In reply to Ksharp.

I'm not sure what you mean by "magic." Perhaps you noticed that I fit the quantiles (not the mean) but still got a good agreement between the sample mean and the mean of the fitted distribution? I think this will happen with many distributions. If you fit the quantiles, then moments (mean, standard deviation, skewness,...) also tend to be approximately preserved.

With a long-tailed distribution, there is always a chance that a random sample might get several extreme values. If so, the sample mean might be extreme for that sample. That did not happen in this random sample..