What can be inferred about a population if its distribution is lognormal?

Question

If I have a population that is lognormally distributed, what does that say about the population? If I log-transform the population, can I draw conclusions about it based on the normally distributed transformed data? If so, what conclusions can I draw?

For example, can I call a given sample an outlier if its natural log is greater than the 3rd quartile + (1.5 * IQR) of the log-transformed population? Does that distinction translate back to the original distribution?

Answer 1

You can call anything whatever you want. Whether that is a sensible thing to do is another matter, and will depend on the circumstances.

• With a normal distribution with mean $\mu$ and standard deviation $\sigma$, anything more than about $\mu+2.698\sigma$ would meet your definition of "outlier", and that would cover about $0.35\%$ of the distribution. It is an actual part of the distribution and cannot be rejected simply for being too far away from the centre of the distribution. (Perhaps you have the equivalent on the the lower tail.)

• For a log-normal distribution, what happens depends on the value of $\sigma$, but in any case you get a positive skew and would expect more upper "outliers". With parameters $\mu=0, \sigma=1$ your upper "outliers" would be about $7.76\%$ of the distribution. With $\mu=0, \sigma=6$ your upper "outliers" would be about $20.4\%$ of the distribution, so the large majority of the upper quarter of the distribution.

Here is an example using R of a box plot of the first case of a lognormal taking a sample size $100$ (so we might hope to see perhaps about $7$ or $8$ "outliers" and actually see $7$ here though two are very close so look like the same point just below $10$ on the chart, not bad for a simulation). These are part of the log-normal distribution and should not be rejected just because they are above the top whisker.

set.seed(2023)
X <- exp(rnorm(100, mean=0, sd=1))
boxplot(X)