Contents - Index

Lognormal Family of Distributions


The Lognormal family of distributions is made up of three distributions: lognormal, negative lognormal and normal.  It covers any specified average, standard deviation and skewness.  Together they form a 3-parameter family of distributions that is represented by a curve on a skewness-kurtosis plot as shown below.  The lognormal distribution covers the positive skewness portion of the curve.  The negative lognormal distribution covers the negative skewness portion of the curve.  The normal distribution handles the remaining case of zero skewness.

        Missing image: lognormal skewness-kurtosis plot.bmp

Density Function:

The density function of the lognormal distribution is shown below:

        Missing image: lognormal density function plot.bmp

The equation, parameters and bounds of the density function are:

         Missing image: lognormal density function.bmp


The moments of the lognormal distribution can be calculated from the parameters as shown below:

        Missing image: lognormal moments.bmp


As the skewness goes to zero, both the lognormal and negative lognormal distributions limit to the normal distribution.  This means that in some cases the lognormal and normal distributions can be difficult to distinguish between.  As a results, some sets of data may fit both the lognormal and normal distributions.

The lognormal distribution is the distribution of multiplication and division.  The central limit theorem states that as positive items are multiplied and divided, under certain restrictions, the result will tend to the lognormal distribution.  To see the central limit theory in practice, go to the Dice Experiments dialog box and specify 4 dice be multiplied together as shown below.

        Missing image: lognormal dice experiment.bmp

The lognormal distribution fits the resulting data as shown below.

        Missing image: lognormal dice experiment results.bmp