Lognormal(m,s)

Lognormal equations (format 1)

Crystal Ball parameter restrictions

Examples of two Lognormal distribution are given below:

#### Uses

The Lognormal distribution is useful for modeling naturally occurring variables that are the *product* of a number of other naturally occurring variables. Central Limit Theorem shows that the product of a large number of independent random variables is Lognormally distributed. For example, the volume of gas in a petroleum reserve is often Lognormally distributed because it is the product of the area of the formation, its thickness, formation pressure, porosity and the gas:liquid ratio.

Lognormal distributions often provide a good representation for a physical quantity that extend from zero to + infinity and is positively skewed, perhaps because some Central limit Theorem type of process is determining the variable's size. Lognormal distributions are also very useful for representing quantities that are thought of in orders of magnitude. For example, if a variable can be estimated to within a factor of 2 or to within an order of magnitude, the Lognormal distribution is often a reasonable model.

Lognormal distributions have also been used to model lengths of words and sentences in a document, particle sizes in aggregates, critical doses in pharmacy and incubation periods of infectious diseases, but one reason the Lognormal distribution appears so frequently is because it is easy to fit and test (one simply transforms the data to logs and manipulate as a Normal distribution), and so observing its use in your field does not necessarily mean it is a good model: it may just have been a convenient one. Modern software and statistical techniques have removed much of the need for assumptions of normality, so be cautious about using the Lognormal because it has always been that way.

This version of ModelAssist offers some finance model examples where the Lognormal distribution is used in credit risk modeling and in modeling stock prices.

#### Comments

A variable is Lognormally distributed when the natural log of the variable is Normally distributed i.e. *X* is Lognormally distributed if ln[*X*] is Normally distributed. The Lognormal2 distribution has as its parameters the mean and standard deviation of the Normal distribution of ln[*X*]. It is very occasionally known as the *Galton-McAlister* distribution and, in economics, as the *Cobb-Douglas* distribution where it is applied to production data.