# Pareto principle

The Pareto principle (also known as the 80-20 rule,[1] the law of the vital few and the principle of factor sparsity) states that, for many events, roughly 80% of the effects come from 20% of the causes.[2][3] Business management thinker Joseph M. Juran suggested the principle and named it after Italian economist Vilfredo Pareto, who observed that 80% of the land in Italy was owned by 20% of the population.[3] It is a common rule of thumb in business; e.g., "80% of your sales come from 20% of your clients." Mathematically, where something is shared among a sufficiently large set of participants, there will always be a number k between 50 and 100 such that k% is taken by (100 − k)% of the participants. However, k may vary from 50 in the case of equal distribution to nearly 100 when a tiny number of participants account for almost all of the resource. There is nothing particularly special about the number 80, but many systems will have k somewhere around this region of intermediate imbalance in distribution.

The Pareto principle is only tangentially related to Pareto efficiency, which was also introduced by the same economist, Vilfredo Pareto. Pareto developed both concepts in the context of the distribution of income and wealth among the population.

## In economics

The original observation was in connection with income and wealth. Pareto noticed that 80% of Italy's wealth was owned by 20% of the population.[4] He then carried out surveys on a variety of other countries and found to his surprise that a similar distribution applied.

Because of the scale-invariant nature of the power law relationship, the relationship applies also to subsets of the income range. Even if we take the ten wealthiest individuals in the world, we see that the top three (Warren Buffett, Carlos Slim Helú, and Bill Gates) own as much as the next seven put together. [5]

A chart that gave the inequality a very visible and comprehensible form, the so-called 'champagne glass' effect,[6] was contained in the 1992 United Nations Development Program Report, which showed the distribution of global income to be very uneven, with the richest 20% of the world's population controlling 82.7% of the world's income.[7]

Distribution of world GDP, 1989[8]
Quintile of Population Income
Richest 20% 82.7%
Second 20% 11.7%
Third 20% 2.3%
Fourth 20% 1.4%
Poorest 20% 1.2%

The Pareto Principle has also been used to attribute the widening economic inequality in the USA to 'skill-biased technical change' - i.e. income growth accrues to those with the education and skills required to take advantage of new technology and globalisation. However, Paul Krugman in the New York Times dismissed this "80-20 fallacy" as being cited "not because it's true, but because it's comforting." He asserts that the benefits of economic growth over the last 30 years have largely been concentrated in the top 1%, rather than the top 20%[9].

## Other applications

The Pareto Principle also applies to a variety of more mundane matters: one might guess approximately that we wear our 20% most favoured clothes about 80% of the time, perhaps we spend 80% of the time with 20% of our acquaintances, etc.

The Pareto principle has many applications in quality control. It is the basis for the Pareto chart, one of the key tools used in total quality control and six sigma. The Pareto principle serves as a baseline for ABC-analysis and XYZ-analysis, widely used in logistics and procurement for the purpose of optimizing stock of goods, as well as costs of keeping and replenishing that stock (Rushton et al. 2000, pp. 107-108).

In computer science and engineering control theory such as for electromechanical energy converters, the Pareto principle can be applied to optimization efforts. (M. Gen & R. Cheng, Generic Algorithms and Engineering Optimisation. New York, Wiley, 2002)

Microsoft also noted that by fixing the top 20% of the most reported bugs, 80% percent of the errors and crashes would be eliminated.[10]

In computer graphics the Pareto principle is used for ray-tracing: 80% of rays intersect 20% of geometry.[11]

The Pareto principle was a prominent part of the 2007 bestseller The 4-Hour Workweek by Tim Ferriss. Ferriss recommended focusing one's activities to those 20% that contribute to 80% of the income. More notably, he also recommends firing the 20% customers who take up the majority of one's time and cause most trouble.[12]

## Mathematical notes

The idea has rule-of-thumb application in many places, but it is commonly misused. For example, it is a misuse to state that a solution to a problem "fits the 80-20 rule" just because it fits 80% of the cases; it must be implied that this solution requires only 20% of the resources needed to solve all cases.

Mathematically, where something is shared among a sufficiently large set of participants, there will always be a number k between 50 and 100 such that k% is taken by (100 − k)% of the participants; however, k may vary from 50 in the case of equal distribution (e.g. exactly 50% of the people take 50% of the resources) to nearly 100 in the case of a tiny number of participants taking almost all of the resources. There is nothing special about the number 80, but many systems will have k somewhere around this region of intermediate imbalance in distribution.

This is a special case of the wider phenomenon of Pareto distributions. If the parameters in the Pareto distribution are suitably chosen, then one would have not only 80% of effects coming from 20% of causes, but also 80% of that top 80% of effects coming from 20% of that top 20% of causes, and so on (80% of 80% is 64%; 20% of 20% is 4%, so this implies a "64-4 law").

80-20 is only a shorthand for the general principle at work. In individual cases, the distribution could just as well be, say, 80-10 or 80-30. (There is no need for the two numbers to add up to 100%, as they are measures of different things, e.g., 'number of customers' vs 'amount spent'). The classic 80-20 distribution occurs when the gradient of the line is −1 when plotted on log-log axes of equal scaling. Pareto rules are not mutually exclusive. Indeed, the 0-0 and 100-100 rules always hold.

Adding up to 100 leads to a nice symmetry. For example, if 80% of effects come from the top 20% of sources, then the remaining 20% of effects come from the lower 80% of sources. This is called the "joint ratio", and can be used to measure the degree of imbalance: a joint ratio of 96:4 is very imbalanced, 80:20 is significantly imbalanced (Gini index: 60%), 70:30 is moderately imbalanced (Gini index: 40%), and 55:45 is just slightly imbalanced.

The Pareto Principle is an illustration of a "Power law" relationship, which also occurs in phenomena such as brush-fires and earthquakes. Because it is self-similar over a wide range of magnitudes, it produces outcomes completely different from 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—for example—stock movement sizes.[13]

## Inequality measures

### Gini coefficient and Hoover index

Using the “A:B” notation, (example : 0,8:0,2) and with  A + B = 1, inequality measures like the Gini index and the Hoover index can be computed. In this case both are the same.

$H=G=\left|2A-1 \right|=\left|2B-1 \right|$
$A:B = \left( \frac{1+H}{2} \right): \left( \frac{1-H}{2} \right)$

### Theil index

The Theil index is an entropy measure used to quantify inequities. The measure is 0 for 50:50 distributions and reaches 1 at a Pareto distribution of 82:18. Higher inequities yield Theil indices above 1.[14]

$T_T=T_L=T_s = 2 H \, \operatorname{arctanh} \left( H \right).\,$

## Notes

1. ^ The Pareto principle has several name variations, including: Pareto's Law, the 80/20 rule, the 80:20 rule, and 80 20 rule.
2. ^ "Joseph Juran, 103, Pioneer in Quality Control, Dies" New York times, 2008-03-03, webpage: NYTimes-Juran-obit.
3. ^ a b "What is 80/20 Rule, Pareto’s Law, Pareto Principle", www.80-20presentationrule.com, 2008, webpage: 8020p-what.
4. ^ Translation of Manuale di economia politica ("Manual of political economy"), By Vilfredo Pareto, Alfred N Page, Contributor Alfred N Page Publisher: A.M. Kelley, 1971, ISBN 0678008817, 9780678008812.
5. ^ http://www.thisismoney.co.uk/news/article.html?in_article_id=418243&in_page_id=3
6. ^ Xabier Gorostiaga,"World has become a 'champagne glass' globalization will fill it fuller for a wealthy few' National Catholic Reporter, Jan 27, 1995 '
7. ^ United Nations Development Program. 1992 Human Development Report, 1992 (New York, Oxford University Press)
8. ^ "Human Development Report 1992, Chapter 3". Retrieved on 2007-07-08.
9. ^ Krugman, Paul (February 27, 2006). "Graduates versus Oligarchs". New York Times: pp. A19.
10. ^ Microsoft's CEO: 80-20 Rule Applies To Bugs, Not Just Features
11. ^ Philipp Slusallek, Ray Tracing Dynamic Scenes
12. ^ Ferris, Tim (2006.) The 4-Hour Workweek. Crown Publishing
13. ^ "The Black Swan" (2007) Nassim Taleb pp228-252 and 274-285.
14. ^ On Line Calculator: Inequality

## References

• Bookstein, Abraham (1990). "Informetric distributions, part I: Unified overview". Journal of the American Society for Information Science 41: 368–375. doi:10.1002/(SICI)1097-4571(199007)41:5<368::AID-ASI8>3.0.CO;2-C.
• Klass, O. S.; Biham, O.; Levy, M.; Malcai, O.; Soloman, S. (2006). "The Forbes 400 and the Pareto wealth distribution". Economics Letters 90 (2): 290–295. doi:10.1016/j.econlet.2005.08.020.
• Koch, R. (2001). The 80/20 Pinciple: The Secret of Achieving More with Less. London: Nicholas Brealey Publishing.
• Koch, R. (2004). Living the 80/20 Way: Work Less, Worry Less, Succeed More, Enjoy More. London: Nicholas Brealey Publishing. ISBN 1857883314.
• Reed, W. J. (2001). "The Pareto, Zipf and other power laws". Economics Letters 74 (1): 15–19. doi:10.1016/S0165-1765(01)00524-9.
• Rosen, K. T.; Resnick, M. (1980). "The size distribution of cities: an examination of the Pareto law and primacy". Journal of Urban Economics 8: 165–186.
• Rushton, A.; Oxley, J.; Croucher, P. (2000). The handbook of logistics and distribution management (2nd ed. ed.). London: Kogan Page. ISBN 9780749433659.