The log-normal distribution is a statistical model used to describe continuous random variables that are meaningful in the positive real numbers $\mathbb{R}^+$. Its shape is determined by the parameters $\alpha$ and $\beta$.

As shown above, a log-normal distribution can take on a wide range of shapes.

Generally, the shape changes significantly when the ratio of $\alpha$ to $\beta$ changes. For an increased $\beta$, relative to a constant $\alpha$, the density function becomes positively skewed. When the opposite is the case, so $\beta$ is small in comparison to $\alpha$, the distribution becomes symmetrical and approximates the normal distribution.

The effect of increased symmetry would also occur if $\alpha \rightarrow \infty$. In such cases, the whole distribution is moved far away from the lower boundary and therefore behaves as unconstrained.

A random variable $x$ is log-normally distributed if $\log(x)$ is normally distributed. Aitchison & Brown, 1957

Note: The log-normal distribution is only defined for all $X : x \in \mathbb{R}^+$, since the logarithm of 0 is $-\infty$ and not defined for $x < 0$.
In other words: log-normally distributed variables lie on a purely positive ratio scale.

The central tendency ($1^{st}$ moment) and dispersion ($2^{nd}$ moment) of the log-normal distribution are the geometric mean $$ \mu_{\text{geo}} = \frac{1}{n} \sum_{i=1}^{n} \log x_i= \left(\prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}$$ and geometric standard deviation $$\sigma_{\text{geo}} =\sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (\log x_i - \mu_{\text{geo}})^2}\, .$$

The density function of any log-normal distribution is defined as:

$$ f_X(x)=\frac{1}{x\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left[\frac{\log x-\mu}{\sigma}\right]^2}\, . $$

For ratio-scaled variables, the central tendency and dispersion can be estimated using the geometric mean and the geometric standard deviation. They operate by multiplication and division, which inherently avoids negative results, for $x \in \mathbb{R}^+$. Therefore, we can freely operate, without leaving the $\mathbb{R}^+$ data space. McAllister (1879) & Galton (1879).

Sigma intervals for log-normal distribution are reported as:

$$ \bar{x}_{geo} \,/\, sd_{geo} $$

and the three-sigma intervals are defined by:

$$ 1\sigma = \bar{x}_{geo} \cdot sd_{geo}^{1} $$ $$ 2\sigma = \bar{x}_{geo} \cdot sd_{geo}^{2} $$ $$ 3\sigma = \bar{x}_{geo} \cdot sd_{geo}^{3} $$

They contain approximately 68% ($\sigma^1$), 95% ($\sigma^2$), and 99.7% ($\sigma^3$) of the data.

For further explanation, see: The Uses of the Mean and Standard Deviation

The relevance of log-normal distributions

Log-normally distributed variables show a characteristic probability density function. This is commonly a positively skewed shape as well as a lower boundary, below which the probability $p$ of an event $x$ becomes exactly $p(x)=0$ Aitchison & Brown, 1957. As this might sound much more specific than the widely known normal distribution, the log-normal distribution is actually much more common to observe in empirical data. Aitchison & Brown, 1957 described this effect for economic sciences, which was later more validated across a variety of sciences by Limpert et. al., 2001.

This is the case because many physical quantities cannot be negative.

This is the case for e.g.: The height of physical objects, absolute concentrations of substances, or absolute mass. This causes the distribution to skew towards its lower boundary and commonly results in a log-normal distribution. Some may even be double constrained, meaning they inherit an upper boundary, in addition to the lower boundary. Such variables are better described by other distribution models.

The relationship between the normal and log-normal

The log-normal distribution and the normal distribution are very closely related to each other. We can translate any log-normally distributed variable $x$ to a normal distribution $z=f(x)$ by:

$$ z_i=\log(x_i) $$

and transform it back to a log-normal by:

$$ x_i=e^{z_i}. $$

This becomes especially useful when we need to perform statistical methods for $x$, which commonly require normality. The simple transformation of all values $x_i$ opens the data space to $\mathbb{R}$ and allows us to operate freely with sums and subtractions. The results of our method can then be transformed back to the original scale to give a meaningful result. Johnson, 1949.

In the words of Kirkwood Thomas, 1979, p.909, we may conclude:

"Non-statisticians are already familiar with means, standard deviations, standard errors, and coefficients of variation, and it should be relatively simple to extend this familiarity to their geometric counterparts. It is, of course, a little more complicated to perform the geometric calculations than the corresponding arithmetic ones, but this ought not to be a serious obstacle; nor is there any way such a difficulty can be wholly avoided if the multiplicative nature of the variation is to be acknowledged."

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The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.