To help clarify my understanding of this statistic, I'd appreciate feedback on the rationale presented here.
Assume we have a distribution that seems potentially lognormal. Checking the median against geometric mean can be an indication of lognormality (though I don't know if this is a reliable statistical test).
x = ggplot2::diamonds$price median(x) / exp(mean(log(x))) #>  0.996877
But I'm wondering about use of Kurtosis approach. The following function uses Pearson's measure.
require(moments) #> Loading required package: moments kurtosis(x) #>  5.177383
As I understand this tells us the tails are not a normal distribution. So checking the kurtosis of the log gives us:
kurtosis(log(x)) #>  1.903206
Does less than 3 indicates less tail than we would expect with a lognormal distribution?
In the general case (exploring lognormality) is this a sensible approach? Would we also be wanting to apply skewness methods to robustly pin this down?