Statistics

== Introduction ==
{{main|Outline of statistics}}

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of [[data]],<ref>Moses, Lincoln E. (1986) ''Think and Explain with Statistics'', Addison-Wesley, {{isbn|978-0-201-15619-5}}. pp. 1–3</ref> or as a branch of [[mathematics]].<ref>Hays, William Lee, (1973) ''Statistics for the Social Sciences'', Holt, Rinehart and Winston, p. xii, {{isbn|978-0-03-077945-9}}</ref> Some consider statistics to be a distinct [[mathematical sciences|mathematical science]] rather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally concerned with the use of data in the context of uncertainty and decision-making in the face of uncertainty.<ref>{{cite book |last=Moore |first=David |title=Statistics for the Twenty-First Century |publisher=The Mathematical Association of America |editor=F. Gordon |editor2=S. Gordon |location=Washington, DC |year=1992 |pages=[https://archive.org/details/statisticsfortwe0000unse/page/14 14–25] |chapter=Teaching Statistics as a Respectable Subject |isbn=978-0-88385-078-7 |chapter-url=https://archive.org/details/statisticsfortwe0000unse/page/14 }}
</ref><ref>{{cite book |last=Chance |first=Beth L. |author1-link=Beth Chance |author2=Rossman, Allan J. |title=Investigating Statistical Concepts, Applications, and Methods |publisher=Duxbury Press |year=2005 |chapter=Preface |isbn=978-0-495-05064-3 |chapter-url=http://www.rossmanchance.com/iscam/preface.pdf |access-date=2009-12-06 |archive-date=2020-11-22 |archive-url=https://web.archive.org/web/20201122092901/http://www.rossmanchance.com/iscam/preface.pdf |url-status=live }}</ref>

In applying statistics to a problem, it is common practice to start with a [[statistical population|population]] or process to be studied. Populations can be diverse topics, such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population (an operation called a [[census]]). This may be organized by governmental statistical institutes. ''[[Descriptive statistics]]'' can be used to summarize the population data. Numerical descriptors include [[mean]] and [[standard deviation]] for [[Continuous probability distribution|continuous data]] (like income), while frequency and percentage are more useful in terms of describing [[categorical data]] (like education).

When a census is not feasible, a chosen subset of the population called a [[sampling (statistics)|sample]] is studied. Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or [[experiment]]al setting. Again, descriptive statistics can be used to summarize the sample data. However, drawing the sample contains an element of randomness; hence, the numerical descriptors from the sample are also prone to uncertainty. To draw meaningful conclusions about the entire population, ''[[inferential statistics]]'' are needed. It uses patterns in the sample data to draw inferences about the population represented while accounting for randomness. These inferences may take the form of answering yes/no questions about the data ([[hypothesis testing]]), estimating numerical characteristics of the data ([[Estimation theory|estimation]]), describing [[Association (statistics)|associations]] within the data ([[correlation and dependence|correlation]]), and modeling relationships within the data (for example, using [[regression analysis]]). Inference can extend to the [[forecasting]], [[prediction]], and estimation of unobserved values either in or associated with the population being studied. It can include [[extrapolation]] and [[interpolation]] of [[time series]] or [[spatial statistics|spatial data]], as well as [[data mining]].

===Mathematical statistics===
{{main|Mathematical statistics}}

Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include [[mathematical analysis]], [[linear algebra]], [[stochastic analysis]], [[differential equations]], and [[measure-theoretic probability theory]].<ref>{{cite book|last1=Lakshmikantham|first1=D. |last2=Kannan|first2= V.|title=Handbook of stochastic analysis and applications|date=2002|publisher=M. Dekker|location=New York|isbn=0824706609}}</ref><ref>{{cite book|last=Schervish|first=Mark J.|title=Theory of statistics|date=1995|publisher=Springer|location=New York|isbn=0387945466|edition=Corr. 2nd print.}}</ref>