LogNormal Distribution

The LogNormal distribution is based on the normal distribution.  It is describes a variable, x, where log(x) is normally distributed.  It is valid for values of x which are greater than zero.  The lognormal distribution describes many naturally occurring populations.  In the mining and extraction industries, it has been observed that where the value of an item is proportional to size, the population is probably lognormally distributed, with few valuable items and lots of uncommercial items, the biosciences may have a different perception.

There are three ways of defining a lognormal distribution:

LogNormal - A

Mean and Standard Deviation of x.

LogNormal - B

Mean of log(x) and Standard Deviation of log(x)

LogNormal - C

Median of x and Standard Deviation of log(x)

The relationship between the types of input is:

The table below shows a lognormal distribution with parameter values for the A, B and C schemes described above:

	LogNormal A	Lognormal B	LogNormal C
Mean(x)	10
StDev(x)	5
Mean (log(x))		2.19
StDev (log(x))		0.47	0.47
Median(x)			8.93

The PDF of the distribution is shown below:

The reason the three schemes are presented is to reflect the inputs requested by different software packages, e.g. Excel requires the mean(ln(x)) and stdev(ln(x)), i.e. LogNormal B

Example

The lognormal distribution is often used to model the distribution of reserves in oilfields within a province.  The example below is for the United Kingdom's East Midlands basin:

In this case the data represents a sample of 20 known fields, whilst an imperfect fit, the lognormal could be used as the basis for modelling an exploration program.

Profile

Parameters

LogNormal A

Parameter	Description	Characteristics
mean	Population mean	A float > 0 and < ∞
std. dev.	Standard Deviation of Population	A float > 0

LogNormal B

Parameter	Description	Characteristics
mean (log)	Mean of log(x)	A float > 0 and < ∞
std. dev. (log)	Standard Deviation of log(x)	A float > 0

LogNormal C

Parameter	Description	Characteristics
median	Median of x	A float > 0 and < ∞
std. dev. (log)	Standard Deviation of log(x)	A float > 0

Range

The range of random numbers generated for the LogNormal distribution is from greater than zero to positive infinity.

Properties

Random Number Generation

The Box-Muller method described in section for the Normal distribution can be adapted for the LogNormal distribution:

Step 1

Assuming the required distribution is defined in terms of the mean and standard deviation of the population, use equations 122.1 and 122.2 to derive mean_log and stdev_log.

Step 2

Generate standard normally distributed random number:

r1=Sqr(-2 * Log(rnd())) * Sin(2 * PI * rnd())

Step 3

Scale the standard value with mean_log and stdev_log

r1=mean_log+r1*stdev_log

Step 4

Exponentiate r1 to create a lognormal value:

r1=exp(r1)

Parameter Estimation

The mean and standard deviation are derived formulas adapted from the standard normal distribution: