Editing Median (section)

==History==
Scientific researchers in the ancient near east appear not to have used summary statistics altogether, instead choosing values that offered maximal consistency with a broader theory that integrated a wide variety of phenomena.<ref name=":0">{{Cite journal|last1=Bakker|first1=Arthur|last2=Gravemeijer|first2=Koeno P. E.|s2cid=143708116|date=2006-06-01|title=An Historical Phenomenology of Mean and Median|journal=Educational Studies in Mathematics|language=en|volume=62|issue=2|pages=149–168|doi=10.1007/s10649-006-7099-8|issn=1573-0816}}</ref>  Within the Mediterranean (and, later, European) scholarly community, statistics like the mean are fundamentally a medieval and early modern development.  (The history of the median outside Europe and its predecessors remains relatively unstudied.)

The idea of the median appeared in the 6th century in the [[Talmud]], in order to fairly analyze divergent [[Economic appraisal|appraisals]].<ref>{{Cite web|url=http://danadler.com/blog/2014/12/31/talmud-and-modern-economics/|title=Talmud and Modern Economics|last=Adler|first=Dan|date=31 December 2014|website=Jewish American and Israeli Issues|url-status=dead|archive-url=https://web.archive.org/web/20151206134315/http://danadler.com/blog/2014/12/31/talmud-and-modern-economics/|archive-date=6 December 2015|access-date=22 February 2020}}</ref><ref>[http://www.wisdom.weizmann.ac.il/math/AABeyond12/presentations/Aumann.pdf Modern Economic Theory in the Talmud] by [[Yisrael Aumann]]</ref>  However, the concept did not spread to the broader scientific community.

Instead, the closest ancestor of the modern median is the [[mid-range]], invented by [[Al-Biruni]]<ref name="Eisenhart">{{Cite speech|last=Eisenhart|first=Churchill|author-link=Churchill Eisenhart|event=131st Annual Meeting of the American Statistical Association|location=Colorado State University|date=24 August 1971|url=https://www.stat.uchicago.edu/~stigler/eisenhart.pdf|title=The Development of the Concept of the Best Mean of a Set of Measurements from Antiquity to the Present Day|format=PDF}}</ref>{{Rp|31}}<ref name=":2">{{Cite web|url=http://priceonomics.com/how-the-average-triumphed-over-the-median/|title=How the Average Triumphed Over the Median|website=Priceonomics|date=5 April 2016|language=en|access-date=2020-02-23}}</ref>  Transmission of his work to later scholars is unclear. He applied his technique to [[assay]]ing currency metals, but, after he published his work, most assayers still adopted the most unfavorable value from their results, lest they appear to [[Debasement|cheat]].<ref name="Eisenhart" />{{Rp|35–8}} <ref>{{Cite journal |last=Sangster |first=Alan |date=March 2021 |title=The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator |url=https://onlinelibrary.wiley.com/doi/10.1111/abac.12218 |journal=Abacus |language=en |volume=57 |issue=1 |pages=126–152 |doi=10.1111/abac.12218 |hdl=2164/16100 |s2cid=233917744 |issn=0001-3072|hdl-access=free }}</ref> However, increased navigation at sea during the [[Age of Discovery]] meant that ship's navigators increasingly had to attempt to determine latitude in unfavorable weather against hostile shores, leading to renewed interest in summary statistics.  Whether rediscovered or independently invented, the mid-range is recommended to nautical navigators in Harriot's "Instructions for Raleigh's Voyage to Guiana, 1595".<ref name="Eisenhart" />{{Rp|45–8}}

The idea of the median may have first appeared in [[Edward Wright (mathematician)|Edward Wright]]'s 1599 book ''Certaine Errors in Navigation'' on a section about [[compass]] navigation.<ref>{{Cite journal |last1=Wright |first1=Edward |last2=Parsons |first2=E. J. S. |last3=Morris |first3=W. F. |date=1939 |title=Edward Wright and His Work |url=https://www.jstor.org/stable/1149920 |journal=Imago Mundi |volume=3 |issue=1 |pages=61–71 |doi=10.1080/03085693908591862 |jstor=1149920 |issn=0308-5694}}</ref> Wright was reluctant to discard measured values, and may have felt that the median — incorporating a greater proportion of the dataset than the [[mid-range]] — was more likely to be correct.  However, Wright did not give examples of his technique's use, making it hard to verify that he described the modern notion of median.<ref name=":0" /><ref name=":2" />{{Efn|Subsequent scholars appear to concur with Eisenhart that Boroughs' 1580 figures, while suggestive of the median, in fact describe an arithmetic mean.;<ref name="Eisenhart" />{{rp|62–3}} Boroughs is mentioned in no other work.}}  The median (in the context of probability) certainly appeared in the correspondence of [[Christiaan Huygens]], but as an example of a statistic that was inappropriate for [[Actuarial science|actuarial practice]].<ref name=":0" />

The earliest recommendation of the median dates to 1757, when [[Roger Joseph Boscovich]] developed a regression method based on the [[L1 norm|''L''<sup>1</sup> norm]] and therefore implicitly on the median.<ref name=":0" /><ref name="Stigler1986">{{cite book|last=Stigler|first=S. M.|url=https://archive.org/details/historyofstatist00stig|title=The History of Statistics: The Measurement of Uncertainty Before 1900|publisher=Harvard University Press|year=1986|isbn=0674403401}}</ref>  In 1774, [[Pierre-Simon Laplace|Laplace]] made this desire explicit: he suggested the median be used as the standard estimator of the value of a posterior [[Probability density function|PDF]]. The specific criterion was to minimize the expected magnitude of the error; <math>|\alpha - \alpha^{*}|</math> where <math>\alpha^{*}</math> is the estimate and <math>\alpha</math> is the true value.  To this end, Laplace determined the distributions of both the sample mean and the sample median in the early 1800s.<ref name="Stigler1973" /><ref name="Laplace1818">Laplace PS de (1818) ''Deuxième supplément à la Théorie Analytique des Probabilités'', Paris, Courcier</ref>  However, a decade later, [[Carl Friedrich Gauss|Gauss]] and [[Adrien-Marie Legendre|Legendre]] developed the [[least squares]] method, which minimizes <math>(\alpha - \alpha^{*})^{2}</math> to obtain the mean; the strong justification of this estimator by reference to [[maximum likelihood estimation]] based on a [[normal distribution]] means it has mostly replaced Laplace's original suggestion.<ref>{{cite book|last1=Jaynes|first1=E.T.|title=Probability theory : the logic of science|date=2007|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0-521-59271-0|page=172|edition=5. print.}}</ref>

[[Antoine Augustin Cournot]] in 1843 was the first<ref>{{Cite book|title=Dictionary of Mathematical Geosciences: With Historical Notes|last=Howarth|first=Richard|publisher=Springer|year=2017|pages=374}}</ref> to use the term ''median'' (''valeur médiane'') for the value that divides a probability distribution into two equal halves. [[Gustav Theodor Fechner]] used the median (''Centralwerth'') in sociological and psychological phenomena.<ref name="Keynes1921">Keynes, J.M. (1921) ''[[A Treatise on Probability]]''. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, {{isbn|9781596055308}} : multiple other reprints)</ref> It had earlier been used only in astronomy and related fields. [[Gustav Theodor Fechner|Gustav Fechner]] popularized the median into the formal analysis of data, although it had been used previously by Laplace,<ref name="Keynes1921" /> and the median appeared in a textbook by [[Francis Ysidro Edgeworth|F. Y. Edgeworth]].<ref>{{Cite book|last=Stigler|first=Stephen M.|url=https://books.google.com/books?id=qQusWukdPa4C&q=stigler+%22statistics+on+the+table%22|title=Statistics on the Table: The History of Statistical Concepts and Methods|date=2002|publisher=Harvard University Press|isbn=978-0-674-00979-0|pages=105–7|language=en}}</ref>  [[Francis Galton]] used the term ''median'' in 1881,<ref name=Galton1881>Galton F (1881) "Report of the Anthropometric Committee" pp 245–260. [https://www.biodiversitylibrary.org/item/94448 ''Report of the 51st Meeting of the British Association for the Advancement of Science'']</ref><ref>{{Cite journal|last=David|first=H. A.|date=1995|title=First (?) Occurrence of Common Terms in Mathematical Statistics|journal=The American Statistician|volume=49|issue=2|pages=121–133|doi=10.2307/2684625|jstor=2684625|issn=0003-1305}}</ref> having earlier used the terms ''middle-most value'' in 1869, and the ''medium'' in 1880.<ref>[https://www.encyclopediaofmath.org/index.php/Galton,_Francis ''encyclopediaofmath.org'']</ref><ref>[http://www.personal.psu.edu/users/e/c/ecb5/Courses/M475W/WeeklyReadings/Week%2013/DevelopmentOfModernStatistics.pdf  ''personal.psu.edu'']</ref>

<!-- this isn't why it replaced the median—the reason why is the central limit theorem and ubiquity of normally-distributed data --><!--Statisticians encouraged the use of medians intensely throughout the 19th century for its intuitive clarity. However, the notion of median does not lend itself to the theory of higher moments as well as the [[arithmetic mean]] does, and is much harder to compute.  As a result, the median was steadily supplanted as a notion of generic average by the arithmetic mean during the 20th century.<ref name=":0" /><ref name=":2" />-->  <!--(Ironically, the same time period saw the rise of term "average" to describe any location statistic, not merely the arithmetic mean.)<ref name="Eisenhart" />{{Rp|7}} -->