Statistics

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation.[1][2] In applying statistics to, for example, 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 every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.[1] See glossary of probability and statistics.

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 an 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.

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. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.[5]

The Normal Distribution
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, Z-scores, and T-scores.
Scatterplot
Scatter plots are used in descriptive statistics to show the observed relationships between different variables.

Scope

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."[6]
  • Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."[7]

Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data,[8] or as a branch of mathematics.[9] 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.[10][11]

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.[12][13]

Overview

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 people 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

Sampling

When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through statistical models. The idea of making inferences based on sampled data began around the mid-1600s in connection with estimating populations and developing precursors of life insurance.[14]

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

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[15]—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

The basic steps of a statistical experiment are:

  1. Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size 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.
  2. 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.
  3. Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
  4. Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
  5. Documenting and presenting the results of the study.

Experiments on human behavior have special concerns. The 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.[16]

Observational study

An example of an observational study is one that explores the association between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a cohort study, and then look for the number of cases of lung cancer in each group.[17] A case-control study is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.

Types of data

Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case 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)[18] distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)[19] described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998),[20] van den Berg (1991).[21]

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).[22]

Terminology and theory of inferential statistics

Statistics, estimators and pivotal quantities

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.[23] 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

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.[24][25]

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

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.

Linear least squares(2)
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.[26]

Interval estimation

NYW-confidence-interval
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

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).

P-value in statistical significance testing
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[23] 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.

Although 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 test is logically equivalent to saying that the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. 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 the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result. An alternative to this approach is offered by Bayesian inference, although it requires establishing a prior probability.[27]
  • 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.

Examples

Some well-known statistical tests and procedures are:

Misuse

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.[28] 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[28] 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)).[29]

Ways to avoid misuse of statistics include using proper diagrams and avoiding bias.[30] 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.[31] Bar graphs are arguably the easiest diagrams to use and understand, and they can be made either by hand or with simple computer programs.[30] 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.[31] To make data gathered from statistics believable and accurate, the sample taken must be representative of the whole.[32] According to Huff, "The dependability of a sample can be destroyed by [bias]... allow yourself some degree of skepticism."[33]

To assist in the understanding of statistics Huff proposed a series of questions to be asked in each case:[34]

  • 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?)
Simple Confounding Case
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

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

Jerôme Cardan
Gerolamo Cardano, a pioneer on the mathematics of probability.

The earliest writing on statistics was found in a 9th-century book entitled Manuscript on Deciphering Cryptographic Messages, written by Arab scholar Al-Kindi (801–873). In his book, Al-Kindi gave a detailed description of how to use statistics and frequency analysis to decipher encrypted messages. This text laid the foundations for statistics and cryptanalysis.[35][36] Al-Kindi also made the earliest known use of statistical inference, while he and other Arab cryptologists developed the early statistical methods for decoding encrypted messages. Arab mathematicians including Al-Kindi, Al-Khalil (717–786) and Ibn Adlan (1187–1268) used forms of probability and statistics, with one of Ibn Adlan's most important contributions being on sample size for use of frequency analysis.[37]

The earliest European writing on statistics dates back to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by John Graunt.[38] Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences.

The mathematical foundations of modern statistics 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.[39] The method of least squares was first described by Adrien-Marie Legendre in 1805.

Karl Pearson, 1910
Karl Pearson, a founder of mathematical statistics.

The modern field of statistics emerged in the late 19th and early 20th century in three stages.[40] 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.[41] Pearson developed the Pearson product-moment correlation coefficient, defined as a product-moment,[42] the method of moments for the fitting of distributions to samples and the Pearson distribution, among many other things.[43] 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.[44]

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".[45][46]

The second wave of the 1910s and 20s was initiated by William Sealy 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,[47][48][49][50] where he developed rigorous design of experiments models. He originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information.[51] In his 1930 book The Genetical Theory of Natural Selection he applied statistics to various biological concepts such as Fisher's principle[52]). Nevertheless, A.W.F. Edwards has remarked that it is "probably the most celebrated argument in evolutionary biology".[52] (about the sex ratio), the Fisherian runaway,[53][54][55][56][57][58] 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.[59]

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.[60]

Applications

Applied statistics, theoretical statistics and mathematical statistics

Applied statistics comprises descriptive statistics and the application of inferential statistics.[61][62] Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as 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

Machine Learning models are statistical and probabilistic models that captures patterns in the data through use of computational algorithms.

Statistics in society

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

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. Examples of available software capable of complex statistical computation include programs such as Mathematica, SAS, SPSS, and R.

Statistics applied to mathematics or the arts

Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This tradition has changed with use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically. Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.

  • In number theory, 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.
  • The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed. With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art.
  • Methods of statistics may be used predicatively in performance art, 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

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:

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

Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles, it is a key tool, and perhaps the only reliable tool.

See also

Foundations and major areas of statistics

References

  1. ^ a b Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, Oxford University Press. ISBN 0-19-920613-9
  2. ^ Romijn, Jan-Willem (2014). "Philosophy of statistics". Stanford Encyclopedia of Philosophy.
  3. ^ Lund Research Ltd. "Descriptive and Inferential Statistics". statistics.laerd.com. Retrieved 2014-03-23.
  4. ^ "What Is the Difference Between Type I and Type II Hypothesis Testing Errors?". About.com Education. Retrieved 2015-11-27.
  5. ^ "How to Calculate Descriptive Statistics". Answers Consulting. 2018-02-03.
  6. ^ "Definition of STATISTICS". www.merriam-webster.com. Retrieved 2016-05-28.
  7. ^ "Essay on Statistics: Meaning and Definition of Statistics". Economics Discussion. 2014-12-02. Retrieved 2016-05-28.
  8. ^ Moses, Lincoln E. (1986) Think and Explain with Statistics, Addison-Wesley, ISBN 978-0-201-15619-5. pp. 1–3
  9. ^ Hays, William Lee, (1973) Statistics for the Social Sciences, Holt, Rinehart and Winston, p.xii, ISBN 978-0-03-077945-9
  10. ^ Moore, David (1992). "Teaching Statistics as a Respectable Subject". In F. Gordon and S. Gordon (eds.). 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)
  11. ^ Chance, Beth L.; Rossman, Allan J. (2005). "Preface". Investigating Statistical Concepts, Applications, and Methods (PDF). Duxbury Press. ISBN 978-0-495-05064-3.
  12. ^ 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)
  13. ^ Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.
  14. ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1082. ISBN 1-57955-008-8.
  15. ^ Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7
  16. ^ 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. doi:10.1186/1471-2288-7-30. PMC 1936999. PMID 17608932.
  17. ^ Rothman, Kenneth J; Greenland, Sander; Lash, Timothy, eds. (2008). "7". Modern Epidemiology (3rd ed.). Lippincott Williams & Wilkins. p. 100.
  18. ^ Mosteller, F., & Tukey, J.W. (1977). Data analysis and regression. Boston: Addison-Wesley.
  19. ^ 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.
  20. ^ Chrisman, Nicholas R (1998). "Rethinking Levels of Measurement for Cartography". Cartography and Geographic Information Science. 25 (4): 231–242. doi:10.1559/152304098782383043.
  21. ^ van den Berg, G. (1991). Choosing an analysis method. Leiden: DSWO Press
  22. ^ Hand, D.J. (2004). Measurement theory and practice: The world through quantification. London: Arnold.
  23. ^ a b Piazza Elio, Probabilità e Statistica, Esculapio 2007
  24. ^ Everitt, Brian (1998). The Cambridge Dictionary of Statistics. Cambridge, UK New York: Cambridge University Press. ISBN 0521593468.
  25. ^ "Cohen (1994) The Earth Is Round (p < .05)". YourStatsGuru.com.
  26. ^ Rubin, Donald B.; Little, Roderick J.A., Statistical analysis with missing data, New York: Wiley 2002
  27. ^ Ioannidis, J.P.A. (2005). "Why Most Published Research Findings Are False". PLoS Medicine. 2 (8): e124. doi:10.1371/journal.pmed.0020124. PMC 1182327. PMID 16060722.
  28. ^ a b Huff, Darrell (1954) How to Lie with Statistics, WW Norton & Company, Inc. New York. ISBN 0-393-31072-8
  29. ^ 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.
  30. ^ a b Drennan, Robert D. (2008). "Statistics in archaeology". In Pearsall, Deborah M. (ed.). Encyclopedia of Archaeology. Elsevier Inc. pp. 2093–2100. ISBN 978-0-12-373962-9.
  31. ^ 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.
  32. ^ Freund, J.E. (1988). "Modern Elementary Statistics". Credo Reference.
  33. ^ 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.
  34. ^ Huff, Darrell; Irving Geis (1954). How to Lie with Statistics. New York: Norton.
  35. ^ Singh, Simon (2000). The code book : the science of secrecy from ancient Egypt to quantum cryptography (1st Anchor Books ed.). New York: Anchor Books. ISBN 978-0-385-49532-5.
  36. ^ Ibrahim A. Al-Kadi "The origins of cryptology: The Arab contributions", Cryptologia, 16(2) (April 1992) pp. 97–126.
  37. ^ Broemeling, Lyle D. (1 November 2011). "An Account of Early Statistical Inference in Arab Cryptology". The American Statistician. 65 (4): 255–257. doi:10.1198/tas.2011.10191.
  38. ^ Willcox, Walter (1938) "The Founder of Statistics". Review of the International Statistical Institute 5(4): 321–328. JSTOR 1400906
  39. ^ J. Franklin, The Science of Conjecture: Evidence and Probability before Pascal, Johns Hopkins Univ Pr 2002
  40. ^ Helen Mary Walker (1975). Studies in the history of statistical method. Arno Press.
  41. ^ Galton, F (1877). "Typical laws of heredity". Nature. 15 (388): 492–553. Bibcode:1877Natur..15..492.. doi:10.1038/015492a0.
  42. ^ Stigler, S.M. (1989). "Francis Galton's Account of the Invention of Correlation". Statistical Science. 4 (2): 73–79. doi:10.1214/ss/1177012580.
  43. ^ 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.
  44. ^ "Karl Pearson (1857–1936)". Department of Statistical Science – University College London. Archived from the original on 2008-09-25.
  45. ^ Fisher|1971|loc=Chapter II. The Principles of Experimentation, Illustrated by a Psycho-physical Experiment, Section 8. The Null Hypothesis
  46. ^ OED quote: 1935 R.A. Fisher, The Design of Experiments ii. 19, "We may speak of this hypothesis as the 'null hypothesis', and the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation."
  47. ^ 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.
  48. ^ Box, JF (February 1980). "R.A. Fisher and the Design of Experiments, 1922–1926". The American Statistician. 34 (1): 1–7. doi:10.2307/2682986. JSTOR 2682986.
  49. ^ Yates, F (June 1964). "Sir Ronald Fisher and the Design of Experiments". Biometrics. 20 (2): 307–321. doi:10.2307/2528399. JSTOR 2528399.
  50. ^ 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. doi:10.3102/00028312003003223. JSTOR 1161806.
  51. ^ 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.
  52. ^ a b Edwards, A.W.F. (1998). "Natural Selection and the Sex Ratio: Fisher's Sources". American Naturalist. 151 (6): 564–569. doi:10.1086/286141. PMID 18811377.
  53. ^ Fisher, R.A. (1915) The evolution of sexual preference. Eugenics Review (7) 184:192
  54. ^ Fisher, R.A. (1930) The Genetical Theory of Natural Selection. ISBN 0-19-850440-3
  55. ^ Edwards, A.W.F. (2000) Perspectives: Anecdotal, Historial and Critical Commentaries on Genetics. The Genetics Society of America (154) 1419:1426
  56. ^ Andersson, M. (1994) Sexual selection. ISBN 0-691-00057-3
  57. ^ Andersson, M. and Simmons, L.W. (2006) Sexual selection and mate choice. Trends, Ecology and Evolution (21) 296:302
  58. ^ Gayon, J. (2010) Sexual selection: Another Darwinian process. Comptes Rendus Biologies (333) 134:144
  59. ^ 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.
  60. ^ "Science in a Complex World – Big Data: Opportunity or Threat?". Santa Fe Institute.
  61. ^ Nikoletseas, M.M. (2014) "Statistics: Concepts and Examples." ISBN 978-1500815684
  62. ^ 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

External links

Automotive industry

The automotive industry is a wide range of companies and organizations involved in the design, development, manufacturing, marketing, and selling of motor vehicles. It is one of the world's largest economic sectors by revenue. The automotive industry does not include industries dedicated to the maintenance of automobiles following delivery to the end-user, such as automobile repair shops and motor fuel filling stations.

The word automotive is from the Greek autos (self), and Latin motivus (of motion) to refer to any form of self-powered vehicle. This term, as proposed by Elmer Sperry

(1860-1930), first came into use with reference to automobiles in 1898.

Batting average

Batting average is a statistic in cricket, baseball, and softball that measures the performance of batsmen in cricket and batters in baseball and softball. The development of the baseball statistic was influenced by the cricket statistic.

Census

A census is the procedure of systematically acquiring and recording information about the members of a given population. The term is used mostly in connection with national population and housing censuses; other common censuses include agriculture, business, and traffic censuses. The United Nations defines the essential features of population and housing censuses as "individual enumeration, universality within a defined territory, simultaneity and defined periodicity", and recommends that population censuses be taken at least every 10 years. United Nations recommendations also cover census topics to be collected, official definitions, classifications and other useful information to co-ordinate international practice.The word is of Latin origin: during the Roman Republic, the census was a list that kept track of all adult males fit for military service. The modern census is essential to international comparisons of any kind of statistics, and censuses collect data on many attributes of a population, not just how many people there are. Censuses typically began as the only method of collecting national demographic data, and are now part of a larger system of different surveys. Although population estimates remain an important function of a census, including exactly the geographic distribution of the population, statistics can be produced about combinations of attributes e.g. education by age and sex in different regions. Current administrative data systems allow for other approaches to enumeration with the same level of detail but raise concerns about privacy and the possibility of biasing estimates.A census can be contrasted with sampling in which information is obtained only from a subset of a population; typically main population estimates are updated by such intercensal estimates. Modern census data are commonly used for research, business marketing, and planning, and as a baseline for designing sample surveys by providing a sampling frame such as an address register. Census counts are necessary to adjust samples to be representative of a population by weighting them as is common in opinion polling. Similarly, stratification requires knowledge of the relative sizes of different population strata which can be derived from census enumerations. In some countries, the census provides the official counts used to apportion the number of elected representatives to regions (sometimes controversially – e.g., Utah v. Evans). In many cases, a carefully chosen random sample can provide more accurate information than attempts to get a population census.

Chi-squared test

A chi-squared test, also written as χ2 test, is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

In the standard applications of this test, the observations are classified into mutually exclusive classes, and there is some theory, or say null hypothesis, which gives the probability that any observation falls into the corresponding class. The purpose of the test is to evaluate how likely the observations that are made would be, assuming the null hypothesis is true.

Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.

Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-squared distribution as closely as desired by making the sample size large enough.

Confidence interval

In statistics, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter. The interval has an associated confidence level that, loosely speaking, quantifies the level of confidence that the parameter lies in the interval. More strictly speaking, the confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value of the unknown population parameter. In other words, if confidence intervals are constructed using a given confidence level from an infinite number of independent sample statistics, the proportion of those intervals that contain the true value of the parameter will be equal to the confidence level.Confidence intervals consist of a range of potential values of the unknown population parameter. However, the interval computed from a particular sample does not necessarily include the true value of the parameter. Based on the (usually taken) assumption that observed data are random samples from a true population, the confidence interval obtained from the data is also random.

The confidence level is designated prior to examining the data. Most commonly, the 95% confidence level is used. However, other confidence levels can be used, for example, 90% and 99%.

Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample will tend to produce a better estimate of the population parameter, when all other factors are equal. A higher confidence level will tend to produce a broader confidence interval.

Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.

Correlation and dependence

In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense correlation is any statistical association, though in common usage it most often refers to how close two variables are to having a linear relationship with each other. Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a limited supply product and its price.

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).

Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of relationship between mean values.[clarification needed] There are several correlation coefficients, often denoted or , measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients have been developed to be more robust than the Pearson correlation – that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.

Earned run average

In baseball statistics, earned run average (ERA) is the mean of earned runs given up by a pitcher per nine innings pitched (i.e. the traditional length of a game). It is determined by dividing the number of earned runs allowed by the number of innings pitched and multiplying by nine. Runs resulting from defensive errors (including pitchers' defensive errors) are recorded as unearned runs and omitted from ERA calculations.

List of Indian Premier League records and statistics

This is an overall list of statistics and records of the Indian Premier League, a Twenty20 cricket competition based in India.

List of busiest airports by passenger traffic

The world's busiest airports by passenger traffic are measured by total passengers (data from Airports Council International), defined as passengers enplaned plus passengers deplaned plus direct-transit passengers. Hartsfield–Jackson Atlanta International Airport has been the world's busiest airport every year since 2000; with all airports combined London has the world's busiest city airport system by passenger count. As of 2018, six countries have at least two airports in the top 50: the United States has 15, Greater China has 10, and the United Kingdom, Germany, India and Spain have two airports each.

List of cities in India by population

The following tables are the list of cities in India by population. Often cities are bifurcated into multiple regions (municipalities) which results in creation of cities within cities which may figure in the list. The entire work of this article is based on Census of India, 2011, conducted by the Office of the Registrar General and Census Commissioner, under Ministry of Home Affairs (India), Government of India.

List of religious populations

This is a list of religious populations by number of adherents and countries.

Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity, structure, space, and change.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

Mean

There are several kinds of means in various branches of mathematics (especially statistics).

For a data set, the arithmetic mean, also called the mathematical expectation or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted ) to distinguish it from the mean of the underlying distribution, the population mean (denoted or ). Pronounced "mew" /'mjuː/.

In probability and statistics, the population mean, or expected value, are a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving . An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.

Outside probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.

Median

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution). For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fifth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

The median is a commonly used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets which vary greatly, a mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is.

Because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.

Probability

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

Standard score

In statistics, the standard score is the signed fractional number of standard deviations by which the value of an observation or data point is above the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores.

It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. It is a dimensionless quantity. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization for more).

Standard scores are also called z-values, z-scores, normal scores, and standardized variables. They are most frequently used to compare an observation to a theoretical deviate, such as a standard normal deviate.

Computing a z-score requires knowing the mean and standard deviation of the complete population to which a data point belongs; if one only has a sample of observations from the population, then the analogous computation with sample mean and sample standard deviation yields the t-statistic.

Unemployment

Unemployment or joblessness is a situation in which able-bodied people who are looking for a job cannot find a job.

The causes of unemployment are heavily debated. Classical economics, new classical economics, and the Austrian School of economics argued that market mechanisms are reliable means of resolving unemployment. These theories argue against interventions imposed on the labor market from the outside, such as unionization, bureaucratic work rules, minimum wage laws, taxes, and other regulations that they claim discourage the hiring of workers. Keynesian economics emphasizes the cyclical nature of unemployment and recommends government interventions in the economy that it claims will reduce unemployment during recessions. This theory focuses on recurrent shocks that suddenly reduce aggregate demand for goods and services and thus reduce demand for workers. Keynesian models recommend government interventions designed to increase demand for workers; these can include financial stimuli, publicly funded job creation, and expansionist monetary policies. Its namesake economist, John Maynard Keynes, believed that the root cause of unemployment is the desire of investors to receive more money rather than produce more products, which is not possible without public bodies producing new money. A third group of theories emphasize the need for a stable supply of capital and investment to maintain full employment. On this view, government should guarantee full employment through fiscal policy, monetary policy and trade policy as stated, for example, in the US Employment Act of 1946, by counteracting private sector or trade investment volatility, and reducing inequality.In addition to theories of unemployment, there are a few categorizations of unemployment that are used to more precisely model the effects of unemployment within the economic system. Some of the main types of unemployment include structural unemployment and frictional unemployment, as well as cyclical unemployment, involuntary unemployment, and classical unemployment. Structural unemployment focuses on foundational problems in the economy and inefficiencies inherent in labor markets, including a mismatch between the supply and demand of laborers with necessary skill sets. Structural arguments emphasize causes and solutions related to disruptive technologies and globalization. Discussions of frictional unemployment focus on voluntary decisions to work based on each individuals' valuation of their own work and how that compares to current wage rates plus the time and effort required to find a job. Causes and solutions for frictional unemployment often address job entry threshold and wage rates.

The unemployment rate is a measure of the prevalence of unemployment and it is calculated as a percentage by dividing the number of unemployed individuals by all individuals currently in the labor force. During periods of recession, an economy usually experiences a relatively high unemployment rate. Millions of people globally or 6% of the world's workforce were without a job in 2012.

Urban area

An urban area or urban agglomeration is a human settlement with high population density and infrastructure of built environment. Urban areas are created through urbanization and are categorized by urban morphology as cities, towns, conurbations or suburbs. In urbanism, the term contrasts to rural areas such as villages and hamlets and in urban sociology or urban anthropology it contrasts with natural environment. The creation of early predecessors of urban areas during the urban revolution led to the creation of human civilization with modern urban planning, which along with other human activities such as exploitation of natural resources leads to human impact on the environment.

The world's urban population in 1950 of just 746 million has increased to 3.9 billion in the decades since. In 2009, the number of people living in urban areas (3.42 billion) surpassed the number living in rural areas (3.41 billion) and since then the world has become more urban than rural. This was the first time that the majority of the world's population lived in a city. In 2014 there were 7.2 billion people living on the planet, of which the global urban population comprised 3.9 billion. The Population Division of the United Nations Department of Economic and Social Affairs at that time predicted the urban population would grow to 6.4 billion by 2050, with 37% of that growth to come from three countries: China, India and Nigeria.Urban areas are created and further developed by the process of urbanization. Urban areas are measured for various purposes, including analyzing population density and urban sprawl.

Unlike an urban area, a metropolitan area includes not only the urban area, but also satellite cities plus intervening rural land that is socio-economically connected to the urban core city, typically by employment ties through commuting, with the urban core city being the primary labor market.

Western Asia

Western Asia, West Asia, Southwestern Asia or Southwest Asia is the westernmost subregion of Asia. The concept is in limited use, as it significantly overlaps with the Middle East (or the Near East), the main difference usually being the exclusion of the majority of Egypt (which would be counted as part of North Africa) and the inclusion of the Caucasus. The term is sometimes used for the purposes of grouping countries in statistics. The total population of Western Asia is an estimated 300 million as of 2015. Although the term "Western Asia" is mostly used as a convenient division of contemporary sovereign states into a manageable number of world regions for statistical purposes, it is sometimes used instead of the more geopolitical term "Middle East".

In an unrelated context, the term is also used in ancient history and archaeology to divide the Fertile Crescent into the "Asiatic" or "Western Asian" cultures as opposed to ancient Egypt. As a geographic concept, Western Asia includes the Levant, Mesopotamia, Anatolia, Iran, the Armenian Highlands, the South Caucasus, the Arabian peninsula as well as the Sinai Peninsula, making Egypt a transcontinental country.

The term is used pragmatically and has no "correct" or generally agreed-upon definition. The National Geographic Style Manual as well as Maddison's The World Economy: Historical Statistics (2003) by the Organisation for Economic Co-operation and Development (OECD) only includes Bahrain, Iran, Iraq, Israel, Jordan, Kuwait, Lebanon, Oman, Qatar, Palestinian territories (called West Bank and Gaza in the latter), Saudi Arabia, Syria, Turkey, UAE, and Yemen as West Asian countries. In contrast to this definition, the United Nations Industrial Development Organization (UNIDO) in its 2015 yearbook also includes Armenia and Azerbaijan, and excludes Israel (as Other) and Turkey (as Europe).

Unlike the UNIDO, the United Nations Statistics Division (UNSD) excludes Iran from Western Asia and includes Turkey, Georgia, and Cyprus in the region. In the United Nation's geopolitical Eastern European Group, Armenia and Georgia are included in Eastern Europe, whereas Cyprus and East Thracian Turkey are in Southern Europe. These three nations are listed in the European category of the United Nations Educational, Scientific, and Cultural Organisation (UNESCO).

National members of West Asian sports governing bodies are limited to Bahrain, Iran, Iraq, Jordan, Kuwait, Lebanon, Syria, Oman, Palestine, Qatar, Saudi Arabia, United Arab Emirates, and Yemen. The Olympic Council of Asia's multi-sport event West Asian Games are contested by athletes representing these thirteen countries. Among the region's sports organisations are the West Asia Basketball Association, West Asian Billiards and Snooker Federation, West Asian Football Federation, and the West Asian Tennis Federation.

Statistics
Algebra
Analysis
Number theory
Discrete
Geometry
Foundations
Topology
Computational
Applied
Others
Glossaries of science and engineering

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.