The task for a general and useful classification of the heaviness of the
tails of probability distributions still has no satisfactory solution. Due to
lack of information outside the range of the data the tails of the distribution
should be described via many characteristics. Index of regular variation is a
good characteristic, but it puts too many distributions with very different
tail behavior in one and the same class. One can consider for example
Pareto($\alpha$), Fr$\acute{e}$chet($\alpha$) and Hill-horror($\alpha$) with
one and the same fixed parameter $\alpha > 0$. The main disadvantage of VaR,
expectiles, and hazard functions, when we speak about the tails of the
distribution, is that they depend on the center of the distribution and on the
scaling factor. Therefore they are very appropriate for predicting "big
losses", but after a right characterization of the distributional type of "the
payoff". When analyzing the heaviness of the tail of the observed distribution
we need some characteristic which does not depend on the moments because in the
most important cases of the heavy-tailed distributions theoretical moments do
not exist and the corresponding empirical moments fluctuate too much. In this
paper, we show that probabilities for different types of outliers can be very
appropriate characteristics of the heaviness of the tails of the observed
distribution. They do not depend on increasing affine transformations and do
not need the existence of the moments. The idea origins from Tukey's box plots,
and allows us to obtain one and the same characteristic of the heaviness of the
tail of the observed distribution within the whole distributional type with
respect to all affine transformations. These characteristics answer the
question: At what extent we can observe "unexpected" values?