### Some basic facts and formulas about the lognormal distribution

Definition. If X has a lognormal distribution, then Z=log(X) has a normal distribution. (Here, as usually, log is taken to be the natural logarithm.)

Parameters. The normal variable Z is best characterized by mean mu and variance sigma^2 or standard deviation sigma.
If you transform to the lognormal X by X=exp(Z), then

• mu.star=exp(mu) is the median and
• sigma.star=exp(sigma) is what we call the multiplicative standard deviation.

Moments and other characteristics.
The mean (expectation) of X is
E(X) = exp( mu + sigma^2/2 )
the variance,
var(X) = exp( sigma^2 + 2 mu) ( exp(sigma^2)-1 )
the median,
med(X) = exp(mu) = mu.star
thus, coefficient of variation = sqrt( exp(sigma^2)-1 )
mean/median = exp(sigma^2/2)

skewness = eta^3 + 3 eta , eta^2 := exp(sigma^2)-1
kurtosis = eta^8 + 6 eta^6 + 15 eta^4 + 16 eta^2

see Aitchinson and Brown (1957)
author = {Aitchinson, J. and Brown, J. A. C.},
title = {The Lognormal Distribution with special reference to its uses ineconomics},
publisher = {Cambridge UP},
year = 1957

Estimation of the parameters.
From the mean   mean.z   and the standard deviation   sd.z   of the logarithmized observations, as
mu.star.est = exp(mean.z), sigma.star.est = exp(sd.z)

Obtaining the parameters from expectation and variance:
exp(sigma^2) = 1 + cv^2 = 1 + var(X) / [E(X)^2] =: omega
sigma = sqrt(log(omega))
mu = log(E(X)) - sigma^2/2
mu.star = E(X) / sqrt(omega)
sigma.star = exp(sigma)

How do quartiles lead to an estimate of s*?
Explanation of the formula given after (4) on p. 345 of Limpert, Stahel, Abbt, Bioscience 51.
Quantiles transform to quantiles in monotonic transformations. Therefore, log(q_1) and log(q_2) are the lower and upper quartiles of log(X) . log(X) has a normal distribution.
Now, phi^-1(0.75)=0.674 is the upper quartile of a standard normal distribution, and 2*0.674=1.349 is the distance between the two quartiles, also called the interquartile range (IQR). Therefore, IQR/1.349 is an estimator of the standard deviation of a normal distribution.
Now, the interquartile range of the log(X) data is log(q_2) - log(q_1) = log(q_2/q_1), and log(q_2/q_1)/1.349 estimates the standard deviation of log(X). Transform back (take exp of this expression) to obtain (q_2/q_1)^(1/1.349).
This corresponds to the formula given after (4) on p. 345 -- up to a typo: q_1 and q_2 are mixed up in the paper.
(Thanks to Susan Budge from Alberta, CA for pointing this out to us.)

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