text:
Statistics
From Wikipedia, the free encyclopedia
Jump to: navigation, search
More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.[1][2] In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal." Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.[1]
When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study 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.
Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).[3] Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.
A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null 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 falsely rejected giving a "false positive") and Type II errors (null hypothesis fails to be rejected and an actual difference between populations is missed giving a "false negative").[4] Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.[citation needed]
Measurement processes that generate statistical data are also subject to error. 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 be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.
Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from calculus and probability theory.
Contents
[hide]
1 Scope
1.1 Mathematical statistics
2 Overview
3 Data collection
3.1 Sampling
3.2 Experimental and observational studies
3.2.1 Experiments
3.2.2 Observational study
4 Types of data
5 Terminology and theory of inferential statistics
5.1 Statistics, estimators and pivotal quantities
5.2 Null hypothesis and alternative hypothesis
5.3 Error
5.4 Interval estimation
5.5 Significance
5.6 Examples
6 Misuse
6.1 Misinterpretation: correlation
7 History of statistical science
8 Applications
8.1 Applied statistics, theoretical statistics and mathematical statistics
8.2 Machine learning and data mining
8.3 Statistics in society
8.4 Statistical computing
8.5 Statistics applied to mathematics or the arts
9 Specialized disciplines
10 See also
11 References
12 Further reading
13 External links
Scope[edit]
Some definitions are:
Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data."[5]
Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."[6]
Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data,[7] or as a branch of mathematics.[8] Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty.[9][10]
Mathematical statistics[edit]
Main article: Mathematical statistics
Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory.[11][12]
Overview[edit]
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal".
Ideally, statisticians compile data about the entire population (an operation called census). This may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types (like income), while frequency and percentage are more useful in terms of describing categorical data (like race).
When a census is not feasible, a chosen subset of the population called a sample is studied. Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are also due to uncertainty. To still draw meaningful conclusions about the entire population, inferential statistics is needed. It 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), describing associations within the data (correlation) and modeling relationships within the data (for example, using regression analysis). Inference can extend to forecasting, prediction and estimation of unobserved values either in or associated with the population being studied; it can include extrapolation and interpolation of time series or spatial data, and can also include data mining.
Data collection[edit]
Sampling[edit]
When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through statistical models.
To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures. There are also methods of experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population.
Sampling theory is part of the mathematical discipline of probability theory. Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total population.
Experimental and observational studies[edit]
A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables. There are two major types of causal statistical studies: experimental studies and observational 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 additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between predictors and response are investigated. While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data – like natural experiments and observational studies[13] – for which a statistician would use a modified, more structured estimation method (e.g., Difference in differences estimation and instrumental variables, among many others) that produce consistent estimators.
Experiments[edit]
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, using blocking to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the experimental protocol that will guide the performance of the experiment and which specifies the primary analysis of the experimental data.
Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
Documenting and presenting the results of the study.
Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed.[14]
Observational study[edit]
An example of an observational study is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study, and then look for the number of cases of lung cancer in each group.[15] A case-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.
Types of data[edit]
Main articles: Statistical data type and Levels of measurement
Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation.
Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the integral data type, and continuous variables with the real data type involving floating point computation. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented.
Other categorizations have been proposed. For example, Mosteller and Tukey (1977)[16] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)[17] described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998),[18] van den Berg (1991).[19]
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" (Hand, 2004, p. 82).[20]
Terminology and theory of inferential statistics[edit]
Statistics, estimators and pivotal quantities[edit]
Consider independent identically distributed (IID) random variables with a given probability distribution: standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables.[21] The population being examined is described by a probability distribution that may have unknown parameters.
A statistic is a random variable that is a function of the random sample, but not a function of unknown parameters. The probability distribution of the statistic, though, may have unknown parameters.
Consider now a function of the unknown parameter: an estimator is a statistic used to estimate such function. Commonly used estimators include sample mean, unbiased sample variance and sample covariance.
A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score, the chi square statistic and Student's t-value.
Between two estimators of a given parameter, the one with lower mean squared error is said to be more efficient. Furthermore, an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the limit to the true value of such parameter.
Other desirable properties for estimators include: UMVUE estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and consistent estimators which converges in probability to the true value of such parameter.
This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the method of moments, the maximum likelihood method, the least squares method and the more recent method of estimating equations.
Null hypothesis and alternative hypothesis[edit]
Interpretation of statistical information can often involve the development of a null hypothesis which is usually (but not necessarily) that no relationship exists among variables or that no change occurred over time.[22][23]
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 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 necessarily accept H0 but fails to reject H0. While one can not "prove" a null hypothesis, one can test how close it is to being true with a power test, which tests for type II errors.
What statisticians call an alternative hypothesis is simply a hypothesis that contradicts the null hypothesis.
Error[edit]
Working from a null hypothesis, 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".
Standard deviation refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean, while Standard error refers to an estimate of difference between sample mean and population mean.
A statistical error is the amount by which an observation differs from its expected value, a residual is the amount an observation differs from the value the estimator of the expected value assumes on a given sample (also called prediction).
Mean squared error is used for obtaining efficient estimators, a widely used class of estimators. Root mean square error is simply the square root of mean squared error.
A least squares fit: in red the points to be fitted, in blue the fitted line.
Many statistical methods seek to minimize the residual sum of squares, and these are called "methods of least squares" in contrast to Least absolute deviations. The latter gives equal weight to small and big errors, while the former gives more weight to large errors. Residual sum of squares is also differentiable, which provides a handy property for doing regression. Least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called non-linear least squares. Also in a linear regression model the non deterministic part of the model is called error term, disturbance or more simply noise. Both linear regression and non-linear regression are addressed in polynomial least squares, which also describes the variance in a prediction of the dependent variable (y axis) as a function of the independent variable (x axis) and the deviations (errors, noise, disturbances) from the estimated (fitted) curve.
Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.[24]
Interval estimation[edit]
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 don't fully represent the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the true (population) value in 95% of all possible cases. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not even make sense, as the true value is not a random variable. Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed random variables. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from Bayesian statistics: this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability.
In principle confidence intervals can be symmetrical or asymmetrical. An interval can be asymmetrical because it works as lower or upper bound for a parameter (left-sided interval or right sided interval), but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate. Sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds.
Significance[edit]
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 the p-value).
In this graph the black line is probability distribution for the test statistic, the critical region is the set of values to the right of the observed data point (observed value of the test statistic) and the p-value is represented by the green area.
The standard approach[21] is to test a null hypothesis against an alternative hypothesis. A critical region is the set of values of the estimator that leads to refuting the null hypothesis. The probability of type I error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true (statistical significance) and the probability of type II error is the probability that the estimator doesn't belong to the critical region given that the alternative hypothesis is true. The statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false.
Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms. For example, in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably.
While in principle the acceptable level of statistical significance may be subject to debate, the p-value is the smallest significance level that allows the test to reject the null hypothesis. This is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. Therefore, the smaller the p-value, the lower the probability of committing type I error.
Some problems are usually associated with this framework (See criticism of hypothesis testing):
A difference that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests to account for this. One response involves going beyond reporting only the significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the size or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies. A better and increasingly common approach is to report confidence intervals. Although these are produced from the same calculations as those of hypothesis tests or p-values, they describe both the size of the effect and the uncertainty surrounding it.
Fallacy of the transposed conditional, aka prosecutor's fallacy: criticisms arise because the hypothesis testing approach forces one hypothesis (the null hypothesis) to be favored, since what is being evaluated is probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered by Bayesian inference, although it requires establishing a prior probability.[25]
Rejecting the null hypothesis does not automatically prove the alternative hypothesis.
As everything in inferential statistics it relies on sample size, and therefore under fat tails p-values may be seriously mis-computed.[clarification needed]
Examples[edit]
Some well-known statistical tests and procedures are:
Analysis of variance (ANOVA)
Chi-squared test
Correlation
Factor analysis
Mann–Whitney U
Mean square weighted deviation (MSWD)
Pearson product-moment correlation coefficient
Regression analysis
Spearman's rank correlation coefficient
Student's t-test
Time series analysis
Conjoint Analysis
Misuse[edit]
Main article: Misuse of statistics
Misuse of statistics can produce subtle, but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics.
Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data—which measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy.
There is a general perception that statistical knowledge is all-too-frequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.[26] A mistrust and misunderstanding of statistics is associated with the quotation, "There are three kinds of lies: lies, damned lies, and statistics". Misuse of statistics can be both inadvertent and intentional, and the book How to Lie with Statistics[26] 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)).[27]
Ways to avoid misuse of statistics include using proper diagrams and avoiding bias.[28] Misuse can occur when conclusions are overgeneralized and claimed to be representative of more than they really are, often by either deliberately or unconsciously overlooking sampling bias.[29] Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs.[28] Unfortunately, most people do not look for bias or errors, so they are not noticed. Thus, people may often believe that something is true even if it is not well represented.[29] To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.[30] According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."[31]
To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:[32]
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?)
The confounding variable problem: X and Y may be correlated, not because there is causal relationship between them, but because both depend on a third variable Z. Z is called a confounding factor.
Misinterpretation: correlation[edit]
The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set often reveals that two variables (properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable or confounding variable. 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.)
History of statistical science[edit]
Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Main articles: History of statistics and Founders of statistics
Statistical methods date back at least to the 5th century BC.
Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt.[33] Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences.
Its mathematical foundations were laid in the 17th century with the development of the probability theory by Gerolamo Cardano, Blaise Pascal and Pierre de Fermat. Mathematical probability theory arose from the study of games of chance, although the concept of probability was already examined in medieval law and by philosophers such as Juan Caramuel.[34] The method of least squares was first described by Adrien-Marie Legendre in 1805.
Karl Pearson, a founder of mathematical statistics.
The modern field of statistics emerged in the late 19th and early 20th century in three stages.[35] The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. Galton's contributions included introducing the concepts of standard deviation, correlation, regression analysis and the application of these methods to the study of the variety of human characteristics – height, weight, eyelash length among others.[36] Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment,[37] the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things.[38] Galton and Pearson founded Biometrika as the first journal of mathematical statistics and biostatistics (then called biometry), and the latter founded the world's first university statistics department at University College London.[39]
Ronald Fisher coined the term null hypothesis during the Lady tasting tea experiment, which "is never proved or established, but is possibly disproved, in the course of experimentation".[40][41]
The second wave of the 1910s and 20s was initiated by William Gosset, and reached its culmination in the insights of Ronald 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 paper The Correlation between Relatives on the Supposition of Mendelian Inheritance, which was the first to use the statistical term, variance, his classic 1925 work Statistical Methods for Research Workers and his 1935 The Design of Experiments,[42][43][44][45] where he developed rigorous design of experiments models. He originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information.[46] In his 1930 book The Genetical Theory of Natural Selection he applied statistics to various biological concepts such as Fisher's principle[47]). Nevertheless, A. W. F. Edwards has remarked that it is "probably the most celebrated argument in evolutionary biology".[47] (about the sex ratio), the Fisherian runaway,[48][49][50][51][52][53] a concept in sexual selection about a positive feedback runaway affect found in evolution.
The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. They introduced the concepts of "Type II" error, power of a test and confidence intervals. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.[54]
Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. The use of modern computers has expedited large-scale statistical computations, and has also made possible new methods that are impractical to perform manually. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data.[55]
Applications[edit]
Applied statistics, theoretical statistics and mathematical statistics[edit]
"Applied statistics" comprises descriptive statistics and the application of inferential statistics.[56][57] Theoretical statistics concerns both the logical arguments underlying justification of approaches to statistical inference, as well encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.
Machine learning and data mining[edit]
There are two applications for machine learning and data mining: data management and data analysis. Statistics tools are necessary for the data analysis.
Statistics in society[edit]
Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical consultants can help organizations and companies that don't have in-house expertise relevant to their particular questions.
Statistical computing[edit]
gretl, an example of an open 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 of linear models, but powerful computers, coupled with suitable numerical algorithms, caused an increased interest in nonlinear models (such as neural networks) as well as the creation of new types, such as generalized linear models and multilevel models.
Increased computing power has also led to the growing popularity of computationally intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs sampling have made use of Bayesian models more feasible. The computer revolution has implications for the future of statistics with new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose statistical software are now available.
Statistics applied to mathematics or the arts[edit]
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.[according to whom?] Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.
In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
Methods of statistics including predictive methods in forecasting are combined with chaos theory and fractal geometry to create video works that are considered to have great beauty.[citation needed]
The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed.[citation needed] With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art.[citation needed]
Methods of statistics may be used predicatively in performance art, as in a card trick based on a Markov process that only works some of the time, the occasion of which can be predicted using statistical methodology.
Statistics can be used to predicatively create art, as in the statistical or stochastic music invented by Iannis Xenakis, where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tunable using statistics.
Specialized disciplines[edit]
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, sociology and social research. Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:
Actuarial science (assesses risk in the insurance and finance industries)
Applied information economics
Astrostatistics (statistical evaluation of astronomical data)
Biostatistics
Business statistics
Chemometrics (for analysis of data from chemistry)
Data mining (applying statistics and pattern recognition to discover knowledge from data)
Data science
Demography
Econometrics (statistical analysis of economic data)
Energy statistics
Engineering statistics
Epidemiology (statistical analysis of disease)
Geography and Geographic Information Systems, specifically in Spatial analysis
Image processing
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 (statistics)
Structural equation modelling
Survey methodology
Survival analysis
Statistics in various sports, particularly baseball - known as Sabermetrics - and cricket
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles, it is a key tool, and perhaps the only reliable tool.
See also[edit]
Library resources about
Statistics
Resources in your library
Main article: Outline of statistics
Abundance estimation
Data science
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
Foundations and major areas of statistics
Foundations of statistics
List of statisticians
Official statistics
Multivariate analysis of variance
References[edit]
^ Jump up to: a b Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
Jump up ^ Romijn, Jan-Willem (2014). "Philosophy of statistics". Stanford Encyclopedia of Philosophy.
Jump up ^ Lund Research Ltd. "Descriptive and Inferential Statistics". statistics.laerd.com. Retrieved 2014-03-23.
Jump up ^ "What Is the Difference Between Type I and Type II Hypothesis Testing Errors?". About.com Education. Retrieved 2015-11-27.
Jump up ^ "Definition of STATISTICS". www.merriam-webster.com. Retrieved 2016-05-28.
Jump up ^ "Essay on Statistics: Meaning and Definition of Statistics". Economics Discussion. 2014-12-02. Retrieved 2016-05-28.
Jump up ^ Moses, Lincoln E. (1986) Think and Explain with Statistics, Addison-Wesley, ISBN 978-0-201-15619-5 . pp. 1–3
Jump up ^ Hays, William Lee, (1973) Statistics for the Social Sciences, Holt, Rinehart and Winston, p.xii, ISBN 978-0-03-077945-9
Jump up ^ Moore, David (1992). "Teaching Statistics as a Respectable Subject". In F. Gordon and S. Gordon. Statistics for the Twenty-First Century. Washington, DC: The Mathematical Association of America. pp. 14–25. ISBN 978-0-88385-078-7. CS1 maint: Uses editors parameter (link)
Jump up ^ Chance, Beth L.; Rossman, Allan J. (2005). "Preface". Investigating Statistical Concepts, Applications, and Methods (PDF). Duxbury Press. ISBN 978-0-495-05064-3.
Jump up ^ Lakshmikantham,, ed. by D. Kannan,... V. (2002). Handbook of stochastic analysis and applications. New York: M. Dekker. ISBN 0824706609. CS1 maint: Extra text: authors list (link)
Jump up ^ Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.
Jump up ^ Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7
Jump up ^ McCarney R, Warner J, Iliffe S, van Haselen R, Griffin M, Fisher P (2007). "The Hawthorne Effect: a randomised, controlled trial". BMC Med Res Methodol. 7 (1): 30. PMC 1936999  . PMID 17608932. doi:10.1186/1471-2288-7-30.
Jump up ^ Rothman, Kenneth J; Greenland, Sander; Lash, Timothy, eds. (2008). "7". Modern Epidemiology (3rd ed.). Lippincott Williams & Wilkins. p. 100.
Jump up ^ Mosteller, F., & Tukey, J. W. (1977). Data analysis and regression. Boston: Addison-Wesley.
Jump up ^ Nelder, J. A. (1990). The knowledge needed to computerise the analysis and interpretation of statistical information. In Expert systems and artificial intelligence: the need for information about data. Library Association Report, London, March, 23–27.
Jump up ^ Chrisman, Nicholas R (1998). "Rethinking Levels of Measurement for Cartography". Cartography and Geographic Information Science. 25 (4): 231–242. doi:10.1559/152304098782383043.
Jump up ^ van den Berg, G. (1991). Choosing an analysis method. Leiden: DSWO Press
Jump up ^ Hand, D. J. (2004). Measurement theory and practice: The world through quantification. London, UK: Arnold.
^ Jump up to: a b Piazza Elio, Probabilità e Statistica, Esculapio 2007
Jump up ^ Everitt, Brian (1998). The Cambridge Dictionary of Statistics. Cambridge, UK New York: Cambridge University Press. ISBN 0521593468.
Jump up ^ "Cohen (1994) The Earth Is Round (p < .05) - YourStatsGuru.com - YourStatsGuru.com".
Jump up ^ Rubin, Donald B.; Little, Roderick J. A., Statistical analysis with missing data, New York: Wiley 2002
Jump up ^ Ioannidis, J. P. A. (2005). "Why Most Published Research Findings Are False". PLoS Medicine. 2 (8): e124. PMC 1182327  . PMID 16060722. doi:10.1371/journal.pmed.0020124.
^ Jump up to: a b Huff, Darrell (1954) How to Lie with Statistics, WW Norton & Company, Inc. New York, NY. ISBN 0-393-31072-8
Jump up ^ Warne, R. Lazo; Ramos, T.; Ritter, N. (2012). "Statistical Methods Used in Gifted Education Journals, 2006–2010". Gifted Child Quarterly. 56 (3): 134–149. doi:10.1177/0016986212444122.
^ Jump up to: a b Drennan, Robert D. (2008). "Statistics in archaeology". In Pearsall, Deborah M. Encyclopedia of Archaeology. Elsevier Inc. pp. 2093–2100. ISBN 978-0-12-373962-9.
^ Jump up to: a b Cohen, Jerome B. (December 1938). "Misuse of Statistics". Journal of the American Statistical Association. JSTOR. 33 (204): 657–674. doi:10.1080/01621459.1938.10502344.
Jump up ^ Freund, J. E. (1988). "Modern Elementary Statistics". Credo Reference.
Jump up ^ Huff, Darrell; Irving Geis (1954). How to Lie with Statistics. New York: Norton. The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism.
Jump up ^ Huff, Darrell; Irving Geis (1954). How to Lie with Statistics. New York: Norton.
Jump up ^ Willcox, Walter (1938) "The Founder of Statistics". Review of the International Statistical Institute 5(4):321–328. JSTOR 1400906
Jump up ^ J. Franklin, The Science of Conjecture: Evidence and Probability before Pascal, Johns Hopkins Univ Pr 2002
Jump up ^ Helen Mary Walker (1975). Studies in the history of statistical method. Arno Press.
Jump up ^ Galton, F (1877). "Typical laws of heredity". Nature. 15: 492–553. doi:10.1038/015492a0.
Jump up ^ Stigler, S. M. (1989). "Francis Galton's Account of the Invention of Correlation". Statistical Science. 4 (2): 73–79. doi:10.1214/ss/1177012580.
Jump up ^ Pearson, K. (1900). "On the Criterion that a given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling". Philosophical Magazine Series 5. 50 (302): 157–175. doi:10.1080/14786440009463897.
Jump up ^ "Karl Pearson (1857–1936)". Department of Statistical Science – University College London.
Jump up ^ Fisher|1971|loc=Chapter II. The Principles of Experimentation, Illustrated by a Psycho-physical Experiment, Section 8. The Null Hypothesis
Jump up ^ OED quote: 1935 R. A. Fisher, The Design of Experiments ii. 19, "We may speak of this hypothesis as the 'null hypothesis', and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation."
Jump up ^ Stanley, J. C. (1966). "The Influence of Fisher's "The Design of Experiments" on Educational Research Thirty Years Later". American Educational Research Journal. 3 (3): 223. doi:10.3102/00028312003003223.
Jump up ^ Box, JF (February 1980). "R. A. Fisher and the Design of Experiments, 1922-1926". The American Statistician. 34 (1): 1–7. JSTOR 2682986. doi:10.2307/2682986.
Jump up ^ Yates, F (June 1964). "Sir Ronald Fisher and the Design of Experiments". Biometrics. 20 (2): 307–321. JSTOR 2528399. doi:10.2307/2528399.
Jump up ^ Stanley, Julian C. (1966). "The Influence of Fisher's "The Design of Experiments" on Educational Research Thirty Years Later". American Educational Research Journal. 3 (3): 223–229. JSTOR 1161806. doi:10.3102/00028312003003223.
Jump up ^ Agresti, Alan; David B. Hichcock (2005). "Bayesian Inference for Categorical Data Analysis" (PDF). Statistical Methods & Applications. 14 (14): 298. doi:10.1007/s10260-005-0121-y.
^ Jump up to: a b Edwards, A.W.F. (1998). "Natural Selection and the Sex Ratio: Fisher's Sources". American Naturalist. 151 (6): 564–569. PMID 18811377. doi:10.1086/286141.
Jump up ^ Fisher, R.A. (1915) The evolution of sexual preference. Eugenics Review (7) 184:192
Jump up ^ Fisher, R.A. (1930) The Genetical Theory of Natural Selection. ISBN 0-19-850440-3
Jump up ^ Edwards, A.W.F. (2000) Perspectives: Anecdotal, Historial and Critical Commentaries on Genetics. The Genetics Society of America (154) 1419:1426
Jump up ^ Andersson, M. (1994) Sexual selection. ISBN 0-691-00057-3
Jump up ^ Andersson, M. and Simmons, L.W. (2006) Sexual selection and mate choice. Trends, Ecology and Evolution (21) 296:302
Jump up ^ Gayon, J. (2010) Sexual selection: Another Darwinian process. Comptes Rendus Biologies (333) 134:144
Jump up ^ Neyman, J (1934). "On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection". Journal of the Royal Statistical Society. 97 (4): 557–625. JSTOR 2342192.
Jump up ^ "Science in a Complex World - Big Data: Opportunity or Threat?". Santa Fe Institute.
Jump up ^ Nikoletseas, M. M. (2014) "Statistics: Concepts and Examples." ISBN 978-1500815684
Jump up ^ Anderson, D.R.; Sweeney, D.J.; Williams, T.A. (1994) Introduction to Statistics: Concepts and Applications, pp. 5–9. West Group. ISBN 978-0-314-03309-3
Further reading[edit]
Barbara Illowsky; Susan Dean (2014). Introductory Statistics. OpenStax CNX. ISBN 9781938168208.
Stephen Jones, 2010. Statistics in Psychology: Explanations without Equations. Palgrave Macmillan. ISBN 9781137282392.
Cohen, J. (1990). Things I have learned (so far). American Psychologist, 45, 1304-1312.
External links[edit]
(Electronic Version): StatSoft, Inc. (2013). Electronic Statistics Textbook. Tulsa, OK: StatSoft.
UCLA Statistical Computing Resources
Template:Sisters project links
[hide]
v
t
e
Statistics
Outline
Index
[show]
Descriptive statistics
Continuous data
Center
Mean
arithmetic
geometric
harmonic
Median
Mode
Dispersion
Variance
Standard deviation
Coefficient of variation
Percentile
Range
Interquartile range
Shape
Central limit theorem
Moments
Skewness
Kurtosis
L-moments
Count data
Index of dispersion
Summary tables
Grouped data
Frequency distribution
Contingency table
Dependence
Pearson product-moment correlation
Rank correlation
Spearman's rho
Kendall's tau
Partial correlation
Scatter plot
Graphics
Bar chart
Biplot
Box plot
Control chart
Correlogram
Fan chart
Forest plot
Histogram
Pie chart
Q–Q plot
Run chart
Scatter plot
Stem-and-leaf display
Radar chart
[show]
Data collection
Study design
Population
Statistic
Effect size
Statistical power
Sample size determination
Missing data
Survey methodology
Sampling
stratified
cluster
Standard error
Opinion poll
Questionnaire
Controlled experiments
Design
control
optimal
Controlled trial
Randomized
Random assignment
Replication
Blocking
Interaction
Factorial experiment
Uncontrolled studies
Observational study
Natural experiment
Quasi-experiment
[show]
Statistical inference
Statistical theory
Population
Statistic
Probability distribution
Sampling distribution
Order statistic
Empirical distribution
Density estimation
Statistical model
Lp space
Parameter
location
scale
shape
Parametric family
Likelihood (monotone)
Location–scale family
Exponential family
Completeness
Sufficiency
Statistical functional
Bootstrap
U
V
Optimal decision
loss function
Efficiency
Statistical distance
divergence
Asymptotics
Robustness
Frequentist inference
Point estimation
Estimating equations
Maximum likelihood
Method of moments
M-estimator
Minimum distance
Unbiased estimators
Mean-unbiased minimum-variance
Rao–Blackwellization
Lehmann–Scheffé theorem
Median unbiased
Plug-in
Interval estimation
Confidence interval
Pivot
Likelihood interval
Prediction interval
Tolerance interval
Resampling
Bootstrap
Jackknife
Testing hypotheses
1- & 2-tails
Power
Uniformly most powerful test
Permutation test
Randomization test
Multiple comparisons
Parametric tests
Likelihood-ratio
Wald
Score
Specific tests
Z (normal)
Student's t-test
F
Goodness of fit
Chi-squared
Kolmogorov–Smirnov
Anderson–Darling
Lilliefors
Jarque–Bera
Normality (Shapiro–Wilk)
Likelihood-ratio test
Model selection
Cross validation
AIC
BIC
Rank statistics
Sign
Sample median
Signed rank (Wilcoxon)
Hodges–Lehmann estimator
Rank sum (Mann–Whitney)
Nonparametric anova
1-way (Kruskal–Wallis)
2-way (Friedman)
Ordered alternative (Jonckheere–Terpstra)
Bayesian inference
Bayesian probability
prior
posterior
Credible interval
Bayes factor
Bayesian estimator
Maximum posterior estimator
[show]
Correlation
Regression analysis
Correlation
Pearson product-moment
Partial correlation
Confounding variable
Coefficient of determination
Regression analysis
Errors and residuals
Regression model validation
Mixed effects models
Simultaneous equations models
Multivariate adaptive regression splines (MARS)
Linear regression
Simple linear regression
Ordinary least squares
General linear model
Bayesian regression
Non-standard predictors
Nonlinear regression
Nonparametric
Semiparametric
Isotonic
Robust
Heteroscedasticity
Homoscedasticity
Generalized linear model
Exponential families
Logistic (Bernoulli) / Binomial / Poisson regressions
Partition of variance
Analysis of variance (ANOVA, anova)
Analysis of covariance
Multivariate ANOVA
Degrees of freedom
[show]
Categorical / Multivariate / Time-series / Survival analysis
Categorical
Cohen's kappa
Contingency table
Graphical model
Log-linear model
McNemar's test
Multivariate
Regression
Manova
Principal components
Canonical correlation
Discriminant analysis
Cluster analysis
Classification
Structural equation model
Factor analysis
Multivariate distributions
Elliptical distributions
Normal
Time-series
General
Decomposition
Trend
Stationarity
Seasonal adjustment
Exponential smoothing
Cointegration
Structural break
Granger causality
Specific tests
Dickey–Fuller
Johansen
Q-statistic (Ljung–Box)
Durbin–Watson
Breusch–Godfrey
Time domain
Autocorrelation (ACF)
partial (PACF)
Cross-correlation (XCF)
ARMA model
ARIMA model (Box–Jenkins)
Autoregressive conditional heteroskedasticity (ARCH)
Vector autoregression (VAR)
Frequency domain
Spectral density estimation
Fourier analysis
Wavelet
Survival
Survival function
Kaplan–Meier estimator (product limit)
Proportional hazards models
Accelerated failure time (AFT) model
First hitting time
Hazard function
Nelson–Aalen estimator
Test
Log-rank test
[show]
Applications
Biostatistics
Bioinformatics
Clinical trials / studies
Epidemiology
Medical statistics
Engineering statistics
Chemometrics
Methods engineering
Probabilistic design
Process / quality control
Reliability
System identification
Social statistics
Actuarial science
Census
Crime statistics
Demography
Econometrics
National accounts
Official statistics
Population statistics
Psychometrics
Spatial statistics
Cartography
Environmental statistics
Geographic information system
Geostatistics
Kriging
Category
Portal
Commons
WikiProject
[show]
v
t
e
Areas of mathematics
outline
topic lists
Areas
Algebra
elementary
linear
multilinear
abstract
Arithmetic / Number theory
Calculus / Analysis
Category theory
Combinatorics
Group theory
Computation
Control theory
Differential equations / Dynamical systems
Functional analysis
Game theory
Geometry
discrete
algebraic
analytic
differential
finite
Graph theory
History of mathematics
Information theory
Lie theory
Mathematics and art
Mathematics education
Mathematical logic
Mathematical physics
Mathematical statistics
Numerical analysis
Optimization
Order theory
Philosophy of mathematics
Philosophy of mathematics education
Probability
Recreational mathematics
Representation theory
Set theory
Statistics
Topology
Trigonometry
Divisions
Pure
Applied
Discrete
Computational
Category
Portal
Commons
WikiProject
Statistics portal
Mathematics portal
Authority control
GND: 4056995-0
NDL: 00573173
Retrieved from "https://en.wikipedia.org/w/index.php?title=Statistics&oldid=804851008"
Categories:
Statistics
Data
Formal sciences
Information
Mathematical and quantitative methods (economics)
Research methods
Hidden categories:
CS1 maint: Uses editors parameter
CS1 maint: Extra text: authors list
All articles with unsourced statements
Articles with unsourced statements from April 2015
Wikipedia articles needing clarification from October 2016
All articles with specifically marked weasel-worded phrases
Articles with specifically marked weasel-worded phrases from April 2014
Articles with unsourced statements from February 2015
Articles with unsourced statements from March 2013
Wikipedia articles with GND identifiers
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Wikibooks
Wikiquote
Wikiversity
Languages
Afrikaans
Alemannisch
አማርኛ
العربية
Aragonés
Asturianu
Azərbaycanca
تۆرکجه
বাংলা
Bân-lâm-gú
Башҡортса
Беларуская
Беларуская (тарашкевіца)
Bislama
Български
Bosanski
Brezhoneg
Català
Čeština
Cymraeg
Dansk
Deutsch
ދިވެހިބަސް
Eesti
Ελληνικά
Español
Esperanto
Estremeñu
Euskara
فارسی
Føroyskt
Français
Frysk
Furlan
Gaeilge
Gaelg
Gàidhlig
Galego
贛語
한국어
Հայերեն
हिन्दी
Hrvatski
Ido
Bahasa Indonesia
Interlingua
ᐃᓄᒃᑎᑐᑦ/inuktitut
Íslenska
Italiano
עברית
Basa Jawa
ಕನ್ನಡ
ქართული
Қазақша
Kiswahili
Kurdî
Кыргызча
Ladino
ລາວ
Latina
Latviešu
Lëtzebuergesch
Lietuvių
Limburgs
Magyar
Македонски
Malagasy
മലയാളം
मराठी
Bahasa Melayu
Mirandés
မြန်မာဘာသာ
Nederlands
नेपाली
नेपाल भाषा
日本語
Norfuk / Pitkern
Norsk
Norsk nynorsk
Occitan
ଓଡ଼ିଆ
Oʻzbekcha/ўзбекча
ਪੰਜਾਬੀ
پنجابی
پښتو
Patois
Piemontèis
Polski
Português
Română
Русиньскый
Русский
Scots
Seeltersk
Shqip
Sicilianu
සිංහල
Simple English
سنڌي
Slovenčina
Slovenščina
کوردی
Српски / srpski
Srpskohrvatski / српскохрватски
Basa Sunda
Suomi
Svenska
Tagalog
தமிழ்
Татарча/tatarça
తెలుగు
ไทย
Тоҷикӣ
ತುಳು
Türkçe
Türkmençe
Українська
اردو
Vèneto
Tiếng Việt
Võro
Winaray
ייִדיש
Yorùbá
粵語
Zazaki
Žemaitėška
中文
Edit links
This page was last edited on 11 October 2017, at 14:40.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

attrs: alt src width height class srcset data-file-width data-file-height

attrs: href class title

values:
/wiki/Standardized_testing_(statistics)
mw-redirect
Standardized testing (statistics)

values:
/wiki/Standard_deviations
mw-redirect
Standard deviations

attrs: class style

values:
/wiki/Statistical_survey
mw-redirect
Statistical survey

values:
/wiki/Experimental_design
mw-redirect
Experimental design

values:
/wiki/Experimental_study
mw-redirect
Experimental study

values:
/wiki/Type_I_error
mw-redirect
Type I error

values:
/wiki/Type_II_error
mw-redirect
Type II error

values: This claim needs references to reliable sources. (April 2015)

values:
noprint Inline-Template Template-Fact
white-space:nowrap;

values:
/wiki/Stochastic_analysis
mw-redirect
Stochastic analysis

values:
/wiki/Differential_equations
mw-redirect
Differential equations

values:
/wiki/Categorical_data
mw-redirect
Categorical data

values:
/wiki/Inferential_statistics
mw-redirect
Inferential statistics

values:
/wiki/Hypothesis_testing
mw-redirect
Hypothesis testing

values:
/wiki/Association_(statistics)
mw-redirect
Association (statistics)

values:
/wiki/Spatial_data_analysis
mw-redirect
Spatial data analysis

values:
/wiki/Sample_statistic
mw-redirect
Sample statistic

values:
/wiki/Statistical_decision_theory
mw-redirect
Statistical decision theory

values: Edit section: Experimental and observational studies

values:
/wiki/Instrumental_variable
mw-redirect
Instrumental variable

values:
/wiki/Experimental_error
mw-redirect
Experimental error

values:
/wiki/Confounding_variable
mw-redirect
Confounding variable

values:
/wiki/Randomized_assignment
mw-redirect
Randomized assignment

values:
/wiki/Protocol_(natural_sciences)
mw-redirect
Protocol (natural sciences)

values:
/wiki/Hawthorne_study
mw-redirect
Hawthorne study

values:
/wiki/Western_Electric_Company
mw-redirect
Western Electric Company

values:
/wiki/Control_group
mw-redirect
Control group

values:
/wiki/Levels_of_measurement
mw-redirect
Levels of measurement

values:
/wiki/Floating_point
mw-redirect
Floating point

values: Edit section: Terminology and theory of inferential statistics

values: Edit section: Statistics, estimators and pivotal quantities

values:
/wiki/Random_sample
mw-redirect
Random sample

values:
/wiki/Random_vector
mw-redirect
Random vector

values:
/wiki/Column_vector
mw-redirect
Column vector

values:
/wiki/Sample_mean#Sample_mean
mw-redirect
Sample mean

values:
/wiki/Sample_variance
mw-redirect
Sample variance

values:
/wiki/Unbiased_estimator
mw-redirect
Unbiased estimator

values:
/wiki/Converges_in_probability
mw-redirect
Converges in probability

values:
/wiki/Maximum_likelihood
mw-redirect
Maximum likelihood

values: Edit section: Null hypothesis and alternative hypothesis

values:
/wiki/Efficient_estimators
mw-redirect
Efficient estimators

values:
/wiki/Root_mean_square_error
mw-redirect
Root mean square error

values:
/wiki/Random_error
mw-redirect
Random error

values:
/wiki/Systematic_error
mw-redirect
Systematic error

values:
/wiki/Confidence_intervals
mw-redirect
Confidence intervals

values:
/wiki/Significance_level
mw-redirect
Significance level

values: The text near this tag may need clarification or removal of jargon. (October 2016)

values:
noprint Inline-Template
margin-left:0.1em; white-space:nowrap;

values:
/wiki/Mann%E2%80%93Whitney_U
mw-redirect
Mann–Whitney U

values:
/wiki/Mean_square_weighted_deviation
mw-redirect
Mean square weighted deviation

values: Pearson product-moment correlation coefficient

values:
/wiki/Time_series_analysis
mw-redirect
Time series analysis

values:
/wiki/Conjoint_Analysis
mw-redirect
Conjoint Analysis

values: div-col columns column-count column-count-2

values: -moz-column-count: 2; -webkit-column-count: 2; column-count: 2;

values: Edit section: Misinterpretation: correlation

values: Edit section: History of statistical science

values:
/wiki/Juan_Caramuel
mw-redirect
Juan Caramuel

values:
/wiki/Method_of_least_squares
mw-redirect
Method of least squares

values:
/wiki/William_Gosset
mw-redirect
William Gosset

values: Statistical Methods for Research Workers

values:
/wiki/Sufficiency_(statistics)
mw-redirect
Sufficiency (statistics)

values: The Genetical Theory of Natural Selection

values: Edit section: Applied statistics, theoretical statistics and mathematical statistics

values: Edit section: Machine learning and data mining

values:
/wiki/Academic_discipline
mw-redirect
Academic discipline

values:
/wiki/Statistical_consultant
mw-redirect
Statistical consultant

values:
/wiki/Open_source
mw-redirect
Open source

values:
/wiki/Neural_networks
mw-redirect
Neural networks

values: Edit section: Statistics applied to mathematics or the arts

values: The material near this tag may use weasel words or too-vague attribution. (April 2014)

values: Wikipedia:Manual of Style/Words to watch

values:
/wiki/Fractal_geometry
mw-redirect
Fractal geometry

values: This claim needs references to reliable sources. (February 2015)

values: This claim needs references to reliable sources. (March 2013)

values:
/wiki/Markov_process
mw-redirect
Markov process

values:
/wiki/Stochastic_music
mw-redirect
Stochastic music

values: List of fields of application of statistics

values:
/wiki/Network_biology
mw-redirect
Network biology

values:
/wiki/Specialized_terminology
mw-redirect
Specialized terminology

values:
/wiki/Geographic_Information_Systems
mw-redirect
Geographic Information Systems

values:
/wiki/Image_processing
mw-redirect
Image processing

values:
/wiki/Medical_Statistics
mw-redirect
Medical Statistics

values:
/wiki/Political_Science
mw-redirect
Political Science

values:
/wiki/Statistical_Mechanics
mw-redirect
Statistical Mechanics

values:
/wiki/Structural_equation_modelling
mw-redirect
Structural equation modelling

values:
plainlist
padding:0 0.1em 0.4em;text-align:left;

values: float:right;clear:right;width:auto;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%

values: List of academic statistical associations

values: List of important publications in statistics

values: List of national and international statistical services

values: List of university statistical consulting centers

values: -moz-column-width: 20em; -webkit-column-width: 20em; column-width: 20em;

values: Category:CS1 maint: Uses editors parameter

values: Category:CS1 maint: Extra text: authors list

values:
/wiki/David_A._Freedman_(statistician)
mw-redirect
David A. Freedman (statistician)

values:
/wiki/PubMed_Identifier
mw-redirect
PubMed Identifier

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/17608932

values:
nofollow
external text
//doi.org/10.1186%2F1471-2288-7-30

values:
/wiki/John_P._A._Ioannidis
mw-redirect
John P. A. Ioannidis

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/16060722

values:
nofollow
external text
https://www.jstor.org/stable/1400906

values:
nofollow
external text
//doi.org/10.1038%2F015492a0

values:
nofollow
external text
//doi.org/10.1214%2Fss%2F1177012580

values:
nofollow
external text
//www.jstor.org/stable/2682986

values:
nofollow
external text
//doi.org/10.2307%2F2682986

values:
nofollow
external text
//www.jstor.org/stable/2528399

values:
nofollow
external text
//doi.org/10.2307%2F2528399

values:
nofollow
external text
//www.jstor.org/stable/1161806

values:
nofollow
external text
//doi.org/10.1007%2Fs10260-005-0121-y

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/18811377

values:
nofollow
external text
//doi.org/10.1086%2F286141

values: Journal of the Royal Statistical Society

values:
nofollow
external text
//www.jstor.org/stable/2342192

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values:
nofollow
http://www.statsoft.com/textbook/
external text

values: Template:Sisters project links (page does not exist)

values:
/wiki/Shape_of_the_distribution
mw-redirect
Shape of the distribution

values: Kendall tau rank correlation coefficient

values: nowraplinks collapsible collapsed navbox-subgroup

values:
/wiki/Population_(statistics)
mw-redirect
Population (statistics)

values:
/wiki/Median-unbiased_estimator
mw-redirect
Median-unbiased estimator

values:
/wiki/Likelihood_interval
mw-redirect
Likelihood interval

values:
/wiki/Power_(statistics)
mw-redirect
Power (statistics)

values:
/wiki/Permutation_test
mw-redirect
Permutation test

values:
/wiki/Randomization_test
mw-redirect
Randomization test

values:
/wiki/Multiple_comparisons
mw-redirect
Multiple comparisons

values:
/wiki/Rank_statistics
mw-redirect
Rank statistics

values:
/wiki/Sample_median
mw-redirect
Sample median

values: Kruskal–Wallis one-way analysis of variance

values:
CorrelationRegression_analysis
font-size:114%;margin:0 4em
hlist hlist-separated

values:
/wiki/Regression_model_validation
mw-redirect
Regression model validation

values: Multivariate adaptive regression splines

values:
/wiki/Multivariate_distribution
mw-redirect
Multivariate distribution

values:
/wiki/Trend_estimation
mw-redirect
Trend estimation

values: nowraplinks hlist collapsible expanded navbox-inner

values: navigation navbox Statistics padding:3px

values: nowraplinks hlist collapsible autocollapse navbox-inner

values:
noprint metadata navbox
navigation
Portals
font-weight:bold;padding:0.4em 2em

values:
nofollow
external text
http://d-nb.info/gnd/4056995-0

values: Category:Mathematical and quantitative methods (economics)

values: Category:All articles with unsourced statements

values: Category:Articles with unsourced statements from April 2015

values: Category:Wikipedia articles needing clarification from October 2016

values: Category:All articles with specifically marked weasel-worded phrases

values: Category:Articles with specifically marked weasel-worded phrases from April 2014

values: Category:Articles with unsourced statements from February 2015

values: Category:Articles with unsourced statements from March 2013

values: Category:Wikipedia articles with GND identifiers

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Statistics
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label

values:
uls-settings-trigger
Language settings
0
button
true

values: የዝርዝር ሂሳብ (እስታቲስቲክስ) – Amharic

values: إحصاء – Arabic

values: পরিসংখ্যান – Bangla

values: Статистика – Bashkir

values: Статыстыка – Belarusian

values: Статыстыка – Belarusian (Taraškievica orthography)

values: Статистика – Bulgarian

values:
https://da.wikipedia.org/wiki/Statistik
Statistik – Danish
da
da
interlanguage-link-target

values:
https://de.wikipedia.org/wiki/Statistik
Statistik – German
de
de
interlanguage-link-target

values: ތަފާސް ހިސާބު – Divehi

values: Στατιστική – Greek

values: interlanguage-link interwiki-eu badge-Q17437796 badge-featuredarticle

values:
https://fy.wikipedia.org/wiki/Statistyk
Statistyk – Western Frisian
fy
fy
interlanguage-link-target

values:
https://gv.wikipedia.org/wiki/Staydraa
Staydraa – Manx
gv
gv
interlanguage-link-target

values: Վիճակագրություն – Armenian

values: सांख्यिकी – Hindi

values: ᑭᓯᑦᓯᓯᖕᖑᕐᓗᒋᑦ ᐹᓯᔅᓱᑎᔅᓴᑦ – Inuktitut

values: סטטיסטיקה – Hebrew

values: ಸಂಖ್ಯಾಶಾಸ್ತ್ರ – Kannada

values: სტატისტიკა (მეცნიერება) – Georgian

values: Статистика – Kazakh

values:
https://sw.wikipedia.org/wiki/Takwimu
Takwimu – Swahili
sw
sw
interlanguage-link-target

values:
https://ku.wikipedia.org/wiki/Amar
Amar – Kurdish
ku
ku
interlanguage-link-target

values: Статистика – Kyrgyz

values: ສະຖິຕິສາດ – Lao

values:
https://lb.wikipedia.org/wiki/Statistik
Statistik – Luxembourgish
lb
lb
interlanguage-link-target

values: Статистика – Macedonian

values: സ്ഥിതിഗണിതം – Malayalam

values: संख्याशास्त्र – Marathi

values:
https://ms.wikipedia.org/wiki/Statistik
Statistik – Malay
ms
ms
interlanguage-link-target

values: စာရင်းအင်း ပညာ – Burmese

values: तथ्याङ्क – Nepali

values: तथ्याङ्क – Newari

values: ପରିସଂଖ୍ୟାନ – Odia

values: ਅੰਕੜਾ ਵਿਗਿਆਨ – Punjabi

values: سرشمیرنه – Pashto

values: Штатістіка – Rusyn

values: Статистика – Russian

values: සංඛ්‍යානය – Sinhala

values: اِسٽيٽسٽڪس – Sindhi

values: Статистика – Serbian

values:
https://su.wikipedia.org/wiki/Statistik
Statistik – Sundanese
su
su
interlanguage-link-target

values:
https://sv.wikipedia.org/wiki/Statistik
Statistik – Swedish
sv
sv
interlanguage-link-target

values: புள்ளியியல் – Tamil

values: Статистика – Tatar

values: సాంఖ్యక శాస్త్రం – Telugu

values: สถิติศาสตร์ – Thai

values: Омор – Tajik

values: ಸಂಖ್ಯಾಶಾಸ್ತ್ರ – Tulu

values: Статистика – Ukrainian

values: شماریات – Urdu

values: סטאטיסטיק – Yiddish

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Statistics rootpage-Statistics vector-nav-directionality skin-vector action-view