Editing Pareto principle (section)

== Mathematical explanation ==
The demonstration of the Pareto principle is explained by a large proportion of process variation being associated with a small proportion of process variables.<ref name=":0" /> This is a special case of the wider phenomenon of [[Pareto distribution]]s. If the [[Pareto index]] '''α''', which is one of the parameters characterizing a Pareto distribution, is chosen as '''α''' = log<sub>4</sub>5 ≈ 1.16, then one has 80% of effects coming from 20% of causes.<ref>{{Citation |last=Dunford |title=The Pareto Principle |url=https://pearl.plymouth.ac.uk/bitstream/handle/10026.1/14054/TPSS-2014-Vol7n1_140-148Dunford.pdf |journal=The Plymouth Student Scientist |year=2014}}. Internet Archive [https://web.archive.org/web/20220122121809/https://pearl.plymouth.ac.uk/bitstream/handle/10026.1/14054/TPSS-2014-Vol7n1_140-148Dunford.pdf copy] of 22.10.2022. </ref>

The term 80/20 is only a shorthand for the general principle at work. In individual cases, the distribution could be nearer to 90/5 or 70/40. Note that there is no need for the two numbers to add up to the number 100, as they are measures of different things. The Pareto principle is an illustration of a "[[power law]]" relationship, which also occurs in phenomena such as [[bush fire]]s and earthquakes.<ref>{{Citation |last=Bak |first=Per |title=How Nature Works: the science of self-organized criticality |page=89 |year=1999 |publisher=Springer |isbn=0-387-94791-4 |author-link=Per Bak}}</ref> Because it is self-similar over a wide range of magnitudes, it produces outcomes completely different from [[Normal distribution|Normal or Gaussian distribution]] phenomena. This fact explains the frequent breakdowns of sophisticated financial instruments, which are modeled on the assumption that a Gaussian relationship is appropriate to something like stock price movements.<ref>{{Citation |last=Taleb |first=Nassim |title=The Black Swan |pages=229–252, 274–285 |year=2007 |author-link=Nassim Taleb |title-link=The Black Swan (Taleb book)}}</ref>

=== Gini coefficient and Hoover index ===
Using the "''A'':''B''" notation (for example, 0.8:0.2) and with ''A'' + ''B'' = 1, [[Income inequality metrics|inequality measures]] like the [[Gini index]] (G) ''and'' the [[Hoover index]] (H) can be computed. In this case both are the same:

: <math>H=G=|2A-1|=|1-2B| \, </math>

: <math>A:B = \left( \frac{1+H} 2 \right): \left( \frac{1-H} 2 \right)</math>