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
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
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.)