Limpert, and 2 co-authors, reported in 2020 that as the need grows for conceptualization, formalization, and abstraction in sciences, so too does mathematics’ relevance to the field (Fagerström et al. 1996). 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. 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 (Swoboda 1974). 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 (Aitchison and Brown 1957, Crow and Shimizu 1988, Lee 1992, Johnson et al. 1994, Sachs 1997). Examples fitting the normal distribution, which is symmetrical, and the lognormal distribution, which is skewed, are given in Figure 1. Note that body height fits both distributions.
The authors concluded that 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 (Table 3), the proposed characterization by ¯x * and s* offers several advantages, some of which have been described before (Sartwell 1950, Ahrens 1954, Limpert 1993). Both ¯x * 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.
On the first page of their book, Aitchison and Brown (1957) 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 lognormal 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 lognormal, or multiplicative normal, distribution over the Gaussian distribution when describing original data.
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