Limpert, et al., (2001) reported as the need grows for conceptualization, formalization, and interpretations across the sciences, so too does mathematics' relevance to various scientific fields. Mathematics is particularly important in biology 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.
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 x and the standard deviation s, so that data sets are commonly described by the expression x ± 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. In addition, such disparate phenomena as milk production by cows and random deviations from target values in industrial processes fit a normal distribution. 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. Examples fitting the normal distribution, which is symmetrical, and the log-normal distribution, which is skewed, are examined Note that body height fits both distributions.
Often, biological mechanisms induce log-normal distributions, as when, for instance, exponential growth is combined with further symmetrical variation: With a mean concentration of, say, 106 bacteria, one cell division more—or less—will lead to 2 × 106—or 5 — 105—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.
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. 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.
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. 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.
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. 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.
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, the authors present a novel physical model for generating log-normal distributions, thus filling a 100-year-old gap. They also demonstrate the evolution and use of parameters allowing characterization of the data at the original scale. Moreover, they 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.
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