Statistics

==Applications==
===Applied statistics, theoretical statistics and mathematical statistics===
''Applied statistics,'' sometimes referred to as ''Statistical science,''<ref>{{Cite journal|last=Nelder|first=John A.|date=1999|title=From Statistics to Statistical Science|url=https://www.jstor.org/stable/2681191|journal=Journal of the Royal Statistical Society. Series D (The Statistician)|volume=48|issue=2|pages=257–269|doi=10.1111/1467-9884.00187|jstor=2681191|issn=0039-0526|access-date=2022-01-15|archive-date=2022-01-15|archive-url=https://web.archive.org/web/20220115160959/https://www.jstor.org/stable/2681191|url-status=live}}</ref> comprises descriptive statistics and the application of inferential statistics.<ref>Nikoletseas, M.M. (2014) "Statistics: Concepts and Examples." {{isbn|978-1500815684}}</ref><ref>Anderson, D.R.; Sweeney, D.J.; Williams, T.A. (1994) ''Introduction to Statistics: Concepts and Applications'', pp. 5–9. West Group. {{isbn|978-0-314-03309-3}}</ref> ''Theoretical statistics'' concerns the logical arguments underlying justification of approaches to [[statistical inference]], as well as encompassing ''mathematical statistics''. Mathematical statistics includes not only the manipulation of [[probability distribution]]s necessary for deriving results related to methods of estimation and inference, but also various aspects of [[computational statistics]] and the [[design of experiments]].

[[Statistical consultant]]s can help organizations and companies that do not have in-house expertise relevant to their particular questions.

===Machine learning and data mining===
[[Machine learning]] models are statistical and probabilistic models that capture patterns in the data through use of computational algorithms.

===Statistics in academia===
Statistics is applicable to a wide variety of [[academic discipline]]s, including [[Natural science|natural]] and [[social science]]s, government, and business. Business statistics applies statistical methods in [[econometrics]], [[auditing]] and production and operations, including services improvement and marketing research.<ref>{{cite web|url=https://amstat.tandfonline.com/loi/jbes|title=Journal of Business & Economic Statistics|website=Journal of Business & Economic Statistics|publisher=Taylor & Francis|access-date=16 March 2020|archive-date=27 July 2020|archive-url=https://web.archive.org/web/20200727052958/https://amstat.tandfonline.com/loi/jbes|url-status=live}}</ref> A study of two journals in tropical biology found that the 12 most frequent statistical tests are: [[analysis of variance]] (ANOVA), [[chi-squared test]], [[Student's t-test]], [[linear regression]], [[Pearson's correlation coefficient]], [[Mann-Whitney U test]], [[Kruskal-Wallis test]], [[Diversity index#Shannon index|Shannon's diversity index]], [[Tukey's range test]], [[cluster analysis]], [[Spearman's rank correlation coefficient]] and [[principal component analysis]].<ref name=":0">{{Cite journal|last=Natalia Loaiza Velásquez, María Isabel González Lutz & Julián Monge-Nájera|date=2011|title=Which statistics should tropical biologists learn?|url=https://investiga.uned.ac.cr/ecologiaurbana/wp-content/uploads/sites/30/2017/09/JMN-2011-statistics-should-learn.pdf|journal=Revista Biología Tropical|volume=59|pages=983–992|access-date=2020-04-26|archive-date=2020-10-19|archive-url=https://web.archive.org/web/20201019160957/https://investiga.uned.ac.cr/ecologiaurbana/wp-content/uploads/sites/30/2017/09/JMN-2011-statistics-should-learn.pdf|url-status=live}}</ref>

A typical statistics course covers descriptive statistics, probability, binomial and [[normal distribution]]s, test of hypotheses and confidence intervals, [[linear regression]], and correlation.<ref>{{cite book|last=Pekoz|first=Erol|title=The Manager's Guide to Statistics|date=2009|publisher=Erol Pekoz|isbn=978-0979570438}}</ref> Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use of [[free statistics software]].<ref name=":0" />

===Statistical computing===

[[File:Gretl screenshot.png|thumb|upright=1.15|right|[[gretl]], an example of an [[List of open source statistical packages|open source statistical package]]]]

{{main|Computational statistics}}

The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models were almost always from the class of [[linear model]]s, but powerful computers, coupled with suitable numerical [[algorithms]], caused an increased interest in [[Nonlinear regression|nonlinear models]] (such as [[Artificial neural network|neural networks]]) as well as the creation of new types, such as [[generalized linear model]]s and [[multilevel model]]s.

Increased computing power has also led to the growing popularity of computationally intensive methods based on [[Resampling (statistics)|resampling]], such as [[permutation test]]s and the [[Bootstrapping (statistics)|bootstrap]], while techniques such as [[Gibbs sampling]] have made use of [[Bayesian model]]s more feasible. The computer revolution has implications for the future of statistics with a new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose [[List of statistical packages|statistical software]] are now available. Examples of available software capable of complex statistical computation include programs such as [[Mathematica]], [[SAS (software)|SAS]], [[SPSS]], and [[R (programming language)|R]].

===Business statistics===
In business, "statistics" is a widely used [[Management#Nature of work|management-]] and [[decision support]] tool. It is particularly applied in [[financial management]], [[marketing management]], and [[Manufacturing process management|production]], [[operations management for services|services]] and [[operations management]] .<ref>{{cite web |url=https://amstat.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=ubes20 |title=Aims and scope |website=Journal of Business & Economic Statistics |publisher=Taylor & Francis |access-date=16 March 2020 |archive-date=23 June 2021 |archive-url=https://web.archive.org/web/20210623194835/https://amstat.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=ubes20 |url-status=live }}</ref><ref>{{cite web |url=https://amstat.tandfonline.com/loi/jbes |title=Journal of Business & Economic Statistics |website=Journal of Business & Economic Statistics |publisher=Taylor & Francis |access-date=16 March 2020 |archive-date=27 July 2020 |archive-url=https://web.archive.org/web/20200727052958/https://amstat.tandfonline.com/loi/jbes |url-status=live }}</ref> Statistics is also heavily used in [[management accounting]] and [[auditing]]. The discipline of [[Management Science]] formalizes the use of statistics, and other mathematics, in business. ([[Econometrics]] is the application of statistical methods to [[economic data]] in order to give empirical content to [[economic theory|economic relationships]].)

A typical "Business Statistics" course is intended for [[Business education#Undergraduate education|business majors]], and covers <ref>Numerous texts are available, reflecting the scope and reach of the discipline in the business world:
* Sharpe, N. (2014). ''Business Statistics'', Pearson. {{ISBN|978-0134705217}}
* Wegner, T. (2010). ''Applied Business Statistics: Methods and Excel-Based Applications,'' Juta Academic. {{ISBN|0702172863}}
Two [[open textbook]]s are:
* Holmes, L., Illowsky, B., Dean, S. (2017).  [https://open.umn.edu/opentextbooks/textbooks/509 ''Introductory Business Statistics''] {{Webarchive|url=https://web.archive.org/web/20210616084059/https://open.umn.edu/opentextbooks/textbooks/509 |date=2021-06-16 }}
* Nica, M. (2013). [https://open.umn.edu/opentextbooks/textbooks/384 ''Principles of Business Statistics''] {{Webarchive|url=https://web.archive.org/web/20210518151122/https://open.umn.edu/opentextbooks/textbooks/384 |date=2021-05-18 }} </ref> [[descriptive statistics]] ([[Data collection|collection]], description, analysis, and summary of data), probability (typically the [[binomial distribution|binomial]] and [[normal distribution]]s), test of hypotheses and confidence intervals, [[linear regression]], and correlation; (follow-on) courses may include [[forecasting]], [[time series]], [[decision trees]], [[multiple linear regression]], and other topics from [[business analytics]] more generally. See also {{sectionlink|Business mathematics#University level}}. [[Professional certification in financial services|Professional certification programs]], such as the [[Chartered Financial Analyst|CFA]], often include topics in statistics.

===Statistics applied to mathematics or the arts===
Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This tradition has changed with the use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically.{{according to whom|date=April 2014}} Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.
* In [[number theory]], [[scatter plot]]s of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
* Predictive methods of statistics in [[forecasting]] combining [[chaos theory]] and [[fractal geometry]] can be used to create video works.<ref>{{Cite book|last=Cline|first=Graysen|url=https://www.worldcat.org/oclc/1132348139|title=Nonparametric Statistical Methods Using R|date=2019|publisher=EDTECH|isbn=978-1-83947-325-8|oclc=1132348139|access-date=2021-09-16|archive-date=2022-05-15|archive-url=https://web.archive.org/web/20220515012840/https://www.worldcat.org/title/nonparametric-statistical-methods-using-r/oclc/1132348139|url-status=live}}</ref>
* The [[process art]] of [[Jackson Pollock]] relied on artistic experiments whereby underlying distributions in nature were artistically revealed.<ref>{{Cite journal|last1=Palacios|first1=Bernardo|last2=Rosario|first2=Alfonso|last3=Wilhelmus|first3=Monica M.|last4=Zetina|first4=Sandra|last5=Zenit|first5=Roberto|date=2019-10-30|title=Pollock avoided hydrodynamic instabilities to paint with his dripping technique|journal=PLOS ONE|language=en|volume=14|issue=10|pages=e0223706|doi=10.1371/journal.pone.0223706|issn=1932-6203|pmc=6821064|pmid=31665191|bibcode=2019PLoSO..1423706P|doi-access=free}}</ref> With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art.{{Citation needed|date=March 2013}}
* Methods of statistics may be used predicatively in [[performance art]], as in a card trick based on a [[Markov process]] that only works some of the time, the occasion of which can be predicted using statistical methodology.
* Statistics can be used to predicatively create art, as in the statistical or [[stochastic music]] invented by [[Iannis Xenakis]], where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tunable using statistics.