Editing Long tail (section)

{{Short description|Feature of some statistical distributions}}
{{Other uses|Long tail (disambiguation)}}
{{Redirect|Long-tailed distribution|the term used in statistics|Heavy-tailed distribution}}
[[File:Long tail.svg|thumb|right|An example of a [[power law]] graph showing popularity ranking. To the right (yellow) is the long tail; to the left (green) are the few that dominate.]]
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In [[statistics]] and [[business]], a '''long tail''' of some [[probability distribution|distributions]] of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various [[probabilities]], etc.<ref>
{{cite book
 | title = The Long Tail of Expertise
 | author = Alpheus Bingham and Dwayne Spradlin
 | publisher = Pearson Education
 | year = 2011
 | isbn = 9780132823135
 | page = 5
 | url = https://books.google.com/books?id=9qLfsonmwhAC&pg=PT5
 }}</ref> The term is often used loosely, with no definition or an arbitrary definition, but precise definitions are possible.

In statistics, the term ''long-tailed distribution'' has a narrow technical meaning, and is a subtype of [[heavy-tailed distribution]].<ref name="Asmussen">{{Cite book | doi = 10.1007/0-387-21525-5_10 | first = S. R. | last = Asmussen| chapter = Steady-State Properties of GI/G/1 | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 266–301 | year = 2003 | isbn = 978-0-387-00211-8 }}</ref><ref>Levine, David M.; Stephan, David; Krehbiel, Timothy C.; Berenson, Mark L. ''Statistics for Managers using Microsoft Excel''. 3rd edition. Prentice Hall, 2002, p. 124.</ref><ref>{{cite book
 | title = Using R for Introductory Statistics
 | author = John Verzani
 | publisher = CRC Press
 | year = 2004
 | isbn = 978-1-58488450-7
 | page = [https://books.google.com/books?id=RAjNY11rJXIC&pg=PA62 62]
 }}</ref> Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: large quantities are probably even larger.{{efn|1=Formally, <math>\lim_{x \to \infty} \Pr[X>x+t|X>x] =1,</math> equivalently <math>\overline{F}(x+t) \sim \overline{F}(x) \quad \mbox{as } x \to \infty.</math>}} Note that there is no sense of ''the'' "long tail" of a distribution, but only the ''property'' of a distribution being long-tailed.

In business, the term ''long tail'' is applied to [[rank-size distribution]]s or [[rank-frequency distribution]]s (primarily of popularity), which often form [[power law]]s and are thus long-tailed distributions in the statistical sense. This is used to describe the retailing strategy of selling many unique items with relatively small quantities sold of each (the "long tail")—usually in addition to selling fewer popular items in large quantities (the "head"). Sometimes an intermediate category is also included, variously called the ''body'', ''belly'', ''torso'', or ''middle''. The specific cutoff of what part of a distribution is ''the'' "long tail" is often arbitrary, but in some cases may be specified objectively; see [[Rank-size distribution#Segmentation|segmentation of rank-size distributions]].

The long tail concept has found some ground for application, research, and experimentation. It is a term used in online business, [[mass media]], [[micro-finance]] ([[Grameen Bank]], for example), user-driven innovation ([[Eric von Hippel]]), knowledge management, and social network mechanisms (e.g. [[crowdsourcing]], [[crowdcasting]], [[Peer-to-Peer (meme)|peer-to-peer]]), economic models, marketing ([[viral marketing]]), and IT Security threat hunting within a SOC ([[Information security operations center]]).