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[[Statistics]], in the modern sense of the word, began evolving in the 18th century in response to the novel needs of industrializing [[Westphalian sovereignty|sovereign states]].

In early times, the meaning was restricted to information about states, particularly [[demographics]] such as population. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as in [[national accounts]] and [[temperature record]], and analytical work which requires [[statistical inference]]. Statistical activities are often associated with models expressed using [[probability|probabilities]], hence the connection with probability theory. The large requirements of data processing have made statistics a key application of computing. A number of statistical concepts have an important impact on a wide range of sciences. These include the [[design of experiments]] and approaches to statistical inference such as [[Bayesian inference]], each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics.

==Introduction==
By the 18th century, the term "[[statistics]]" designated the [[official statistics|systematic collection]] of [[demographic]] and [[economics|economic]] data by states. For at least two millennia, these data were mainly tabulations of human and material resources that might be taxed or put to military use. In the early 19th century, collection intensified, and the meaning of "statistics" broadened to include the discipline concerned with the collection, summary, and analysis of data. Today, data is collected and statistics are computed and widely distributed in government, business, most of the sciences and sports, and even for many pastimes. Electronic [[computer]]s have expedited more elaborate [[computational statistics|statistical computation]] even as they have facilitated the collection and aggregation of data. A single data analyst may have available a set of data-files with millions of records, each with dozens or hundreds of separate measurements. These were collected over time from computer activity (for example, a stock exchange) or from computerized sensors, point-of-sale registers, and so on. Computers then produce simple, accurate summaries, and allow more tedious analyses, such as those that require inverting a large matrix or perform hundreds of steps of iteration, that would never be attempted by hand. Faster computing has allowed statisticians to develop "computer-intensive" methods which may look at all permutations, or use randomization to look at 10,000 permutations of a problem, to estimate answers that are not easy to quantify by theory alone.

The term "[[mathematical statistics]]" designates the mathematical theories of [[probability theory|probability]] and [[statistical inference]], which are used in [[applied statistics|statistical practice]]. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly used [[probability theory]], whose initial results were found in the 17th and 18th centuries, particularly in the analysis of [[games of chance]] (gambling). By 1800, astronomy used probability models and statistical theories, particularly the [[method of least squares]]. Early probability theory and statistics was systematized in the 19th century and statistical reasoning and probability models were used by social scientists to advance the new sciences of [[experimental psychology]] and [[sociology]], and by physical scientists in [[thermodynamics]] and [[statistical mechanics]]. The development of statistical reasoning was closely associated with the development of [[inductive logic]] and the [[scientific method]], which are concerns that move statisticians away from the narrower area of mathematical statistics. Much of the theoretical work was readily available by the time computers were available to exploit them. By the 1970s, [[Samuel Kotz|Johnson and Kotz]] produced a four-volume ''Compendium on Statistical Distributions'' (1st ed., 1969–1972), which is still an invaluable resource.

Applied statistics can be regarded as not a field of [[mathematics]] but an autonomous [[mathematical science]], like [[computer science]] and [[operations research]]. Unlike mathematics, statistics had its origins in [[public administration]]. Applications arose early in [[demography]] and [[economics]]; large areas of micro- and macro-economics today are "statistics" with an emphasis on time-series analyses. With its emphasis on learning from data and making best predictions, statistics also has been shaped by areas of academic research including psychological testing, medicine and [[epidemiology]]. The ideas of statistical testing have considerable overlap with [[decision science]]. With its concerns with searching and effectively presenting [[data]], statistics has overlap with [[information science]] and [[computer science]].

==Etymology==
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The term ''statistics'' is ultimately derived from the [[Neo-Latin]] {{Lang|la|statisticum collegium}} ("council of state") and the [[Italian language|Italian]] word {{Lang|it|statista}} ("statesman" or "[[politician]]"). The [[German language|German]] {{Lang|de|Statistik}}, first introduced by [[Gottfried Achenwall]] (1749), originally designated the analysis of [[data]] about the [[State (polity)|state]], signifying the "science of state" (then called ''political arithmetic'' in English). It acquired the meaning of the collection and classification of data generally in the early 19th century. It was introduced into English in 1791 by [[Sir John Sinclair]] when he published the first of 21 volumes titled ''[[Statistical Account of Scotland]]''.<ref>{{cite book|last1 = Ball|first1 = Philip|title = Critical Mass|publisher =  Farrar, Straus and Giroux|year = 2004|isbn=978-0-374-53041-9|page=53 }}</ref>

==Origins in probability theory==
{{Main|History of probability}}
{{See also|Timeline of probability and statistics}}

Basic forms of statistics have been used since the beginning of civilization. Early empires often collated censuses of the population or recorded the trade in various commodities. The [[Han dynasty]] and the [[Roman Empire]] were some of the first states to extensively gather data on the size of the empire's population, geographical area and wealth.

The use of statistical methods dates back to at least the 5th century BCE. The historian [[Thucydides]] in his ''[[History of the Peloponnesian War]]''<ref>{{cite book|last=Thucydides|title=History of the Peloponnesian War|year=1985|publisher=Penguin Books, Ltd.|location=New York|pages=204}}</ref> describes how the Athenians calculated the height of the wall of [[Plataea|Platea]] by counting the number of bricks in an unplastered section of the wall sufficiently near them to be able to count them. The count was repeated several times by a number of soldiers. The most frequent value (in modern terminology – the [[Mode (statistics)|mode]]) so determined was taken to be the most likely value of the number of bricks. Multiplying this value by the height of the bricks used in the wall allowed the Athenians to determine the height of the ladders necessary to scale the walls.{{Citation needed|date=September 2019}}

The [[Trial of the Pyx]] is a test of the purity of the coinage of the [[Royal Mint]] which has been held on a regular basis since the 12th century. The Trial itself is based on statistical sampling methods. After minting a series of coins – originally from ten pounds of silver – a single coin was placed in the Pyx – a box in [[Westminster Abbey]]. After a given period – now once a year – the coins are removed and weighed. A sample of coins removed from the box are then tested for purity.

The ''[[Nuova Cronica]]'', a 14th-century [[history of Florence]] by the Florentine banker and official [[Giovanni Villani]], includes much statistical information on population, ordinances, commerce and trade, education, and religious facilities and has been described as the first introduction of statistics as a positive element in history,<ref name="Villani2008">Villani, Giovanni. Encyclopædia Britannica. [[Encyclopædia Britannica 2006 Ultimate Reference Suite DVD]]. Retrieved on 2008-03-04.</ref> though neither the term nor the concept of statistics as a specific field yet existed.

The arithmetic [[mean]], although a concept known to the Greeks, was not generalised to more than two values until the 16th century. The invention of the decimal system by [[Simon Stevin]] in 1585 seems likely to have facilitated these calculations. This method was first adopted in astronomy by [[Tycho Brahe]] who was attempting to reduce the errors in his estimates of the locations of various celestial bodies.

The idea of the [[median]] originated in [[Edward Wright (mathematician)|Edward Wright]]'s book on navigation (''Certaine Errors in Navigation'') in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations. The difference between the mean and the median was noticed in 1669 by Chistiaan Huygens in the context of using Graunt's tables.<ref>{{Cite journal |last1=Bakker |first1=Arthur |last2=Gravemeijer |first2=Koeno P. E. |date=2006-06-01 |title=An Historical Phenomenology of Mean and Median |url=https://doi.org/10.1007/s10649-006-7099-8 |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>

[[File:Sir William Petty.jpg|thumb|200px|Sir [[William Petty]], a 17th-century economist who used early statistical methods to analyse demographic data]]

The term 'statistic' was introduced by the Italian scholar [[Girolamo Ghilini]] in 1589 with reference to this science.<ref>{{cite journal |last=Ostasiewicz |first=Walenty |year=2014 |title=The emergence of statistical science |journal=Śląski Przegląd Statystyczny |volume=12 |issue=18 |doi=10.15611/sps.2014.12.04 |pages=76–77|doi-access=free }}</ref><ref>{{cite book |last=Bruneau |first=Quentin |url=https://books.google.com/books?id=63RnEAAAQBAJ&pg=PT64 |title=States and the Masters of Capital: Sovereign Lending, Old and New |publisher=[[Columbia University Press]] |year=2022 |isbn=978-0231555647}}</ref> The birth of statistics is often dated to 1662, when [[John Graunt]], along with [[William Petty]], developed early human statistical and [[census]] methods that provided a framework for modern [[demography]]. He produced the first [[life table]], giving probabilities of survival to each age. His book ''Natural and Political Observations Made upon the Bills of Mortality'' used analysis of the [[Mortality rate|mortality]] rolls to make the first statistically based estimation of the population of [[London]]. He knew that there were around 13,000 funerals per year in London and that three people died per eleven families per year. He estimated from the parish records that the average family size was 8 and calculated that the population of London was about 384,000; this is the first known use of a [[ratio estimator]]. [[Laplace]] in 1802 estimated the population of France with a similar method; see {{slink|Ratio estimator|History}} for details.

Although the original scope of statistics was limited to data useful for governance, the approach was extended to many fields of a scientific or commercial nature during the 19th century. The mathematical foundations for the subject heavily drew on the new [[probability theory]], pioneered in the 16th century by [[Gerolamo Cardano]], [[Pierre de Fermat]] and [[Blaise Pascal]]. [[Christiaan Huygens]] (1657) gave the earliest known scientific treatment of the subject. [[Jakob Bernoulli]]'s ''[[Ars Conjectandi]]'' (posthumous, 1713) and [[Abraham de Moivre]]'s ''[[The Doctrine of Chances]]'' (1718) treated the subject as a branch of mathematics. In his book Bernoulli introduced the idea of representing complete certainty as one and probability as a number between zero and one.

A key early application of statistics in the 18th century was to the [[human sex ratio]] at birth.<ref>{{cite book |title=The Descent of Human Sex Ratio at Birth |first1=Éric |last1=Brian |first2=Marie |last2=Jaisson |chapter=Physico-Theology and Mathematics (1710–1794) |pages=1–25 |year=2007 |publisher=Springer Science & Business Media |isbn=978-1-4020-6036-6}}</ref> [[John Arbuthnot]] studied this question in 1710.<ref>{{cite journal|author=John Arbuthnot |title=An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes|journal=[[Philosophical Transactions of the Royal Society of London]] | volume=27| pages=186–190 | year=1710 | url=http://www.york.ac.uk/depts/maths/histstat/arbuthnot.pdf|doi=10.1098/rstl.1710.0011|issue=325–336|s2cid=186209819|doi-access=free}}</ref><ref name="Conover1999">{{Citation
|last=Conover
|first=W.J.
|title=Practical Nonparametric Statistics
|edition=Third
|year=1999
|publisher=Wiley
|isbn=978-0-471-16068-7
|pages=157–176
|chapter=Chapter 3.4: The Sign Test
}} 
</ref><ref name="Sprent1989">{{Citation
|last=Sprent
|first=P.
|title=Applied Nonparametric Statistical Methods
|edition=Second
|year=1989
|publisher=Chapman & Hall
|isbn=978-0-412-44980-2
}}
</ref><ref>{{cite book |title=The History of Statistics: The Measurement of Uncertainty Before 1900 |first=Stephen M. |last=Stigler |publisher=Harvard University Press |year=1986 |isbn=978-0-67440341-3 |pages=[https://archive.org/details/historyofstatist00stig/page/225 225–226]}}</ref> Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 0.5^82, or about 1 in 4,8360,0000,0000,0000,0000,0000; in modern terms, the [[p-value|''p''-value]]. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." This is and other work by Arbuthnot is credited as "the first use of [[significance test]]s"<ref name="Bellhouse2001">{{Citation
|last=Bellhouse
|first=P.
|title=in Statisticians of the Centuries 
|editor1-link=Chris Heyde
|editor1=C.C. Heyde
|editor2-link=Eugene Seneta
|editor2= E. Seneta
|year=2001
|publisher=Springer
|isbn=978-0-387-95329-8
|pages=39–42
|chapter=John Arbuthnot}}
</ref> the first example of reasoning about [[statistical significance]] and moral certainty,{{sfn|Hald|1998|p=65}} and "... perhaps the first published report of a [[non-parametric test|nonparametric test]] ...",<ref name="Conover1999" /> specifically the [[sign test]]; see details at {{section link|Sign test|History}}.

The formal study of [[theory of errors]] may be traced back to [[Roger Cotes]]' ''Opera Miscellanea'' (posthumous, 1722), but a memoir prepared by [[Thomas Simpson]] in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the [[axiom]]s that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given. Simpson discussed several possible distributions of error. He first considered the [[Discrete uniform distribution|uniform distribution]] and then the discrete symmetric [[triangular distribution]] followed by the continuous symmetric triangle distribution. [[Tobias Mayer]], in his study of the [[libration]] of the [[moon]] (''{{Lang|de|Kosmographische Nachrichten}}'', Nuremberg, 1750), invented the first formal method for estimating the unknown quantities by generalized the averaging of observations under identical circumstances to the averaging of groups of similar equations.

[[Roger Joseph Boscovich]] in 1755 based in his work on the shape of the earth proposed in his book ''De Litteraria expeditione per pontificiam ditionem ad dimetiendos duos meridiani gradus a PP. Maire et Boscovicli'' that the true value of a series of observations would be that which minimises the sum of absolute errors. In modern terminology this value is the median. The first example of what later became known as the normal curve was studied by [[Abraham de Moivre]] who plotted this curve on November 12, 1733.<ref name="deMoive1738">de Moivre, A. (1738) The doctrine of chances. Woodfall</ref> de Moivre was studying the number of heads that occurred when a 'fair' coin was tossed.

In 1763 Richard Price transmitted to the Royal Society [[Thomas Bayes]] proof of a rule for using a binomial distribution to calculate a posterior probability on a prior event. 

In 1765 [[Joseph Priestley]] invented the first [[timeline]] charts.

[[Johann Heinrich Lambert]] in his 1765 book ''Anlage zur Architectonic'' proposed the [[semicircle]] as a distribution of errors:

: <math> f(x) = \frac{ 1 }{ 2 } \sqrt{ ( 1 - x^2 ) } </math>

with -1 < ''x'' < 1.

[[File:Laplace distribution pdf.png|325px|thumb|left|Probability density plots for the [[Laplace distribution]]]]
[[Pierre-Simon Laplace]] (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve and deduced a formula for the mean of three observations.

Laplace in 1774 noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded.<ref name="Laplace1774">{{cite journal | last1 = Laplace | first1 = P-S | year = 1774 | title = Mémoire sur la probabilité des causes par les évènements | journal = Mémoires de l'Académie Royale des Sciences Présentés par Divers Savants | volume = 6 | pages = 621–656 }}</ref><ref name="Wilson1923">Wilson, Edwin Bidwell (1923) "First and second laws of error", ''[[Journal of the American Statistical Association]]'', 18 (143), 841-851 {{JSTOR|2965467}}</ref> This distribution is now known as the [[Laplace distribution]]. Lagrange proposed a [[parabolic fractal distribution]] of errors in 1776.

Laplace in 1778 published his second law of errors wherein he noted that the frequency of an error was proportional to the exponential of the square of its magnitude. This was subsequently rediscovered by [[Gauss]] (possibly in 1795) and is now best known as the [[normal distribution]] which is of central importance in statistics.<ref name="Havil2003">Havil J (2003) ''Gamma: Exploring Euler's Constant''. Princeton, NJ: Princeton University Press, p. 157</ref> This distribution was first referred to as the ''normal'' distribution by [[C. S. Peirce]] in 1873 who was studying measurement errors when an object was dropped onto a wooden base.<ref name="Peirce1873">[[C. S. Peirce]]  (1873) Theory of errors of observations. Report of the Superintendent US Coast Survey, Washington, Government Printing Office. Appendix no. 21: 200-224</ref> He chose the term ''normal'' because of its frequent occurrence in naturally occurring variables.

Lagrange also suggested in 1781 two other distributions for errors – a [[raised cosine distribution]] and a [[logarithmic distribution]].

Laplace gave (1781) a formula for the law of facility of error (a term due to [[Joseph Louis Lagrange]], 1774), but one which led to unmanageable equations. [[Daniel Bernoulli]] (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

In 1786 [[William Playfair]] (1759–1823) introduced the idea of graphical representation into statistics. He invented the [[line chart]], [[bar chart]] and [[histogram]] and incorporated them into his works on [[economics]], the ''Commercial and Political Atlas''. This was followed in 1795 by his invention of the [[pie chart]] and circle chart which he used to display the evolution of England's imports and exports. These latter charts came to general attention when he published examples in his ''Statistical Breviary'' in 1801.

Laplace, in an investigation of the motions of [[Saturn]] and [[Jupiter]] in 1787, generalized Mayer's method by using different linear combinations of a single group of equations.

In 1791 [[Sir John Sinclair, 1st Baronet|Sir John Sinclair]] introduced the term 'statistics' into English in his [[Statistical Accounts of Scotland]].

In 1802 Laplace estimated the population of France to be 28,328,612.<ref name="Cochran1978">Cochran W.G. (1978) "Laplace's ratio estimators". pp 3-10. In David H.A., (ed). ''Contributions to Survey Sampling and Applied Statistics: papers in honor of H. O. Hartley''. Academic Press, New York  {{ISBN|978-1483237930}}</ref> He calculated this figure using the number of births in the previous year and census data for three communities. The census data of these communities showed that they had 2,037,615 persons and that the number of births were 71,866. Assuming that these samples were representative of France, Laplace produced his estimate for the entire population.

[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|[[Carl Friedrich Gauss]], mathematician who developed the [[method of least squares]] in 1809]]
The [[method of least squares]], which was used to minimize errors in data [[measurement]], was published independently by [[Adrien-Marie Legendre]] (1805), [[Robert Adrain]] (1808), and [[Carl Friedrich Gauss]] (1809). Gauss had used the method in his famous 1801 prediction of the location of the [[dwarf planet]] [[Ceres (dwarf planet)|Ceres]]. The observations that Gauss based his calculations on were made by the Italian monk Piazzi.

The method of least squares was preceded by the use a median regression slope. This method minimizing the sum of the absolute deviances. A method of estimating this slope was invented by [[Roger Joseph Boscovich]] in 1760 which he applied to astronomy.

The term ''probable error'' (''{{Lang|de|der wahrscheinliche Fehler}}'') – the median deviation from the mean – was introduced in 1815 by the German astronomer [[Friedrich Bessel|Frederik Wilhelm Bessel]]. [[Antoine Augustin Cournot]] in 1843 was the first to use the term ''median'' (''{{Lang|fr|valeur médiane}}'') for the value that divides a probability distribution into two equal halves.

Other contributors to the theory of errors were Ellis (1844), [[Augustus De Morgan|De Morgan]] (1864), [[James Whitbread Lee Glaisher|Glaisher]] (1872), and [[Giovanni Schiaparelli]] (1875).{{citation needed|date=July 2012}} Peters's (1856) formula for <math>r</math>, the "probable error" of a single observation was widely used and inspired early [[robust statistics]] (resistant to [[outlier]]s: see [[Peirce's criterion]]).

In the 19th century authors on [[statistical theory]] included Laplace, [[Sylvestre Lacroix|S. Lacroix]] (1816), Littrow (1833), [[Richard Dedekind|Dedekind]] (1860), Helmert (1872), [[Hermann Laurent|Laurent]] (1873), Liagre, Didion, [[Augustus De Morgan|De Morgan]] and [[George Boole|Boole]].

[[Gustav Theodor Fechner]] used the median (''Centralwerth'') in sociological and psychological phenomena.<ref name="Keynes1921">Keynes, JM (1921) A treatise on probability. Pt II Ch XVII §5 (p 201)</ref> It had earlier been used only in astronomy and related fields. [[Francis Galton]] used the English term ''median'' for the first time in 1881 having earlier used the terms ''middle-most value'' in 1869 and the ''medium'' in 1880.<ref name="Galton1881">Galton F (1881) Report of the Anthropometric Committee pp 245-260. Report of the 51st Meeting of the British Association for the Advancement of Science</ref>

[[Adolphe Quetelet]] (1796–1874), another important founder of statistics, introduced the notion of the "average man" (''l'homme moyen'') as a means of understanding complex social phenomena such as [[crime rates]], [[marriage rates]], and [[suicide rates]].<ref>Stigler (1986, Chapter 5: Quetelet's Two Attempts)</ref>

The first tests of the normal distribution were invented by the German statistician [[Wilhelm Lexis]] in the 1870s. The only data sets available to him that he was able to show were normally distributed were birth rates.

===Development of modern statistics===

Although the origins of statistical theory lie in the 18th-century advances in probability, the modern field of statistics only emerged in the late-19th and early-20th century in three stages. The first wave, at the turn of the century, was led by the work of [[Francis Galton]] and [[Karl Pearson]], who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. The second wave of the 1910s and 20s was initiated by [[William Sealy Gosset]], and reached its culmination in the insights of [[Ronald Fisher]]. This involved the development of better [[design of experiments]] models, hypothesis testing and techniques for use with small data samples. The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between [[Egon Pearson]] and [[Jerzy Neyman]] in the 1930s.<ref>{{cite book|url=https://books.google.com/books?id=jYFRAAAAMAAJ|title=Studies in the history of statistical method|author=Helen Mary Walker|year=1975|publisher=Arno Press|isbn=9780405066283}}</ref> Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology.

[[File:Statistical Society of London - 1837 logo.png|thumb|140px|right|The original logo of the [[Royal Statistical Society]], founded in 1834]]
The first statistical bodies were established in the early 19th century. The [[Royal Statistical Society]] was founded in 1834 and [[Florence Nightingale]], its first female member, pioneered the application of statistical analysis to health problems for the furtherance of epidemiological understanding and public health practice. However, the methods then used would not be considered as modern statistics today.

The [[Oxford University|Oxford]] scholar [[Francis Ysidro Edgeworth]]'s book, ''Metretike: or The Method of Measuring Probability and Utility'' (1887) dealt with probability as the basis of inductive reasoning, and his later works focused on the 'philosophy of chance'.<ref>(Stigler 1986, Chapter 9: The Next Generation: Edgeworth)</ref> His first paper on statistics (1883) explored the law of error ([[normal distribution]]), and his ''Methods of Statistics'' (1885) introduced an early version of the [[Student's t-distribution|t distribution]], the [[Edgeworth expansion]], the [[Edgeworth series]], the method of variate transformation and the asymptotic theory of maximum likelihood estimates.

The Norwegian [[Anders Nicolai Kiær]] introduced the concept of [[stratified sampling]] in 1895.<ref name="Bellhouse1988">Bellhouse DR (1988) A brief history of random sampling methods. Handbook of statistics. Vol 6 pp 1-14 Elsevier</ref> [[Arthur Lyon Bowley]] introduced new methods of data sampling in 1906 when working on social statistics. Although statistical surveys of social conditions had started with [[Charles Booth (philanthropist)|Charles Booth]]'s "Life and Labour of the People in London" (1889–1903) and [[Seebohm Rowntree]]'s "Poverty, A Study of Town Life" (1901), Bowley's, key innovation consisted of the use of [[random sampling]] techniques. His efforts culminated in his ''New Survey of London Life and Labour''.<ref name="Bowley1906">{{cite journal | last1 = Bowley | first1 = AL | year = 1906 | title = Address to the Economic Science and Statistics Section of the British Association for the Advancement of Science | url = https://zenodo.org/record/2152740| journal = J R Stat Soc | volume = 69 | pages = 548–557 | doi = 10.2307/2339344 | jstor = 2339344 }}</ref>

[[Francis Galton]] is credited as one of the principal founders of statistical theory. His contributions to the field included introducing the concepts of [[standard deviation]], [[correlation]],  [[regression analysis|regression]] and the application of these methods to the study of the variety of human characteristics – height, weight, eyelash length among others. He found that many of these could be fitted to a normal curve distribution.<ref name="Galton1877">{{cite journal | last1 = Galton | first1 = F | year = 1877 | title = Typical laws of heredity | journal = Nature | volume = 15 | issue = 388| pages = 492–553 | doi = 10.1038/015492a0 | bibcode = 1877Natur..15..492. | doi-access = free }}</ref>

Galton submitted a paper to ''Nature'' in 1907 on the usefulness of the median.<ref name="Galton1907">{{cite journal | last1 = Galton | first1 = F | year = 1907 | title = One Vote, One Value | url = https://zenodo.org/record/2125755| journal = Nature | volume = 75 | issue = 1948| page = 414 | doi = 10.1038/075414a0 | bibcode = 1907Natur..75Q.414G | s2cid = 4053860 | doi-access = free }}</ref> He examined the accuracy of 787 guesses of the weight of an ox at a country fair. The actual weight was 1208 pounds: the median guess was 1198. The guesses were markedly non-normally distributed (cf. [[Wisdom of the crowd|Wisdom of the Crowd]]).

[[File:Karl Pearson, 1910.jpg|thumb|left|200px|[[Karl Pearson]], the founder of [[mathematical statistics]]]]
Galton's publication of ''Natural Inheritance'' in 1889 sparked the interest of a brilliant mathematician, [[Karl Pearson]],<ref>Stigler (1986, Chapter 10: Pearson and Yule)</ref> then working at [[University College London]], and he went on to found the discipline of mathematical statistics.<ref>{{Cite journal|jstor=27956805|title=The development of modern statistics|journal=The Mathematics Teacher|volume=56|issue=4|pages=252–257|last1=Varberg|first1=Dale E.|year=1963|doi=10.5951/MT.56.4.0252 }}</ref> He emphasised the statistical foundation of scientific laws and promoted its study and his laboratory attracted students from around the world attracted by his new methods of analysis, including [[Udny Yule]]. His work grew to encompass the fields of [[biology]], [[epidemiology]], anthropometry, [[medicine]] and social [[history]]. In 1901, with [[Walter Frank Raphael Weldon|Walter Weldon]], founder of [[biometry]], and Galton, he founded the journal ''[[Biometrika]]'' as the first journal of mathematical statistics and biometry.

His work, and that of Galton, underpins many of the 'classical' statistical methods which are in common use today, including the [[Pearson product-moment correlation coefficient|Correlation coefficient]], defined as a product-moment;<ref>{{Cite journal | doi = 10.1214/ss/1177012580 | last1 = Stigler | first1 = S. M. | year = 1989 | title = Francis Galton's Account of the Invention of Correlation | journal = Statistical Science | volume = 4 | issue = 2| pages = 73–79 | doi-access = free }}</ref> the [[Method of moments (statistics)|method of moments]] for the fitting of distributions to samples; [[Pearson distribution|Pearson's system of continuous curves]] that forms the basis of the now conventional continuous probability distributions; [[Mahalanobis distance|Chi distance]] a precursor and special case of the [[Mahalanobis distance]]<ref name="Pearson, On the criterion">{{Cite journal|last1=Pearson|first1=K.|year=1900|title=On the Criterion that a given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling|url=https://zenodo.org/record/1430618|journal=Philosophical Magazine |series=Series 5|volume=50|issue=302|pages=157–175|doi=10.1080/14786440009463897}}</ref> and [[P-value]], defined as the probability measure of the complement of the [[Ball (mathematics)|ball]] with the hypothesized value as center point and chi distance as radius.<ref name="Pearson, On the criterion"/> He also introduced the term  'standard deviation'.

He also founded the [[Statistical hypothesis testing|statistical hypothesis testing theory]],<ref name="Pearson, On the criterion"/> [[Pearson's chi-squared test]] and [[principal component analysis]].<ref>{{Cite journal|last1=Pearson|first1=K.|year=1901|title=On Lines and Planes of Closest Fit to Systems of Points is Space|url=https://zenodo.org/record/1430636|journal=Philosophical Magazine |series=Series 6|volume=2|issue=11|pages=559–572|doi=10.1080/14786440109462720}}</ref><ref>Jolliffe, I. T. (2002). ''Principal Component Analysis, 2nd ed''. New York: Springer-Verlag.</ref> In 1911 he founded the world's first university statistics department at [[University College London]].

The second wave of mathematical statistics was pioneered by [[Ronald Fisher]] who wrote two textbooks, ''[[Statistical Methods for Research Workers]]'', published in 1925 and ''[[The Design of Experiments]]'' in 1935, that were to define the academic discipline in universities around the world. He also systematized previous results, putting them on a firm mathematical footing. In his 1918 seminal paper ''[[The Correlation between Relatives on the Supposition of Mendelian Inheritance]]'', the first use to use the statistical term, [[variance]]. In 1919, at [[Rothamsted Experimental Station]] he started a major study of the extensive collections of data recorded over many years. This resulted in a series of reports under the general title ''Studies in Crop Variation.'' In 1930 he published ''[[The Genetical Theory of Natural Selection]]'' where he applied statistics to [[evolution]].

<!-- Deleted image removed: [[File:Anil Kumar Gain.png|thumb|left|[[Anil Kumar Gain]], famous Indian statistician]] -->
Over the next seven years, he pioneered the principles of the [[design of experiments]] (see below) and elaborated his studies of analysis of variance. He furthered his studies of the statistics of small samples. Perhaps even more important, he began his systematic approach of the analysis of real data as the springboard for the development of new statistical methods. He developed computational algorithms for analyzing data from his balanced experimental designs. In 1925, this work resulted in the publication of his first book, ''[[Statistical Methods for Research Workers]]''.<ref>Box, ''R. A. Fisher'', pp 93–166</ref> This book went through many editions and translations in later years, and it became the standard reference work for scientists in many disciplines. In 1935, this book was followed by ''[[The Design of Experiments]]'', which was also widely used.

In addition to analysis of variance, Fisher named and promoted the method of [[maximum likelihood]] estimation. Fisher also originated the concepts of [[sufficiency (statistics)|sufficiency]], [[ancillary statistic]]s, [[linear discriminant analysis|Fisher's linear discriminator]] and [[Fisher information]]. His article ''On a distribution yielding the error functions of several well known statistics'' (1924) presented [[Pearson's chi-squared test]] and [[William Sealy Gosset]]'s [[Student's t-distribution|t]] in the same framework as the [[Gaussian distribution]], and his own parameter in the analysis of variance [[Fisher's z-distribution]] (more commonly used decades later in the form of the [[F distribution]]).<ref>{{cite journal|last1=Agresti|first1=Alan|author2=David B. Hichcock|year=2005|title=Bayesian Inference for Categorical Data Analysis|journal=Statistical Methods & Applications|issue=3|page=298|url=http://www.stat.ufl.edu/~aa/articles/agresti_hitchcock_2005.pdf|doi=10.1007/s10260-005-0121-y|volume=14|s2cid=18896230}}</ref>
The 5% level of significance appears to have been introduced by Fisher in 1925.<ref name="Fisher1925">Fisher RA (1925) Statistical methods for research workers, Edinburgh: Oliver & Boyd</ref> Fisher stated that deviations exceeding twice the standard deviation are regarded as significant. Before this deviations exceeding three times the probable error were considered significant. For a symmetrical distribution the probable error is half the interquartile range. For a normal distribution the probable error is approximately 2/3 the standard deviation. It appears that Fisher's 5% criterion was rooted in previous practice.

Other important contributions at this time included [[Charles Spearman]]'s [[Spearman's rank correlation coefficient|rank correlation coefficient]] that was a useful extension of the Pearson correlation coefficient. [[William Sealy Gosset]], the English statistician better known under his pseudonym of ''Student'', introduced [[Student's t-distribution]], a continuous probability distribution useful in situations where the sample size is small and population standard deviation is unknown.

[[Egon Pearson]] (Karl's son) and [[Jerzy Neyman]] introduced the concepts of "[[Type I and type II errors|Type II]]" error, power of a test and [[confidence interval]]s. [[Jerzy Neyman]] in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.<ref>Neyman, J (1934) On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection. ''[[Journal of the Royal Statistical Society]]'' 97 (4) 557-625 {{JSTOR|2342192}}</ref>

==Design of experiments==
{{Primary sources|section|date=February 2012}}
[[File:James lind.jpg|thumb|[[James Lind (physician)|James Lind]] carried out the first ever clinical trial in 1747, in an effort to find a treatment for [[scurvy]].]]
In 1747, while serving as surgeon on HM Bark ''Salisbury'', [[James Lind (physician)|James Lind]] carried out a controlled experiment to develop a cure for [[scurvy]].<ref name="ADC1997">{{Cite journal| last =Dunn | first=Peter |title=James Lind (1716-94) of Edinburgh and the treatment of scurvy  | journal =Archives of Disease in Childhood: Fetal and Neonatal Edition | volume =76 | issue = 1| pages =64–65 |date = January 1997| doi = 10.1136/fn.76.1.F64| pmc =1720613 | pmid=9059193}}</ref> In this study his subjects' cases "were as similar as I could have them", that is he provided strict entry requirements to reduce extraneous variation. The men were paired, which provided [[Blocking (statistics)|blocking]]. From a modern perspective, the main thing that is missing is randomized allocation of subjects to treatments.

Lind is today often described as a one-factor-at-a-time experimenter.<ref name="Hinkelmann">{{cite book|url=https://books.google.com/books?id=8Y71m9yoK6UC|title=Design and Analysis of Experiments, Special Designs and Applications|author=Klaus Hinkelmann|year=2012|publisher=John Wiley & Sons|page=xvii|isbn=9780470530689}}</ref> Similar one-factor-at-a-time (OFAT) experimentation was performed at the [[Rothamsted Research Station]] in the 1840s by Sir [[John Lawes]] to determine the optimal inorganic fertilizer for use on wheat.<ref name="Hinkelmann" />

A theory of statistical inference was developed by [[Charles Sanders Peirce|Charles S. Peirce]] in "[[Charles Sanders Peirce bibliography#illus|Illustrations of the Logic of Science]]" (1877–1878) and "[[Charles Sanders Peirce bibliography#SIL|A Theory of Probable Inference]]" (1883), two publications that emphasized the importance of randomization-based inference in statistics. In another study, Peirce randomly assigned volunteers to a [[blinding (medicine)|blinded]], [[repeated measures design|repeated-measures design]] to evaluate their ability to discriminate weights.<ref name="smalldiff">{{cite journal| author=[[Charles Sanders Peirce]] and [[Joseph Jastrow]]|year=1885|title=On Small Differences in Sensation|url=http://psychclassics.yorku.ca/Peirce/small-diffs.htm| journal=Memoirs of the National Academy of Sciences|volume=3|pages=73–83}}</ref><ref name="telepathy">{{cite journal|first=Ian |last=Hacking|  author-link=Ian Hacking | title=Telepathy: Origins of Randomization in Experimental Design|journal=[[Isis (journal)|Isis]]|issue=A Special Issue on Artifact and Experiment, number 3 |volume=79|date=September 1988 |pages=427–451|jstor=234674 | mr = 1013489|doi=10.1086/354775|s2cid=52201011 }}</ref><ref name="stigler">{{cite journal|author=[[Stephen M. Stigler]]|title=A Historical View of Statistical Concepts in Psychology and Educational Research| journal=American Journal of Education| volume=101|date=November 1992|pages=60–70|issue=1|doi=10.1086/444032|s2cid=143685203 }}</ref><ref name="dehue">{{cite journal|author=Trudy Dehue|title=Deception, Efficiency, and Random Groups: Psychology and the Gradual Origination of the Random Group Design|journal=[[Isis (journal)|Isis]]|volume=88|date=December 1997|pages=653–673|issue=4|doi=10.1086/383850|pmid=9519574|s2cid=23526321 |url=https://www.rug.nl/research/portal/files/71855616/237831.pdf}}</ref>

Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s.<ref name="smalldiff"/><ref name="telepathy"/><ref name="stigler"/><ref name="dehue"/> Peirce also contributed the first English-language publication on an [[optimal design]] for [[Regression analysis|regression]]-[[statistical model|models]] in 1876.<ref>{{cite journal| author=[[Charles Sanders Peirce|Peirce, C. S.]] | year=1876| title=Note on the Theory of the Economy of Research | journal=Coast Survey Report | pages=197–201}}, actually published 1879, NOAA [http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0025.PDF#page=222 PDF Eprint].<br /> Reprinted in ''[[Charles Sanders Peirce bibliography#CP|Collected Papers]]'' '''7''', paragraphs 139–157, also in ''[[Charles Sanders Peirce bibliography#W|Writings]]'' '''4''', pp. 72–78, and in {{Cite journal|author-link=Charles Sanders Peirce|last1=Peirce|first1=C.S.|title=Note on the Theory of the Economy of Research|url=http://or.journal.informs.org/cgi/content/abstract/15/4/643|journal=Operations Research|volume=15|date=July–August 1967|pages=643–648|jstor=168276|doi=10.1287/opre.15.4.643|issue=4|access-date=2010-11-18|archive-date=2011-07-26|archive-url=https://web.archive.org/web/20110726181131/http://or.journal.informs.org/cgi/content/abstract/15/4/643|url-status=dead}}</ref> A pioneering [[optimal design]] for [[polynomial regression]] was suggested by [[Joseph Diaz Gergonne|Gergonne]] in 1815.{{citation needed|date=July 2012}} In 1918 [[Kirstine Smith]] published optimal designs for polynomials of degree six (and less).<ref>{{cite journal|last=Smith|first=Kirstine|title=On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance they give Towards a Proper Choice of the Distribution of Observations|journal=[[Biometrika]]|volume=12|issue=1/2|pages=1–85|year=1918|doi=10.2307/2331929|jstor=2331929|url=https://zenodo.org/record/1431591}}</ref>

The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, was pioneered<ref>Johnson, N.L. (1961). "Sequential analysis: a survey." ''[[Journal of the Royal Statistical Society]]'', Series A. Vol. 124 (3), 372&ndash;411. (pages 375&ndash;376)</ref> by [[Abraham Wald]] in the context of sequential tests of statistical hypotheses.<ref>Wald, A. (1945) "Sequential Tests of Statistical Hypotheses", [[Annals of Mathematical Statistics]], 16 (2), 117&ndash;186.</ref> Surveys are available of optimal [[sequential analysis|sequential designs]],<ref name="ref3">[[Herman Chernoff|Chernoff, H.]] (1972) ''Sequential Analysis and Optimal Design'', [[Society for Industrial and Applied Mathematics|SIAM]] Monograph. {{isbn|978-0898710069}}</ref> and of [[Minimisation (clinical trials)|adaptive designs]].<ref>Zacks, S. (1996) "Adaptive Designs for Parametric Models". In: Ghosh, S. and Rao, C. R., (Eds) (1996). "Design and Analysis of Experiments," ''Handbook of Statistics'', Volume 13. North-Holland. {{isbn|0-444-82061-2}}.  (pages 151&ndash;180)</ref> One specific type of sequential design is the "two-armed bandit", generalized to the [[multi-armed bandit]], on which early work was done by [[Herbert Robbins]] in 1952.<ref>{{cite journal | doi = 10.1090/S0002-9904-1952-09620-8 | last1 = Robbins | first1 = H. | year = 1952 | title = Some Aspects of the Sequential Design of Experiments | journal = Bulletin of the American Mathematical Society | volume = 58 | issue = 5| pages = 527–535 | citeseerx = 10.1.1.335.3232 }}</ref>

The term "design of experiments" (DOE) derives from early statistical work performed by [[Ronald Fisher|Sir Ronald Fisher]]. He was described by [[Anders Hald]] as "a genius who almost single-handedly created the foundations for modern statistical science."{{sfn|Hald|1998|p={{Page needed|date=May 2012}}}} Fisher initiated the principles of [[design of experiments]] and elaborated on his studies of "[[analysis of variance]]". Perhaps even more important, Fisher began his systematic approach to the analysis of real data as the springboard for the development of new statistical methods.  He began to pay particular attention to the labour involved in the necessary computations performed by hand, and developed methods that were as practical as they were founded in rigour.  In 1925, this work culminated in the publication of his first book, ''[[Statistical Methods for Research Workers]]''.<ref>Box, Joan Fisher (1978) ''R. A. Fisher: The Life of a Scientist'', Wiley. {{isbn|0-471-09300-9}} (pp 93–166)</ref>  This went into many editions and translations in later years, and became a standard reference work for scientists in many disciplines.<ref>{{cite book|last=Edwards|first=A.W.F.|contribution=R. A. Fisher, Statistical Methods for Research Workers, 1925|editor-last=Grattan-Guinness|editor-first=Ivor|title=Landmark writings in Western mathematics 1640-1940|publisher=Elsevier|location=Amsterdam Boston|year=2005|isbn=9780444508713}}</ref>

A methodology for designing experiments was proposed by [[Ronald Fisher|Ronald A. Fisher]], in his innovative book ''[[The Design of Experiments]]'' (1935) which also became a standard.<ref>{{cite journal | doi = 10.3102/00028312003003223 | title = The Influence of Fisher's "The Design of Experiments" on Educational Research Thirty Years Later | year = 1966 | author = Stanley, J. C. | journal = American Educational Research Journal | volume = 3 | pages = 223–229 | issue = 3| s2cid = 145725524 }}</ref><ref>{{cite journal | author = Box, JF | title = R. A. Fisher and the Design of Experiments, 1922-1926 | jstor = 2682986 | journal = [[The American Statistician]] | volume = 34 | issue = 1 |date=February 1980 | pages = 1–7 | doi = 10.2307/2682986}}</ref><ref>{{cite journal | last1= Yates |first1=Frank |author-link=Frank Yates| title = Sir Ronald Fisher and the Design of Experiments | jstor = 2528399 | journal = [[Biometrics (journal)|Biometrics]] | volume = 20 | issue = 2 |date= June 1964 | pages = 307–321 |doi = 10.2307/2528399}}</ref><ref>{{cite journal
|title=The Influence of Fisher's "The Design of Experiments" on Educational Research Thirty Years Later|first1=Julian C. |last1=Stanley|journal=American Educational Research Journal|volume=3|issue=3|year=1966|pages= 223–229|jstor=1161806|doi=10.3102/00028312003003223|s2cid=145725524 }}</ref> As an example, he described how to test the [[hypothesis]] that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. While this sounds like a frivolous application, it allowed him to illustrate the most important ideas of experimental design: see [[Lady tasting tea]].

[[Agricultural science]] advances served to meet the combination of larger city populations and fewer farms. But for crop scientists to take due account of widely differing geographical growing climates and needs, it was important to differentiate local growing conditions. To extrapolate experiments on local crops to a national scale, they had to extend crop sample testing economically to overall populations. As statistical methods advanced (primarily the efficacy of designed experiments instead of one-factor-at-a-time experimentation), representative factorial design of experiments began to enable the meaningful extension, by inference, of experimental sampling results to the population as a whole.{{Citation needed|date=May 2012}} But it was hard to decide how representative was the crop sample chosen.{{Citation needed|date=May 2012}} Factorial design methodology showed how to estimate and correct for any random variation within the sample and also in the data collection procedures.

==Bayesian statistics==
[[File:Pierre-Simon Laplace.jpg|right|thumb|Pierre-Simon, marquis de Laplace, the main early developer of Bayesian statistics]]
The term ''Bayesian'' refers to [[Thomas Bayes]] (1702&ndash;1761), who proved that probabilistic limits could be placed on an unknown event.  However it was [[Pierre-Simon Laplace]] (1749&ndash;1827) who introduced (as principle VI) what is now called [[Bayes' theorem]] and applied it to [[celestial mechanics]], medical statistics, [[Reliability (statistics)|reliability]], and [[jurisprudence]].<ref name="Stigler1986">Stigler (1986, Chapter 3: Inverse Probability)</ref>  When insufficient knowledge was available to specify an informed prior, Laplace used [[uniform distribution (continuous)|uniform]] priors, according to his "[[principle of insufficient reason]]".<ref name=Stigler1986/><ref>Hald (1998){{Page needed|date=May 2012}}</ref> Laplace assumed uniform priors for mathematical simplicity rather than for philosophical reasons.<ref name=Stigler1986/> Laplace also introduced{{Citation needed|date=May 2012}} primitive versions of [[conjugate prior]]s and the [[Bernstein–von Mises theorem|theorem]] of [[Richard von Mises|von Mises]] and [[S. N. Bernstein|Bernstein]], according to which the posteriors corresponding to initially differing priors ultimately agree, as the number of observations increases.<ref>
Lucien Le Cam (1986) ''Asymptotic Methods in Statistical Decision Theory'': Pages 336 and 618&ndash;621 (von Mises and Bernstein).</ref> This early Bayesian inference, which used uniform priors following Laplace's [[principle of insufficient reason]], was called "[[inverse probability]]" (because it [[Inductive reasoning|infer]]s backwards from observations to parameters, or from effects to causes<ref name="Fienberg2006">Stephen. E. Fienberg, (2006) [http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf When did Bayesian Inference become "Bayesian"?] {{webarchive|url=https://web.archive.org/web/20140910070556/http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf |date=2014-09-10 }} ''Bayesian Analysis'', 1 (1), 1&ndash;40. See page 5.</ref>).

After the 1920s, [[inverse probability]] <!-- (Bayesian inference with uniform priors following Laplace) --> was largely supplanted{{Citation needed|date=May 2012}}  by a collection of methods that were developed by [[Ronald A. Fisher]], [[Jerzy Neyman]] and [[Egon Pearson]]. Their methods came to be called [[frequentist statistics]].<ref name=Fienberg2006/> Fisher rejected the Bayesian view, writing that "the theory of inverse probability is founded upon an error, and must be wholly rejected".<ref name="ba.stat.cmu.edu">{{cite journal | last1 = Aldrich | first1 = A | year = 2008 | title = R. A. Fisher on Bayes and Bayes' Theorem | journal = Bayesian Analysis | volume = 3 | issue = 1 | pages = 161–170 | doi = 10.1214/08-ba306 | doi-access = free }}</ref> At the end of his life, however, Fisher expressed greater respect for the essay of Bayes, which Fisher believed to have anticipated his own, [[fiducial inference|fiducial]] approach to probability; Fisher still maintained that Laplace's views on probability were "fallacious rubbish".<ref name="ba.stat.cmu.edu" /> Neyman started out as a "quasi-Bayesian", but subsequently developed [[confidence interval]]s (a key method in frequentist statistics) because "the whole theory would look nicer if it were built from the start without reference to Bayesianism and priors".<ref>{{cite journal | last1 = Neyman | first1 = J. | year = 1977 | title = Frequentist probability and frequentist statistics | journal = Synthese | volume = 36 | issue = 1| pages = 97–131 | doi = 10.1007/BF00485695 | s2cid = 46968744 }}</ref>
The word ''Bayesian'' appeared around 1950, and by the 1960s it became the term preferred by those dissatisfied with the limitations of frequentist statistics.<ref name="Fienberg2006" /><ref name="Miller Earliest Uses">Jeff Miller, [http://jeff560.tripod.com/b.html "Earliest Known Uses of Some of the Words of Mathematics (B)"]
"The term Bayesian entered circulation around 1950. R. A. Fisher used it in the notes he wrote to accompany the papers in his Contributions to Mathematical Statistics  (1950). Fisher thought Bayes's argument was all but extinct for the only recent work to take it seriously was [[Harold Jeffreys]]'s Theory of Probability (1939). In 1951 L. J. Savage, reviewing Wald's Statistical Decisions Functions, referred to "modern, or unBayesian, statistical theory" ("The Theory of Statistical Decision," ''Journal of the American Statistical Association'', 46, p. 58.). Soon after, however, Savage changed from being an unBayesian to being a Bayesian."</ref>

In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to ''objective'' and ''subjective'' currents in Bayesian practice. In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.<ref name="Bernardo">{{cite book | doi = 10.1016/S0169-7161(05)25002-2 | last1 = Bernardo | first1 = JM. | title = Bayesian Thinking - Modeling and Computation | author-link = José-Miguel Bernardo |name-list-style=vanc | year = 2005 | chapter = Reference analysis | volume = 25 | pages = 17–90 | series = Handbook of Statistics | isbn = 9780444515391 }}</ref> No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.

In the further development of Laplace's ideas, subjective ideas predate objectivist positions. The idea that 'probability' should be interpreted as 'subjective degree of belief in a proposition' was  proposed, for example, by [[John Maynard Keynes]] in the early 1920s.{{Citation needed|date=May 2012}} This idea was taken further by [[Bruno de Finetti]] in Italy (''{{Lang|it|Fondamenti Logici del Ragionamento Probabilistico}}'', 1930) and [[Frank P. Ramsey|Frank Ramsey]] in Cambridge (''The Foundations of Mathematics'', 1931).<ref>Gillies, D. (2000), ''Philosophical Theories of Probability''. Routledge. {{isbn|0-415-18276-X}}  pp&nbsp;50&ndash;1</ref> The approach was devised to solve problems with the [[frequentist| frequentist definition of probability]] but also with the earlier, objectivist approach of Laplace.<ref name=Bernardo/> The subjective Bayesian methods were further developed and popularized in the 1950s by [[Leonard Jimmie Savage|L.J. Savage]].{{Citation needed|date=September 2010}}

Objective Bayesian inference was further developed by [[Harold Jeffreys]] at the [[University of Cambridge]]. His book ''Theory of Probability'' first appeared in 1939 and played an important role in the revival of the [[Bayesian probability|Bayesian view of probability]].<ref>E. T. Jaynes. ''Probability Theory: The Logic of Science'' Cambridge University Press, (2003). {{isbn|0-521-59271-2}}</ref><ref name="mactutor">{{MacTutor Biography|id=Jeffreys}}</ref> In 1957, [[Edwin Thompson Jaynes|Edwin Jaynes]] promoted the concept of [[Principle of maximum entropy|maximum entropy]] for constructing priors, which is an important principle in the formulation of objective methods, mainly for discrete problems. In 1965, [[Dennis Lindley]]'s two-volume work "Introduction to Probability and Statistics from a Bayesian Viewpoint" brought Bayesian methods to a wide audience. In 1979, [[José-Miguel Bernardo]] introduced [[Prior probability#Uninformative priors|reference analysis]],<ref name=Bernardo/> which offers a general applicable framework for objective analysis.<ref>Bernardo, J. M. and Smith, A. F. M. (1994). "Bayesian Theory". Chichester: Wiley.</ref> Other well-known proponents of Bayesian probability theory include [[I.J. Good]], [[B.O. Koopman]], [[Howard Raiffa]], [[Robert Schlaifer]] and [[Alan Turing]].

<!--The Bayesian viewpoint has an axiomatic basis. In 1946, [[Richard Threlkeld Cox|Richard T. Cox]] showed that the rules of Bayesian inference follow from a set of constraints, including the representation of degrees of belief by real numbers, the need for consistency, and two [[functional equations]].<ref name = "vkdmsn"/> Another fundamental justification of the Bayesian approach is [[De Finetti's theorem]], which was formulated in 1930.<ref>{{cite journal | last1 = de Finetti | first1 = B | year = 1930 | title = Funzione caratteristica di un fenomeno aleatorio | journal = Mem. Acad. Naz. Lincei | volume = 4 | pages = 86–133 }}</ref>
-->
In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of [[Markov chain Monte Carlo]] methods, which removed many of the [[computational problem]]s, and an increasing interest in nonstandard, complex applications.<ref>{{cite journal | last1 = Wolpert | first1 = RL | year = 2004 | title = A conversation with James O. Berger | url = http://projecteuclid.org/euclid.ss/1089808283 | journal = Statistical Science | volume = 9 | pages = 205–218 | doi = 10.1214/088342304000000053 | mr = 2082155 | doi-access = free }}</ref> Despite growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.<ref>{{cite conference|first=J. M.|last=Bernardo|title=A Bayesian Mathematical Statistics Primer|book-title=Proceedings of the Seventh International Conference on Teaching Statistics [CDROM]|publisher=International Association for Statistical Education|year=2006|location=Salvador (Bahia), Brazil|url=http://www.stat.auckland.ac.nz/~iase/publications/17/3I2_BERN.pdf}}</ref> Nonetheless, Bayesian methods are widely accepted and used, such as for example in the field of [[machine learning]].<ref name="ReferenceA">Bishop, C.M. (2007) ''Pattern Recognition and Machine Learning''. Springer {{isbn|978-0387310732}}</ref>

==Important contributors to statistics==
{{See also|List of statisticians|Founders of statistics}}
{{col-begin}}
{{col-break}}
* [[Raghu Raj Bahadur]]
* [[Thomas Bayes]]
* [[George E. P. Box]]
* [[Pafnuty Chebyshev]]
* [[David R. Cox]]
* [[Gertrude Mary Cox|Gertrude Cox]]
* [[Harald Cramér]]
* [[Francis Ysidro Edgeworth]]
* [[Bradley Efron]]
* [[Bruno de Finetti]]
* [[Ronald A. Fisher]]
* [[Francis Galton]]
{{col-break}}
* [[Carl Friedrich Gauss]]
* [[William Sealey Gosset]] (pen name: "Student")
* [[Al-Kindi]]
* [[Andrey Kolmogorov]]
* [[Pierre-Simon Laplace]]
* [[Erich Leo Lehmann|Erich L. Lehmann]]
* [[Aleksandr Lyapunov]]
* [[Anil Kumar Gain]]
* [[Prasanta Chandra Mahalanobis]]
* [[Abraham De Moivre]]
* [[Jerzy Neyman]]
* [[Florence Nightingale]]
{{col-break}}
* [[Blaise Pascal]]
* [[Karl Pearson]]
* [[Charles Sanders Peirce|Charles S. Peirce]]
* [[Adolphe Quetelet]]
* [[C. R. Rao]]
* [[Walter A. Shewhart]]
* [[Charles Spearman]]
* [[Charles Stein (statistician)|Charles Stein]]
* [[Thorvald N. Thiele]]
* [[John Tukey]]
* [[Abraham Wald]]
{{col-end}}

== References ==
{{Reflist}}

==Bibliography==
*{{Cite journal
| last1 = Freedman | first1 = D. 
| title = From association to causation: Some remarks on the history of statistics 
| doi = 10.1214/ss/1009212409 
| journal = Statistical Science 
| volume = 14 
| issue = 3 
| pages = 243–258 
| year = 1999 
| doi-access = free 
}} ([http://www.stat.berkeley.edu/~census/521.pdf Revised version, 2002])
*{{Cite book |title=A History of Probability and Statistics and Their Applications before 1750 |last=Hald |first=Anders |author-link=Anders Hald |year=2003 |publisher=Wiley |location=Hoboken, NJ |isbn=978-0-471-47129-5 }}
*{{Cite book |title=A History of Mathematical Statistics from 1750 to 1930 |last=Hald |first=Anders |author-link=Anders Hald |year=1998 |publisher=Wiley |location=New York |isbn=0-471-17912-4 }}
* Kotz, S., Johnson, N.L. (1992,1992,1997). ''Breakthroughs in Statistics'', Vols I, II, III. Springer {{isbn|0-387-94037-5}}, {{isbn|0-387-94039-1}}, {{isbn|0-387-94989-5}}
*{{Cite book |title=The History of Statistics in the 17th and 18th Centuries against the changing background of intellectual, scientific and religious thought (Lectures by Karl Pearson given at University College London during the academic sessions 1921-1933) |last=Pearson |first=Egon |author-link=Egon Pearson |year=1978 |publisher=MacMillan Publishing Co., Inc. |location=New York |isbn=978-0-02-850120-8 |pages=744 }}
* [[David Salsburg|Salsburg, David]] (2001). ''[[The Lady Tasting Tea|The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century]]''. {{isbn|0-7167-4106-7}}
*{{Cite book
 | author = Stigler, Stephen M.
 | author-link = Stephen Stigler
 | year = 1986
 | title = The History of Statistics: The Measurement of Uncertainty before 1900
 | publisher = Belknap Press/Harvard University Press
 | isbn = 978-0-674-40341-3
}}
* Stigler, Stephen M. (1999) ''Statistics on the Table: The History of Statistical Concepts and Methods''. Harvard University Press. {{isbn|0-674-83601-4}}
* {{Cite journal | last1 = David | first1 = H. A. | title = First (?) Occurrence of Common Terms in Mathematical Statistics | doi = 10.2307/2684625 | jstor = 2684625 | journal = [[The American Statistician]]| volume = 49 | issue = 2 | pages = 121–133 | year = 1995 }}

== External links ==
{{Commons category|History of statistics}}
* [https://www.jehps.net/publications.htm JEHPS: Recent publications in the history of probability and statistics]
* [https://www.jehps.net/indexang.html Electronic Journ@l for History of Probability and Statistics/Journ@l Electronique d'Histoire des Probabilités et de la Statistique]
* [https://web.archive.org/web/20220316084055/http://www.economics.soton.ac.uk/staff/aldrich/Figures.htm Figures from the History of Probability and Statistics (Univ. of Southampton)]
* [https://www.york.ac.uk/depts/maths/histstat/ Materials for the History of Statistics (Univ. of York)]
* [https://web.archive.org/web/20220330113904/http://www.economics.soton.ac.uk/staff/aldrich/Probability%20Earliest%20Uses.htm Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)]
*[https://mathshistory.st-andrews.ac.uk/Miller/mathsym/stat/ Earliest Uses of Symbols in Probability and Statistics] on [https://mathshistory.st-andrews.ac.uk/Miller/mathsym/ Earliest Uses of Various Mathematical Symbols]

{{Statistics}}
{{History of science}}
{{History of mathematics}}

{{DEFAULTSORT:History Of Statistics}}
[[Category:History of probability and statistics| ]]
[[Category:History of science by discipline|Statistics]]