Quality & Quantity 32: 213–227, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Reﬂections on the Structure and Origin of Pareto
H. C. KOPPERER
9 Silvertree Street, Noordwyk, P.O. Box 207, Halfway House, 1685 Gauteng, South Africa
Abstract. The essay takes the reader on a voyage of exploration with the aim of discovering the
origin of Pareto-curves. It shows with thought-experiments backed up by computer-simulation the
generation of log-normal curves in detail. Extending forward this conceptual trajectory, it arrives via a
quasi-Newtonian “ﬂuxion”-insight – inﬁnitessimal differential integration – at a novel mathematical
concept: Pareto-curves are simply special log-normal curves where a large number of random-factors
interacted and impacted at their genesis (the author called it the “Kopp-effect”).
Key words: Pareto-curves, Pareto-distributions, log-normal curves, generation of Pareto-Curves,
Vilfredo Pareto discovered the Pareto Distribution over one hundred years ago.
He gathered and collated income-class frequency distributions from industrially
advanced countries of the time, including France, England, Germany and distant,
underdeveloped countries such as Peru, and found striking similarities in the struc-
ture of the income distribution patterns of their respective populations. Although
this brilliant engineer, economist and sociologist was not awarded what would have
been a well-deserved Nobel Prize, the impact of his discovery is today stronger
than ever and its ramiﬁcations can be found in many social and and managerial-
science applications and tenets (e.g. the 20/80 rule, ABC stock management). The
basic phenomenon is, of course, that there are always a few big ﬁsh and many
unimportant small-fry in a typical ﬁshing ground. Pareto himself tried to provide
a formula which would yield the basic model-shape for his empirically obtained,
characteristic, “shifted hump”, lopsided, bell-shaped frequency distribution curves.
In its cumulative or dense form, it can be called an inverted J curve and he proposed,
with some of his own caveats, the formula
F(x) = 1 − x
as a reasonable representation of empirical distributions.
In the ﬁeld of metal conversion involving particle creating processes, several
factors inﬂuence simultaneously the size/frequency distribution of the created par-
ticles. Engineers associated with this ﬁeld discerned in their statistics a similar
phenomenon which they were able to deﬁne and correctly call the “Log-Normal”