  We study distributions which have both fractal and non-fractal scale regions
by introducing a typical scale into a scale invariant system. As one of models
in which distributions follow power law in the large scale region and deviate
further from the power law in the smaller scale region, we employ 2-dim quantum
gravity modified by the $R^2$ term. As examples of distributions in the real
world which have similar property to this model, we consider those of personal
income in Japan over latest twenty fiscal years. We find relations between the
typical scale and several kinds of averages in this model, and observe that
these relations are also valid in recent personal income distributions in Japan
with sufficient accuracy. We show the existence of the fiscal years so called
bubble term in which the gap has arisen in power law, by observing that the
data are away from one of these relations. We confirm, therefore, that the
distribution of this model has close similarity to those of personal income. In
addition, we can estimate the value of Pareto index and whether a big gap
exists in power law by using only these relations. As a result, we point out
that the typical scale is an useful concept different from average value and
that the distribution function derived in this model is an effective tool to
investigate these kinds of distributions.
