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Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of
data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...

data
. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a
statistical population In statistics, a population is a Set (mathematics), set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way ga ...
or a
statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system ...
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
surveys Survey may refer to: Statistics and human research * Statistical survey, a method for collecting quantitative information about items in a population * Survey (human research), including opinion polls Spatial measurement * Surveying, the techniqu ...
and
experiments An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When
census A census is the procedure of systematically calculating, acquiring and recording information Information is processed, organised and structured data Data (; ) are individual facts, statistics, or items of information, often numeric. In ...

census
data cannot be collected,
statistician A statistician is a person who works with Theory, theoretical or applied statistics. The profession exists in both the private sector, private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, a ...
s collect data by developing specific experiment designs and survey
samples Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population - Complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of somet ...
. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An
experimental study An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary ...
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 In fields such as epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined populations. It is a cornerstone of public hea ...
does not involve experimental manipulation. Two main statistical methods are used in
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modelling In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century Eng ...
:
descriptive statistics A descriptive statistic (in the count noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The trad ...
, which summarize data from a sample using
indexes Index may refer to: Arts, entertainment, and media Fictional entities * Index (A Certain Magical Index), Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo (megastr ...
such as the
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
or
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

standard deviation
, and
inferential statistics Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a ''distribution'' (sample or population): ''
central tendency In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mo ...
'' (or ''location'') seeks to characterize the distribution's central or typical value, while ''
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variation ...
'' (or ''variability'') characterizes the extent to which members of the distribution depart from its center and each other. Inferences on
mathematical statistics Mathematical statistics is the application of probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related ...
are made under the framework of
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
, which deals with the analysis of random phenomena. A standard statistical procedure involves the collection of data leading to 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 ''AlterNative: An International Journal of Indigenous Peoples'' (formerly ''AlterNative: An International Journal of Indigenous Scholarship'') is a quarterly peer-reviewed Peer review is the evaluation of work by one or more people with simi ...
to an idealized
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
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 In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference Statistical inference is the process of using data analysis Data analysis is a process of inspecting, Data cleansing, cleansing, Data tr ...
s (null hypothesis is falsely rejected giving a "false positive") and
Type II error In statistical hypothesis testing A statistical hypothesis is a hypothesis that is testable on the basis of observed data modelled as the realised values taken by a collection of random variables. A set of data is modelled as being realised ...
s (null hypothesis fails to be rejected and an actual relationship between populations is missed giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded Open-mindedness is receptiveness to new ideas. Open-mindedness relates to the way in which people approach the views and kn ...
), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of
missing data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
or censoring may result in biased estimates and specific techniques have been developed to address these problems.


Introduction

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of
data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...

data
, or as a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. 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. In applying statistics to a problem, it is common practice to start with a
population Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...
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 A census is the procedure of systematically calculating, acquiring and recording information Information is processed, organised and structured data Data (; ) are individual facts, statistics, or items of information, often numeric. In ...

census
). This may be organized by governmental statistical institutes. ''
Descriptive statistics A descriptive statistic (in the count noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The trad ...
'' can be used to summarize the population data. Numerical descriptors include
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

standard deviation
for continuous data (like income), while frequency and percentage are more useful in terms of describing
categorical data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
(like education). When a census is not feasible, a chosen subset of the population called a 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 An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...

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 Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
'' is 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 A statistical hypothesis test is a method of statistical inference Statistical inference is the process of using data analysis Data analysis is a process of inspecting, Data cleansing, cleansing, Data transformation, transforming, and Data ...
), estimating numerical characteristics of the data (
estimation Estimation (or estimating) is the process of finding an estimate, or approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived ...
), describing associations within the data (
correlation In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a m ...
), and modeling relationships within the data (for example, using
regression analysis In statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a u ...
). Inference can extend to
forecasting Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, ...
,
prediction A prediction (Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in ...
, and estimation of unobserved values either in or associated with the population being studied. It can include
extrapolation In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known ...

extrapolation
and
interpolation In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

interpolation
of
time series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

time series
or
spatial data Geographic data and information is defined in the ISO/TC 211 series of standards as data and information having an implicit or explicit association with a location relative to Earth Earth is the third planet from the Sun and the only astr ...
, and
data mining Data mining is a process of extracting and discovering patterns in large data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...
.


Mathematical statistics

Mathematical statistics is the application of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
to statistics. Mathematical techniques used for this include
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
,
stochastic analysis Stochastic calculus is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...
,
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
, and measure-theoretic probability theory.


History

The early writings on statistical inference date back to Arab mathematicians and cryptographers, during the
Islamic Golden Age The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam The history of Islam concerns the political, social, economic, and cultural developments of Muslim world, Islamic civilization. M ...
between the 8th and 13th centuries.
Al-Khalil Hebron ( ar, الخليل أو الخليل الرحمن ; he, חֶבְרוֹן ) is a State of Palestine, Palestinian. city in the southern West Bank, south of Jerusalem. Nestled in the Judaean Mountains, it lies 930 meters (3,050 ft) Abo ...
(717–786) wrote the ''Book of Cryptographic Messages'', which contains the first use of
permutations and combinations In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting Counting is the process of determining the number of Element (math ...
, to list all possible
Arabic Arabic (, ' or , ' or ) is a Semitic language The Semitic languages are a branch of the Afroasiatic language family originating in the Middle East The Middle East is a list of transcontinental countries, transcontinental region ...

Arabic
words with and without vowels. In his book, ''Manuscript on Deciphering Cryptographic Messages,'' Al-Kindi gave a detailed description of how to use
frequency analysis In cryptanalysis, frequency analysis (also known as counting letters) is the study of the letter frequencies, frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers. Frequency analy ...
to decipher
encrypted In cryptography, encryption is the process of Code, encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can ...

encrypted
messages. Al-Kindi also made the earliest known use of
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
, while he and later Arab cryptographers developed the early statistical methods for encrypted messages.
Ibn Adlan ʻAfīf al-Dīn ʻAlī ibn ʻAdlān al-Mawsilī ( ar, عفيف لدين علي بن عدلان الموصلي ; 1187–1268 CE), born in Mosul Nineveh - Mashki Gate Mosul ( ar, الموصل, al-Mawṣil, ku, Mûsil ,مووسڵ, syr, ܡܘ ...
(1187–1268) later made an important contribution, on the use of
sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample In statistics and quantitative research methodology, a sample is a set of individuals or objects collected or selected ...
in frequency analysis. 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 John Graunt (24 April 1620 – 18 April 1674) has been regarded as the founder of demography. Graunt was one of the first demographers, and perhaps the first epidemiologist, though by profession he was a haberdasher. He was bankrupted later in lif ...
. Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its ''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 mathematical foundations of modern statistics were laid in the 17th century with the development of the
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
by
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, , "having learned much"; ...

Gerolamo Cardano
,
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...

Blaise Pascal
and
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of suc ...

Pierre de Fermat
. Mathematical probability theory arose from the study of
games of chance A game of chance is a game with separate sliding drawer, from 1390 to 1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New York City) '', 1560, Pieter Bruegel the Elder File:Paul Cézanne, ...
, although the concept of probability was already examined in
medieval law In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
and by philosophers such as Juan Caramuel. The
method of least squares The method of least squares is a standard approach in regression analysis In ing, regression analysis is a set of statistical processes for the relationships between a (often called the 'outcome' or 'response' variable) and one or more s ...
was first described by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials In physical science and mathematic ...
in 1805. The modern field of statistics 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 Sir Francis Galton, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family R ...

Francis Galton
and
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...

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 In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

standard deviation
,
correlation In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a m ...

correlation
,
regression analysis In statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a u ...
and the application of these methods to the study of the variety of human characteristics—height, weight, eyelash length among others. Pearson developed the
Pearson product-moment correlation coefficient In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin w ...
, defined as a product-moment, the method of moments for the fitting of distributions to samples and the
Pearson distribution The Pearson distribution is a family of continuous probability distribution, continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostati ...
, among many other things. Galton and Pearson founded ''
Biometrika ''Biometrika'' is a peer-reviewed Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work ( peers). It functions as a form of self-regulation by qualified members of a profession with ...

Biometrika
'' as the first journal of mathematical statistics and
biostatistics Biostatistics (also known as biometry) are the development and application of statistical Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data are units of in ...
(then called biometry), and the latter founded the world's first university statistics department at
University College London University College London, which Trade name, operates as UCL, is a major public university , public research university located in London, United Kingdom. UCL is a Member institutions of the University of London, member institution of the Federa ...
.
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a subs ...
coined the term
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
during the
Lady tasting tea In the design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variat ...
experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation".OED quote: 1935 R.A. Fisher, ''
The Design of Experiments ''The Design of Experiments'' is a 1935 book by the English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval Englan ...
'' ii. 19, "We may speak of this hypothesis as the 'null hypothesis', and the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation."
The second wave of the 1910s and 20s was initiated by
William Sealy Gosset William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small samp ...

William Sealy Gosset
, and reached its culmination in the insights of
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a subs ...
, 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 "The Correlation between Relatives on the Supposition of Mendelian Inheritance" is a science, scientific paper by Ronald Fisher which was published in the ''Transactions of the Royal Society of Edinburgh'' in 1918, (volume 52, pages 399–433). ...
'' (which was the first to use the statistical term,
variance In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

variance
), his classic 1925 work ''
Statistical Methods for Research Workers ''Statistical Methods for Research Workers'' is a classic book on statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, ...
'' and his 1935 ''
The Design of Experiments ''The Design of Experiments'' is a 1935 book by the English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval Englan ...
'', where he developed rigorous
design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
models. He originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and
Fisher information In mathematical statistics Mathematical statistics is the application of probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number t ...
. In his 1930 book ''
The Genetical Theory of Natural Selection ''The Genetical Theory of Natural Selection'' is a book by Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British statistician, geneticist, and academic. For his work in statistics, he has been describ ...
'', he applied statistics to various
biological Biology is the natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowl ...

biological
concepts such as
Fisher's principleFisher's principle is an evolution Evolution is change in the Heredity, heritable Phenotypic trait, characteristics of biological populations over successive generations. These characteristics are the Gene expression, expressions of genes that ...
(which A. W. F. Edwards called "probably the most celebrated argument in
evolutionary biology Evolutionary biology is the subfield of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interacti ...
") and
Fisherian runaway Fisherian runaway or runaway selection is a sexual selection Sexual selection is a mode of natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype right ...
,Fisher, R.A. (1915) The evolution of sexual preference. Eugenics Review (7) 184:192Fisher, R.A. (1930)
The Genetical Theory of Natural Selection ''The Genetical Theory of Natural Selection'' is a book by Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British statistician, geneticist, and academic. For his work in statistics, he has been describ ...
.
Edwards, A.W.F. (2000) Perspectives: Anecdotal, Historial and Critical Commentaries on Genetics. The Genetics Society of America (154) 1419:1426Andersson, M. and Simmons, L.W. (2006) Sexual selection and mate choice. Trends, Ecology and Evolution (21) 296:302Gayon, J. (2010) Sexual selection: Another Darwinian process. Comptes Rendus Biologies (333) 134:144 a concept in
sexual selection Sexual selection is a mode of natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype right , Here the relation between genotype and phenotype is illustrat ...
about a positive feedback runaway effect found in
evolution Evolution is change in the heritable Heredity, also called inheritance or biological inheritance, is the passing on of Phenotypic trait, traits from parents to their offspring; either through asexual reproduction or sexual reproduction, ...

evolution
. The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between
Egon Pearson Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician A mathematician is someone who uses an ...
and
Jerzy Neyman Jerzy Neyman (April 16, 1894 – August 5, 1981; born Jerzy Spława-Neyman; ) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, ...

Jerzy Neyman
in the 1930s. They introduced the concepts of " Type II" error,
power of a test The statistical power of a binary hypothesis test A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test ...
and
confidence interval In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

confidence interval
s. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling. 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 A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

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 Big data is a field that treats ways to analyze, systematically extract information from, or otherwise deal with data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of informati ...

big data
.


Statistical data


Data collection


Sampling

When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through
statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system ...
s. To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative
sampling Sampling may refer to: *Sampling (signal processing), converting a continuous signal into a discrete signal *Sample (graphics), Sampling (graphics), converting continuous colors into discrete color components *Sampling (music), the reuse of a sound ...
assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures. There are also methods of experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Sampling theory is part of the mathematical discipline of
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
. Probability is used in
mathematical statistics Mathematical statistics is the application of probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related ...
to study the
sampling distribution In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
s of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction— inductively inferring from samples to the parameters of a larger or total population.


Experimental and observational studies

A common goal for a statistical research project is to investigate
causality Causality (also referred to as causation, or cause and effect) is influence by which one Event (relativity), event, process, state or object (a ''cause'') contributes to the production of another event, process, state or object (an ''effect'') ...
, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables. There are two major types of causal statistical studies: and
observational studies In fields such as epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics ...
. In both types of studies, the effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. 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
experimental manipulation An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary ...
. Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data—like
natural experiment A natural experiment is an empirical study in which individuals (or clusters of individuals) are exposed to the experimental and control conditions that are determined by nature or by other factors outside the control of the investigators. The proce ...
s and
observational studies In fields such as epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics ...
—for which a statistician would use a modified, more structured estimation method (e.g., Difference in differences estimation and
instrumental variable In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
s, among many others) that produce
consistent estimator Image:Consistency of estimator.svg, 250px, is a sequence of estimators for parameter ''θ''0, the true value of which is 4. This sequence is consistent: the estimators are getting more and more concentrated near the true value ''θ''0; at the same ...
s.


=Experiments

= The basic steps of a statistical experiment are: # Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability. Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects. #
Design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
, using
blocking Blocking may refer to: Science, technology, and mathematics Computing and telecommunications *Blacklist (computing) *Blocking (computing), holding up a task until an event occurs *Blocking probability, for calls in a telecommunications system *H ...
to reduce the influence of
confounding variable In statistics, a confounder (also confounding variable, confounding factor, extraneous determinant or lurking variable) is a variable that influences both the dependent variable and independent variable, causing a spurious association. Conf ...
s, and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the '' experimental protocol'' that will guide the performance of the experiment and which specifies the'' primary analysis'' of the experimental data. # Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol. # Further examining the data set in secondary analyses, to suggest new hypotheses for future study. # Documenting and presenting the results of the study. Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the
Western Electric Company The Western Electric Company was an American electrical engineering and manufacturing company officially founded in 1869. A wholly owned subsidiary of American Telephone & Telegraph, AT&T for most of its lifespan, it served as the primary equipmen ...
. The researchers were interested in determining whether increased illumination would increase the productivity of the workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a
control group In the design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation ...

control group
and
blindness Visual impairment, also known as vision impairment or vision loss, is a decreased ability to see to a degree that causes problems not fixable by usual means, such as glasses Glasses, also known as eyeglasses or spectacles, are vision eyew ...
. The
Hawthorne effect The Hawthorne effect refers to a type of reactivity in which individuals modify an aspect of their behavior in response to their awareness of being observed. Some scholars feel descriptions of this well-known and remarkable effect, which was disco ...
refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed.


=Observational study

= An example of an observational study is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a
cohort study A cohort study is a particular form of longitudinal study A longitudinal study (or longitudinal survey, or panel study) is a research design Research design refers to the overall strategy utilized to carry out research Research is "crea ...
, and then look for the number of cases of lung cancer in each group. A case-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.


Types of data

Various attempts have been made to produce a taxonomy of
levels of measurement Level or levels may refer to: Engineering *Level (instrument), a device used to measure true horizontal or relative heights *Canal pound or level *Regrading or levelling, the process of raising and/or lowering the levels of land *Storey or level, ...
. The psychophysicist
Stanley Smith Stevens Stanley Smith Stevens (November 4, 1906 – January 18, 1973) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of ...
defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

temperature
measurements in
Celsius The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...

Celsius
or
Fahrenheit The Fahrenheit scale ( or ) is a temperature scale Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to convenient and stable parameters, such a ...

Fahrenheit
), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation. Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as
categorical variable In statistics, a categorical variable is a variable (research), variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal catego ...
s, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either
discrete Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...

discrete
or
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, due to their numerical nature. Such distinctions can often be loosely correlated with
data type In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...
in computer science, in that dichotomous categorical variables may be represented with the
Boolean data type In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , an ...
, polytomous categorical variables with arbitrarily assigned
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s in the
integral data type In computer science, an integer is a data, datum of integral data type, a data type that represents some interval (mathematics), range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain n ...
, and continuous variables with the
real data type A real data type is a data type used in a computer program A computer program is a collection of instructions that can be executed by a computer to perform a specific task. A computer program is usually written by a computer programmer in a ...
involving
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
computation. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented. Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998), van den Berg (1991).) The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer."


Methods


Descriptive statistics

A descriptive statistic (in the
count noun In linguistics, a count noun (also countable noun) is a noun that can be modified by a Numeral (linguistics), numeral and that occurs in both grammatical number, singular and plural forms, and that can co-occur with quantificational determiners ...
sense) is a
summary statistic In descriptive statistics A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the ...
that quantitatively describes or summarizes features of a collection of
information Information is processed, organised and structured data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...

information
, while descriptive statistics in the
mass noun In linguistics Linguistics is the scientific study of language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic ...
sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from
inferential statistics Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
(or inductive statistics), in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the
population Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...
that the sample of data is thought to represent.


Inferential statistics

Statistical inference is the process of using
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modelling In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century Eng ...
to deduce properties of an underlying
probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a
population Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...
, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is
sampled Sample or samples may refer to: Base meaning * Sample (statistics) In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a sci ...
from a larger population. Inferential statistics can be contrasted with
descriptive statistics A descriptive statistic (in the count noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The trad ...
. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.


Terminology and theory of inferential statistics


=Statistics, estimators and pivotal quantities

= Consider independent identically distributed (IID) random variables with a given
probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
: standard
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
and
estimation theory Estimation theory is a branch of statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of inf ...
defines a
random sample In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin w ...
as the
random vector In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
given by the
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
of these IID variables.Piazza Elio, Probabilità e Statistica, Esculapio 2007 The
population Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...
being examined is described by a probability distribution that may have unknown parameters. A statistic is a random variable that is a function of the random sample, but . The probability distribution of the statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: an
estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguishe ...

estimator
is a statistic used to estimate such function. Commonly used estimators include
sample mean The sample mean (or "empirical mean") and the sample covariance are statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes incl ...
, unbiased
sample variance In probability theory Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ax ...
and sample covariance. A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution ''does not depend on the unknown parameter'' is called a
pivotal quantityIn statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...
or pivot. Widely used pivots include the
z-score In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a ...
, the and Student's t-value. Between two estimators of a given parameter, the one with lower
mean squared error In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
is said to be more efficient. Furthermore, an estimator is said to be
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
if its
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
to the true value of such parameter. Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and
consistent estimator Image:Consistency of estimator.svg, 250px, is a sequence of estimators for parameter ''θ''0, the true value of which is 4. This sequence is consistent: the estimators are getting more and more concentrated near the true value ''θ''0; at the same ...
s which
converges in probability In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressi ...
to the true value of such parameter. This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the method of moments, the
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating Estimation (or estimating) is the process of finding an estimate, or approximation An approximation is anything that is intentionally similar but not exactly equa ...
method, the
least squares The method of least squares is a standard approach in regression analysis In ing, regression analysis is a set of statistical processes for the relationships between a (often called the 'outcome' or 'response' variable) and one or more s ...
method and the more recent method of
estimating equationsIn statistics, the method of estimating equations is a way of specifying how the parameters of a statistical modelA statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, ...
.


=Null hypothesis and alternative hypothesis

= Interpretation of statistical information can often involve the development of a
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time. The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H0, asserts that the defendant is innocent, whereas the alternative hypothesis, H1, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H0 (status quo) stands in opposition to H1 and is maintained unless H1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H0" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily ''accept'' H0 but ''fails to reject'' H0. While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test, which tests for
type II error In statistical hypothesis testing A statistical hypothesis is a hypothesis that is testable on the basis of observed data modelled as the realised values taken by a collection of random variables. A set of data is modelled as being realised ...
s. What
statisticians A statistician is a person who works with theoretical or applied statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, ...
call an
alternative hypothesis In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypoth ...
is simply a hypothesis that contradicts the
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
.


=Error

= Working from a
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
, two broad categories of error are recognized: *
Type I errors Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * Typ ...
where the null hypothesis is falsely rejected, giving a "false positive". *
Type II errors In statistical hypothesis testing, a type I error is the rejection of a true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the non-rejection of a fals ...
where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".
Standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

Standard deviation
refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while
Standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpret ...
refers to an estimate of difference between sample mean and population mean. A
statistical error In statistics and mathematical optimization, optimization, errors and residuals are two closely related and easily confused measures of the deviation (statistics), deviation of an observed value of an element of a Sample (statistics), statistical s ...

statistical error
is the amount by which an observation differs from its
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
, a is the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).
Mean squared error In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
is used for obtaining efficient estimators, a widely used class of estimators. Root mean square error is simply the square root of mean squared error. Many statistical methods seek to minimize the
residual sum of squares In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
, and these are called " methods of least squares" in contrast to
Least absolute deviations Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute value (LAV), least absolute residual (LAR), sum of absolute deviations, or the ''L''1 norm condition, is a statistical optimality criterion In statistics, a ...
. The latter gives equal weight to small and big errors, while the former gives more weight to large errors. Residual sum of squares is also
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
, which provides a handy property for doing regression. Least squares applied to
linear regression In statistics, linear regression is a Linearity, linear approach for modelling the relationship between a Scalar (mathematics), scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of ...

linear regression
is called
ordinary least squares In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...
method and least squares applied to
nonlinear regression for details In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The d ...
is called
non-linear least squares Non-linear least squares is the form of least squares The method of least squares is a standard approach in regression analysis In ing, regression analysis is a set of statistical processes for the relationships between a (often called t ...
. Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in
polynomial least squares In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
, which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve. Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as
random In common parlance, randomness is the apparent or actual lack of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric p ...
(noise) or
systematic Systematic may refer to: * Something related to systematics or Taxonomy (biology), taxonomy, a sub-discipline of biology * Short for systematic error * Systematic (band), an American hard rock band * Systematic Paris-Region, a French business clust ...
(
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded Open-mindedness is receptiveness to new ideas. Open-mindedness relates to the way in which people approach the views and kn ...

bias
), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of
missing data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
or censoring may result in biased estimates and specific techniques have been developed to address these problems.


=Interval estimation

= Most studies only sample part of a population, so results don't fully represent the whole population. Any estimates obtained from the sample only approximate the population value.
Confidence intervals In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This does ''not'' imply that the probability that the true value is in the confidence interval is 95%. From the
frequentist Frequentist inference is a type of statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxfo ...
perspective, such a claim does not even make sense, as the true value is not a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
. Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a
credible interval In Bayesian statistics, a credible interval is an interval within which an unobserved parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''para'': "beside", "subsidiary"; and wikt:μέ ...
from
Bayesian statistics Bayesian statistics is a theory in the field of statistics based on the Bayesian probability, Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an Event (probability theory), event. The degree of belief m ...
: this approach depends on a different way of interpreting what is meant by "probability", that is as a
Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequentist probability, frequency or propensity probability, propensity of some phenomenon, probability is interpreted as reasonable expectation represe ...
. In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds.


=Significance

= Statistics rarely give a simple Yes/No type answer to the question under analysis. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the
p-value In null hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis In inferential statistics, the null hypo ...
). The standard approach is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (
statistical significance In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypothe ...
) and the probability of type II error is the probability that the estimator doesn't belong to the critical region given that the alternative hypothesis is true. The
statistical power The statistical power of a binary hypothesis test A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis tes ...
of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false. Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably. Although in principle the acceptable level of
statistical significance In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypothe ...
may be subject to debate, the
significance level In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis. More precisely, a study's defined significance level, denoted by \alpha, is the probability of the study r ...
is the largest p-value that allows the test to reject the null hypothesis. This test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the
test statistic A test statistic is a statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population Population t ...
. Therefore, the smaller the significance level, the lower the probability of committing type I error. Some problems are usually associated with this framework (See criticism of hypothesis testing): * A difference that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests to account for this. One response involves going beyond reporting only the
significance level In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis. More precisely, a study's defined significance level, denoted by \alpha, is the probability of the study r ...
to include the ''p''-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the
size Size in general is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term fo ...
or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to report
confidence interval In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mor ...

confidence interval
s. Although these are produced from the same calculations as those of hypothesis tests or ''p''-values, they describe both the size of the effect and the uncertainty surrounding it. * Fallacy of the transposed conditional, aka
prosecutor's fallacy The prosecutor's fallacy is a fallacy A fallacy is the use of Validity (logic), invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument. A fallacious argument may be deceptive by appearing to be better than it r ...
: criticisms arise because the hypothesis testing approach forces one hypothesis (the
null hypothesis In inferential statistics, the null hypothesis (often denoted ''H''0) is that there is no difference between two possibilities. The null hypothesis is that the observed difference is due to chance alone. Using statistical tests it is possible to ...
) 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 Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and e ...
, although it requires establishing a
prior probability In Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers to a range of concepts and approaches that are ultimately based on a degree-of-belief interpretation o ...
. * Rejecting the null hypothesis does not automatically prove the alternative hypothesis. * As everything in
inferential statistics Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
it relies on sample size, and therefore under
fat tails A fat-tailed distribution is a probability distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statist ...
p-values may be seriously mis-computed.


=Examples

= Some well-known statistical
tests Test(s), testing, or TEST may refer to: * Test (assessment) A test or examination (exam or evaluation) is an educational assessment Educational assessment or educational evaluation is the systematic process of documenting and using em ...
and procedures are:


Exploratory data analysis

Exploratory data analysis (EDA) is an approach to
analyzing Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Ari ...
data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantit ...
s to summarize their main characteristics, often with visual methods. A
statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system ...
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

Misuse of statistics Statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...
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. Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The
statistical significance In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypothe ...
of a trend in the data—which measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as
statistical literacyStatistical literacy is the ability to understand and reason with statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, in ...
. There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.Huff, Darrell (1954) '' How to Lie with Statistics'', WW Norton & Company, Inc. New York. A mistrust and misunderstanding of statistics is associated with the quotation, "". Misuse of statistics can be both inadvertent and intentional, and the book '' How to Lie with Statistics'' 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)). Ways to avoid misuse of statistics include using proper diagrams and avoiding
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded Open-mindedness is receptiveness to new ideas. Open-mindedness relates to the way in which people approach the views and kn ...
. Misuse can occur when conclusions are overgeneralized and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias. Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs. 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 represented. To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole. According to Huff, "The dependability of a sample can be destroyed by
ias IAS may refer to: Science * Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has serv ...

ias
.. allow yourself some degree of skepticism." To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case: * Who says so? (Does he/she have an axe to grind?) * How does he/she know? (Does he/she have the resources to know the facts?) * What's missing? (Does he/she give us a complete picture?) * Did someone change the subject? (Does he/she offer us the right answer to the wrong problem?) * Does it make sense? (Is his/her conclusion logical and consistent with what we already know?)


Misinterpretation: correlation

The concept of
correlation In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a m ...

correlation
is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a
data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantit ...
often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable or
confounding variable In statistics, a confounder (also confounding variable, confounding factor, extraneous determinant or lurking variable) is a variable that influences both the dependent variable and independent variable, causing a spurious association. Conf ...
. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.


Applications


Applied statistics, theoretical statistics and mathematical statistics

''Applied statistics'' comprises descriptive statistics and the application of inferential statistics. ''Theoretical statistics'' concerns the logical arguments underlying justification of approaches to
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
, as well as encompassing ''mathematical statistics''. Mathematical statistics includes not only the manipulation of
probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
s necessary for deriving results related to methods of estimation and inference, but also various aspects of
computational statistics Computational statistics, or statistical computing, is the interface between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific com ...
and the
design of experiments The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
. Statistical consultants can help organizations and companies that don't have in-house expertise relevant to their particular questions.


Machine learning and data mining

Machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

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 An academic discipline or academic field is a subdivision of knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. The usual test for a statement of fact is ...
s, including
natural Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...

natural
and
social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

social science
s, government, and business. Business statistics applies statistical methods in
econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Econ ...

econometrics
,
auditing An audit is an "independent examination of financial information of any entity, whether profit oriented or not, irrespective of its size or legal form when such an examination is conducted with a view to express an opinion thereon.” Auditin ...
and production and operations, including services improvement and marketing research. A study of two journals in tropical biology found that the 12 most frequent statistical tests are:
Analysis of Variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
(ANOVA),
Chi-Square Test A chi-squared test (also chi-square or test) is a statistical hypothesis test A statistical hypothesis is a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hy ...
, Student’s T Test,
Linear Regression In statistics, linear regression is a Linearity, linear approach for modelling the relationship between a Scalar (mathematics), scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of ...

Linear Regression
, Pearson’s Correlation Coefficient, Mann-Whitney U Test, Kruskal-Wallis Test, Shannon’s Diversity Index, Tukey's Test,
Cluster Analysis Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of ...
, Spearman’s Rank Correlation Test and Principal Component Analysis. A typical statistics course covers descriptive statistics, probability, binomial and normal distributions, test of hypotheses and confidence intervals,
linear regression In statistics, linear regression is a Linearity, linear approach for modelling the relationship between a Scalar (mathematics), scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of ...

linear regression
, and correlation. Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use of free statistics software.


Statistical computing

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 models, 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 models and multilevel models. 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 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 . Statistics is also heavily used in management accounting and
auditing An audit is an "independent examination of financial information of any entity, whether profit oriented or not, irrespective of its size or legal form when such an examination is conducted with a view to express an opinion thereon.” Auditin ...
. 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
descriptive statistics A descriptive statistic (in the count noun In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The trad ...
(Data collection, collection, description, analysis, and summary of data), probability (typically the binomial distribution, binomial and normal distributions), test of hypotheses and confidence intervals,
linear regression In statistics, linear regression is a Linearity, linear approach for modelling the relationship between a Scalar (mathematics), scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of ...

linear regression
, and correlation; (follow-on) courses may include
forecasting Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, ...
,
time series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

time series
, decision trees, multiple linear regression, and other topics from business analytics more generally. See also . 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. Initially derided by some mathematical purists, it is now considered essential methodology in certain areas. * In number theory, scatter plots 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 Forecasting is the process of making predictions based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, ...
combining chaos theory and fractal geometry can be used to create video works. * The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed. With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art. * 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.


Specialized disciplines

Statistical techniques are used in a wide range of types of scientific and social research, including:
biostatistics Biostatistics (also known as biometry) are the development and application of statistical Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data are units of in ...
, computational biology, computational sociology, network biology,
social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

social science
, sociology and social research. Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include: In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology: Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles, it is a key tool, and perhaps the only reliable tool.


See also

;Foundations and major areas of statistics


References


Further reading

* Lydia Denworth, "A Significant Problem: Standard scientific methods are under fire. Will anything change?", ''Scientific American'', vol. 321, no. 4 (October 2019), pp. 62–67. "The use of p value, ''p'' values for nearly a century [since 1925] to determine
statistical significance In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypothe ...
of
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...

experiment
al results has contributed to an illusion of certainty and [to] Replication crisis, reproducibility crises in many science, scientific fields. There is growing determination to reform statistical analysis... Some [researchers] suggest changing statistical methods, whereas others would do away with a threshold for defining "significant" results." (p. 63.) * *
''OpenIntro Statistics''
, 3rd edition by Diez, Barr, and Cetinkaya-Rundel * Stephen Jones, 2010
''Statistics in Psychology: Explanations without Equations''
Palgrave Macmillan. . * * *


External links

* (Electronic Version): TIBCO Software Inc. (2020)
Data Science Textbook


Developed by Rice University (Lead Developer), University of Houston Clear Lake, Tufts University, and National Science Foundation.
UCLA Statistical Computing Resources

Philosophy of Statistics
from the Stanford Encyclopedia of Philosophy {{Authority control Statistics, Data Formal sciences Information Mathematical and quantitative methods (economics) Research methods Arab inventions