Pareto 80/20 Law: Derivation via Random Partitioning
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| Title: | Pareto 80/20 Law: Derivation via Random Partitioning |
|---|---|
| Language: | English |
| Authors: | Lipovetsky, Stan |
| Source: | International Journal of Mathematical Education in Science and Technology. Jan 2009 40(2):271-277. |
| Availability: | Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals |
| Peer Reviewed: | Y |
| Physical Description: | |
| Page Count: | 7 |
| Publication Date: | 2009 |
| Document Type: | Journal Articles Reports - Descriptive |
| Descriptors: | Computation, Monte Carlo Methods, Mathematical Concepts, Nonparametric Statistics, Mathematical Logic, Funding Formulas |
| DOI: | 10.1080/00207390802213609 |
| ISSN: | 0020-739X |
| Abstract: | The Pareto 80/20 Rule, also known as the Pareto principle or law, states that a small number of causes (20%) is responsible for a large percentage (80%) of the effect. Although widely recognized as a heuristic rule, this proportion has not been theoretically based. The article considers derivation of this 80/20 rule and some other standard quotients from the mean and its interval estimation for the total value defined by the product of two variables in the random partitioning model. (Contains 2 figures.) |
| Abstractor: | As Provided |
| Number of References: | 22 |
| Entry Date: | 2009 |
| Accession Number: | EJ829592 |
| Database: | ERIC |
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| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwEWA452P0Rt8pyuRhEf8vwYAAAA4jCB3wYJKoZIhvcNAQcGoIHRMIHOAgEAMIHIBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDOnMTbjt02772qTrBwIBEICBmnSFLk8BHilL0Ak6Q_nci97krPZSZU418MTZBEJLdz47TFfFRwh6rw_cjx9gjqbU-Do0Mk1yuI8IvB3xx2eF0yXLIg8Nk0l_Oyx5Gg1pczo0WqvfH3TPp6KUboUs_8bh-XN5YgPWjXB2cmrPyTAHake6criutmsYnp_w-bni5d4yu3iN_vbNwK6Q6VEKx6B8CnWg-0UKgBLEYSs= Text: Availability: 1 Value: <anid>AN0036591844;imt15mar.09;2019Feb27.14:40;v2.2.500</anid> <title id="AN0036591844-1">Pareto 80/20 law: derivation via random partitioning. </title> <p>The Pareto 80/20 Rule, also known as the Pareto principle or law, states that a small number of causes (20%) is responsible for a large percentage (80%) of the effect. Although widely recognized as a heuristic rule, this proportion has not been theoretically based. The article considers derivation of this 80/20 rule and some other standard quotients from the mean and its interval estimation for the total value defined by the product of two variables in the random partitioning model.</p> <p>Keywords: Pareto 80-20 rule; random partitioning; mean; SD</p> <hd id="AN0036591844-2">1. Introduction</hd> <p>About a 100 years ago, the Italian economist Vilfredo Pareto first noticed that 80% of the land in Italy was owned by 20% of the population. Later he discovered that the same partitioning was applicable in other phenomena of life, for instance, in gardening, 80% of peas were produced by 20% of the peapods. He formulated this relation as follows: 'In any series of elements to be controlled, a selected small fraction in terms of number of elements almost always accounts for a large fraction in terms of effect' [<reflink idref="bib1" id="ref1">1</reflink>], [<reflink idref="bib2" id="ref2">2</reflink>]. Lorenz observed in countries of various sizes the similar shares of wealth distribution across the population groups [<reflink idref="bib3" id="ref3">3</reflink>]. The phenomenon of about 80% of the total wealth belonging approximately to the 20% of the individuals was widely discussed in the literature. Lorenz also found that a large percentage of crime was committed by a small percentage of the population. The American quality management pioneer Joseph Juran recognized such a proportion as a universal principle of 'vital few and trivial many', and called it the Pareto Principle [[<reflink idref="bib4" id="ref4">4</reflink>]]. He advocated using this principle in management that permits to focus on the main 20% of efforts providing 80% of the benefits. Similarly, 20% of products or services produce 80% of the revenues, 20% of customers account for 80% of the turnover, or 20% of the vendors supply with 80% of the merchandise, or 20% of the components accrue 80% of the cost, etc. Due to Juran, the Pareto principle or 80/20 law, can serve as a daily reminder to focus 80% of time and energy on the 20% of work that is really important. Juran also introduced this rule in the quality control, suggesting that 20% of sources are causing 80% of the problems. For instance, 80% of the defects and rejects arise from 20% of the process issues, 80% of delays in a schedule happen due to 20% of causes. Juran's works on total quality control and six sigma principles were originated from the 80/20 Principle. Graphical presentation of the Pareto principle is given in Figure 1.</p> <p>Graph: Figure 1. Pareto 80/20 law.</p> <p>The Pareto 80/20 rule is an empirical approach widely used nowadays in various aspects of human activity. For instance, in horse racing, if 80% of races are won by just 20% of jockeys, it makes sense to increase chances of winning by picking a jockey in the 20% bucket. Another example: 20% of carpeting gets 80% of the foot traffic, so only these 20% actually need any fixing, or people wear 20% most favoured clothes about 80% of the time, so only they need cleaning. We spend 80% of the time with 20% of our acquaintances, and 80% of decisions come from 20% of time spent on their elaboration. Computers use 20% of the resources to perform 80% of the operations, and 20% of the bugs cause 80% of the crashes. Tons of such examples can be easily found in Internet and in literature [[<reflink idref="bib7" id="ref5">7</reflink>]].</p> <p>Of course, many phenomena correspond to this rule only approximately. Two numbers present different characteristics, each with its own distribution, so they are not necessarily summing to 100%. But for the easier and more convenient comparison, people usually express them as the shares of total hundred. The actual normalized proportion can be closer to 90/10 or to 60/40 or to another quotient [<reflink idref="bib3" id="ref6">3</reflink>]. For instance, in computer science, the law of 90/10 is known–it states that about 90% of the execution time of a program is spent executing 10% of the script. Global Forum for Health Research uses the 10/90 gap referring to the statistical estimation that only 10% of worldwide expenditure on health research and development is devoted to the problems that primarily affect the poorest 90% of the world's population, which needs new and affordable medicines for neglected diseases. The 90/10 rule of project schedules states: 'The first 90% of the task takes 10% of the time, and the last 10% takes the other 90%' [<reflink idref="bib10" id="ref7">10</reflink>], p. 70]. On the other hand, the 60/40 rule is also known, for instance, in advertizing, marketing or political campaigns: 20% of most prospects are avid supporters and 20% are strict rejecters, so no amount of persuasion will change the view or attitude of about 40% of population. The remaining 60% represent the interested prospects, which should be convinced or 'sold', so this segment needs a special attention in answering its concerns. The described above proportions correspond to the LaCombe's rule of percentages: 'The incidence of anything worthwhile is either 15–25 percent or 80–90 percent', and the related Dudenhoefer's corollary: 'An answer of 50 percent will suffice for the 40–60 range' [<reflink idref="bib10" id="ref8">10</reflink>], p. 81].</p> <p>Although some efforts of its proof can be found in the literature, for instance, [<reflink idref="bib11" id="ref9">11</reflink>], the Pareto 80/20 principle has been mostly used as a rule of thumb not based on a theoretical derivation but approved in practical trying. In the current work, such a proof is attained. The article is arranged as follows. Section 2 describes the Pareto distribution, and Section 3 suggests a model for two variables mean partitioning, which yields the 80/20 proportion. Section 4 considers this mean in interval estimation, which produces the 90/10 and 60/40 quotients, and Section 5 summarizes.</p> <hd id="AN0036591844-3">2. Pareto distribution</hd> <p>The Pareto power distribution can be presented as:</p> <p>Graph</p> <p>where <emph>x</emph> and <emph>y</emph> are the percentiles of population and wealth, respectively, and <emph>a</emph> and <emph>b</emph> are the distribution parameters. Similar empirical dependencies given in logarithmic or power rank-size distributions are known in the natural and social sciences–particularly, the Lorenz law, Benford's and Heap's laws, Lotka's and Bode's laws, Zipf 's law and its generalization in Zipf–Mandelbrot law [[<reflink idref="bib12" id="ref10">12</reflink>]]. In the Pareto law (<reflink idref="bib1" id="ref11">1</reflink>), 100% of population possesses all 100% of the wealth, so the expression (<reflink idref="bib1" id="ref12">1</reflink>) reduces to</p> <p>Graph</p> <p>Dividing (<reflink idref="bib1" id="ref13">1</reflink>) by (<reflink idref="bib2" id="ref14">2</reflink>) yields the one parameter distribution:</p> <p>Graph</p> <p>In assumption of the two variables normalized to total constant of 100%,</p> <p>Graph</p> <p>let us substitute one of them, <emph>x</emph> = 100 − <emph>y</emph>, into the expression (<reflink idref="bib3" id="ref15">3</reflink>) and take logarithm by the base 10. It yields the equation for finding this parameter:</p> <p>Graph</p> <p>With the value (<reflink idref="bib5" id="ref16">5</reflink>), from the expression (<reflink idref="bib2" id="ref17">2</reflink>), the value of the other parameter is:</p> <p>Graph</p> <p>The Pareto 80/20 rule corresponds to <emph>y</emph> = 80% of wealth distributed within <emph>x</emph> = 20% of population. For this value <emph>y</emph> = 80%, the parameters (<reflink idref="bib5" id="ref18">5</reflink>) and (<reflink idref="bib6" id="ref19">6</reflink>) equal to <emph>b</emph> = 0.139 and <emph>a</emph> = 52.809. For another proportion 90/10, so for the value <emph>y</emph> = 90%, these parameters are <emph>b</emph> = 0.0458 and <emph>a</emph> = 81. With proportion 60/40 or the value <emph>y</emph> = 60%, these parameters become <emph>b</emph> = 0.557 and <emph>a</emph> = 7.674. If to take proportion 50/50, so <emph>y</emph> = 50%, the parameters are equal, <emph>b</emph> = <emph>a</emph> = 1, so the Equation (<reflink idref="bib1" id="ref20">1</reflink>) degenerates into the straight line. Figure 2 presents these examples of the Pareto 80/20 and several other power distributions.</p> <p>Graph: Figure 2. Power law distributions.</p> <hd id="AN0036591844-4">3. Mean segments in the random partitioning model for 80/20 rule</hd> <p>Consider now derivation of the Pareto 80/20 principle from a characteristic of the total value, which is defined as the product of two variables used in the rule. By any of 80/20 rule applications, it is easy to notice that within the two variables, <emph>x</emph> and <emph>y</emph>, one measures the 'cost of a unit' and another measures the 'quantity', so their product gives the total value for the observable phenomenon. For instance, 'capital per head' (<emph>y</emph>) and 'number of people' (<emph>x</emph>) yield in their product (<emph>S</emph> = <emph>xy</emph>), the total volume of wealth in all the strata of population [<reflink idref="bib19" id="ref21">19</reflink>]. Taking values of the two variables normalized to total constant (<reflink idref="bib4" id="ref22">4</reflink>), the probability of belonging to each stratum is estimated as the area under the distribution curve:</p> <p>Graph</p> <p>Assuming the uniform distribution of picking any <emph>x</emph> value in the closed interval [0, <emph>p</emph>], let us find the expectation of the area (<reflink idref="bib7" id="ref23">7</reflink>).</p> <p>Graph</p> <p>Geometrically, the expression obtained in (<reflink idref="bib8" id="ref24">8</reflink>) corresponds to the mean of all possible rectangular areas with the lengths of sides <emph>x</emph> and <emph>y</emph>, and with the same semi-perimeter (<reflink idref="bib4" id="ref25">4</reflink>). Naturally, the mean area <emph>p</emph><sups>2</sups>/6 is less than the maximum possible area <emph>p</emph><sups>2</sups>/4 of the square with the side <emph>x</emph> = <emph>p</emph>/2.</p> <p>Let us estimate the sides of the rectangular with the mean area (<reflink idref="bib8" id="ref26">8</reflink>)–it can be done from the quadratic Equation (<reflink idref="bib7" id="ref27">7</reflink>) with the given value :</p> <p>Graph</p> <p>The roots of (<reflink idref="bib9" id="ref28">9</reflink>) are</p> <p>Graph</p> <p>With <emph>p</emph> = 100%, the rounded values of the two roots are:</p> <p>Graph</p> <p>It is the Pareto proportion of 80/20 segments in total 100%.</p> <hd id="AN0036591844-5">4. Margins in the random partitioning model for 90/10 and 60/40 rules</hd> <p>Consider now derivation of the other 90/10 and 60/40 proportions discussed above. With the same model, let us estimate the variance of <emph>S</emph> (<reflink idref="bib7" id="ref29">7</reflink>):</p> <p>Graph</p> <p>Then the SD equals:</p> <p>Graph</p> <p>so the margins around the mean value (<reflink idref="bib8" id="ref30">8</reflink>) are:</p> <p>Graph</p> <p>It is interesting to note that the terms in the parentheses (<reflink idref="bib14" id="ref31">14</reflink>) coincide with the Golden Section partitioning [[<reflink idref="bib20" id="ref32">20</reflink>]], so the left and right margins are proportional to the Phidias numbers .</p> <p>Repeating the derivations (<reflink idref="bib9" id="ref33">9</reflink>) and (<reflink idref="bib10" id="ref34">10</reflink>) for the segments corresponding to the areas (<reflink idref="bib14" id="ref35">14</reflink>), yield the quadratic equations:</p> <p>Graph</p> <p>For the margin <emph>S</emph><subs>+</subs> (<reflink idref="bib14" id="ref36">14</reflink>), the solution of (<reflink idref="bib15" id="ref37">15</reflink>) is:</p> <p>Graph</p> <p>so with <emph>p</emph> = 100%, the rounded values of the roots are:</p> <p>Graph</p> <p>It is the proportion of 60/40 discussed above on the example of advertizing, marketing or political campaigns. With another margin <emph>S</emph><subs>−</subs> (<reflink idref="bib14" id="ref38">14</reflink>), the solution of (<reflink idref="bib15" id="ref39">15</reflink>) yields:</p> <p>Graph</p> <p>and for <emph>p</emph> = 100%, the rounded values of the roots are:</p> <p>Graph</p> <p>It is the proportion of 90/10 discussed above on the couple of examples.</p> <hd id="AN0036591844-6">5. Summary</hd> <p>A random partitioning model with estimation of two complimentary to 100% segments is applied to find the mean value and SD, or the point and interval means of the variables' product. Then two segments are found from the values of the mean product. These segments yield the quotient of the Pareto 80/20 rule, as well as two other standard 60/40 and 90/10 proportions. The model helps to understand the process of evaluation of the factors that influence managerial decision making.</p> <ref id="AN0036591844-7"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Pareto, V. 1916. Trattato di Sociologia Generale, Firenze: G. Barbera.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> 1935. The Mind and Society, New York: Dover.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> Lorenz, MO. 1905. Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc., 9: 209–219.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> Juran, JM. 1944. Bureaucracy, A Challenge to Better Management, New York, London: Harper Bros.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref16" type="bt">5</bibl> <bibtext> 1945. Management of Inspection and Quality Control, New York, , London: Harper Bros.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref19" type="bt">6</bibl> <bibtext> Juran, JM and Gryna, FM. 1951. Juran's Quality Control Handbook, New York: McGraw-Hill.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref5" type="bt">7</bibl> <bibtext> Koch, R. 2004. Living the 80/20 Way, London: Nicholas Brealey Publishing.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref24" type="bt">8</bibl> <bibtext> Rigg, A. 2004. How To Beat 80/20 Rule in Selling, Londo: Lightning Source Inc..</bibtext> </blist> <blist> <bibl id="bib9" idref="ref28" type="bt">9</bibl> <bibtext> Parmenter, D. 2006. Pareto's 80/20 Rule for Corporate Accountants, New York: John Wiley &amp; Sons Inc..</bibtext> </blist> <blist> <bibtext> Bloch, A. 1980. Murphy's Law and Other Reasons Why Things Go Wrong, Los Angeles, CA: Price/Stern/Sloan Publishers Inc..</bibtext> </blist> <blist> <bibtext> Ultsch, A. 2002. Proof of Pareto's 80/20 law and precise limits for ABC-analysis, Marburg, , Germany: Data Bionics Research Group, University of Marburg. Technical Report 2002c</bibtext> </blist> <blist> <bibtext> Benford, F. 1938. The law of anomalous numbers. PAPS,, 78: 551–572.</bibtext> </blist> <blist> <bibtext> Zipf, GK. 1949. Human Behaviour and the Principle of Least-Effort, Cambridge, MA: Addison-Wesley.</bibtext> </blist> <blist> <bibtext> Mandelbrot, B. 1965. "Information theory and psycholinguistics". In Scientific Psychology: Principles and Approaches, Edited by: Wolman, BB and Nagel, E. New York: Basic Books.</bibtext> </blist> <blist> <bibtext> Hill, TP. 1995. The significant-digit phenomenon. Am. Math. Mon., 102: 322–327.</bibtext> </blist> <blist> <bibtext> 1996. A statistical derivation of the significant-digit law. Stat. Sci., 10: 354–363.</bibtext> </blist> <blist> <bibtext> Nigrini, MJ. 2000. 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Res., 121: 213–216.</bibtext> </blist> </ref> <aug> <p>By Stan Lipovetsky</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref7"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref9"></nolink> <nolink nlid="nl3" bibid="bib12" firstref="ref10"></nolink> <nolink nlid="nl4" bibid="bib19" firstref="ref21"></nolink> <nolink nlid="nl5" bibid="bib14" firstref="ref31"></nolink> <nolink nlid="nl6" bibid="bib20" firstref="ref32"></nolink> <nolink nlid="nl7" bibid="bib15" firstref="ref37"></nolink> |
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| Items | – Name: Title Label: Title Group: Ti Data: Pareto 80/20 Law: Derivation via Random Partitioning – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lipovetsky%2C+Stan%22">Lipovetsky, Stan</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. Jan 2009 40(2):271-277. – Name: Avail Label: Availability Group: Avail Data: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: PhysDesc Label: Physical Description Group: PhysDesc Data: PDF – Name: Pages Label: Page Count Group: Src Data: 7 – Name: DatePubCY Label: Publication Date Group: Date Data: 2009 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Descriptive – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Monte+Carlo+Methods%22">Monte Carlo Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Nonparametric+Statistics%22">Nonparametric Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Funding+Formulas%22">Funding Formulas</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1080/00207390802213609 – Name: ISSN Label: ISSN Group: ISSN Data: 0020-739X – Name: Abstract Label: Abstract Group: Ab Data: The Pareto 80/20 Rule, also known as the Pareto principle or law, states that a small number of causes (20%) is responsible for a large percentage (80%) of the effect. Although widely recognized as a heuristic rule, this proportion has not been theoretically based. The article considers derivation of this 80/20 rule and some other standard quotients from the mean and its interval estimation for the total value defined by the product of two variables in the random partitioning model. (Contains 2 figures.) – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: Ref Label: Number of References Group: RefInfo Data: 22 – Name: DateEntry Label: Entry Date Group: Date Data: 2009 – Name: AN Label: Accession Number Group: ID Data: EJ829592 |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1080/00207390802213609 Languages: – Text: English PhysicalDescription: Pagination: PageCount: 7 StartPage: 271 Subjects: – SubjectFull: Computation Type: general – SubjectFull: Monte Carlo Methods Type: general – SubjectFull: Mathematical Concepts Type: general – SubjectFull: Nonparametric Statistics Type: general – SubjectFull: Mathematical Logic Type: general – SubjectFull: Funding Formulas Type: general Titles: – TitleFull: Pareto 80/20 Law: Derivation via Random Partitioning Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lipovetsky, Stan IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2009 Identifiers: – Type: issn-print Value: 0020-739X Numbering: – Type: volume Value: 40 – Type: issue Value: 2 Titles: – TitleFull: International Journal of Mathematical Education in Science and Technology Type: main |
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