Statistics (fromGerman:Statistik,orig. "description of astate, a country"[1]) is the discipline that concerns the collection, organization, analysis, interpretation, and presentation ofdata.[2] In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with astatistical population or astatistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design ofsurveys andexperiments.[3]
Whencensus data (comprising every member of the target population) cannot be collected,statisticians collect 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 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, anobservational study does not involve experimental manipulation.
Two main statistical methods are used indata analysis:descriptive statistics, which summarize data from a sample usingindexes such as themean orstandard deviation, andinferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).[4] 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 made usingmathematical statistics employ 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 relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, analternative to an idealizednull hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized:Type I 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 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.[4]
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 data orcensoring may result in biased estimates and specific techniques have been developed to address these problems.
"Statistics is both the science of uncertainty and the technology of extracting information from data." - featured in the International Encyclopedia of Statistical Science.[5]
Statistics is the discipline that deals withdata, facts and figures with which meaningful information is inferred. Data may represent a numerical value, in form of quantitative data, or a label, as with qualitative data. Data may be collected, presented and summarised, in one of two methods called descriptive statistics. Two elementary summaries of data, singularly called a statistic, are the mean and dispersion. Whereas inferential statistics interprets data from a population sample to induce statements and predictions about a population.[6][7][5]
Statistics is regarded as a body of science[8] or a branch of mathematics.[9] It is based on probability, a branch of mathematics that studies random events. Statistics is considered the science of uncertainty. This arises from the ways to cope with measurement and sampling error as well as dealing with uncertanties in modelling. Although probability and statistics were once paired together as a single subject, they are conceptually distinct from one another. The former is based on deducing answers to specific situations from a general theory of probability, meanwhile statistics induces statements about a population based on a data set. Statistics serves to bridge the gap between probability and applied mathematical fields.[10][5][11]
Some consider statistics to be a distinctmathematical science rather 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.[12][13] Statistics is indexed at 62, a subclass of probability theory and stochastic processes, in the Mathematics Subject Classification.[14] Mathematical statistics is covered in the range 276-280 of subclass QA (science > mathematics) in the Library of Congress Classification.[15]
The word statistics ultimately comes from the Latin word Status, meaning "situation" or "condition" in society, which in late Latin adopted the meaning "state". Derived from this, political scientist Gottfried Achenwall, coined the German word statistik (a summary of how things stand). In 1770, the term entered the English language through German and referred to the study of political arrangements. The term gained its modern meaning in the 1790s in John Sinclair's works.[16][17] In modern German, the term statistik is synonymous with mathematical statistics. The term statistic, in singular form, is used to describe a function that returns its value of the same name.[18]
When full census data cannot be collected, statisticians collect sample data by developing specificexperiment designs andsurvey 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. Representativesampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures. There are also methods of experimental design 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 discipline ofprobability theory. Probability is used inmathematical statistics to study thesampling distributions ofsample statistics and, 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 todeduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.
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 studies andobservational 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 levels using 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 experiments andobservational studies[19]—for which a statistician would use a modified, more structured estimation method (e.g.,difference in differences estimation andinstrumental variables, among many others) that produceconsistent estimators.
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.
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 famousHawthorne study examined 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 line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of acontrol group andblindness. TheHawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed.[20]
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.[21] Acase-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.
Various attempts have been made to produce a taxonomy oflevels of measurement. The psychophysicistStanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case withlongitude andtemperature measurements inCelsius orFahrenheit), 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 eitherdiscrete orcontinuous, due to their numerical nature. Such distinctions can often be loosely correlated withdata type in computer science, in that dichotomous categorical variables may be represented with theBoolean data type, polytomous categorical variables with arbitrarily assignedintegers in theintegral data type, and continuous variables with thereal data type involvingfloating-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)[22] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)[23] described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),[24] van den Berg (1991).[25])
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."[26]: 82
Adescriptive statistic (in thecount noun sense) is asummary statistic that quantitatively describes or summarizes features of a collection ofinformation,[27] whiledescriptive statistics in themass noun sense 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 thepopulation that the sample of data is thought to represent.[28]
Statistical inference is the process of usingdata analysis to deduce properties of an underlyingprobability distribution.[29] Inferential statistical analysis infers properties of apopulation, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set issampled from 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.[30]
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: anestimator is a statistic used to estimate such function. Commonly used estimators includesample mean, unbiasedsample variance andsample 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 parameter is called apivotal quantity or pivot. Widely used pivots include thez-score, thechi square statistic and Student'st-value.
Between two estimators of a given parameter, the one with lowermean squared error is said to be moreefficient. Furthermore, an estimator is said to beunbiased if itsexpected value is equal to thetrue value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at thelimit to the true value of such parameter.
Other desirable properties for estimators include:UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) andconsistent estimators whichconverges in probability to the true value of such parameter.
Interpretation of statistical information can often involve the development of anull hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time.[32][33] Thealternative hypothesis is the name of the hypothesis that contradicts the null hypothesis.
The best illustration for a novice is the predicament encountered by acriminal trial. The null hypothesis, H0, asserts that the defendant is innocent, whereas the alternative hypothesis, H1, asserts that the defendant isguilty. Theindictment comes because of suspicion of the guilt. The H0 (thestatus quo) stands in opposition to H1 and is maintained unless H1 is supported by evidence "beyond a reasonable doubt". However, "failure to reject H0" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarilyaccept H0 butfails to reject H0. 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.
Working from anull hypothesis, two broad categories of error are recognized:
Type I errors where the null hypothesis is falsely rejected, giving a "false positive".
Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed, giving a "false negative".
Standard deviation refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, whileStandard error refers to an estimate of difference between sample mean and population mean.
Astatistical error is the amount by which an observation differs from itsexpected value. Aresidual is the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).
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 regression is calledordinary least squares method and least squares applied tononlinear regression is 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 data orcensoring may result inbiased estimates and specific techniques have been developed to address these problems.[34]
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 intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This doesnot imply that the probability that the true value is in the confidence interval is 95%. From thefrequentist perspective, 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 interval fromBayesian 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.
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 region is the set of values to the right of the observed data point (observed value of the test statistic) and thep-value is represented by the green area.
The standard approach[31] is to test a null hypothesis against an alternative hypothesis. Acritical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (statistical significance) and the probability of type II error is the probability that the estimator does not belong to the critical region given that the alternative hypothesis is true. Thestatistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false.
Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably.
Although in principle the acceptable level of statistical significance may be subject to debate, thesignificance level is the largest p-value that allows the test to reject the null hypothesis. This test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as thetest statistic. Therefore, the smaller the significance level, the lower the probability of committing type I error.
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 level to include thep-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate thesize or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to reportconfidence 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.[35]
Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
For statistically modelling purposes, Bayesian models tend to behierarchical, for example, one could model eachYouTube channel as having video views distributed as a normal distribution with channel dependent mean and variance, while modeling the channel means as themselves coming from a normal distribution representing the distribution of average video view counts per channel, and the variances as coming from another distribution.
Exploratory data analysis (EDA) is an approach toanalyzingdata sets to summarize their main characteristics, often with visual methods. Astatistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task.
Although the termstatistic was introduced by the Italian scholarGirolamo Ghilini in 1589 with reference to a collection of facts and information about a state, it was the GermanGottfried Achenwall in 1749 who started using the term as a collection of quantitative information, in the modern use for this science.[37][38] The earliest writing containing statistics in Europe dates back to 1663, with the publication ofNatural and Political Observations upon the Bills of Mortality byJohn Graunt.[39] 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 Gauss made major contributions to probabilistic methods leading to statistics.
Karl Pearson, a founder of mathematical statistics
The modern field of statistics emerged in the late 19th and early 20th century in three stages.[43] The first wave, at the turn of the century, was led by the work ofFrancis Galton andKarl 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 analysis and the application of these methods to the study of the variety of human characteristics—height, weight and eyelash length among others.[44] Pearson developed thePearson product-moment correlation coefficient, defined as a product-moment,[45] themethod of moments for the fitting of distributions to samples and thePearson distribution, among many other things.[46] Galton and Pearson foundedBiometrika as the first journal of mathematical statistics andbiostatistics (then calledbiometry), and the latter founded the world's first university statistics department atUniversity College London.[47]
The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work betweenEgon Pearson andJerzy Neyman in the 1930s. They introduced the concepts of "Type II" error,power of a test andconfidence intervals. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.[61]
Among the early attempts to measure national economic activity were those ofWilliam Petty in the 17th century. In the 20th century the uniformSystem of National Accounts was developed.[62]
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 moderncomputers has expedited large-scale statistical computations and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on the problem of how to analyzebig data.[63]
Applied statistics, sometimes referred to asStatistical science,[64] comprises descriptive statistics and the application of inferential statistics.[65][66]Theoretical statistics concerns the logical arguments underlying justification of approaches tostatistical inference, as well as encompassingmathematical statistics. Mathematical statistics includes not only the manipulation ofprobability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects ofcomputational statistics and thedesign of experiments.
Statistical consultants can help organizations and companies that do not have in-house expertise relevant to their particular questions.
A typical statistics course covers descriptive statistics, probability, binomial andnormal distributions, test of hypotheses and confidence intervals,linear regression, and correlation.[69] Modern fundamental statistical courses for undergraduate students focus on correct test selection, results interpretation, and use offree statistics software.[68]
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 models andmultilevel models.
Increased computing power has also led to the growing popularity of computationally intensive methods based onresampling, such aspermutation tests and thebootstrap, while techniques such asGibbs sampling have made use ofBayesian models more feasible. The computer revolution has implications for the future of statistics with a new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purposestatistical software are now available. Examples of available software capable of complex statistical computation include programs such asMathematica,SAS,SPSS, andR.
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 control or SPC), for summarizing data, and to make data-driven decisions.
Misuse of statistics can produce subtle but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.
Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. Thestatistical significance of a trend in the data—which measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to asstatistical literacy.
There is a general perception that statistical knowledge is all-too-frequently intentionallymisused by finding ways to interpret only the data that are favorable to the presenter.[73] 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,[73] 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)).[74]
Ways to avoid misuse of statistics include using proper diagrams and avoidingbias.[75] Misuse can occur when conclusions areovergeneralized and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias.[76] Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs.[75] 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.[76] To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.[77] According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."[78]
To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:[73]
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 variable problem:X andY may be correlated, not because there is causal relationship between them, but because both depend on a third variableZ.Z is called a confounding factor.
The concept ofcorrelation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of adata set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death, might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable orconfounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables.
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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 ofp values for nearly a century [since 1925] to determinestatistical significance ofexperimental results has contributed to an illusion ofcertainty and [to]reproducibility crises in 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.)
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