Log-normal Distributions across the Sciences: Keys and Clues

As the need grows for conceptualization, formalization, and abstraction in biology, 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.Often, biological mechanisms induce log-normal distributions (Koch 1966), as when, for instance, exponential growth is combined with further symmetrical variation: With a mean concentration of, say, 10 6 bacteria, one cell division moreor less-will lead to 2 × 10 6 -or 5 × 10 5 -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 (Groth 1914, Powers 1936, Sinnot 1937).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.