# When is the log-normal distribution appropriate?

Iv'e read the Wikipedia entry about the log-normal distribution, as well as a few other sources online, and still do not understand what sort of natural processes are expected to produce a log-normal distribution.

I understand how this distribution arises in processes with many independent factors whose effect is multiplicative, but not which processes are expected to behave in this way.

Both the Wikipedia entry and this review supply several examples of log-normally distributed phenomena, but the only one (aside from the multiplicative Galton board) for which I understand why the effect is multiplicative, is the distribution of bacteria colony sizes - The colonies double in number at each successive division, and the log of the colony size, the number of divisions, should be normally distributed.

Question:

Could anyone explain why the many examples of log-normally distributed data are multiplicative in nature, and more generally, how one comes to suspect , a-priori, such multiplicative phenomena as opposed to additive?

• I wouldn't over-emphasise any process or mechanistic rationale. The lognormal can be a plausible candidate distribution for heterogeneous mixtures regardless of how they were produced. Ditto for positive but right skewed distributions. Also, a distribution doesn't have to fit closely to be worth taking seriously, just more convincing than leading alternatives. May 7, 2019 at 14:18
• The leading question across much of statistical sciences is modelling a response in terms of selected predictors. The marginal distribution of the response can be messy if the predictors are; conversely, toy schemes that show how particular distributions can arise are often highly idealised. May 7, 2019 at 14:21
• NIST Handbook:"The lognormal distribution is used extensively in reliability applications to model failure times. The lognormal and Weibull distributions are probably the most commonly used distributions in reliability applications." Also google. // Many applications refer to empirical 'fitting' rather then theoretical derivations. // Often used to model numbers of earthquakes at various magnitude; sparse left tail may be due simply to lack of detection. May 7, 2019 at 17:05
• @NickCox: I agree that process considerations shouldn't be over-emphasized. But conversely, they can be under-emphasized. There is always a temptation to just fit lots of distributions and go with the one that fits best. I prefer having at least some kind of argument why a particular distribution makes sense. May 8, 2019 at 8:23
• A paper answering the question and a similar Q with answer: stats.stackexchange.com/questions/204578/… May 8, 2019 at 9:47

I can give one example where one might suspect multiplicative effects, leading to a lognormal distribution.

Retailers like supermarkets have to forecast their demand (Fildes et al, 2018). Demand is influenced by many factors, like seasonality (intra-weekly and intra-yearly), calendar events, promotions and prices.

A promotion on ice cream will have a much higher additive uplift in summer than in winter. The effect will also be higher on Saturday than on Wednesday (Saturdays are usually much higher selling days than Wednesdays, at least in Europe and the US).

This motivates multiplicative models. Yes, sales are usually count data, so a continuous distribution is not really appropriate, but especially on aggregate data, the approximation is often good enough.

And as a matter of fact, sales forecasting in retail often does use models on logged data. A random example would be autoregressive distributed lags (ADL) models as, e.g., in Huang, Fildes & Soopramanien (2014): (Sorry for just posting a screenshot, but this is for illustration only, anyway.)

• This kind of thinking has been used for modeling alcohol consumption (I first learned about that under the final exam of my first probability course). Here is a ncbi paper criticizing that empirically, and concluding that Gamma distributions work better. May 8, 2019 at 10:01