I’ve recently shown some histograms for site ring widths and opined that they loooked somewhat gamma-ish. Louis Hissink said that they looked log-normal to him. Louis is right for Gotland anyway.
Esper stated that the variance in Gotland was "heteroskedastic" and they did exercised to eliminate the supposed heteroskedasticity by a power-transformation by a power of 0.32. I’ll discuss heteroskedasticity in a minute, but first here are qqnorm- plots for:
Left – ring widths rw ; middle – with a power transformation of rw^0.32 and right – log(rw+1,2). Pretty convincing log-normality.
If this data set is representative, then there is pretty wide scale treatment of log-normally distributed ring widths as though they were normally distributed. This stuff is endless.
I’m not sure that Esper has correctly distinguished between heteroskedasticity and non-normality. The ring width distribution is obviously non-normal; the power transformation mitigates the non-normality, but nearly as well as a log-transformation. "Heteroskedasticity" is different than non-normality: it’s a situation where the variance is not uniform and, in this context, the variance is related to the mean: high average widths are associated in any given year with higher variance. I haven’t fully worked through this, but intuitively, it seems to me that either of these transformations will mitigate non-normality but not necessarily heteroskedasticity.
Mentioning logarithms and Gotland reminds me of an ancient joke, back from the time when we still used slide rules – yes, when I was at university, we still used slide rules. It was something about God telling the snakes to multiply, but they didn’t know how; so he gave them log tables and then the adders could multiply.