I have random population sample data that I would like to describe using a distribution. If I plot the estimated kernel density, the data appear positively skewed and using functions in R such as fitdistr and fitdistrplus, it looks like a gamma or log-normal distribution could fit (by QQ plot). I have read the numerous discussions (sometimes in comments so I won't link the questions) that real data is rarely ever well modeled by something so simple as a two-parameter distribution, but I was hoping that it might still be appropriate, if I acknowledge that I accept it as an approximation.
There has also been discussion about distinguishing between a gamma and log-normal distribution. I trust this will not be a duplicate however, because in deciding between the two, I took a suggestion I saw in a comment (the post to which I can no longer find, as I forget what I searched at the time) that if one plots the log transformation of the data, it should flip the skew if gamma and become normal (by definition) if log-normal.
I took multiple 10,000 point samples of my data, fit gamma and log-normal parameters, plotted the original and the log normal. The sample data definitely changes from positive to negative skew, the modeled log-normal looks normal, and the modeled gamma probably switches, but it's very slight. The normality test comes in (much maligned I see) where I thought to try Anderson-Darling on the log-transformed data to suggest that the sample data may be more appropriately described with gamma parameters than log-normal. Resulting p-values for the log transformed data: data sample, 4.7e-5; lognormal, 0.18; gamma 3.6e-22
From the point of view of someone more knowledgeable, is it fair to say that by AD test, log-transformation of the respective datasets indicates that the sample data and the gamma modeled data are not consistent with a normal distribution, while that for the lognormal is not inconsistent with normality, but that it cannot be proven normal? (except, I suppose, by definition of log-normality) And thus provide some (certainly not be all, end all) justification for choosing gamma between the two?