Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of

_{0}, asserts that the defendant is innocent, whereas the alternative hypothesis, H_{1}, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H_{0} (status quo) stands in opposition to H_{1} and is maintained unless H_{1} is supported by evidence "beyond a reasonable doubt". However, "failure to reject H_{0}" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily ''accept'' H_{0} but ''fails to reject'' H_{0}. While one can not "prove" a null hypothesis, one can test how close it is to being true with a Statistical power, power test, which tests for type II errors.
What statisticians call an alternative hypothesis 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 ...

.

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 and type II errors#Type I error, Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
* Type I and type II errors#Type II error, Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".
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 (statistics)#Standard error of the mean, Standard error refers to an estimate of difference between sample mean and population mean.
A Errors and residuals in statistics#Introduction, statistical error is the amount by which an observation differs from its expected value, a Errors and residuals in statistics#Introduction, residual 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 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, and these are called "least squares, methods of least squares" in contrast to Least absolute deviations. 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 function, differentiable, which provides a handy property for doing regression analysis, regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. 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, 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 error, random (noise) or Systematic error, systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of

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, although it requires establishing a prior probability.
* 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 p-values may be seriously mis-computed.

''OpenIntro Statistics''

, 3rd edition by Diez, Barr, and Cetinkaya-Rundel * Stephen Jones, 2010

''Statistics in Psychology: Explanations without Equations''

Palgrave Macmillan. . * * *

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

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 ...

. 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
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that on ...

.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 ...

data cannot be collected, statistician
A statistician is a person who works with theoretical
A theory is a rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of th ...

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 determinants
In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics), ...

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 ...

, 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. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wit ...

'' (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 type I error is the mistaken rejection of an actually 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 mist ...

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. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

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 ofdata
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 ...

, 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 ...

). 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 ...

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. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

(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 ...

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 is a hypothesis
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method
The scientific method is an Empirical evidence ...

), 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 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 (often called 'predictors', 'covariates', 'explanatory variables' or 'features' ...

). 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
Estimation (or estimating) is the process of finding an estimate, or approximatio ...

, prediction
Image:Old Farmer's Almanac 1793 cover.jpg, frame, ''The Old Farmer's Almanac'' is famous in the US for its (not necessarily accurate) long-range weather predictions.
A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecas ...

, 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 ...

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 ...

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 ...

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 ofmathematics
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 theIslamic 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 ...

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 ...

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
Demography (from prefix ''demo-'' from Ancient Greek δῆμος (''dēmos'') meaning 'the people', and ''-graphy'' from γράφω (''graphō'') m ...

. 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 (also Girolamo or Geronimo) Cardano (; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)– 21 September 1576 (O. S.)) was an Italian polymath
A polymath ( el, πολυμαθής, ', "having learn ...

, Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic Church, Catholic theologian.
He was a child prodigy who was educated by his father, a tax collector i ...

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 ...

. 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 to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the resid ...

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 and Legendre transformation are named a ...

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 Re ...

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 ...

, 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 ...

, 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 ...

, 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 (often called 'predictors', 'covariates', 'explanatory variables' or 'features' ...

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 ...

'' 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 variatio ...

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 ...

, 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 ...

), 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 300px, Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
Mathematical statistics is the application of probability theory
Probability theory is the br ...

. 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 ...

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, ...

.
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 and, like his father Karl Pearson
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician
A mathematician is some ...

and Jerzy Neyman
Jerzy Neyman (April 16, 1894 – August 5, 1981; born Jerzy Spława-Neyman; ) was a Polish mathematician and statistician who spent the first part of his professional career at various institutions in Warsaw
Warsaw ( ; pl, Warszawa ; see ...

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. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

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 ...

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 ...

.
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 throughstatistical 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. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

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 investigatecausality
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 variableIn 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 ...

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 researchers were interested in determining whether increased illumination would increase the productivity of the assembly line 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 and double-blind, blindness. The Hawthorne effect 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, 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 level of measurement, levels of measurement. The psychophysicist Stanley Smith Stevens 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 and temperature measurements in Celsius or 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 variables, whereas ratio and interval measurements are grouped together as Variable (mathematics)#Applied statistics, quantitative variables, which can be either Probability distribution#Discrete probability distribution, discrete or Probability distribution#Continuous probability distribution, continuous, due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the integer (computer science), integral data type, and continuous variables with the real data type involving floating point 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 sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished frominferential 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 (statistics), 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 usingdata 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.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a Statistical population, population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is Sampling (statistics), sampled 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, independent identically distributed (IID) random variables with a given probability distribution: standardstatistical 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 defines a random sample as the random vector given by the column vector of these IID variables.Piazza Elio, Probabilità e Statistica, Esculapio 2007 The Statistical population, population 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 is a statistic used to estimate such function. Commonly used estimators include sample mean, unbiased sample variance 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 quantity or pivot. Widely used pivots include the z-score, the Chi-squared distribution#Applications, chi square statistic and Student's Student's t-distribution#How the t-distribution arises, t-value.
Between two estimators of a given parameter, the one with lower mean squared error is said to be more Efficient estimator, efficient. Furthermore, an estimator is said to be Unbiased estimator, unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the Limit (mathematics), limit 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 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 (statistics), method of moments, the maximum likelihood method, the least squares method and the more recent method of estimating equations.
=Null hypothesis and alternative hypothesis

= Interpretation of statistical information can often involve the development of anull 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, H=Error

= Working from amissing data
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 ...

or censoring may result in bias (statistics), 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 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 inference, frequentist perspective, such a claim does not even make sense, as the true value is not a random variable. 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 variables. 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 from Bayesian statistics: this approach depends on a different way of Probability interpretations, interpreting what is meant by "probability", that is as a Bayesian probability. 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). The standard approach is to test a null hypothesis against an alternative hypothesis. A Critical region#Definition of terms, 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) 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 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 may be subject to debate, the significance level 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. Therefore, the smaller the significance level, the lower the probability of committing type I error. Some problems are usually associated with this framework (See Statistical hypothesis testing#Criticism, 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 to include the p-value, ''p''-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the effect size, size 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 reportconfidence interval
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 ...

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: criticisms arise because the hypothesis testing approach forces one hypothesis (the =Examples

= Some well-known statistical Statistical hypothesis testing, tests and procedures are:Exploratory data analysis

Exploratory data analysis (EDA) is an approach to data analysis, analyzing data sets to summarize their main characteristics, often with visual methods. Astatistical 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 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 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 literacy. There is a general perception that statistical knowledge is all-too-frequently intentionally Misuse of statistics, misused by finding ways to interpret only the data that are favorable to the presenter.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, "Lies, damned lies, and statistics, There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional, and the book ''How to Lie with Statistics'' 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 avoidingbias
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 Hasty generalization, 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 Sampling (statistics), 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 [bias]... 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 ofcorrelation
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 ...

is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set 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 tostatistical 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 distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics 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 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 disciplines, including Natural science, natural and social sciences, government, and business. Business statistics applies statistical methods in econometrics, auditing 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 (ANOVA), Chi-Square Test, Student’s T Test, Linear Regression, Pearson’s Correlation Coefficient, Mann-Whitney U Test, Kruskal-Wallis Test, Shannon’s Diversity Index, Tukey's range test, Tukey's Test, Cluster Analysis, 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, 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. 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 coversdescriptive 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, 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
Estimation (or estimating) is the process of finding an estimate, or approximatio ...

, 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 ...

, 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 inforecasting
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
Estimation (or estimating) is the process of finding an estimate, or approximatio ...

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, 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 statisticsReferences

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 ofexperiment
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 ...

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