em>x</em></div> and the standard deviation s of a sample are available, i.e. the data is summarized in the form <div class="overline"><em>x</em></div> ± s, the parameters µ* and s* can be estimated from them by using <div class="overline"><em>x</em></div>/√ω and exp(√log(ω), respectively, with ω = 1 +(s/<div class="overline"><em>x</em></div>)<sup>2</sup> = 1 + cv<sup>2</sup>, cv = coefficient of variation. Thus, this estimate of s* is determined only by the cv (eq. 2).</p>                <a id="37941601" scrollto-destination="37941601"></a>
<div id="I0006-3568-51-5-341-T01" data-id="I0006-3568-51-5-341-T01" class="fig fig-section"><div class="label fig-label">Table 1.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T01" data-open="I0006-3568-51-5-341-T01" class="fig-link"><img class="content-image" src="" alt="A bridge between normal and log-normal distributions" path-from-xml="i0006-3568-51-5-341-t01.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t01.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t01.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>A bridge between normal and log-normal distributions</p></div></div><div content-id="I0006-3568-51-5-341-T01" class="fig fig-modal reveal-modal"><div class="label fig-label">Table 1.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T01" data-open="I0006-3568-51-5-341-T01" class="fig-link"><img class="content-image" src="" alt="A bridge between normal and log-normal distributions" path-from-xml="i0006-3568-51-5-341-t01.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t01.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t01.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>A bridge between normal and log-normal distributions</p></div></div>                <a id="37941602" scrollto-destination="37941602"></a>
<div id="I0006-3568-51-5-341-T201" data-id="I0006-3568-51-5-341-T201" class="fig fig-section"><div class="label fig-label">Table 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="fig-link"><img class="content-image" src="" alt="Comparing log-normal distributions across the sciences in terms of the original data. x* is an estimator of the median of the distribution, usually the geometric mean of the observed data, and s*estimates the multiplicative standard deviation, the shape parameter of the distribution; 68% of the data are within the range of x*×/ s*, and 95% within x*×/ (s*)2. In general, values of s*and some of x* were obtained by transformation from the parameters given in the literature (cf. Table 3). The goodness of fit was tested either by the original authors or by us" path-from-xml="i0006-3568-51-5-341-t201.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t201.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t201.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Comparing log-normal distributions across the sciences in terms of the original data. <div class="overline"><em>x</em></div>* is an estimator of the median of the distribution, usually the geometric mean of the observed data, and s*estimates the multiplicative standard deviation, the shape parameter of the distribution; 68% of the data are within the range of <div class="overline"><em>x</em></div>*×/ s*, and 95% within <div class="overline"><em>x</em></div>*×/ (s*)<sup>2</sup>. In general, values of s*and some of <div class="overline"><em>x</em></div>* were obtained by transformation from the parameters given in the literature (cf. <a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="link link-reveal link-table xref-fig">Table 3</a>). The goodness of fit was tested either by the original authors or by us</p></div></div><div content-id="I0006-3568-51-5-341-T201" class="fig fig-modal reveal-modal"><div class="label fig-label">Table 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="fig-link"><img class="content-image" src="" alt="Comparing log-normal distributions across the sciences in terms of the original data. x* is an estimator of the median of the distribution, usually the geometric mean of the observed data, and s*estimates the multiplicative standard deviation, the shape parameter of the distribution; 68% of the data are within the range of x*×/ s*, and 95% within x*×/ (s*)2. In general, values of s*and some of x* were obtained by transformation from the parameters given in the literature (cf. Table 3). The goodness of fit was tested either by the original authors or by us" path-from-xml="i0006-3568-51-5-341-t201.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t201.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t201.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Comparing log-normal distributions across the sciences in terms of the original data. <div class="overline"><em>x</em></div>* is an estimator of the median of the distribution, usually the geometric mean of the observed data, and s*estimates the multiplicative standard deviation, the shape parameter of the distribution; 68% of the data are within the range of <div class="overline"><em>x</em></div>*×/ s*, and 95% within <div class="overline"><em>x</em></div>*×/ (s*)<sup>2</sup>. In general, values of s*and some of <div class="overline"><em>x</em></div>* were obtained by transformation from the parameters given in the literature (cf. <a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="link link-reveal link-table xref-fig">Table 3</a>). The goodness of fit was tested either by the original authors or by us</p></div></div>                <a id="37941604" scrollto-destination="37941604"></a>
<div id="I0006-3568-51-5-341-T202" data-id="I0006-3568-51-5-341-T202" class="fig fig-section"><div class="label fig-label">Table 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T202" data-open="I0006-3568-51-5-341-T202" class="fig-link"><img class="content-image" src="" alt="(continued from previous page)" path-from-xml="i0006-3568-51-5-341-t202.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t202.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t202.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>(continued from previous page)</p></div></div><div content-id="I0006-3568-51-5-341-T202" class="fig fig-modal reveal-modal"><div class="label fig-label">Table 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T202" data-open="I0006-3568-51-5-341-T202" class="fig-link"><img class="content-image" src="" alt="(continued from previous page)" path-from-xml="i0006-3568-51-5-341-t202.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t202.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t202.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>(continued from previous page)</p></div></div>                <a id="37941605" scrollto-destination="37941605"></a>
<div id="I0006-3568-51-5-341-T03" data-id="I0006-3568-51-5-341-T03" class="fig fig-section"><div class="label fig-label">Table 3.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="fig-link"><img class="content-image" src="" alt="Established methods for describing log-normal distributions" path-from-xml="i0006-3568-51-5-341-t03.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t03.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t03.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Established methods for describing log-normal distributions</p></div></div><div content-id="I0006-3568-51-5-341-T03" class="fig fig-modal reveal-modal"><div class="label fig-label">Table 3.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-T03" data-open="I0006-3568-51-5-341-T03" class="fig-link"><img class="content-image" src="" alt="Established methods for describing log-normal distributions" path-from-xml="i0006-3568-51-5-341-t03.gif" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig"  path-from-xml="i0006-3568-51-5-341-t03.gif" target="_blank">View large</a><a section=""  class="download-slide" path-from-xml="i0006-3568-51-5-341-t03.gif" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Established methods for describing log-normal distributions</p></div></div>                <a id="37941606" scrollto-destination="37941606"></a>
<div id="I0006-3568-51-5-341-F01" data-id="I0006-3568-51-5-341-F01" class="fig fig-section"><div class="label fig-label">Figure 1.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=DGNfdGwtu-KynEyUBaNdSWIBbcu0DOXyH64ACVcyBnv2-KMlKtwSpoA9Sre2pRe~yF5~vCxufcNkm5a2o3-WLFa7aBUbabz5iFUvNymvj1cL-T24c3S-VdH814AXFcJCChLwAMTb5sAhsQNv0bSovw3tEqBmauECnYN3rEeFvmd2i4YDynQ~1wPQZP4ZvJKmhGAwuoGlmm-gzButgdNRM9mRd3D2ChZlDYY7Hoadi9N~eVCcdmMm37Ko1yHYCFpDXCtISBiVKPuiBf-D9yI6mhC4MVxnVCiIg0kWFxwLnjYQVT~afEkeCHdIC-Wr5fqgd30HAzQIEAvWrS8Y9CKbtw__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Examples of normal and log-normal distributions. While the distribution of the heights of 1052 women (a, in inches; Snedecor and Cochran 1989) fits the normal distribution, with a goodness of fit p value of 0.75, that of the content of hydroxymethylfurfurol (HMF, mg·kg−1) in 1573 honey samples (b; Renner 1970) fits the log-normal (p = 0.41) but not the normal (p = 0.0000). Interestingly, the distribution of the heights of women fits the log-normal distribution equally well (p = 0.74)" path-from-xml="i0006-3568-51-5-341-f01.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=HcNAl4bC0MppenlIZRgssArvs4MQoNJbgLc6~7~zWgrRfZOwWQxcPebZOjoGzQfldbE-L~ymtms1vYUiUVcru9TogV4xQaNPOFTWT8QXqJnsMq-eSpX3e10zSLutemBUcTaGPbQpKtXosl1fcTtbMTCqRT2mTNJQYD9dAIYOqeB9FFjAs3XT7VdC9aICDCW9A4RaxMxFWNqYgKDWPU6wuf4jtPNBqnCbrF20iSG2jjm8Ud8gzWrSzbLDMI55Top8pt9bw0Fluebb0PhGIMSGO4E8GbnJceqDIdefnnCCaLREwih4ru4CATTVdpCiK4g3nDXYf509jhO~lkcNxOjKJQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f01.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=HcNAl4bC0MppenlIZRgssArvs4MQoNJbgLc6~7~zWgrRfZOwWQxcPebZOjoGzQfldbE-L~ymtms1vYUiUVcru9TogV4xQaNPOFTWT8QXqJnsMq-eSpX3e10zSLutemBUcTaGPbQpKtXosl1fcTtbMTCqRT2mTNJQYD9dAIYOqeB9FFjAs3XT7VdC9aICDCW9A4RaxMxFWNqYgKDWPU6wuf4jtPNBqnCbrF20iSG2jjm8Ud8gzWrSzbLDMI55Top8pt9bw0Fluebb0PhGIMSGO4E8GbnJceqDIdefnnCCaLREwih4ru4CATTVdpCiK4g3nDXYf509jhO~lkcNxOjKJQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941606&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f01.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p><em>Examples of normal and log-normal distributions. While the distribution of the heights of 1052 women (a, in inches; <a reveal-id="I0006-3568-51-5-341-SNEDECOR1" data-open="I0006-3568-51-5-341-SNEDECOR1" class="link link-ref link-reveal xref-bibr">Snedecor and Cochran 1989</a>) fits the normal distribution, with a goodness of fit</em> p <em>value of 0.75, that of the content of hydroxymethylfurfurol (HMF, mg·kg<sup>−1</sup>) in 1573 honey samples (b; <a reveal-id="I0006-3568-51-5-341-RENNER1" data-open="I0006-3568-51-5-341-RENNER1" class="link link-ref link-reveal xref-bibr">Renner 1970</a>) fits the log-normal</em> (p = 0.41) <em>but not the normal</em> (p = 0.0000). <em>Interestingly, the distribution of the heights of women fits the log-normal distribution equally well</em> (p = 0.74)</p></div></div><div content-id="I0006-3568-51-5-341-F01" class="fig fig-modal reveal-modal"><div class="label fig-label">Figure 1.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F01" data-open="I0006-3568-51-5-341-F01" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=DGNfdGwtu-KynEyUBaNdSWIBbcu0DOXyH64ACVcyBnv2-KMlKtwSpoA9Sre2pRe~yF5~vCxufcNkm5a2o3-WLFa7aBUbabz5iFUvNymvj1cL-T24c3S-VdH814AXFcJCChLwAMTb5sAhsQNv0bSovw3tEqBmauECnYN3rEeFvmd2i4YDynQ~1wPQZP4ZvJKmhGAwuoGlmm-gzButgdNRM9mRd3D2ChZlDYY7Hoadi9N~eVCcdmMm37Ko1yHYCFpDXCtISBiVKPuiBf-D9yI6mhC4MVxnVCiIg0kWFxwLnjYQVT~afEkeCHdIC-Wr5fqgd30HAzQIEAvWrS8Y9CKbtw__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Examples of normal and log-normal distributions. While the distribution of the heights of 1052 women (a, in inches; Snedecor and Cochran 1989) fits the normal distribution, with a goodness of fit p value of 0.75, that of the content of hydroxymethylfurfurol (HMF, mg·kg−1) in 1573 honey samples (b; Renner 1970) fits the log-normal (p = 0.41) but not the normal (p = 0.0000). Interestingly, the distribution of the heights of women fits the log-normal distribution equally well (p = 0.74)" path-from-xml="i0006-3568-51-5-341-f01.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=HcNAl4bC0MppenlIZRgssArvs4MQoNJbgLc6~7~zWgrRfZOwWQxcPebZOjoGzQfldbE-L~ymtms1vYUiUVcru9TogV4xQaNPOFTWT8QXqJnsMq-eSpX3e10zSLutemBUcTaGPbQpKtXosl1fcTtbMTCqRT2mTNJQYD9dAIYOqeB9FFjAs3XT7VdC9aICDCW9A4RaxMxFWNqYgKDWPU6wuf4jtPNBqnCbrF20iSG2jjm8Ud8gzWrSzbLDMI55Top8pt9bw0Fluebb0PhGIMSGO4E8GbnJceqDIdefnnCCaLREwih4ru4CATTVdpCiK4g3nDXYf509jhO~lkcNxOjKJQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f01.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f01.jpeg?Expires=1488590074&Signature=HcNAl4bC0MppenlIZRgssArvs4MQoNJbgLc6~7~zWgrRfZOwWQxcPebZOjoGzQfldbE-L~ymtms1vYUiUVcru9TogV4xQaNPOFTWT8QXqJnsMq-eSpX3e10zSLutemBUcTaGPbQpKtXosl1fcTtbMTCqRT2mTNJQYD9dAIYOqeB9FFjAs3XT7VdC9aICDCW9A4RaxMxFWNqYgKDWPU6wuf4jtPNBqnCbrF20iSG2jjm8Ud8gzWrSzbLDMI55Top8pt9bw0Fluebb0PhGIMSGO4E8GbnJceqDIdefnnCCaLREwih4ru4CATTVdpCiK4g3nDXYf509jhO~lkcNxOjKJQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941606&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f01.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p><em>Examples of normal and log-normal distributions. While the distribution of the heights of 1052 women (a, in inches; <a reveal-id="I0006-3568-51-5-341-SNEDECOR1" data-open="I0006-3568-51-5-341-SNEDECOR1" class="link link-ref link-reveal xref-bibr">Snedecor and Cochran 1989</a>) fits the normal distribution, with a goodness of fit</em> p <em>value of 0.75, that of the content of hydroxymethylfurfurol (HMF, mg·kg<sup>−1</sup>) in 1573 honey samples (b; <a reveal-id="I0006-3568-51-5-341-RENNER1" data-open="I0006-3568-51-5-341-RENNER1" class="link link-ref link-reveal xref-bibr">Renner 1970</a>) fits the log-normal</em> (p = 0.41) <em>but not the normal</em> (p = 0.0000). <em>Interestingly, the distribution of the heights of women fits the log-normal distribution equally well</em> (p = 0.74)</p></div></div>                <a id="37941610" scrollto-destination="37941610"></a>
<div id="I0006-3568-51-5-341-F02" data-id="I0006-3568-51-5-341-F02" class="fig fig-section"><div class="label fig-label">Figure 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=GADwCAql4zwVQJx~toGFcStamB4ZCuMVmtFFraGxAdrfVEUstK0uoGEl5XOU3LogomgNNLiv~A7uTrd1SzJk9gBl8FZjC2zUOp1iVZa7mOwYDnWcyxBiF9poFge0n59o5vUHNAspNWTAalgIfg9-9j2CS5-x3JmYraD~CfK8hGKU-0YdlHh2V~hJ~wBsTPZmTpxdjyqLSY~l~AUC2n3XNnSrlX5izJ5x-LzRl2ZG~eb6VFqgViOAQgc1rYGTR~S-l7c13zrzTIQ~ayjCJR4tt6irLPll-u0li5~XNFzC9MEnt2Tj-T2KtuL1uhIBkj6GMntqs6qvTTkPpGMC5PkFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Physical models demonstrating the genesis of normal and log-normal distributions. Particles fall from a funnel onto tips of triangles, where they are deviated to the left or to the right with equal probability (0.5) and finally fall into receptacles. The medians of the distributions remain below the entry points of the particles. If the tip of a triangle is at distance x from the left edge of the board, triangle tips to the right and to the left below it are placed at x + c and x − c for the normal distribution (panel a), and x · ć and x / ć for the log-normal (panel b, patent pending), c and ć being constants. The distributions are generated by many small random effects (according to the central limit theorem) that are additive for the normal distribution and multiplicative for the log-normal. We therefore suggest the alternative name multiplicative normal distribution for the latter" path-from-xml="i0006-3568-51-5-341-f02.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=X80qNW8IUl6NeSa4VqDVcGqIL2mvpAcBUsr9EDhGYLZpLri0n5spGMnvw9ylzHh~4cK-f1tHA1S5aBT~8OfWIuIFw5aB6wt-8W5U1V6~7Sm5iXpGnjyzHw8yzSVKYbJHMs6zR~6k3~FnYk52ByBtswZhYX1mQVHCF4kwkqTnRSHLKbBJmyUBoFDB1FaJMnJSiguZ6rZrD6nj4u7BqBVlrhyOOi74VTWc~h4EA6a7HiXTmgTL~OLcHcDX16X88PXrrTKcU-u6t7YsDpBUOycLD~FAdDEXDWJQKsgyoPhQXZFeXPOKjYU2pSDGAaTuu0x43Dv1u~FkqkYCNrJ8t5by0g__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f02.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=X80qNW8IUl6NeSa4VqDVcGqIL2mvpAcBUsr9EDhGYLZpLri0n5spGMnvw9ylzHh~4cK-f1tHA1S5aBT~8OfWIuIFw5aB6wt-8W5U1V6~7Sm5iXpGnjyzHw8yzSVKYbJHMs6zR~6k3~FnYk52ByBtswZhYX1mQVHCF4kwkqTnRSHLKbBJmyUBoFDB1FaJMnJSiguZ6rZrD6nj4u7BqBVlrhyOOi74VTWc~h4EA6a7HiXTmgTL~OLcHcDX16X88PXrrTKcU-u6t7YsDpBUOycLD~FAdDEXDWJQKsgyoPhQXZFeXPOKjYU2pSDGAaTuu0x43Dv1u~FkqkYCNrJ8t5by0g__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941610&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f02.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Physical models demonstrating the genesis of normal and log-normal distributions. Particles fall from a funnel onto tips of triangles, where they are deviated to the left or to the right with equal probability (0.5) and finally fall into receptacles. The medians of the distributions remain below the entry points of the particles. If the tip of a triangle is at distance x from the left edge of the board, triangle tips to the right and to the left below it are placed at x + c and x − c for the normal distribution (panel a), and x · ć and x / ć for the log-normal (panel b, patent pending), c and ć being constants. The distributions are generated by many small random effects (according to the central limit theorem) that are additive for the normal distribution and multiplicative for the log-normal. We therefore suggest the alternative name multiplicative normal distribution for the latter</p></div></div><div content-id="I0006-3568-51-5-341-F02" class="fig fig-modal reveal-modal"><div class="label fig-label">Figure 2.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F02" data-open="I0006-3568-51-5-341-F02" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=GADwCAql4zwVQJx~toGFcStamB4ZCuMVmtFFraGxAdrfVEUstK0uoGEl5XOU3LogomgNNLiv~A7uTrd1SzJk9gBl8FZjC2zUOp1iVZa7mOwYDnWcyxBiF9poFge0n59o5vUHNAspNWTAalgIfg9-9j2CS5-x3JmYraD~CfK8hGKU-0YdlHh2V~hJ~wBsTPZmTpxdjyqLSY~l~AUC2n3XNnSrlX5izJ5x-LzRl2ZG~eb6VFqgViOAQgc1rYGTR~S-l7c13zrzTIQ~ayjCJR4tt6irLPll-u0li5~XNFzC9MEnt2Tj-T2KtuL1uhIBkj6GMntqs6qvTTkPpGMC5PkFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Physical models demonstrating the genesis of normal and log-normal distributions. Particles fall from a funnel onto tips of triangles, where they are deviated to the left or to the right with equal probability (0.5) and finally fall into receptacles. The medians of the distributions remain below the entry points of the particles. If the tip of a triangle is at distance x from the left edge of the board, triangle tips to the right and to the left below it are placed at x + c and x − c for the normal distribution (panel a), and x · ć and x / ć for the log-normal (panel b, patent pending), c and ć being constants. The distributions are generated by many small random effects (according to the central limit theorem) that are additive for the normal distribution and multiplicative for the log-normal. We therefore suggest the alternative name multiplicative normal distribution for the latter" path-from-xml="i0006-3568-51-5-341-f02.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=X80qNW8IUl6NeSa4VqDVcGqIL2mvpAcBUsr9EDhGYLZpLri0n5spGMnvw9ylzHh~4cK-f1tHA1S5aBT~8OfWIuIFw5aB6wt-8W5U1V6~7Sm5iXpGnjyzHw8yzSVKYbJHMs6zR~6k3~FnYk52ByBtswZhYX1mQVHCF4kwkqTnRSHLKbBJmyUBoFDB1FaJMnJSiguZ6rZrD6nj4u7BqBVlrhyOOi74VTWc~h4EA6a7HiXTmgTL~OLcHcDX16X88PXrrTKcU-u6t7YsDpBUOycLD~FAdDEXDWJQKsgyoPhQXZFeXPOKjYU2pSDGAaTuu0x43Dv1u~FkqkYCNrJ8t5by0g__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f02.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f02.jpeg?Expires=1488590074&Signature=X80qNW8IUl6NeSa4VqDVcGqIL2mvpAcBUsr9EDhGYLZpLri0n5spGMnvw9ylzHh~4cK-f1tHA1S5aBT~8OfWIuIFw5aB6wt-8W5U1V6~7Sm5iXpGnjyzHw8yzSVKYbJHMs6zR~6k3~FnYk52ByBtswZhYX1mQVHCF4kwkqTnRSHLKbBJmyUBoFDB1FaJMnJSiguZ6rZrD6nj4u7BqBVlrhyOOi74VTWc~h4EA6a7HiXTmgTL~OLcHcDX16X88PXrrTKcU-u6t7YsDpBUOycLD~FAdDEXDWJQKsgyoPhQXZFeXPOKjYU2pSDGAaTuu0x43Dv1u~FkqkYCNrJ8t5by0g__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941610&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f02.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Physical models demonstrating the genesis of normal and log-normal distributions. Particles fall from a funnel onto tips of triangles, where they are deviated to the left or to the right with equal probability (0.5) and finally fall into receptacles. The medians of the distributions remain below the entry points of the particles. If the tip of a triangle is at distance x from the left edge of the board, triangle tips to the right and to the left below it are placed at x + c and x − c for the normal distribution (panel a), and x · ć and x / ć for the log-normal (panel b, patent pending), c and ć being constants. The distributions are generated by many small random effects (according to the central limit theorem) that are additive for the normal distribution and multiplicative for the log-normal. We therefore suggest the alternative name multiplicative normal distribution for the latter</p></div></div>                <a id="37941612" scrollto-destination="37941612"></a>
<div id="I0006-3568-51-5-341-F03" data-id="I0006-3568-51-5-341-F03" class="fig fig-section"><div class="label fig-label">Figure 3.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=V0gZtnMbxtcnbrOZbd0xf2Cz1t2P64d3xmPnFCAQefw6OymMsE40jgz6XTpg1aNqe3uvPWmM2q9BmJMbF-Aee667RDiNppP~cVmddKMKDgG1kA8BKkTDxNm5H0LtLFbWy0Ox7OKza8PTuNK8WKoQ40kr6A~J7ZwtAEBDeZJ63uPp8hac5rx9X-uz13EZLOMG~jOPDOIrSksigD7faB7vnWQ25ywKzSjNYJOeLUUvxVNWeTuNz~djTkTzGDsP4DSvHLWJeHgw2LgQyJaNuPQktj847o1~01hOH1SsFI94Tq0aFxV7xPt6YMjqO~jseTj2zqNn6hmEFrT57kH~2nMIGQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="A log-normal distribution with original scale (a) and with logarithmic scale (b). Areas under the curve, from the median to both sides, correspond to one and two standard deviation ranges of the normal distribution" path-from-xml="i0006-3568-51-5-341-f03.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=ZwdxdL8Nv-ottTfr6FfyCqXFrPKl9kTdCWfDnVwQSsoW9GSY4m9eM78xLYLKyv8PJKgL4-LXIMK9OBmeuYjmTHU9bbqj74hD3~~mcU33cVlLL35ylvayuEcZr9RnBlhvKs~RNfP5BiFo2cfzIjkZiTRORF4JQTgLIcsfV2hgjC~ry6sQ~H70gF6~eGLVkkvoV8D6MGAa40pxjt1Fcc74pyZQvtGwjDglcVtuLCQaCJeZSUrrdC9ziV~hDtKM13nEubBzdfiy56gxbfJx5M3MVmI1FZs-0BxDQthwwNYrGcbbgqVzLsU2BuAQFWKowd4TXPWLhzZXmo8S9oBPX3aS7w__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f03.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=ZwdxdL8Nv-ottTfr6FfyCqXFrPKl9kTdCWfDnVwQSsoW9GSY4m9eM78xLYLKyv8PJKgL4-LXIMK9OBmeuYjmTHU9bbqj74hD3~~mcU33cVlLL35ylvayuEcZr9RnBlhvKs~RNfP5BiFo2cfzIjkZiTRORF4JQTgLIcsfV2hgjC~ry6sQ~H70gF6~eGLVkkvoV8D6MGAa40pxjt1Fcc74pyZQvtGwjDglcVtuLCQaCJeZSUrrdC9ziV~hDtKM13nEubBzdfiy56gxbfJx5M3MVmI1FZs-0BxDQthwwNYrGcbbgqVzLsU2BuAQFWKowd4TXPWLhzZXmo8S9oBPX3aS7w__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941612&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f03.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>A log-normal distribution with original scale (a) and with logarithmic scale (b). Areas under the curve, from the median to both sides, correspond to one and two standard deviation ranges of the normal distribution</p></div></div><div content-id="I0006-3568-51-5-341-F03" class="fig fig-modal reveal-modal"><div class="label fig-label">Figure 3.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F03" data-open="I0006-3568-51-5-341-F03" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=V0gZtnMbxtcnbrOZbd0xf2Cz1t2P64d3xmPnFCAQefw6OymMsE40jgz6XTpg1aNqe3uvPWmM2q9BmJMbF-Aee667RDiNppP~cVmddKMKDgG1kA8BKkTDxNm5H0LtLFbWy0Ox7OKza8PTuNK8WKoQ40kr6A~J7ZwtAEBDeZJ63uPp8hac5rx9X-uz13EZLOMG~jOPDOIrSksigD7faB7vnWQ25ywKzSjNYJOeLUUvxVNWeTuNz~djTkTzGDsP4DSvHLWJeHgw2LgQyJaNuPQktj847o1~01hOH1SsFI94Tq0aFxV7xPt6YMjqO~jseTj2zqNn6hmEFrT57kH~2nMIGQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="A log-normal distribution with original scale (a) and with logarithmic scale (b). Areas under the curve, from the median to both sides, correspond to one and two standard deviation ranges of the normal distribution" path-from-xml="i0006-3568-51-5-341-f03.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=ZwdxdL8Nv-ottTfr6FfyCqXFrPKl9kTdCWfDnVwQSsoW9GSY4m9eM78xLYLKyv8PJKgL4-LXIMK9OBmeuYjmTHU9bbqj74hD3~~mcU33cVlLL35ylvayuEcZr9RnBlhvKs~RNfP5BiFo2cfzIjkZiTRORF4JQTgLIcsfV2hgjC~ry6sQ~H70gF6~eGLVkkvoV8D6MGAa40pxjt1Fcc74pyZQvtGwjDglcVtuLCQaCJeZSUrrdC9ziV~hDtKM13nEubBzdfiy56gxbfJx5M3MVmI1FZs-0BxDQthwwNYrGcbbgqVzLsU2BuAQFWKowd4TXPWLhzZXmo8S9oBPX3aS7w__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f03.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f03.jpeg?Expires=1488590074&Signature=ZwdxdL8Nv-ottTfr6FfyCqXFrPKl9kTdCWfDnVwQSsoW9GSY4m9eM78xLYLKyv8PJKgL4-LXIMK9OBmeuYjmTHU9bbqj74hD3~~mcU33cVlLL35ylvayuEcZr9RnBlhvKs~RNfP5BiFo2cfzIjkZiTRORF4JQTgLIcsfV2hgjC~ry6sQ~H70gF6~eGLVkkvoV8D6MGAa40pxjt1Fcc74pyZQvtGwjDglcVtuLCQaCJeZSUrrdC9ziV~hDtKM13nEubBzdfiy56gxbfJx5M3MVmI1FZs-0BxDQthwwNYrGcbbgqVzLsU2BuAQFWKowd4TXPWLhzZXmo8S9oBPX3aS7w__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941612&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f03.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>A log-normal distribution with original scale (a) and with logarithmic scale (b). Areas under the curve, from the median to both sides, correspond to one and two standard deviation ranges of the normal distribution</p></div></div>                <a id="37941613" scrollto-destination="37941613"></a>
<div id="I0006-3568-51-5-341-F04" data-id="I0006-3568-51-5-341-F04" class="fig fig-section"><div class="label fig-label">Figure 4.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F04" data-open="I0006-3568-51-5-341-F04" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=LIYH0PIXaQz9bYM61TkrVQLmc9D2KgvJ4qJbcqZWV7r-RpwoCncxgWPmUYvUX-7CGetzy~kfD-JcYEOPhVEWvUJZsvhhMsJNPWXVI1rCFog603tkdSDAst6piyTPjES1KpjZ8p4YdMFqAjsA~T7dw37sGeVOpMlRtRseiSQDxTBPsHC~9tngT15UulGXIjFI-7xQQ8qPJPHm0Y93NRgLENDM8ebLlLIEpVAkLbb-tOyOHpRjnsc8BLA4oPM2woCUe6re4j0rTyC3p9h2UyhhAP3W4YTyRAlne4IEfOfZeXpt8vbbO~LsYAujkki6oL6OMI8lQXSdDr11knmzbRaktQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Density functions of selected log-normal distributions compared with a normal distribution. Log-normal distributions Λ(µ*,σ*) shown for five values of multiplicative standard deviation, s*, are compared with the normal distribution (100 ± 20, shaded). The σ* values cover most of the range evident in Table 2. While the median µ* is the same for all densities, the modes approach zero with increasing shape parameters σ*. A change in µ* affects the scaling in horizontal and vertical directions, but the essential shape σ* remains the same" path-from-xml="i0006-3568-51-5-341-f04.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=DLvwmQZhh8r8myzKNdx9C3tRp2KJKnwk7YVHltLOM61kdQviLG2M5WEKRwiXudoBVMEDmxpw3q2F1YdRVMZdp~CosjqAqlZvXnga5Ag9caF~YUo7n5dQ9FZF6JqU-2fI3sQucydgXaga9JGqiErYXmb9YwUpyMXvRKj3ATmPLERyS6oHLgpVX3UlLmksoCZltPB5ubkDHGXwbWbmzv-aGJpsu7sWsod0G5iCTSueL1IBSFGrm3~mlbCe8ExPmJxy8iWMcYTXp3Va5y8YjjsI-S1wQ7cJIUN8LvFXKGfaG0vhzb9DxVOEFo6Cm8PVkdindCD-l9wf~1M9Sx0v1XHFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f04.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=DLvwmQZhh8r8myzKNdx9C3tRp2KJKnwk7YVHltLOM61kdQviLG2M5WEKRwiXudoBVMEDmxpw3q2F1YdRVMZdp~CosjqAqlZvXnga5Ag9caF~YUo7n5dQ9FZF6JqU-2fI3sQucydgXaga9JGqiErYXmb9YwUpyMXvRKj3ATmPLERyS6oHLgpVX3UlLmksoCZltPB5ubkDHGXwbWbmzv-aGJpsu7sWsod0G5iCTSueL1IBSFGrm3~mlbCe8ExPmJxy8iWMcYTXp3Va5y8YjjsI-S1wQ7cJIUN8LvFXKGfaG0vhzb9DxVOEFo6Cm8PVkdindCD-l9wf~1M9Sx0v1XHFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941613&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f04.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Density functions of selected log-normal distributions compared with a normal distribution. Log-normal distributions Λ(µ*,σ*) shown for five values of multiplicative standard deviation, s*, are compared with the normal distribution (100 ± 20, shaded). The σ* values cover most of the range evident in <a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup>. While the median µ* is the same for all densities, the modes approach zero with increasing shape parameters σ*. A change in µ* affects the scaling in horizontal and vertical directions, but the essential shape σ* remains the same</p></div></div><div content-id="I0006-3568-51-5-341-F04" class="fig fig-modal reveal-modal"><div class="label fig-label">Figure 4.</div><div class="graphic-wrap"><a reveal-id="I0006-3568-51-5-341-F04" data-open="I0006-3568-51-5-341-F04" class="fig-link"><img class="content-image" src="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051%5B0341_LNDATS%5D2.0.CO;2/3/m_i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=LIYH0PIXaQz9bYM61TkrVQLmc9D2KgvJ4qJbcqZWV7r-RpwoCncxgWPmUYvUX-7CGetzy~kfD-JcYEOPhVEWvUJZsvhhMsJNPWXVI1rCFog603tkdSDAst6piyTPjES1KpjZ8p4YdMFqAjsA~T7dw37sGeVOpMlRtRseiSQDxTBPsHC~9tngT15UulGXIjFI-7xQQ8qPJPHm0Y93NRgLENDM8ebLlLIEpVAkLbb-tOyOHpRjnsc8BLA4oPM2woCUe6re4j0rTyC3p9h2UyhhAP3W4YTyRAlne4IEfOfZeXpt8vbbO~LsYAujkki6oL6OMI8lQXSdDr11knmzbRaktQ__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" alt="Density functions of selected log-normal distributions compared with a normal distribution. Log-normal distributions Λ(µ*,σ*) shown for five values of multiplicative standard deviation, s*, are compared with the normal distribution (100 ± 20, shaded). The σ* values cover most of the range evident in Table 2. While the median µ* is the same for all densities, the modes approach zero with increasing shape parameters σ*. A change in µ* affects the scaling in horizontal and vertical directions, but the essential shape σ* remains the same" path-from-xml="i0006-3568-51-5-341-f04.jpeg" /></a><div class="fig-orig original-slide"><a section="" class="fig-view-orig" href="https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=DLvwmQZhh8r8myzKNdx9C3tRp2KJKnwk7YVHltLOM61kdQviLG2M5WEKRwiXudoBVMEDmxpw3q2F1YdRVMZdp~CosjqAqlZvXnga5Ag9caF~YUo7n5dQ9FZF6JqU-2fI3sQucydgXaga9JGqiErYXmb9YwUpyMXvRKj3ATmPLERyS6oHLgpVX3UlLmksoCZltPB5ubkDHGXwbWbmzv-aGJpsu7sWsod0G5iCTSueL1IBSFGrm3~mlbCe8ExPmJxy8iWMcYTXp3Va5y8YjjsI-S1wQ7cJIUN8LvFXKGfaG0vhzb9DxVOEFo6Cm8PVkdindCD-l9wf~1M9Sx0v1XHFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q" path-from-xml="i0006-3568-51-5-341-f04.jpeg" target="_blank">View large</a><a section="" href="//oup.silverchair-cdn.com/DownloadFile/DownloadImage.aspx?image=https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/bioscience/51/5/10.1641_0006-3568(2001)051[0341_LNDATS]2.0.CO;2/3/i0006-3568-51-5-341-f04.jpeg?Expires=1488590074&Signature=DLvwmQZhh8r8myzKNdx9C3tRp2KJKnwk7YVHltLOM61kdQviLG2M5WEKRwiXudoBVMEDmxpw3q2F1YdRVMZdp~CosjqAqlZvXnga5Ag9caF~YUo7n5dQ9FZF6JqU-2fI3sQucydgXaga9JGqiErYXmb9YwUpyMXvRKj3ATmPLERyS6oHLgpVX3UlLmksoCZltPB5ubkDHGXwbWbmzv-aGJpsu7sWsod0G5iCTSueL1IBSFGrm3~mlbCe8ExPmJxy8iWMcYTXp3Va5y8YjjsI-S1wQ7cJIUN8LvFXKGfaG0vhzb9DxVOEFo6Cm8PVkdindCD-l9wf~1M9Sx0v1XHFwg__&Key-Pair-Id=APKAIUCZBIA4LVPAVW3Q&sec=37941613&ar=243981&xsltPath=~/UI/app/XSLT&imagename=" class="download-slide" path-from-xml="i0006-3568-51-5-341-f04.jpeg" target="_blank">Download slide</a></div></div><div class="caption fig-caption"><p>Density functions of selected log-normal distributions compared with a normal distribution. Log-normal distributions Λ(µ*,σ*) shown for five values of multiplicative standard deviation, s*, are compared with the normal distribution (100 ± 20, shaded). The σ* values cover most of the range evident in <a reveal-id="I0006-3568-51-5-341-T201" data-open="I0006-3568-51-5-341-T201" class="link link-reveal link-table xref-fig">Table 2</a><sup class="sup-zero"><a href="#I0006-3568-51-5-341-T202" class="figlink">202</a></sup>. While the median µ* is the same for all densities, the modes approach zero with increasing shape parameters σ*. A change in µ* affects the scaling in horizontal and vertical directions, but the essential shape σ* remains the same</p></div></div><div content-id="n4"><div class="fn"><div class="label -label">1</div><div class="footnote-content">Species abundance may be described by a log-normal law (<a reveal-id="I0006-3568-51-5-341-PRESTON1" data-open="I0006-3568-51-5-341-PRESTON1" class="link link-ref link-reveal xref-bibr">Preston 1948</a>), usually written in the form S(R) = S<sub>0</sub> · exp(−a<sup>2</sup>R<sup>2</sup>), where S<sub>0</sub> is the number of species at the mode of the distribution. The shape parameter a amounts to approximately 0.2 for all species, which corresponds to s* = 11.6.</div></div></div>        <!-- /foreach -->



                            <h2 class="authorNotes-section-title" id="authorNotesSectionTitle" scrollto-destination="authorNotesSectionTitle">Author notes</h2>
<div content-id="n1"><div class="fn"><div class="label -label">1</div><div class="footnote-content">Eckhard Limpert (email: <a href="mailto:Eckhard.Limpert@ipw.agrl.ethz.ch" target="_blank">Eckhard.Limpert@ipw.agrl.ethz.ch</a>) is a biologist and senior scientist in the Phytopathology Group of the Institute of Plant Sciences in Zurich, Switzerland.</div></div></div><div content-id="n2"><div class="fn"><div class="label -label">2</div><div class="footnote-content">Werner A. Stahel (email: <a href="mailto:stahel@stat.math.ethz.ch" target="_blank">stahel@stat.math.ethz.ch</a>) is a mathematician and head of the Consulting Service at the Statistics Group, Swiss Federal Institute of Technology (ETH), CH-8092 Zürich, Switzerland.</div></div></div><div content-id="n3"><div class="fn"><div class="label -label">3</div><div class="footnote-content">Markus Abbt is a mathematician and consultant at FJA Feilmeier &amp; Junker AG, CH-8008 Zürich, Switzerland.</div></div></div><div class="copyright copyright-statement">© 2001 American Institute of Biological Sciences</div><!-- /foreach -->

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