Statistics

For other uses, seeStatistics (disambiguation).

Thenormal distribution,a very commonprobability density,is used extensively ininferential statistics.

Scatter plotsandline chartsare used indescriptive statisticsto show the observed relationships between different variables, here using theIris flower data set.

Statistics(fromGerman:Statistik,orig."description of astate,a country ")is the discipline that concerns the collection, organization, analysis, interpretation, and presentation ofdata.In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with astatistical populationor astatistical modelto 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 ofsurveysandexperiments.

Whencensusdata cannot be collected,statisticianscollect data by developing specific experiment designs and surveysamples.Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. Anexperimental studyinvolves 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, anobservational studydoes not involve experimental manipulation.

Two main statistical methods are used indata analysis:descriptive statistics,which summarize data from a sample usingindexessuch as themeanorstandard deviation,andinferential statistics,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 adistribution(sample or population):central tendency(orlocation) seeks to characterize the distribution's central or typical value, whiledispersion(orvariability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences onmathematical statisticsare made under the framework ofprobability theory,which deals with the analysis of random phenomena.

A standard statistical procedure involves the collection of data leading to atest of the relationshipbetween 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 analternativeto an idealizednull hypothesisof 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 errors(null hypothesis is rejected when it is in fact true, giving a "false positive" ) andType II errors(null hypothesis fails to be rejected when an it is in fact false, 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.

Statistical measurement processes are also prone to error in regards to the data that they generate. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence ofmissing dataorcensoringmay result in biased estimates and specific techniques have been developed to address these problems.

Introduction

Main article:Outline of statistics

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation ofdata,or as a branch ofmathematics.Some consider statistics to be a distinctmathematical sciencerather than a branch of mathematics. While many scientific investigations make use of data, statistics is generally 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 apopulationor 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 acensus). This may be organized by governmental statistical institutes.Descriptive statisticscan be used to summarize the population data. Numerical descriptors includemeanandstandard deviationforcontinuous data(like income), while frequency and percentage are more useful in terms of describingcategorical data(like education).

When a census is not feasible, a chosen subset of the population called asampleis studied. Once a sample that is representative of the population is determined, data is collected for the sample members in an observational orexperimentalsetting. 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 statisticsare 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), estimating numerical characteristics of the data (estimation), describingassociationswithin the data (correlation), and modeling relationships within the data (for example, usingregression analysis). Inference can extend to theforecasting,prediction,and estimation of unobserved values either in or associated with the population being studied. It can includeextrapolationandinterpolationoftime seriesorspatial data,as well asdata mining.

Mathematical statistics

Main article:Mathematical statistics

Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this includemathematical analysis,linear algebra,stochastic analysis,differential equations,andmeasure-theoretic probability theory.

History

Main articles:History of statisticsandFounders of statistics

Bernoulli'sArs Conjectandiwas the first work that dealt withprobability theoryas currently understood.

Formal discussions on inference date back toArab mathematiciansandcryptographers,during theIslamic Golden Agebetween the 8th and 13th centuries.Al-Khalil(717–786) wrote theBook of Cryptographic Messages,which contains one of the first uses ofpermutationsandcombinations,to list all possibleArabicwords with and without vowels.Al-Kindi'sManuscript on Deciphering Cryptographic Messagesgave a detailed description of how to usefrequency analysisto decipherencryptedmessages, providing an early example ofstatistical inferencefordecoding.Ibn Adlan(1187–1268) later made an important contribution on the use ofsample sizein frequency analysis.

Although the term 'statistic' was introduced by the Italian scholarGirolamo Ghiliniin 1589 with reference to a collection of facts and information about a state, it was the GermanGottfried Achenwallin 1749 who started using the term as a collection of quantitative information, in the modern use for this science.The earliest writing containing statistics in Europe dates back to 1663, with the publication ofNatural and Political Observations upon the Bills of MortalitybyJohn Graunt.Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence itsstat-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.

Carl Friedrich Gaussmade major contributions to probabilistic methods leading to statistics.

The mathematical foundations of statistics developed from discussions concerninggames of chanceamong mathematicians such asGerolamo Cardano,Blaise Pascal,Pierre de Fermat,andChristiaan Huygens.Although the idea of probability was already examined in ancient and medieval law and philosophy (such as the work ofJuan Caramuel),probability theoryas a mathematical discipline only took shape at the very end of the 17th century, particularly inJacob Bernoulli'sposthumous workArs Conjectandi.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.Themethod of least squareswas first described byAdrien-Marie Legendrein 1805, thoughCarl Friedrich Gausspresumably made use of it a decade earlier in 1795.

Karl Pearson,a founder of mathematical statistics

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 ofFrancis GaltonandKarl 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 ofstandard deviation,correlation,regression analysisand the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among others.Pearson developed thePearson product-moment correlation coefficient,defined as a product-moment,themethod of momentsfor the fitting of distributions to samples and thePearson distribution,among many other things.Galton and Pearson foundedBiometrikaas the first journal of mathematical statistics andbiostatistics(then called biometry), and the latter founded the world's first university statistics department atUniversity College London.

The second wave of the 1910s and 20s was initiated byWilliam Sealy Gosset,and reached its culmination in the insights ofRonald Fisher,who wrote the textbooks that were to define the academic discipline in universities around the world. Fisher's most important publications were his 1918 seminal paperThe Correlation between Relatives on the Supposition of Mendelian Inheritance(which was the first to use the statistical term,variance), his classic 1925 workStatistical Methods for Research Workersand his 1935The Design of Experiments,where he developed rigorousdesign of experimentsmodels. He originated the concepts ofsufficiency,ancillary statistics,Fisher's linear discriminatorandFisher information.He also coined the termnull hypothesisduring theLady tasting teaexperiment, which "is never proved or established, but is possibly disproved, in the course of experimentation".In his 1930 bookThe Genetical Theory of Natural Selection,he applied statistics to variousbiologicalconcepts such asFisher's principle(whichA. W. F. Edwardscalled "probably the most celebrated argument inevolutionary biology") andFisherian runaway,a concept insexual selectionabout a positive feedback runaway effect found inevolution.

The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work betweenEgon PearsonandJerzy Neymanin the 1930s. They introduced the concepts of "Type II"error,power of a testandconfidence intervals.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 moderncomputershas 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 analyzebig data.

Statistical data

Main article:Statistical data

Data collection

Sampling

When full census data cannot be collected, statisticians collect sample data by developing specificexperiment designsandsurvey samples.Statistics itself also provides tools for prediction and forecasting throughstatistical models.

To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representativesamplingassures 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 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 themathematical disciplineofprobability theory.Probability is used inmathematical statisticsto study thesampling distributionsofsample statisticsand, more generally, the properties ofstatistical 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 todeduceprobabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferringfrom 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,and in particular to draw a conclusion on the effect of changes in the values of predictors orindependent variables on dependent variables.There are two major types of causal statistical studies:experimental studiesandobservational studies.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 additionalmeasurements with different levelsusing the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involveexperimental manipulation.Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data fromrandomized studies,they are also applied to other kinds of data—likenatural experimentsandobservational studies—for which a statistician would use a modified, more structured estimation method (e.g.,difference in differences estimationandinstrumental variables,among many others) that produceconsistent estimators.

Experiments

The basic steps of a statistical experiment are:

1. Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size oftreatment effects,alternative hypotheses,and the estimatedexperimental 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.
2. Design of experiments,usingblockingto reduce the influence ofconfounding variables,andrandomized assignmentof treatments to subjects to allowunbiased estimatesof treatment effects and experimental error. At this stage, the experimenters and statisticians write theexperimental protocolthat will guide the performance of the experiment and which specifies theprimary analysisof the experimental data.
3. Performing the experiment following theexperimental protocolandanalyzing the datafollowing the experimental protocol.
4. Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
5. Documenting and presenting the results of the study.

Experiments on human behavior have special concerns. The famousHawthorne studyexamined changes to the working environment at the Hawthorne plant of theWestern Electric Company.The researchers were interested in determining whether increased illumination would increase the productivity of theassembly lineworkers. 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 acontrol groupandblindness.TheHawthorne effectrefers 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,and then look for the number of cases of lung cancer in each group.Acase-control studyis 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

Main articles:Statistical data typeandLevels of measurement

Various attempts have been made to produce a taxonomy oflevels of measurement.The psychophysicistStanley Smith Stevensdefined 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 withlongitudeandtemperaturemeasurements inCelsiusorFahrenheit), 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 ascategorical variables,whereas ratio and interval measurements are grouped together asquantitative variables,which can be eitherdiscreteorcontinuous,due to their numerical nature. Such distinctions can often be loosely correlated withdata typein computer science, in that dichotomous categorical variables may be represented with theBoolean data type,polytomous categorical variables with arbitrarily assignedintegersin theintegral data type,and continuous variables with thereal data typeinvolvingfloating-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

Main article:Descriptive statistics

Adescriptive statistic(in thecount nounsense) is asummary statisticthat quantitatively describes or summarizes features of a collection ofinformation,whiledescriptive statisticsin themass nounsense is the process of using and analyzing those statistics. Descriptive statistics is distinguished frominferential statistics(or inductive statistics), in that descriptive statistics aims to summarize asample,rather than use the data to learn about thepopulationthat the sample of data is thought to represent.

Inferential statistics

Main article:Statistical inference

Statistical inferenceis the process of usingdata analysisto deduce properties of an underlyingprobability distribution.Inferential statistical analysis infers properties of apopulation,for example by testing hypotheses and deriving estimates. It is assumed that the observed data set issampledfrom a larger population. Inferential statistics can be contrasted withdescriptive statistics.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

Considerindependent identically distributed (IID) random variableswith a givenprobability distribution:standardstatistical inferenceandestimation theorydefines arandom sampleas therandom vectorgiven by thecolumn vectorof these IID variables.Thepopulationbeing 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, butnot a function of unknown parameters.The probability distribution of the statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: anestimatoris a statistic used to estimate such function. Commonly used estimators includesample mean,unbiasedsample varianceandsample covariance.

A random variable that is a function of the random sample and of the unknown parameter, but whose probability distributiondoes not depend on the unknown parameteris called apivotal quantityor pivot. Widely used pivots include thez-score,thechi square statisticand Student'st-value.

Between two estimators of a given parameter, the one with lowermean squared erroris said to be moreefficient.Furthermore, an estimator is said to beunbiasedif itsexpected valueis equal to thetrue valueof the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at thelimitto the true value of such parameter.

Other desirable properties for estimators include:UMVUEestimators 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) andconsistent estimatorswhichconverges in probabilityto 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: themethod of moments,themaximum likelihoodmethod, theleast squaresmethod and the more recent method ofestimating equations.

Null hypothesis and alternative hypothesis

Interpretation of statistical information can often involve the development of anull hypothesiswhich is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time.

The best illustration for a novice is the predicament encountered by a criminal trial. The null hypothesis, H₀,asserts that the defendant is innocent, whereas the alternative hypothesis, H₁,asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H₀(status quo) stands in opposition to H₁and is maintained unless H₁is supported by evidence "beyond a reasonable doubt". However, "failure to reject H₀"in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarilyacceptH₀butfails to rejectH₀.While one can not "prove" a null hypothesis, one can test how close it is to being true with apower test,which tests fortype II errors.

Whatstatisticianscall analternative hypothesisis simply a hypothesis that contradicts the null hypothesis.

Error

Working from anull hypothesis,two broad categories of error are recognized:

• Type I errorswhere the null hypothesis is falsely rejected, giving a "false positive".
• Type II errorswhere the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".

Standard deviationrefers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, whileStandard errorrefers to an estimate of difference between sample mean and population mean.

Astatistical erroris the amount by which an observation differs from itsexpected value.Aresidualis the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).

Mean squared erroris used for obtainingefficient estimators,a widely used class of estimators.Root mean square erroris simply the square root of mean squared error.

A least squares fit: in red the points to be fitted, in blue the fitted line.

Many statistical methods seek to minimize theresidual sum of squares,and these are called "methods of least squares"in contrast toLeast absolute deviations.The latter gives equal weight to small and big errors, while the former gives more weight to large errors. Residual sum of squares is alsodifferentiable,which provides a handy property for doingregression.Least squares applied tolinear regressionis calledordinary least squaresmethod and least squares applied tononlinear regressionis callednon-linear least squares.Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed inpolynomial least squares,which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve.

Measurement processes that generate statistical data are also subject to error. Many of these errors are classified asrandom(noise) orsystematic(bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence ofmissing dataorcensoringmay result inbiased estimatesand specific techniques have been developed to address these problems.

Interval estimation

Main article:Interval estimation

Confidence intervals:the red line is true value for the mean in this example, the blue lines are random confidence intervals for 100 realizations.

Most studies only sample part of a population, so results do not fully represent the whole population. Any estimates obtained from the sample only approximate the population value.Confidence intervalsallow 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 doesnotimply that the probability that the true value is in the confidence interval is 95%. From thefrequentistperspective, such a claim does not even make sense, as the true value is not arandom variable.Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observedrandom variables.One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use acredible intervalfromBayesian statistics:this approach depends on a different way ofinterpreting what is meant by "probability",that is as aBayesian probability.

In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds.

Significance

Main article:Statistical 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 thep-value).

In this graph the black line is probability distribution for thetest statistic,thecritical regionis the set of values to the right of the observed data point (observed value of the test statistic) and thep-valueis represented by the green area.

The standard approachis to test a null hypothesis against an alternative hypothesis. Acritical regionis 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 does not belong to the critical region given that the alternative hypothesis is true. Thestatistical powerof 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, thesignificance levelis 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 thetest statistic.Therefore, the smaller the significance level, the lower the probability of committing type I error.

Some problems are usually associated with this framework (Seecriticism 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 thesignificance levelto include thep-valuewhen reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate thesizeor importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to reportconfidence intervals.Although these are produced from the same calculations as those of hypothesis tests orp-values, they describe both the size of the effect and the uncertainty surrounding it.
• Fallacy of the transposed conditional, akaprosecutor's fallacy:criticisms arise because the hypothesis testing approach forces one hypothesis (thenull hypothesis) to be favored, since what is being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered byBayesian inference,although it requires establishing aprior probability.
• Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
• As everything ininferential statisticsit relies on sample size, and therefore underfat tailsp-values may be seriously mis-computed.

Examples

Some well-known statisticaltestsand procedures are:

• Analysis of variance(ANOVA)
• Chi-squared test
• Correlation
• Factor analysis
• Mann–WhitneyU
• Mean square weighted deviation(MSWD)
• Pearson product-moment correlation coefficient
• Regression analysis
• Spearman's rank correlation coefficient
• Student'st-test
• Time series analysis
• Conjoint Analysis

Exploratory data analysis

Main article:Exploratory data analysis

Exploratory data analysis(EDA) is an approach toanalyzingdata setsto summarize their main characteristics, often with visual methods. Astatistical modelcan 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

Main article:Misuse of statistics

Misuse of statisticscan 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. Thestatistical significanceof 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 asstatistical literacy.

There is a general perception that statistical knowledge is all-too-frequently intentionallymisusedby finding ways to interpret only the data that are favorable to the presenter.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 bookHow to Lie with Statistics,byDarrell Huff,outlines a range of considerations. In an attempt to shed light on the use and misuse of statistics, reviews of statistical techniques used in particular fields are conducted (e.g. Warne, Lazo, Ramos, and Ritter (2012)).

Ways to avoid misuse of statistics include using proper diagrams and avoidingbias.Misuse can occur when conclusions areovergeneralizedand 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.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 wellrepresented.To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."

To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:

• Who says so? (Does he/she have an axe to grind?)
• How does he/she know? (Does he/she have the resources to know the facts?)
• What's missing? (Does he/she give us a complete picture?)
• Did someone change the subject? (Does he/she offer us the right answer to the wrong problem?)
• Does it make sense? (Is his/her conclusion logical and consistent with what we already know?)

Theconfounding variableproblem:XandYmay be correlated, not because there is causal relationship between them, but because both depend on a third variableZ.Zis called a confounding factor.

Misinterpretation: correlation

See also:Correlation does not imply causation

The concept ofcorrelationis particularly noteworthy for the potential confusion it can cause. Statistical analysis of adata setoften 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 orconfounding variable.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 asStatistical science,comprises descriptive statistics and the application of inferential statistics.Theoretical statisticsconcerns the logical arguments underlying justification of approaches tostatistical inference,as well as encompassingmathematical statistics.Mathematical statistics includes not only the manipulation ofprobability distributionsnecessary for deriving results related to methods of estimation and inference, but also various aspects ofcomputational statisticsand thedesign of experiments.

Statistical consultantscan help organizations and companies that do not have in-house expertise relevant to their particular questions.

Machine learning and data mining

Machine learningmodels 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 disciplines,includingnaturalandsocial sciences,government, and business. Business statistics applies statistical methods ineconometrics,auditingand production and operations, including services improvement and marketing research.A study of two journals in tropical biology found that the 12 most frequent statistical tests are:analysis of variance(ANOVA),chi-squared test,Student's t-test,linear regression,Pearson's correlation coefficient,Mann-Whitney U test,Kruskal-Wallis test,Shannon's diversity index,Tukey's range test,cluster analysis,Spearman's rank correlation coefficientandprincipal component analysis.

A typical statistics course covers descriptive statistics, probability, binomial andnormal distributions,test of hypotheses and confidence intervals,linear regression,and correlation.Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use offree statistics software.

Statistical computing

gretl,an example of anopen source statistical package

Main article:Computational statistics

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 models,but powerful computers, coupled with suitable numericalalgorithms,caused an increased interest innonlinear models(such asneural networks) as well as the creation of new types, such asgeneralized linear modelsandmultilevel models.

Increased computing power has also led to the growing popularity of computationally intensive methods based onresampling,such aspermutation testsand thebootstrap,while techniques such asGibbs samplinghave made use ofBayesian modelsmore 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 purposestatistical softwareare now available. Examples of available software capable of complex statistical computation include programs such asMathematica,SAS,SPSS,andR.

Business statistics

In business, "statistics" is a widely usedmanagement-anddecision supporttool. It is particularly applied infinancial management,marketing management,andproduction,servicesandoperations management.Statistics is also heavily used inmanagement accountingandauditing.The discipline ofManagement Scienceformalizes the use of statistics, and other mathematics, in business. (Econometricsis the application of statistical methods toeconomic datain order to give empirical content toeconomic relationships.)

A typical "Business Statistics" course is intended forbusiness majors,and coversdescriptive statistics(collection,description, analysis, and summary of data), probability (typically thebinomialandnormal distributions), test of hypotheses and confidence intervals,linear regression,and correlation; (follow-on) courses may includeforecasting,time series,decision trees,multiple linear regression,and other topics frombusiness analyticsmore generally. See alsoBusiness mathematics § University level.Professional certification programs,such as theCFA,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,scatter plotsof 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 inforecastingcombiningchaos theoryandfractal geometrycan be used to create video works.
• Theprocess artofJackson Pollockrelied 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 inperformance art,as in a card trick based on aMarkov processthat 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 orstochastic musicinvented byIannis Xenakis,where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tunable using statistics.

Specialized disciplines

Main article:List of fields of application of statistics

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

• Actuarial science(assesses risk in the insurance and finance industries)
• Applied information economics
• Astrostatistics(statistical evaluation of astronomical data)
• Biostatistics
• Chemometrics(for analysis of data fromchemistry)
• Data mining(applying statistics andpattern recognitionto discover knowledge from data)
• Data science(see also:Data science#Relationship to statistics)
• Demography(statistical study of populations)
• Econometrics(statistical analysis of economic data)
• Energy statistics
• Engineering statistics
• Epidemiology(statistical analysis of disease)
• Geographyandgeographic information systems,specifically inspatial analysis
• Image processing
• Jurimetrics(law)
• Medical statistics
• Political science
• Psychological statistics
• Reliability engineering
• Social statistics
• Statistical mechanics

In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:

• Bootstrap/jackknife resampling
• Multivariate statistics
• Statistical classification
• Structured data analysis
• Structural equation modelling
• Survey methodology
• Survival analysis
• Statistics in various sports, particularlybaseball– known assabermetrics– andcricket

Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as instatistical process controlor SPC), for summarizing data, and to make data-driven decisions.

See also

Main article:Outline of statistics

• Abundance estimation
• Glossary of probability and statistics
• List of academic statistical associations
• List of important publications in statistics
• List of national and international statistical services
• List of statistical packages(software)
• List of statistics articles
• List of university statistical consulting centers
• Notation in probability and statistics
• Statistics education
• World Statistics Day
• Robust statistics

Foundations and major areas of statistics

• Philosophy of statistics
• Probability interpretations
• Foundations of statistics
• List of statisticians
• Official statistics
• Multivariate analysis of variance

Further reading

• Lydia Denworth, "A Significant Problem: Standard scientific methods are under fire. Will anything change?",Scientific American,vol. 321, no. 4 (October 2019), pp. 62–67. "The use ofpvaluesfor nearly a century [since 1925] to determinestatistical significanceofexperimentalresults has contributed to an illusion ofcertaintyand [to]reproducibility crisesin manyscientific fields.There is growing determination to reform statistical analysis... Some [researchers] suggest changing statistical methods, whereas others would do away with a threshold for defining "significant" results. "(p. 63.)
• Barbara Illowsky; Susan Dean (2014).Introductory Statistics.OpenStax CNX.ISBN978-1938168208.
• Stockburger, David W."Introductory Statistics: Concepts, Models, and Applications".Missouri State University(3rd Web ed.). Archived fromthe originalon 28 May 2020.
• OpenIntro StatisticsArchived2019-06-16 at theWayback Machine,3rd edition by Diez, Barr, and Cetinkaya-Rundel
• Stephen Jones, 2010.Statistics in Psychology: Explanations without Equations.Palgrave Macmillan.ISBN978-1137282392.
• Cohen, J (1990)."Things I have learned (so far)"(PDF).American Psychologist.45(12): 1304–1312.doi:10.1037/0003-066x.45.12.1304.S2CID7180431.Archived fromthe original(PDF)on 2017-10-18.
• Gigerenzer, G (2004). "Mindless statistics".Journal of Socio-Economics.33(5): 587–606.doi:10.1016/j.socec.2004.09.033.
• Ioannidis, J.P.A. (2005)."Why most published research findings are false".PLOS Medicine.2(4): 696–701.doi:10.1371/journal.pmed.0040168.PMC1855693.PMID17456002.

External links

• (Electronic Version): TIBCO Software Inc. (2020).Data Science Textbook.
• Online Statistics Education: An Interactive Multimedia Course of Study.Developed by Rice University (Lead Developer), University of Houston Clear Lake, Tufts University, and National Science Foundation.
• UCLA Statistical Computing Resources(archived 17 July 2006)
• Philosophy of Statisticsfrom theStanford Encyclopedia of Philosophy