Pareto principle

What is this?

The above graph showcases Pareto distribution (see the math behind the graph) in order to illustrate the Pareto principle (also known as the "80-20 rule").

Pareto principle

The principle was named after economist Vilfredo Pareto, who noticed that approximately 80% of the land in Italy was owned by 20% of the population at the end of the 19th century [1].

Later, similar patterns began to be noticed in other areas. The principle can be broadly defined as “80% of consequences come from 20% of causes” or “80% of results come from 20% of efforts”.

It is important to remember that all these patterns are approximate observations, not proven scientific laws.

What does this mean for me?

This distribution is a mathematical model which means nothing by itself.

There are a lot of different phenomena and patterns which approximately follow this distribution (and this diversity is fascinating!), and there are some assumptions which can be made about similar future events.

Mathematical details of the graph

It is worth noting that the horizontal axis goes from 1 (100%) to 0, so the graph is mirrored relative to the normal direction of the axis.

The graph is the inverse cumulative distribution function (also known as the quantile function [2])

x(p)\,=\,Q(p)\,=\,\inf \left\{x\in {\mathbb {R}}:p\leq F(x)\right\}=\frac{x_\mathrm{m}}{(1-p)^{\frac{1}{\alpha}}}

where $\alpha = log_45$ is a shape parameter of the Pareto distribution (Pareto index),
$x_m = 1$ is a scale parameter.

You could have noticed that the graph is not shown fully (it is bounded on the vertical axis), since closer to 1 it tends to infinity.

The percentage under the graph is the Lorenz (curve) function [3]. It shows part of the area under the graph that is occupied by the selected area:

L(F)={\frac {\int _{0}^{x}t\,f(t)\,dt}{\int _{0}^{\infty }t\,f(t)\,dt}}={\frac {\int _{0}^{F}x(F_{1})\,dF_{1}}{\int _{0}^{1}x(F_{1})\,dF_{1}}}

Links

Original publication by Vilfredo Pareto, in which his observation is mentioned:
Vilfredo Pareto, Cours d'économie politique professé à l'université de Lausanne, 3 volumes, 1896–7
A good explanation of a quantile and quantile functions:
Taboga, Marco (2021). "Quantile of a probability distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/quantile.
Lorenz curve:
Lorenz, M. O. (1905). "Methods of measuring the concentration of wealth". Publications of the American Statistical Association. Publications of the American Statistical Association, Vol. 9, No. 70. 9 (70): 209–219.
Popular science video:
The Zipf Mystery - VSauce - YouTube. https://youtu.be/fCn8zs912OE
Pareto principle - Wikipedia. https://en.wikipedia.org/wiki/Pareto_principle

Made by Vadim Saprykin.

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