Statistics (from

population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using 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

_{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 power test, which tests for null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

.

null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

, two broad categories of error are recognized:
* Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
* Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".

null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

) 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 inferential statistics
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

it relies on sample size, and therefore under

''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

German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...

: ''Statistik
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...

'', "description of a state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...

, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...

. 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 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 galaxy) or a hypoth ...

or a statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...

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 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 cause-and-effect by demonstrating what outcome occurs when ...

.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press.
When census
A census is the procedure of systematically acquiring, recording and calculating information about the members of a given population. This term is used mostly in connection with national population and housing censuses; other common censuses incl ...

data cannot be collected, statistician
A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...

s collect data by developing specific experiment designs and survey samples. 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 or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...

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, social sciences, psychology and statistics, an observational study draws inferences from a sample (statistics), sample to a statistical population, population where the dependent and independent variables, independ ...

does not involve experimental manipulation.
Two main statistical methods are used in data analysis
Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enco ...

: 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 process of using and an ...

, which summarize data from a sample using indexes
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...

such as the mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...

or standard deviation
In statistics, the standard deviation is a measure of the amount of variation or 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 value) of the set, while ...

, and inferential statistics
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

, 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, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...

'' (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 variat ...

'' (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, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...

are made under the framework of 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 expressing it through a set o ...

, which deals with the analysis of random phenomena.
A standard statistical procedure involves the collection of data leading to a 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 or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...

to an idealized null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

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 fa ...

s (null hypothesis is falsely rejected giving a "false positive") and Type II 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 f ...

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, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...

), 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, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Mis ...

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
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...

, or as a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. 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 to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using 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 acquiring, recording and calculating information about the members of a given population. This term is used mostly in connection with national population and housing censuses; other common censuses incl ...

). This may be organized by governmental statistical institutes. ''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 process of using and an ...

'' can be used to summarize the population data. Numerical descriptors include mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...

and standard deviation
In statistics, the standard deviation is a measure of the amount of variation or 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 value) of the set, while ...

for continuous data
In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on all r ...

(like income), while frequency and percentage are more useful in terms of describing categorical data
In statistics, a categorical variable (also called qualitative variable) is a 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 ...

(like education).
When a census is not feasible, a chosen subset of the population called a sample
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 s ...

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 distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

'' 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 used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...

), estimating numerical characteristics of the data (estimation
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...

), describing associations within the data (correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

), and modeling relationships within the data (for example, using regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...

). Inference can extend to forecasting
Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...

, prediction, 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 know ...

and interpolation of time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

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 (a geographic location or geographic position).
It is also call ...

, and data mining.
Mathematical statistics

Mathematical statistics is the application ofmathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

to statistics. Mathematical techniques used for this include mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, 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 matrices.
...

, stochastic analysis
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

, differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

, and measure-theoretic 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 expressing it through a set o ...

.
History

Formal discussions on inference date back to Arab mathematicians andcryptographers
This is a list of cryptographers. Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries.
Pre twentieth century
* Al-Khalil ibn Ahmad al-Farahidi: wrote a (now lost) book ...

, during the Islamic Golden Age
The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...

between the 8th and 13th centuries. Al-Khalil
Hebron ( ar, الخليل or ; he, חֶבְרוֹן ) is a Palestinian. city in the southern West Bank, south of Jerusalem. Nestled in the Judaean Mountains, it lies above sea level. The second-largest city in the West Bank (after East J ...

(717–786) wrote the ''Book of Cryptographic Messages'', which contains one of the first uses of permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s and combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...

s, to list all possible Arabic
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...

words with and without vowels. Al-Kindi's ''Manuscript on Deciphering Cryptographic Messages'' gave a detailed description of how to use frequency analysis
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analysis is based on t ...

to decipher encrypted
In cryptography, encryption is the process of 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 deci ...

messages, providing an early example 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 ...

for decoding. Ibn Adlan
ʻAfīf al-Dīn ʻAlī ibn ʻAdlān al-Mawsilī ( ar, عفيف لدين علي بن عدلان الموصلي ; 1187–1268 CE), born in Mosul, was an Arab cryptologist, linguist and poet who is known for his early contributions to cryptanalysis ...

(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 Replication (statistics), replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make stat ...

in frequency analysis.
The earliest writing containing statistics in Europe dates back to 1663, with the publication of ''Natural and Political Observations upon the Bills of Mortality
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 li ...

'' 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 li ...

. 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 statistics developed from discussions concerning games of chance
A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...

among mathematicians such as Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...

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

, Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

, and Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...

. Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work of Juan Caramuel
Juan Caramuel y Lobkowitz (Juan Caramuel de Lobkowitz, 23 May 1606 in Madrid — 7 or 8 September 1682 in Vigevano) was a Spanish Catholic scholastic philosopher, ecclesiastic, mathematician and writer. He is believed to be a great-grandson of J ...

), 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 expressing it through a set o ...

as a mathematical discipline only took shape at the very end of the 17th century, particularly in Jacob Bernoulli's posthumous work ''Ars Conjectandi
(Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apa ...

''. This was the first book where the realm of games of chance and the realm of the probable (which concerned opinion, evidence, and argument) were combined and submitted to mathematical analysis. 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 res ...

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 name ...

in 1805, though Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

presumably made use of it a decade earlier in 1795.
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 FRAI (; 16 February 1822 – 17 January 1911), was an English Victorian era polymath: a statistician, sociologist, psychologist, anthropologist, tropical explorer, geographer, inventor, meteorologist, proto- ...

and Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing the concepts of standard deviation
In statistics, the standard deviation is a measure of the amount of variation or 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 value) of the set, while ...

, correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

, regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...

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, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...

, 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 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 biostatistics.
History
The Pearson system ...

, among many other things. Galton and Pearson founded '' Biometrika'' as the first journal of mathematical statistics and biostatistics
Biostatistics (also known as biometry) are the development and application of statistical methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experime ...

(then called biometry), and the latter founded the world's first university statistics department at University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public research university
, endowment = £143 million (2020)
, budget = ...

.
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 s ...

, and reached its culmination in the insights of Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...

, 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 #REDIRECT The Correlation between Relatives on the Supposition of Mendelian Inheritance {{R from other capitalisation ...

'' (which was the first to use the statistical term, variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

), his classic 1925 work '' Statistical Methods for Research Workers'' and his 1935 ''The Design of Experiments
''The Design of Experiments'' is a 1935 book by the English statistician Ronald Fisher about the design of experiments and is considered a foundational work in experimental design. Among other contributions, the book introduced the concept of th ...

'', 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 statistic An ancillary statistic is a measure of a sample whose distribution (or whose pmf or pdf) does not depend on the parameters of the model. An ancillary statistic is a pivotal quantity that is also a statistic. Ancillary statistics can be used to c ...

s, Fisher's linear discriminator and Fisher information
In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that model ...

. He also coined the term null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

during the Lady tasting tea
In the design of experiments in statistics, the lady tasting tea is a randomized experiment devised by Ronald Fisher and reported in his book ''The Design of Experiments'' (1935). The experiment is the original exposition of Fisher's notion of ...

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 statistician Ronald Fisher about the design of experiments and is considered a foundational work in experimental design. Among other contributions, the book introduced the concept of th ...

'' 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." In his 1930 book ''The Genetical Theory of Natural Selection
''The Genetical Theory of Natural Selection'' is a book by Ronald Fisher which combines Mendelian genetics with Charles Darwin's theory of natural selection, with Fisher being the first to argue that "Mendelism therefore validates Darwinism" and ...

'', he applied statistics to various biological concepts such as Fisher's principle
Fisher's principle is an evolutionary model that explains why the sex ratio of most species that produce offspring through sexual reproduction is approximately 1:1 between males and females. A. W. F. Edwards has remarked that it is "probably the mo ...

(which A. W. F. Edwards
Anthony William Fairbank Edwards, FRS (born 1935) is a British statistician, geneticist and evolutionary biologist. He is the son of the surgeon Harold C. Edwards, and brother of medical geneticist John H. Edwards. He has sometimes been called ...

called "probably the most celebrated argument in evolutionary biology
Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life fo ...

") and Fisherian runaway
Fisherian runaway or runaway selection is a sexual selection mechanism proposed by the mathematical biologist Ronald Fisher in the early 20th century, to account for the evolution of ostentatious male ornamentation by persistent, directional fem ...

,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 which combines Mendelian genetics with Charles Darwin's theory of natural selection, with Fisher being the first to argue that "Mendelism therefore validates Darwinism" and ...

. Edwards, A.W.F. (2000) Perspectives: Anecdotal, Historical 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 in which members of one biological sex mate choice, choose mates of the other sex to mating, mate with (intersexual selection), and compete with members of the same sex for access to members of t ...

about a positive feedback runaway effect found in evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...

.
The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson 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, Poland and then at University College ...

in the 1930s. They introduced the concepts of " Type II" error, power of a test
In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances o ...

and confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...

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 (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...

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
Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe Big data is the one associated with large body of information that we could not comprehend when used only in smaller am ...

.
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 that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...

s.
To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling 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 concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

. Probability is used in mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...

to study the sampling distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were s ...

s of sample 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 parameter, describing a sample, or evaluating a hypo ...

s 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, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...

, 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: experimental studies
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 cause-and-effect by demonstrating what outcome occurs when ...

and observational studies
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concern ...

. 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. 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 pro ...

s and observational studies
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concern ...

—for which a statistician would use a modified, more structured estimation method (e.g., Difference in differences estimation and instrumental variable
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered t ...

s, among many others) that produce consistent estimator
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...

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 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. Con ...

s, and randomized assignment
Random assignment or random placement is an experimental technique for assigning human participants or animal subjects to different groups in an experiment (e.g., a treatment group versus a control group) using randomization, such as by a cha ...

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
The Hawthorne effect is a type of reactivity in which individuals modify an aspect of their behavior in response to their awareness of being observed. The effect was discovered in the context of research conducted at the Hawthorne Western Electric ...

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 for most of its lifespan, it served as the primary equipment m ...

. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line
An assembly line is a manufacturing process (often called a ''progressive assembly'') in which parts (usually interchangeable parts) are added as the semi-finished assembly moves from workstation to workstation where the parts are added in seq ...

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, hypotheses are applied to experimental units in a treatment group.
In comparative experiments, members of a control group receive a standard treatment, a placebo, or no treatment at all. There may be more than one tr ...

and blindness
Visual impairment, also known as vision impairment, is a medical definition primarily measured based on an individual's better eye visual acuity; in the absence of treatment such as correctable eyewear, assistive devices, and medical treatment ...

. The Hawthorne effect
The Hawthorne effect is a type of reactivity in which individuals modify an aspect of their behavior in response to their awareness of being observed. The effect was discovered in the context of research conducted at the Hawthorne Western Electri ...

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 acohort study
A cohort study is a particular form of longitudinal study that samples a cohort (a group of people who share a defining characteristic, typically those who experienced a common event in a selected period, such as birth or graduation), performing ...

, 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. 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 withlongitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

and temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...

measurements in Celsius
The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The ...

or Fahrenheit
The Fahrenheit scale () is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he originally defined his ...

), 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 quantitative variables, which can be either discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...

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 ...

, due to their numerical nature. Such distinctions can often be loosely correlated with data type
In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler or interpreter how the programmer intends to use the data. Most progra ...

in computer science, in that dichotomous categorical variables may be represented with the Boolean data type
In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is name ...

, polytomous categorical variables with arbitrarily assigned integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s in the integral data type
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are ...

, and continuous variables with the real data type
A real data type is a data type used in a computer program to represent an approximation of a real number.
Because the real numbers are not countable, computers cannot represent them exactly using a finite amount of information.
Most often, a compu ...

involving floating-point arithmetic. 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 thecount noun
In linguistics, a count noun (also countable noun) is a noun that can be modified by a quantity and that occurs in both singular and plural forms, and that can co-occur with quantificational determiners like ''every'', ''each'', ''several'', ...

sense) is a summary statistic
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in
* a measure of ...

that quantitatively describes or summarizes features of a collection of information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...

, while descriptive statistics in the mass noun 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 distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

(or inductive statistics), in that descriptive statistics aims to summarize a sample
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 s ...

, rather than use the data to learn about the population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using 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 modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enco ...

to deduce properties of an underlying probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using 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), 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 so ...

from a larger population. Inferential statistics can be contrasted with 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 process of using and an ...

. 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 givenprobability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

: 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 that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...

defines a random sample
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians atte ...

as the random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...

given by the column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...

of these IID variables.Piazza Elio, Probabilità e Statistica, Esculapio 2007 The estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...

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 statistics computed from a Sample (statistics), sample of data on one or more random variables.
The sample mean is the average value (or mean, mean value) of a sample (statistic ...

, unbiased sample variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

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
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need ...

or pivot. Widely used pivots include the z-score
In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...

, the chi square statistic and Student's t-value
In statistics, the ''t''-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its Standard error (statistics), standard error. It is used in statistical hypothesis testing, hypothesis testin ...

.
Between two estimators of a given parameter, the one with lower mean squared error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...

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, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

is equal to the true value
In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exa ...

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'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...

to the true value of such parameter.
Other desirable properties for estimators include: UMVUE In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.
For pr ...

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
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter ''θ''0—having the property that as the number of data points used increases indefinitely, the result ...

s which converges in probability
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...

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 the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...

method, the least squares method and the more recent method of estimating equations
In statistics, the method of estimating equations is a way of specifying how the parameters of a statistical model should be estimated. This can be thought of as a generalisation of many classical methods—the method of moments, least squares, ...

.
=Null hypothesis and alternative hypothesis

= Interpretation of statistical information can often involve the development of anull hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

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, Htype II 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 f ...

s.
What statisticians
A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...

call an alternative hypothesis is simply a hypothesis that contradicts the =Error

= Working from aStandard deviation
In statistics, the standard deviation is a measure of the amount of variation or 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 value) of the set, while ...

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 or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...

refers to an estimate of difference between sample mean and population mean.
A statistical error
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its " true value" (not necessarily observable). The erro ...

is the amount by which an observation differs from its expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

. A 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
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...

is used for obtaining efficient estimators
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, needs fewer input data or observations than a less efficient one to achie ...

, a widely used class of estimators. Root mean square error
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...

is simply the square root of mean squared error.
Many statistical methods seek to minimize the residual sum of squares
In statistics, the residual sum of squares (RSS), also known as the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepan ...

, 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 residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based minimizing the ''sum o ...

. 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 mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

, which provides a handy property for doing regression. Least squares applied to linear regression is called ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...

method and least squares applied to nonlinear regression
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 data are fi ...

is called non-linear least squares
Non-linear least squares is the form of least squares analysis used to fit a set of ''m'' observations with a model that is non-linear in ''n'' unknown parameters (''m'' ≥ ''n''). It is used in some forms of nonlinear regression. The ...

. 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, polynomial regression is a form of regression analysis in which the relationship between the independent variable ''x'' and the dependent variable ''y'' is modelled as an ''n''th degree polynomial in ''x''. Polynomial regression ...

, 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 usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...

(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, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...

), 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, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Mis ...

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 frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...

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 based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or pro ...

perspective, such a claim does not even make sense, as the true value is not a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

. 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 (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

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 value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. T ...

from Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...

: this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability
Bayesian probability is an Probability interpretations, interpretation of the concept of probability, in which, instead of frequentist probability, frequency or propensity probability, propensity of some phenomenon, probability is interpreted as re ...

.
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. Acritical region
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...

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
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the ...

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 quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifi ...

. 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 (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the ...

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 (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume ...

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 frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...

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 of statistical reasoning involving a test for an occurrence, such as a DNA match. A positive result in the test may paradoxically be more likely to be an erroneous result than an actual occurrence, even i ...

: criticisms arise because the hypothesis testing approach forces one hypothesis (the 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, a ...

, although it requires establishing a prior probability
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...

.
* Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
* As everything in fat tails
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are someti ...

p-values may be seriously mis-computed.
=Examples

= Some well-known statisticaltests
Test(s), testing, or TEST may refer to:
* Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities
Arts and entertainment
* ''Test'' (2013 film), an American film
* ''Test'' (2014 film), ...

and procedures are:
Exploratory data analysis

Exploratory data analysis (EDA) is an approach toanalyzing
Analysis (plural, : analyses) 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 a ...

data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...

s to summarize their main characteristics, often with visual methods. A statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...

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, when used in a misleading fashion, can trick the casual observer into believing something other than what the data shows. That is, a misuse of statistics occurs when a statistical argument asserts a falsehood. In some cases, the misu ...

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 misused by finding ways to interpret only the data that are favorable to the presenter.Huff, Darrell (1954) ''How to Lie with Statistics
''How to Lie with Statistics'' is a book written by Darrell Huff in 1954 presenting an introduction to statistics for the general reader. Not a statistician, Huff was a journalist who wrote many "how to" articles as a freelancer.
The book is a ...

'', WW Norton & Company, Inc. New York. A mistrust and misunderstanding of statistics is associated with the quotation, " 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
''How to Lie with Statistics'' is a book written by Darrell Huff in 1954 presenting an introduction to statistics for the general reader. Not a statistician, Huff was a journalist who wrote many "how to" articles as a freelancer.
The book is a ...

'', by Darrell Huff
Darrell Huff (July 15, 1913 – June 27, 2001) was an United States, American writer, and is best known as the author of ''How to Lie with Statistics'' (1954), the best-selling statistics book of the second half of the twentieth century, and ...

, 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, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...

. 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, in Princeton, New Jersey, United States
* Image Analysis & Stereology, the official journal of the International Society for Stereology & Image Analysis.
* Iowa Archeological Society, Uni ...

.. 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, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

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. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...

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. Con ...

. 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,'' sometimes referred to as ''Statistical science,'' 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 distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

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 bond 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 computi ...

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 consultant
A methodological advisor or statistical consultant provides methodological and statistical advice and guidance to clients interested in making decisions regarding the design of studies, the collection and analysis of data, and the presentation a ...

s 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 a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...

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 ofacademic discipline
An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy ...

s, including natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...

and social science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soc ...

s, government, and business. Business statistics applies statistical methods in econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

, 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.” 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
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 statisticia ...

(ANOVA), Chi-Square Test
A chi-squared test (also chi-square or test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variabl ...

, Student’s T Test, Linear Regression, Pearson’s Correlation Coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the 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
In statistics, Spearman's rank correlation coefficient or Spearman's ''ρ'', named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation ( statistical dependence betwee ...

and Principal Component Analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

.
A typical statistics course covers descriptive statistics, probability, binomial and normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...

s, 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
Free statistical software is a practical alternative to commercial packages. Many of the free to use programs aim to be similar in function to commercial packages, in that they are general statistical packages that perform a variety of statistica ...

.
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 oflinear model
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term ...

s, but powerful computers, coupled with suitable numerical algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing c ...

, caused an increased interest in nonlinear models (such as neural networks
A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

) as well as the creation of new types, such as generalized linear model
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and b ...

s and multilevel model
Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parame ...

s.
Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation test
A permutation test (also called re-randomization test) is an exact statistical hypothesis test making use of the proof by contradiction.
A permutation test involves two or more samples. The null hypothesis is that all samples come from the same di ...

s and the bootstrap, while techniques such as Gibbs sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is dif ...

have made use of Bayesian model
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Ba ...

s more feasible. The computer revolution has implications for the future of statistics with a new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose statistical software are now available. Examples of available software capable of complex statistical computation include programs such as Mathematica, SAS, SPSS
SPSS Statistics is a statistical software suite developed by IBM for data management, advanced analytics, multivariate analysis, business intelligence, and criminal investigation. Long produced by SPSS Inc., it was acquired by IBM in 2009. C ...

, and R.
Business statistics

In business, "statistics" is a widely used management- anddecision support
A decision support system (DSS) is an Information systems, information system that supports business or organizational decision-making activities. DSSs serve the management, operations and planning levels of an organization (usually mid and hig ...

tool. It is particularly applied in financial management
Financial management is the business function concerned with profitability, expenses, cash and credit, so that the "organization may have the means to carry out its objective as satisfactorily as possible;"
the latter often defined as maximizin ...

, marketing management
Marketing management is the Organizational studies, organizational discipline which focuses on the practical application of marketing orientation, techniques and methods inside enterprises and organizations and on the management of a firm's mar ...

, and production
Production may refer to:
Economics and business
* Production (economics)
* Production, the act of manufacturing goods
* Production, in the outline of industrial organization, the act of making products (goods and services)
* Production as a stati ...

, services
Service may refer to:
Activities
* Administrative service, a required part of the workload of university faculty
* Civil service, the body of employees of a government
* Community service, volunteer service for the benefit of a community or a p ...

and operations management . Statistics is also heavily used in management accounting
In management accounting or managerial accounting, managers use accounting information in decision-making and to assist in the management and performance of their control functions.
Definition
One simple definition of management accounting is th ...

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.” Auditing ...

. The discipline of Management Science
Management science (or managerial science) is a wide and interdisciplinary study of solving complex problems and making strategic decisions as it pertains to institutions, corporations, governments and other types of organizational entities. It is ...

formalizes the use of statistics, and other mathematics, in business. (Econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

is the application of statistical methods to economic data
Economic data are data describing an actual economy, past or present. These are typically found in time-series form, that is, covering more than one time period (say the monthly unemployment rate for the last five years) or in cross-sectional data ...

in order to give empirical content to economic relationships.)
A typical "Business Statistics" course is intended for business majors, and covers 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 process of using and an ...

(collection
Collection or Collections may refer to:
* Cash collection, the function of an accounts receivable department
* Collection (church), money donated by the congregation during a church service
* Collection agency, agency to collect cash
* Collectio ...

, description, analysis, and summary of data), probability (typically the binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
* Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
...

and normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...

s), 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. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...

, time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

, decision trees
A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condit ...

, multiple linear regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is cal ...

, and other topics from business analytics
Business analytics (BA) refers to the skills, technologies, and practices for continuous iterative exploration and investigation of past business performance to gain insight and drive business planning. Business analytics focuses on developing ne ...

more generally. See also . Professional certification programs, such as the 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. * Innumber theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...

, scatter plot
A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. ...

s of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
* Predictive methods of statistics in forecasting
Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual ...

combining chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...

and fractal geometry
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

can be used to create video works.
* The process art of Jackson Pollock
Paul Jackson Pollock (; January 28, 1912August 11, 1956) was an American painter and a major figure in the abstract expressionist movement. He was widely noticed for his " drip technique" of pouring or splashing liquid household paint onto a hor ...

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
Performance art is an artwork or art exhibition created through actions executed by the artist or other participants. It may be witnessed live or through documentation, spontaneously developed or written, and is traditionally presented to a pu ...

, as in a card trick based on a Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

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
Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde ...

, 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 methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experime ...

, computational biology
Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has fo ...

, computational sociology
Computational sociology is a branch of sociology that uses computationally intensive methods to analyze and model social phenomena. Using computer simulations, artificial intelligence, complex statistical methods, and analytic approaches like soc ...

, network biology
A biological network is a method of representing systems as complex sets of binary interactions or relations between various biological entities. In general, networks or graphs are used to capture relationships between entities or objects. A typi ...

, social science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soc ...

, sociology
Sociology is a social science that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of Empirical ...

and social research
Social research is a research conducted by social scientists following a systematic plan. Social research methodologies can be classified as quantitative and qualitative.
* Quantitative designs approach social phenomena through quantifiable ...

. 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
Statistical process control (SPC) or statistical quality control (SQC) is the application of statistical methods to monitor and control the quality of a production process. This helps to ensure that the process operates efficiently, producing ...

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
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

'', vol. 321, no. 4 (October 2019), pp. 62–67. "The use of ''p'' values for nearly a century ince 1925 Ince may refer to:
*Ince, Cheshire, a village in Cheshire, UK
*Ince-in-Makerfield in the Metropolitan Borough of Wigan, UK
*Ince (UK Parliament constituency), a former constituency covering Ince-in-Makerfield
*Ince (ward), an electoral ward covering ...

to determine statistical significance 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 ...

al results has contributed to an illusion of certainty
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and o ...

and o reproducibility crises in many scientific fields
The branches of science, also referred to as sciences, scientific fields or scientific disciplines, are commonly divided into three major groups:
*Formal sciences: the study of formal systems, such as those under the branches of logic and ma ...

. There is growing determination to reform statistical analysis... Some esearcherssuggest 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
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

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