Solved – How to calculate a mean and standard deviation for a lognormal distribution using 2 percentiles

lognormal distributionr

I am trying to calculate a mean and standard deviation from 2 percentiles for a lognormal distribution.

I was successful in performing the calculation for a normal distribution using X = mean + sd * Z and solving for mean and sd.

I think I am missing an equation when I try to do the same thing for a lognormal distribution. I looked at wikipedia and trying to use ln(X) = mean + sd * Z but I'm getting confused whether the mean and sd in this case are for the normal distribution or the lognormal.

Which equations should I be using? and will I need more than 2 percentile to solve the calculations?

Best Answer

It seems that you "know" or otherwise assume that you have two quantiles; say you have that 42 and 666 are the 10% and 90% points for a lognormal.

The key is that almost everything is easier to do and understand on the logged (normal) scale; exponentiate as little and as late as possible.

I take as examples quantiles that are symmetrically placed on the cumulative probability scale. Then the mean on the log scale is halfway between them and the standard deviation (sd) on the log scale can be estimated using the normal quantile function.

I used Mata from Stata for these sample calculations. The backslash \ joins elements column-wise.

mean = mean(ln((42 \ 666)))

(ln(666) - mean) / invnormal(0.9)
1.078232092

SD = (ln(666) - mean) / invnormal(0.9)

The mean on the exponentiated scale is then

exp(mean + SD^2/2)
299.0981759

and the variance is left as an exercise.

(Aside: It should be as easy or easier in any other decent software. invnormal() is just qnorm() in R if I recall correctly.)

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Solved – Estimating a distribution based on three percentiles and a mean

if you know the true mean and percentiles, then the fitting approach can be helpful and done by optimizers, but usually you only have estimates for mean and percentiles, like from a data set of 100 points. If you apply now an optimization for the fit the result is highly questionable. For instance a fit to moments should be more stable, best is a fit by MLE. Most approaches need first to make an assumption on the pdf (like data is lognormal), then finding the parameters. Or you may use a very flexible multi-parameter distribution which often ends up that the parameter estimation is not very stable. A good starting point is taking the empirical CDF as model + plus some smoothing (e.g. by kernal densitiy functions). This would be a many-parameter "model". Using less parameters give usually more systematic errors, but more stable parameters & estimations. So a compromize is needed, which e.g. depends on how many data points you have.

Bye Stephan

Solved – Normal Distribution with random mean and standard deviation

Your solution is correct, assuming the two normal random variables are independent. According to the R documentation of rnorm, you can input a vector of means and standard deviations for the mean and sd arguments respectively.

To verify, consider this toy example:

n <- 3 
mean_vector <- c(0,10,100)
sd_vector <- c(1,1,1)

rnorm(3, mean=mean_vector, sd=sd_vector)

Some output:

[1]  1.049676 11.566033 98.481899
[1] -1.374753  9.078215 99.465803
[1]  3.056377  9.837055 98.842553

Clearly the first variate for each simulation is $N(0,1)$ distributed, the second is $N(10,1)$ distributed, and the third is $N(100,1)$ distributed.

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