Lognormal Distribution:
A lognormal (log-normal or Galton) distribution is a probability distribution with a normally
distributed logarithm. A random variable is lognormally distributed if its logarithm is normally
distributed.
Skewed distributions with low mean values, large variance, and all-positive values often fit this
type of distribution. Values must be positive as log(x) exists only for positive values of x.
If log(X) has a normal distribution with mean μ and standard deviation σ , we say that X has
lognormal distribution with parameter μ and σ .
The mean m and variance v of a lognormal random variable are functions of µ and σ are:
m=e(μ+σ 2/2)
,
v=e(2μ+ σ2)(e(σ 2)−1)
A lognormal distribution with mean m and variance v has parameters
μ=log(m2
√
v+m2),σ=
√
log(v
m2+1)
The probability density function is defined by the mean μ and standard deviation, σ:
The shape of the lognormal distribution is defined by three parameters:
•σ, the shape parameter. Also the standard deviation for the lognormal, this affects the general
shape of the distribution. Usually, these parameters are known from historical data.
Sometimes, you might be able to estimate it with current data. The shape parameter doesn’t
change the location or height of the graph; it just affects the overall shape.
•m, the scale parameter (this is also the median). This parameter shrinks or stretches the
graph.
• μ, the location parameter, which tells you where on the x-axis the graph is located.
The standard lognormal distribution has a location parameter of 0 and a scale parameter of 1
(shown in blue in the image below). If μ = 0, the distribution is called a 2-parameter lognormal
distribution.
