**Statistics** is the study of the collection, organization, analysis, and interpretation of

dataThe term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

. It deals with all aspects of this, including the planning of data collection in terms of the design of

surveySurvey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....

s and experiments.

A

statisticianA statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...

is someone who is particularly well versed in the ways of thinking necessary for the successful application of statistical analysis. Such people have often gained this experience through working in any of a

wide number of fields. There is also a discipline called

*mathematical statistics*Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

that studies statistics mathematically.

The word

*statistics*, when referring to the scientific discipline, is singular, as in "Statistics is an art." This should not be confused with the word

*statistic*, referring to a quantity (such as

meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

or

medianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

) calculated from a set of data, whose plural is

*statistics* ("this statistic seems wrong" or "these statistics are misleading").

## Scope

Some consider statistics to be a mathematical body of science pertaining to the collection, analysis, interpretation or explanation, and presentation of

dataThe term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

, while others consider it a branch of

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

concerned with collecting and interpreting data. Because of its empirical roots and its focus on applications, statistics is usually considered to be a distinct mathematical science rather than a branch of mathematics.

Statisticians improve the quality of data with the

design of experimentsIn general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

and

survey samplingIn statistics, survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey.A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to...

. Statistics also provides tools for prediction and forecasting using data and

statistical modelA statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...

s. Statistics is applicable to a wide variety of

academic disciplineAn academic discipline, or field of study, is a branch of knowledge that is taught and researched at the college or university level. Disciplines are defined , and recognized by the academic journals in which research is published, and the learned societies and academic departments or faculties to...

s, including

naturalNatural is an adjective that refers to Nature.Natural may refer too:In science and mathematics:* Natural transformation, category theory in mathematics* Natural foods...

and social sciences, government, and business.

Statistical consultantA statistical consultant provides statistical advice andguidance to clients interested in making decisions through theanalysis or collection of data. Clients often need statistical advice to answer questions in business, medicine, biology, genetics, forestry, agriculture, fisheries, wildlife...

s are available to provide help for organizations and companies without direct access to expertise relevant to their particular problems.

Statistical methods can be used to summarize or describe a collection of data; this is called

*descriptive statistics*Descriptive statistics quantitatively describe the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population that the data are...

. This is useful in research, when communicating the results of experiments. In addition, patterns in the data may be

modeledA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

in a way that accounts for

randomnessRandomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

and uncertainty in the observations, and are then used to draw inferences about the process or population being studied; this is called

*inferential statistics*. Inference is a vital element of scientific advance, since it provides a prediction (based in data) for where a theory logically leads.

To prove the guiding theory further, these predictions are tested as well, as part of the

scientific methodScientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

. If the inference holds true, then the descriptive statistics of the new data increase the soundness of that hypothesis. Descriptive statistics and inferential statistics (a.k.a., predictive statistics) together comprise

*applied statistics*.

Statistics is closely related to

probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, with which it is often grouped; the difference is roughly that in probability theory, one starts from the given parameters of a total population to

deduceDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

probabilities pertaining to samples, but statistical inference moves in the opposite direction,

inductive inferenceInductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

from samples to the parameters of a larger or total population.

## History

The earliest writing on statistics was found in a 9th century book entitled: "Manuscript on Deciphering Cryptographic Messages", written by

Al-Kindi' , known as "the Philosopher of the Arabs", was a Muslim Arab philosopher, mathematician, physician, and musician. Al-Kindi was the first of the Muslim peripatetic philosophers, and is unanimously hailed as the "father of Islamic or Arabic philosophy" for his synthesis, adaptation and promotion...

(801–873 AC). In his book,

Al-Kindi' , known as "the Philosopher of the Arabs", was a Muslim Arab philosopher, mathematician, physician, and musician. Al-Kindi was the first of the Muslim peripatetic philosophers, and is unanimously hailed as the "father of Islamic or Arabic philosophy" for his synthesis, adaptation and promotion...

gave a detailed description of how to use

statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

and

frequency analysisIn cryptanalysis, frequency analysis 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....

to decipher encrypted messages, this was the birth of both statistics and cryptanalysis.

Some scholars pinpoint the origin of statistics to 1663, with the publication of

*Natural and Political Observations upon the Bills of Mortality* by

John GrauntJohn Graunt was one of the first demographers, though by profession he was a haberdasher. Born in London, the eldest of seven or eight children of Henry and Mary Graunt. His father was a draper who had moved to London from Hampshire...

. 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 the natural and social sciences.

Its mathematical foundations were laid in the 17th century with the development of

probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

by

Blaise Pascal Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

and

Pierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

. Probability theory arose from the study of games of chance. The method of least squares was first described by

Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

around 1794. The use of modern

computerA computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

s has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually.

## Overview

In applying statistics to a scientific, industrial, or societal problem, it is necessary to begin with a

populationA statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal". A population can also be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of "population" constitutes what is called a

time seriesIn statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

.

For practical reasons, a chosen subset of the population called a

sampleIn statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....

is studied — as opposed to compiling data about the entire group (an operation called

censusA census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official count of a particular population. The term is used mostly in connection with national population and housing censuses; other common...

). Once a sample that is representative of the population is determined, data are collected for the sample members in an observational or

experimentAn experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

al setting. These data can then be subjected to statistical analysis, serving two related purposes: description and inference.

- Descriptive statistics
Descriptive statistics quantitatively describe the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population that the data are...

summarize the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

and standard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

for continuous data types (like heights or weights), while frequency and percentage are more useful in terms of describing categorical dataIn statistics, categorical data is that part of an observed dataset that consists of categorical variables, or for data that has been converted into that form, for example as grouped data...

(like race).
- Inferential statistics uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. These inferences may take the form of: answering yes/no questions about the data (hypothesis testing), estimating numerical characteristics of the data (estimation
Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...

), describing associationsIn statistics, an association is any relationship between two measured quantities that renders them statistically dependent. The term "association" refers broadly to any such relationship, whereas the narrower term "correlation" refers to a linear relationship between two quantities.There are many...

within the data (correlation) and modeling relationships within the data (for example, using regression analysisIn statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

). Inference can extend to forecastingForecasting is the process of making statements about events whose actual outcomes have not yet been observed. A commonplace example might be estimation for some variable of interest at some specified future date. Prediction is a similar, but more general term...

, predictionA prediction or forecast is a statement about the way things will happen in the future, often but not always based on experience or knowledge...

and estimation of unobserved values either in or associated with the population being studied; it can include extrapolationIn mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...

and interpolationIn the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

of time series or spatial data, and can also include data miningData mining , a relatively young and interdisciplinary field of computer science is the process of discovering new patterns from large data sets involving methods at the intersection of artificial intelligence, machine learning, statistics and database systems...

.

The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a

data setA data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...

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. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables. (See

Correlation does not imply causation"Correlation does not imply causation" is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other "Correlation does not imply causation" (related to "ignoring a common cause" and questionable cause) is a...

.)

For a sample to be used as a guide to an entire population, it is important that it is truly a representative of that overall population. Representative sampling assures that the inferences and conclusions can be safely extended from the sample to the population as a whole. A major problem lies in determining the extent to which the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending 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. Statisticians describe stronger methods as more "robust".

Randomness is studied using the

mathematical disciplineMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

of

probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. Probability is used in "

mathematical statisticsMathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

" (alternatively, "

statistical theoryThe theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that...

") to study the

sampling distributionIn statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample. Sampling distributions are important in statistics because they provide a major simplification on the route to statistical inference...

s of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.

Misuse of statisticsA misuse of statistics occurs when a statistical argument asserts a falsehood. In some cases, the misuse may be accidental. In others, it is purposeful and for the gain of the perpetrator. When the statistical reason involved is false or misapplied, this constitutes a statistical fallacy.The false...

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. See below for further discussion.

Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The

statistical significanceIn statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....

of a trend in the data — which measures the extent to which a trend could be caused by random variation in the sample — may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as

statistical literacyStatistical literacy is a term used to describe an individual's or group's ability to understand statistics. Statistical literacy is necessary for citizens to understand material presented in publications such as newspapers, television, and the Internet. Numeracy is a prerequisite to being...

.

### Experimental and observational studies

A common goal for a statistical research project is to investigate

causalityCausality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....

, and in particular to draw a conclusion on the effect of changes in the values of predictors or

independent variableThe terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

s on dependent variables or response. There are two major types of causal statistical studies: experimental studies and

observational studiesIn epidemiology and statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator...

. 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.

#### 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
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

, using blockingIn the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups that are similar to one another. For example, an experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male...

to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased estimatesIn statistics, bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. Otherwise the estimator is said to be biased.In ordinary English, the term bias is...

of treatment effects and experimental error. At this stage, the experimenters and statisticians write the *experimental protocol*In the natural sciences a protocol is a predefined written procedural method in the design and implementation of experiments. Protocols are written whenever it is desirable to standardize a laboratory method to ensure successful replication of results by others in the same laboratory or by other...

that shall guide the performance of the experiment and that specifies the* primary analysis* of the experimental data.
- Performing the experiment following the experimental protocol
In the natural sciences a protocol is a predefined written procedural method in the design and implementation of experiments. Protocols are written whenever it is desirable to standardize a laboratory method to ensure successful replication of results by others in the same laboratory or by other...

and analyzing the dataIn statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...

following the experimental protocol.
- Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
- Documenting and presenting the results of the study.

Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the

assembly lineAn assembly line is a manufacturing process in which parts are added to a product in a sequential manner using optimally planned logistics to create a finished product much faster than with handcrafting-type methods...

workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and

blindnessA blind or blinded experiment is a scientific experiment where some of the people involved are prevented from knowing certain information that might lead to conscious or subconscious bias on their part, invalidating the results....

. The

Hawthorne effectThe Hawthorne effect is a form of reactivity whereby subjects improve or modify an aspect of their behavior being experimentally measured simply in response to the fact that they know they are being studied, not in response to any particular experimental manipulation.The term was coined in 1950 by...

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 correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a case-control study, and then look for the number of cases of lung cancer in each group.

### Levels of measurement

There are four main

levels of measurementThe "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...

used in statistics: nominal, ordinal, interval, and ratio. Each of these

have different degrees of usefulness in statistical

researchResearch can be defined as the scientific search for knowledge, or as any systematic investigation, to establish novel facts, solve new or existing problems, prove new ideas, or develop new theories, usually using a scientific method...

. Ratio measurements have both a meaningful zero value and the distances between different measurements defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with

longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

and temperature measurements in

CelsiusCelsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

or

FahrenheitFahrenheit is the temperature scale proposed in 1724 by, and named after, the German physicist Daniel Gabriel Fahrenheit . Within this scale, the freezing of water into ice is defined at 32 degrees, while the boiling point of water is defined to be 212 degrees...

). Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values. Nominal measurements have no meaningful rank order among values.

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 or continuous, due to their numerical nature.

#### Null hypothesis

Interpretation of statistical information can often involve the development of a

null hypothesisThe practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

in that the assumption is that whatever is proposed as a cause has no effect on the variable being measured.

The best illustration for a novice is the predicament encountered by a jury trial. The null hypothesis, H

_{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 testThe power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...

, which tests for type II errors.

#### Error

Working from a

null hypothesisThe practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

two basic forms 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".

Error also refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean. Many statistical methods seek to minimize the mean-squared error, and these are called "

methods of least squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

."

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as

randomRandom errors are errors in measurement that lead to measurable values being inconsistent when repeated measures of a constant attribute or quantity are taken...

(noise) or

systematicSystematic errors are biases in measurement which lead to the situation where the mean of many separate measurements differs significantly from the actual value of the measured attribute. All measurements are prone to systematic errors, often of several different types...

(

biasBias is an inclination to present or hold a partial perspective at the expense of alternatives. Bias can come in many forms.-In judgement and decision making:...

), but other important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important.

#### Interval estimation

Most studies will only sample part of a population and so the results are not fully representative of the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value 95% of the time. This does

*not* imply that the probability that the true value is in the confidence interval is 95%. From the

frequentistFrequentist inference is one of a number of possible ways of formulating generally applicable schemes for making statistical inferences: that is, for drawing conclusions from statistical samples. An alternative name is frequentist statistics...

perspective, such a claim does not even make sense, as the true value is not a

random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

. 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 the confidence interval will be constructed, 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 variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

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 intervalIn Bayesian statistics, a credible interval is an interval in the domain of a posterior probability distribution used for interval estimation. The generalisation to multivariate problems is the credible region...

from

Bayesian statisticsBayesian statistics is that subset of the entire field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief or, more specifically, Bayesian probabilities...

: this approach depends on a different way of

interpreting what is meant by "probability"The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we...

, that is as a

Bayesian probabilityBayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

.

#### Significance

Statistics rarely give a simple Yes/No type answer to the question asked of them. Interpretation often comes down to the level of statistical significance applied to the numbers and often refer to the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the

p-valueIn statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the p-value is less than the significance level α ,...

).

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 will be unlikely to help the patient in a noticeable way.

### Examples

Some well-known statistical

testsA statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study . In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold...

and

proceduresA procedure is a sequence of actions or operations which have to be executed in the same manner in order to always obtain the same result under the same circumstances ....

are:

## Specialized disciplines

Statistical techniques are used in a wide range of types of scientific and social research, including:

biostatisticsBiostatistics is the application of statistics to a wide range of topics in biology...

,

computational biologyComputational biology involves the development and application of data-analytical and theoretical methods, mathematical modeling and computational simulation techniques to the study of biological, behavioral, and social systems...

,

computational sociologyComputational 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 new analytic approaches like social network analysis, computational sociology...

,

network biologyBiological network is any network that applies to biological systems. A network is any system with sub-units that are linked into a whole, such as species units linked into a whole food web...

, social science,

sociologySociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...

and

social researchSocial research refers to research conducted by social scientists. Social research methods may be divided into two broad categories:* Quantitative designs approach social phenomena through quantifiable evidence, and often rely on statistical analysis of many cases to create valid and reliable...

. Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:

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 controlStatistical process control is the application of statistical methods to the monitoring and control of a process to ensure that it operates at its full potential to produce conforming product. Under SPC, a process behaves predictably to produce as much conforming product as possible with the least...

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.

## Statistical computing

The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models were almost always from the class of

linear modelIn 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 is also used in time series analysis with a different...

s, but powerful computers, coupled with suitable numerical algorithms, caused an increased interest in

nonlinear modelsIn 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...

(such as

neural networksNeural Networks is the official journal of the three oldest societies dedicated to research in neural networks: International Neural Network Society, European Neural Network Society and Japanese Neural Network Society, published by Elsevier...

) as well as the creation of new types, such as

generalized linear modelIn statistics, the generalized linear model 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 by allowing the magnitude of the variance of each measurement to...

s and

multilevel modelMultilevel models are statistical models of parameters that vary at more than one level...

s.

Increased computing power has also led to the growing popularity of computationally intensive methods based on

resamplingIn statistics, resampling is any of a variety of methods for doing one of the following:# Estimating the precision of sample statistics by using subsets of available data or drawing randomly with replacement from a set of data points # Exchanging labels on data points when performing significance...

, such as permutation tests and the

bootstrapIn statistics, bootstrapping is a computer-based method for assigning measures of accuracy to sample estimates . This technique allows estimation of the sample distribution of almost any statistic using only very simple methods...

, while techniques such as

Gibbs samplingIn statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables...

have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose

statistical software are now available.

## Misuse

There is a general perception that statistical knowledge is all-too-frequently intentionally

misusedA misuse of statistics occurs when a statistical argument asserts a falsehood. In some cases, the misuse may be accidental. In others, it is purposeful and for the gain of the perpetrator. When the statistical reason involved is false or misapplied, this constitutes a statistical fallacy.The false...

by finding ways to interpret only the data that are favorable to the presenter. The famous saying, "

There are three kinds of lies: lies, damned lies, and statistics"Lies, damned lies, and statistics" is a phrase describing the persuasive power of numbers, particularly the use of statistics to bolster weak arguments...

". which was popularized in the USA by

Mark TwainSamuel Langhorne Clemens , better known by his pen name Mark Twain, was an American author and humorist...

and incorrectly attributed by him to Disraeli (1804–1881), has come to represent the general mistrust [and misunderstanding] of statistical science. Harvard President Lawrence Lowell wrote in 1909 that statistics,

*"...like veal pies, are good if you know the person that made them, and are sure of the ingredients."*
If various studies appear to contradict one another, then the public may come to distrust such studies. For example, one study may suggest that a given diet or activity raises

blood pressureBlood pressure is the pressure exerted by circulating blood upon the walls of blood vessels, and is one of the principal vital signs. When used without further specification, "blood pressure" usually refers to the arterial pressure of the systemic circulation. During each heartbeat, BP varies...

, while another may suggest that it lowers blood pressure. The discrepancy can arise from subtle variations in experimental design, such as differences in the patient groups or research protocols, which are not easily understood by the non-expert. (Media reports usually omit this vital contextual information entirely, because of its complexity.)

By choosing (or rejecting, or modifying) a certain sample, results can be manipulated. Such manipulations need not be malicious or devious; they can arise from unintentional biases of the researcher. The graphs used to summarize data can also be misleading.

Deeper criticisms come from the fact that the hypothesis testing approach, widely used and in many cases required by law or regulation, forces one hypothesis (the

null hypothesisThe practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

) to be "favored," and can also seem to exaggerate the importance of minor differences in large studies. A difference that is highly statistically significant can still be of no practical significance. (See criticism of hypothesis testing and controversy over the null hypothesis.)

One response is by giving a greater emphasis on the

*p*-valueIn statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the p-value is less than the significance level α ,...

than simply reporting whether a hypothesis is rejected at the given level of significance. The

*p*-value, however, does not indicate the size of the effect. Another increasingly common approach is to report

confidence intervalIn statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...

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.

## 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 has changed with use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically. Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.

- In number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
- Methods of statistics including predictive methods in forecasting
Forecasting is the process of making statements about events whose actual outcomes have not yet been observed. A commonplace example might be estimation for some variable of interest at some specified future date. Prediction is a similar, but more general term...

, are combined with chaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

and fractal geometry to create video works that are considered to have great beauty.
- The process art
Process art is an artistic movement as well as a creative sentiment and world view where the end product of art and craft, the objet d’art, is not the principal focus. The 'process' in process art refers to the process of the formation of art: the gathering, sorting, collating, associating, and...

of Jackson PollockPaul Jackson Pollock , known as Jackson Pollock, was an influential American painter and a major figure in the abstract expressionist movement. During his lifetime, Pollock enjoyed considerable fame and notoriety. He was regarded as a mostly reclusive artist. He had a volatile personality, and...

relied on artistic experiments whereby underlying distributions in nature were artistically revealed. With the advent of computers, methods of statistics were applied to formalize such distribution driven natural processes, in order to make and analyze moving video art.
- Methods of statistics may be used predicatively in performance art
In art, performance art is a performance presented to an audience, traditionally interdisciplinary. Performance may be either scripted or unscripted, random or carefully orchestrated; spontaneous or otherwise carefully planned with or without audience participation. The performance can be live or...

, as in a card trick based on a Markov processIn probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...

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
Iannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...

, 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 tuneable using statistics.

## See also

- Foundations of statistics
Foundations of statistics is the usual name for the epistemological debate in statistics over how one should conduct inductive inference from data...

- Official statistics
Official statistics are statistics published by government agencies or other public bodies such as international organizations. They provide quantitative or qualitative information on all major areas of citizens' lives, such as economic and social development, living conditions, health, education,...

- List of statisticians

### Online non-commercial textbooks

- "A New View of Statistics", by Will G. Hopkins, AUT University
- "NIST/SEMATECH e-Handbook of Statistical Methods", by U.S. National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

and SEMATECH- Purpose :SEMATECH , a not-for-profit consortium, performs research and development to advance chip manufacturing. SEMATECH has broad engagement with various sectors of the R&D community, including chipmakers, equipment and material suppliers, universities, research institutes, and government...

- "Online Statistics: An Interactive Multimedia Course of Study", by David Lane, Joan Lu, Camille Peres, Emily Zitek, et al.
- "The Little Handbook of Statistical Practice", by Gerard E. Dallal, Tufts University
Tufts University is a private research university located in Medford/Somerville, near Boston, Massachusetts. It is organized into ten schools, including two undergraduate programs and eight graduate divisions, on four campuses in Massachusetts and on the eastern border of France...

- "StatSoft Electronic Textbook", by StatSoft

### Other non-commercial resources

- Statistics (OECD)
- Probability Web (Carleton College
Carleton College is an independent non-sectarian, coeducational, liberal arts college in Northfield, Minnesota, USA. The college enrolls 1,958 undergraduate students, and employs 198 full-time faculty members. In 2012 U.S...

)
- Free online statistics course with interactive practice exercises (Carnegie Mellon University
Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania, United States....

)
- Resources for Teaching and Learning about Probability and Statistics (ERIC)
- Rice Virtual Lab in Statistics (Rice University
William Marsh Rice University, commonly referred to as Rice University or Rice, is a private research university located on a heavily wooded campus in Houston, Texas, United States...

)
- Statistical Science Web (University of Melbourne
The University of Melbourne is a public university located in Melbourne, Victoria. Founded in 1853, it is the second oldest university in Australia and the oldest in Victoria...

)
- Applied statistics applets
- Statlib: data and software archives
- StatProb – peer-reviewed encyclopedia sponsored by statistics and probability societies