Difference between revisions of "Statistics"

{{short description|Study of the collection, analysis, interpretation, and presentation of data}}

{{other uses|Statistics (disambiguation)}}

[[File:Scatterplot.jpg|thumb|upright=1.3|right|[[Scatter plot]]s are used in descriptive statistics to show the observed relationships between different variables.]]

 

<!--PLEASE DO NOT EDIT THE OPENING SENTENCE WITHOUT FIRST PROPOSING YOUR CHANGE AT THE TALK PAGE.-->'''Statistics''' is a branch of [[mathematics]] working with [[data]] collection, organization, analysis, interpretation and presentation.<ref name=Dodge>Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. {{isbn|0-19-920613-9}}</ref><ref>{{cite encyclopedia |first=Jan-Willem |last=Romijn |year=2014 |title=Philosophy of statistics |encyclopedia=Stanford Encyclopedia of Philosophy |url=http://plato.stanford.edu/entries/statistics/}}</ref> In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a [[statistical population]] or a [[statistical model]] to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of [[statistical survey|surveys]] and [[experimental design|experiments]].<ref name=Dodge/>

See [[glossary of probability and statistics]].

Two main statistical methods are used in [[data analysis]]: [[descriptive statistics]], which summarize data from a sample using [[Index (statistics)|indexes]] such as the [[mean]] or [[standard deviation]], and [[statistical inference|inferential statistics]], which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).<ref name=LundResearchLtd>{{cite web|last=Lund Research Ltd. |url=https://statistics.laerd.com/statistical-guides/descriptive-inferential-statistics.php |title=Descriptive and Inferential Statistics |publisher=statistics.laerd.com |accessdate=2014-03-23}}</ref> Descriptive statistics are most often concerned with two sets of properties of a ''distribution'' (sample or population): ''[[central tendency]]'' (or ''location'') seeks to characterize the distribution's central or typical value, while ''[[statistical dispersion|dispersion]]'' (or ''variability'') characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of [[probability theory]], which deals with the analysis of random phenomena.

A standard statistical procedure involves the [[statistical hypothesis testing|test of the relationship]] between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an [[alternative hypothesis|alternative]] to an idealized [[null hypothesis]] of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: [[Type I error]]s (null hypothesis is falsely rejected giving a "false positive") and [[Type II error]]s (null hypothesis fails to be rejected and an actual difference between populations is missed giving a "false negative").<ref>{{Cite web|title = What Is the Difference Between Type I and Type II Hypothesis Testing Errors?|url = http://statistics.about.com/od/Inferential-Statistics/a/Type-I-And-Type-II-Errors.htm|website = About.com Education|accessdate = 2015-11-27}}</ref> Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. {{Citation needed|date=April 2015}}

Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from [[calculus]] and [[probability theory]]. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.<ref>{{cite web|url=https://www.answers.org.za/index.php/tutorials|title= How to Calculate Descriptive Statistics|publisher=Answers Consulting|date=2018-02-03}}</ref>

 

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 science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is 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=14–25 |chapter=Teaching Statistics as a Respectable Subject |isbn=978-0-88385-078-7}}

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 [[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]] types (like income), while frequency and percentage are more useful in terms of describing [[categorical data]] (like education).

The earliest writing on statistics was found in a 9th-century book entitled ''Manuscript on Deciphering Cryptographic Messages'', written by Arab scholar [[Al-Kindi]] (801–873). In his book, Al-Kindi gave a detailed description of how to use statistics and [[frequency analysis]] to decipher [[encrypted]] messages. This text laid the foundations for statistics and [[cryptanalysis]].<ref name=sim2000>{{cite book|last=Singh|first=Simon|authorlink=Simon Singh|title=The code book : the science of secrecy from ancient Egypt to quantum cryptography|year=2000|publisher=Anchor Books|location=New York|isbn=978-0-385-49532-5|edition=1st Anchor Books|title-link=The code book : the science of secrecy from ancient Egypt to quantum cryptography}}</ref><ref name=ibr1992>Ibrahim A. Al-Kadi "The origins of cryptology: The Arab contributions", ''[[Cryptologia]]'', 16(2) (April 1992) pp. 97&ndash;126.</ref> Al-Kindi also made the earliest known use of [[statistical inference]], while he and other Arab cryptologists developed the early statistical methods for [[Code|decoding]] encrypted messages. [[Mathematics in medieval Islam|Arab mathematicians]] including Al-Kindi, [[Al-Khalil ibn Ahmad al-Farahidi|Al-Khalil]] (717–786) and [[Ibn Adlan]] (1187–1268) used forms of [[probability and statistics]], with one of Ibn Adlan's most important contributions being on [[sample size]] for use of frequency analysis.<ref name="LB">{{cite journal|last=Broemeling|first=Lyle D.|title=An Account of Early Statistical Inference in Arab Cryptology|journal=The American Statistician|date=1 November 2011|volume=65|issue=4|pages=255–257|doi=10.1198/tas.2011.10191}}</ref>

The earliest European writing on statistics dates back to 1663, with the publication of ''Natural and Political Observations upon the Bills of Mortality'' by [[John Graunt]].<ref>Willcox, Walter (1938) "The Founder of Statistics". ''Review of the [[International Statistical Institute]]'' 5(4): 321–328. {{jstor|1400906}}</ref> Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its [[History of statistics#Etymology|''stat-'' etymology]]. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences.

The modern field of statistics emerged in the late 19th and early 20th century in three stages.<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}}</ref> 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. Galton's contributions included introducing the concepts of [[standard deviation]], [[correlation]], [[regression analysis]] and the application of these methods to the study of the variety of human characteristics—height, weight, eyelash length among others.<ref name=Galton1877>{{cite journal | last1 = Galton | first1 = F | year = 1877 | title = Typical laws of heredity | url = | journal = Nature | volume = 15 | issue = 388| pages = 492–553 | doi=10.1038/015492a0| bibcode = 1877Natur..15..492. }}</ref> Pearson developed the [[Pearson product-moment 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 | url = | journal = Statistical Science | volume = 4 | issue = 2| pages = 73–79 }}</ref> the [[Method of moments (statistics)|method of moments]] for the fitting of distributions to samples and the [[Pearson distribution]], among many other things.<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> Galton and Pearson founded ''[[Biometrika]]'' as the first journal of mathematical statistics and [[biostatistics]] (then called biometry), and the latter founded the world's first university statistics department at [[University College London]].<ref>{{cite web|year=|title=Karl Pearson (1857–1936)|publisher=Department of Statistical Science&nbsp;– [[University College London]]|url=http://www.ucl.ac.uk/stats/department/pearson.html|deadurl=yes|archiveurl=https://web.archive.org/web/20080925065418/http://www.ucl.ac.uk/stats/department/pearson.html|archivedate=2008-09-25|df=}}</ref>

The second wave of the 1910s and 20s was initiated by [[William Sealy Gosset]], and reached its culmination in the insights of [[Ronald Fisher]], who wrote the textbooks that were to define the academic discipline in universities around the world. Fisher's most important publications were his 1918 seminal paper ''[[The Correlation between Relatives on the Supposition of Mendelian Inheritance]]'', which was the first to use the statistical term, [[variance]], his classic 1925 work ''[[Statistical Methods for Research Workers]]'' and his 1935 ''[[The Design of Experiments]]'',<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}}</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 | author = Yates, F | 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

|jstor=1161806 |doi=10.3102/00028312003003223}}</ref> where he developed rigorous [[design of experiments]] models. He originated the concepts of [[sufficiency (statistics)|sufficiency]], [[ancillary statistic]]s, [[linear discriminant analysis|Fisher's linear discriminator]] and [[Fisher information]].<ref>{{cite journal|last=Agresti|first=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}}</ref> In his 1930 book ''[[The Genetical Theory of Natural Selection]]'' he applied statistics to various [[biology|biological]] concepts such as [[Fisher's principle]]<ref name=Edwards98/>). Nevertheless, [[A.W.F. Edwards]] has remarked that it is "probably the most celebrated argument in [[evolutionary biology]]".<ref name=Edwards98>{{cite journal | doi = 10.1086/286141 | last1 = Edwards | first1 = A.W.F. | year = 1998 | title = Natural Selection and the Sex Ratio: Fisher's Sources | url = | journal = American Naturalist | volume = 151 | issue = 6| pages = 564–569 | pmid = 18811377 }}</ref> (about the [[sex ratio]]), the [[Fisherian runaway]],<ref name ="fisher15">Fisher, R.A. (1915) The evolution of sexual preference. Eugenics Review (7) 184:192</ref><ref name="fisher30">Fisher, R.A. (1930) [[The Genetical Theory of Natural Selection]]. {{isbn|0-19-850440-3}}</ref><ref name="pers00">Edwards, A.W.F. (2000) Perspectives: Anecdotal, Historial and Critical Commentaries on Genetics. The Genetics Society of America (154) 1419:1426</ref><ref name="ander94">Andersson, M. (1994) Sexual selection. {{isbn|0-691-00057-3}}</ref><ref name="ander06">Andersson, M. and Simmons, L.W. (2006) Sexual selection and mate choice. Trends, Ecology and Evolution (21) 296:302</ref><ref name="gayon10">Gayon, J. (2010) Sexual selection: Another Darwinian process. Comptes Rendus Biologies (333) 134:144</ref> a concept in [[sexual selection]] about a positive feedback runaway affect found in [[evolution]].

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. The use of modern [[computer]]s has expedited large-scale statistical computations, and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on the problem of how to analyze [[Big data]].<ref>{{cite web|url=http://www.santafe.edu/news/item/sfnm-wood-big-data/|title=Science in a Complex World – Big Data: Opportunity or Threat?|work=Santa Fe Institute}}</ref>

When full census data cannot be collected, statisticians collect sample data by developing specific [[design of experiments|experiment designs]] and [[survey sampling|survey samples]]. Statistics itself also provides tools for prediction and forecasting through [[statistical model]]s. The idea of making inferences based on sampled data began around the mid-1600s in connection with estimating populations and developing precursors of life insurance.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=1082|isbn=1-57955-008-8}}</ref>

An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve [[Scientific control|experimental manipulation]]. Instead, data are gathered and correlations between predictors and response are investigated.

While the tools of data analysis work best on data from [[Randomized controlled trial|randomized studies]], they are also applied to other kinds of data—like [[natural experiment]]s and [[Observational study|observational studies]]<ref>[[David A. Freedman (statistician)|Freedman, D.A.]] (2005) ''Statistical Models: Theory and Practice'', Cambridge University Press. {{isbn|978-0-521-67105-7}}</ref>—for which a statistician would use a modified, more structured estimation method (e.g., [[Difference in differences|Difference in differences estimation]] and [[instrumental variable]]s, among many others) that produce [[consistent estimator]]s.

Interpretation of statistical information can often involve the development of a [[null hypothesis]] which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time.<ref>{{cite book | last = Everitt | first = Brian | title = The Cambridge Dictionary of Statistics | publisher = Cambridge University Press | location = Cambridge, UK New York | year = 1998 | isbn = 0521593468 }}</ref><ref>{{cite web|url=http://www.yourstatsguru.com/epar/rp-reviewed/cohen1994/|title=Cohen (1994) The Earth Is Round (p < .05) |publisher=YourStatsGuru.com }}</ref>

* Fallacy of the transposed conditional, aka [[prosecutor's fallacy]]: criticisms arise because the hypothesis testing approach forces one hypothesis (the [[null hypothesis]]) to be favored, since what is being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered by [[Bayesian inference]], although it requires establishing a [[prior probability]].<ref name=Ioannidis2005>{{Cite journal | last1 = Ioannidis | first1 = J.P.A. | authorlink1 = John P.A. Ioannidis| title = Why Most Published Research Findings Are False | journal = PLoS Medicine | volume = 2 | issue = 8 | pages = e124 | year = 2005 | pmid = 16060722 | pmc = 1182327 | doi = 10.1371/journal.pmed.0020124}}</ref>

* [[Spearman's rank correlation coefficient]]In statistics, exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.

[[Misuse of statistics]] can produce subtle, but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.

There is a general perception that statistical knowledge is all-too-frequently intentionally [[Misuse of statistics|misused]] by finding ways to interpret only the data that are favorable to the presenter.<ref name=Huff>Huff, Darrell (1954) ''[[How to Lie with Statistics]]'',  WW Norton & Company, Inc. New York.  {{isbn|0-393-31072-8}}</ref> A mistrust and misunderstanding of statistics is associated with the quotation, "[[Lies, damned lies, and statistics|There are three kinds of lies: lies, damned lies, and statistics]]". Misuse of statistics can be both inadvertent and intentional, and the book ''[[How to Lie with Statistics]]''<ref name=Huff/> outlines a range of considerations. In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques used in particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter (2012)).<ref>{{cite journal | last1 = Warne | first1 = R. Lazo | last2 = Ramos | first2 = T. | last3 = Ritter | first3 = N. | year = 2012 | title = Statistical Methods Used in Gifted Education Journals, 2006–2010 | url = | journal = Gifted Child Quarterly | volume = 56 | issue = 3| pages = 134–149 | doi = 10.1177/0016986212444122 }}</ref>

Ways to avoid misuse of statistics include using proper diagrams and avoiding [[Bias (statistics)|bias]].<ref name="Statistics in Archaeology">{{cite book | chapter = Statistics in archaeology | pages = 2093–2100 | first1 = Robert D. | last1 = Drennan | title =  Encyclopedia of Archaeology | year = 2008 | publisher = Elsevier Inc. | editor-first = Deborah M. | editor-last = Pearsall | isbn = 978-0-12-373962-9 }}</ref> Misuse can occur when conclusions are [[Hasty generalization|overgeneralized]] and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias.<ref name="Misuse of Statistics">{{cite journal |last=Cohen |first=Jerome B. |title=Misuse of Statistics |journal=Journal of the American Statistical Association |date=December 1938 |volume=33 |issue=204 |pages=657–674 |location=JSTOR |doi=10.1080/01621459.1938.10502344}}</ref> Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs.<ref name="Statistics in Archaeology" /> Unfortunately, most people do not look for bias or errors, so they are not noticed. Thus, people may often believe that something is true even if it is not well [[Sampling (statistics)|represented]].<ref name="Misuse of Statistics" /> To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.<ref name="Modern Elementary Statistics">{{cite journal|last=Freund|first=J.E.|authorlink = John E. Freund|title=Modern Elementary Statistics|journal=Credo Reference|year=1988}}</ref> According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."<ref>{{cite book|last=Huff|first=Darrell|title=How to Lie with Statistics|year=1954|publisher=Norton|location=New York|author2=Irving Geis |quote=The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism.}}</ref>

Machine Learning models are statistical and probabilistic models that captures patterns in the data through use of computational algorithms.

Statistics is applicable to a wide variety of [[academic discipline]]s, including [[natural]] and [[social science]]s, government, and business. [[Statistical consultant]]s can help organizations and companies that don't have in-house expertise relevant to their particular questions.

Increased computing power has also led to the growing popularity of computationally intensive methods based on [[Resampling (statistics)|resampling]], such as permutation tests and the [[Bootstrapping (statistics)|bootstrap]], while techniques such as [[Gibbs sampling]] have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with 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]].

Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences.{{citation needed|date=September 2018}} This tradition has changed with 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.

Statistical techniques are used in a wide range of types of scientific and social research, including: [[biostatistics]], [[computational biology]], [[computational sociology]], [[network biology]], [[social science]], [[sociology]] and [[social research]]. Some fields of inquiry use applied statistics so extensively that they have [[specialized terminology]]. These disciplines include:

{{Columns-list|colwidth=30em|

* [[Actuarial science]] (assesses risk in the insurance and finance industries)

* [[Statistical mechanics]]

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* Cohen, J. (1990). [http://moityca.com.br/pdfs/Cohen_1990.pdf "Things I have learned (so far)"]. ''American Psychologist'', 45, 1304–1312.

* Gigerenzer, G. (2004). "Mindless statistics". ''Journal of Socio-Economics'', 33, 587–606. {{doi|10.1016/j.socec.2004.09.033}}

* Ioannidis, J.P.A. (2005). "Why most published research findings are false". ''PLoS Medicine'', 2, 696–701. {{doi|10.1371/journal.pmed.0040168}}

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