Comparing Gamma and Log-Normal Distributions

display

pdf

α

1.25

This Demonstration compares the gamma distribution

G

α,β

and the log-normal distribution

L

μ,σ

. Both of these distributions are widely used for describing positively skewed data. Various distribution plots are shown as well as a table comparing the coefficients of skewness and kurtosis, denoted by

γ

1

and

γ

2

, respectively. Plots of the probability density function (pdf) of the distributions are useful in seeing the overall shape of the distribution but other plots provide additional insights. For example, the

q-q

plot and normal probability plot are better for showing small differences in the tails.

Our purpose is to compare the shapes of the gamma and log-normal distributions, so we fix their means to be 1 and constrain their coefficients of variation to be equal. These assumptions require that the log-normal parameter is

2

σ

=log(1+

-1

α

)

and that the second gamma parameter is

β=

1

α

.