# Use of Kurtosis statistic for understanding lognormality

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$price median(x) / exp(mean(log(x))) #>  0.996877  But I'm wondering about use of Kurtosis approach. The following function uses Pearson's measure. require(moments) #> Loading required package: moments kurtosis(x) #>  5.177383  As I understand this tells us the tails are not a normal distribution. So checking the kurtosis of the log gives us: kurtosis(log(x)) #>  1.903206  Does less than 3 indicates less tail than we would expect with a lognormal distribution? In the general case (exploring lognormality) is this a sensible approach? Would we also be wanting to apply skewness methods to robustly pin this down? • 1. Note that statistics that are reasonably consistent with data having been drawn from a lognormal doesn't imply that you have a lognormal; e.g. other distributions can also have median close to geometric mean.$\:$2. Choice of statistic will change depending on what distribution(s) you want good ability to distinguish a lognormal from -- and perhaps what you're doing it for (why you're trying to identify lognormality)$\:\$ 3. "As I understand this tells us the tails are not a normal distribution" - does this imply you're primarily interesting in distinguishing a lognormal from a normal? Jul 12, 2020 at 2:58
• Kurtosis can't be a good indicator of whether a distribution is lognormal for the reasons @Glen_b gives, and others. High kurtosis is consistent with zero or negative skewness, for one. For another, sample kurtosis is limited as a function of sample size and so will often deny the parentage of a lognormal even when that is a fact. The specific example of a lognormal and sample skewness and kurtosis is discussed in detail in stata-journal.com/article.html?article=st0204 (to the references there add jstor.org/stable/2236642) Jul 12, 2020 at 9:26
• The best way to check for lognormal distributions is to take logarithms and check for normal distributions! Jul 12, 2020 at 9:27
• The use of sample kurtosis--and any other high moment--is an exceptionally unreliable way to check for most distributional properties. Even its use to check for normality in the Jarque-Bera test (where the sample kurtosis should be well behaved) has long been deprecated in favor of better methods.
– whuber
Jul 12, 2020 at 17:19
• A different measure of kurtosis can't solve the basic problem. Kurtosis is not a measure of how far distributions are lognormal. Jul 13, 2020 at 11:40