Quality & Quantity 37: 43–69, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Pareto Density Distributions
A Short Exposition of the Concept of their Generation, their Deﬁnitive For-
mulae, an Example of their Synthetic Generation and a Practical Working
Method for the Fine-Fitting of Synthesized Pareto-curves
H. C. KOPPERER
P.O.Box 207, Halfway House, 1685 G.P., Rep. of South Africa, E-mail: email@example.com
Abstract. Building on new insights into the genesis of Pareto-Distributions, (“Kopp”effect etc.)
as published earlier in “Quality and Quantity”, the author gives at least one authentic/deﬁnitive
Pareto-Formula. A practical example of the synthetic generation of Pareto Distributions by means
of spreadsheets. A working D.I.Y-method for ﬁne-ﬁtting Pareto-curves to scattergrams with spread-
sheets using inter alia an indirect method of the least squares of residuals is fully demonstrated.A
comparative test-ﬁt to a cumulative Pareto- Distribution example ,where a simulative curve-formula
evolved by Prof. B. Arnold/Ucla is used for demonstration. Easy to absorb and to retain graph-
ical tableaux are employed to visualize the chain of descent and interconnections between normal
distributions, log-normal distributions and Pareto- Distributions. A quasi-dichotomy of the Pareto-
formulae is presented in tableau-form. One innovative formula for Pareto-distribution is given
F(x) = k
((ln(Integral(ln(x)))) − (ln(Integral(ln(µ)))))
Readers e-mailed constructive opinions &/or inputs are encouraged and welcomed.
Key words: ParetoDistribution, “Kopp”-effect and Pareto Distribution, Pareto-Formula, Pareto
D.-Spreadsheet Generation, DIY Pareto Distribution-spreadsheet-curveﬁtting, Pareto-Distributions,
Normal-, Log-normal distributions, two quasi-clones, Paretoid-formulae.
Pareto Distributions, are the empirical breakthrough-discovery of Prof. Vilfredo
Pareto. He was an Italian who trained as an engineer and who metamorphosed
himself into an outstanding, brilliant economist and sociologist at the University of
Lausanne. This distribution named after him can be perceived as the ﬁnal link in a
propagating chain of closely related class of density distributions.
Pareto himself did not foresee that his empirically discovered distribution would
be mathematically linked to the normal distribution, as an offspring, as it were.
Because the missing link, the log-normal distribution – a distribution which often
occurs in complex metal working-processes had not yet been empirically dis-
covered and classiﬁed. It did not yet enjoy the attention of scientists and engineers,
when Pareto studied his sociological data towards the end of the nineteenth cen-
tury. That essential logical stepping stone (log-normal distributions) was simply