It is a common mistake to think a Lognormal(mu, sigma) distribution has mean mu and standard deviation sigma.

The claim has two components: (a) the mathematical fact that the mu and sigma parameters of a Lognormal(mu, sigma) distribution are not its mean and standard deviation, since the mean is exp(mu + sigma²/2) and the variance is (exp(sigma²) − 1)exp(2mu + sigma²); and (b) the claim that mistaking them for the mean and standard deviation is common. True: (a) holds and there is documented evidence that the confusion is common, for example warnings in textbooks or software documentation, or repeated Q&A threads about it. Mostly true: (a) holds, but the evidence that the mistake is common is only anecdotal or thin. Mostly false: (a) holds, but evidence indicates the mistake is rare. False: the parameters are in fact the mean and standard deviation (this would mean (a) fails).

Set before the research began.

1. E1 The Lognormal Distribution in LCA Springer (book chapter, Springer Nature) · 27 December 2023

   “Carlton and Devore (2017) write “be careful here: the parameters \(\mu \) and \(\sigma \) are not the mean and standard deviation of X but of \(\ln \left( X\right) \)”.”

2. E2 Lognormal Distribution in Environmenal Applications U.S. EPA Office of Research and Development / Office of Solid Waste and Emergency Response (EPA/600/R-97/006) · December 1997

   “It should be noted that fj. and tr2 are not the mean and variance of the lognormal random variable, X, but they are the mean and variance of the log- transformed random variable, Y.”

3. E3 What are the input arguments “s” and “scale” defined in scipy.stats.lognorm? (scipy.stats.lognorm documentation error) GitHub (scipy/scipy issue tracker) · 11 September 2020

   “This paragraph is confusing and seems to state the mean = mu and standard deviation = sigma.”

4. E4 Analysis and remediation of the confusing specification of the lognormal distribution The International Journal of Life Cycle Assessment (Springer), author Reinout Heijungs, VU Research Portal copy · 2024

   “Indeed, we will see that several authors are in doubt, and contradict themselves between their articles, or even within the same article.”

E1 and E2 support part (a): E1 quotes a 2017 probability textbook (Carlton and Devore) and E2 quotes a 1997 EPA issue paper, and both state that the lognormal mu and sigma are the mean and spread of ln(X), not of X itself. For part (b), the criteria count textbook warnings as documented evidence, and E1 shows one. E3 is a 2020 SciPy issue in which a user read the lognormal documentation as saying the mean is mu and the standard deviation is sigma. E4 is a peer-reviewed paper about the confusing way the distribution is specified, and it says several authors contradict themselves on this point. Together, these sources document the confusion beyond a single anecdote, which meets the criteria for true.