Statistics is the science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments.[1]

A statistician Statisticians work with theoretical and applied statistics in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it. The field shares much common history with positivist social science, but often with a greater emphasis on advanced mathematical methods is someone who is particularly versed in the ways of thinking necessary for the successful application of statistical analysis Statistics is the science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments. Often such people have gained this experience after starting work in any of a number of fields Statistics is the mathematical science involving the collection, analysis and interpretation of data. A number of specialties have evolved to apply statistical theory and methods to various disciplines. There is also a discipline called mathematical statistics Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to the term "statistical theory" but also embraces modelling for actuarial science and, which is concerned with the theoretical basis of the subject.

The word statistics can either be singular or plural.[2] In its singular form, a statistic A statistic is the result of applying a function (statistical algorithm) to a set of data is a quantity (such as a mean There are other statistical measures that use samples that some people confuse with averages - including 'median' and 'mode.' Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability) calculated from a set of data,[3] whereas statistics is the mathematical science discussed in this article.

Contents

Scope

Statistics is considered by some to be a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data The term data means groups of information that represent the qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which information and,[4] while others consider it to be a branch of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions[5] concerned with collecting and interpreting data The term data means groups of information that represent the qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which information and.[6] Because of its empirical roots and its focus on applications, statistics is usually considered to be a distinct mathematical science rather than a branch of mathematics.[7][8]

Statisticians improve the quality of data with the design of experiments Design of experiments, or experimental design, is the design of all information-gathering exercises where variation is present, whether under the full control of the experimenter or not. (The latter situation is usually called an observational study.) Often the experimenter is interested in the effect of some process or intervention (the " and survey sampling A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often refers to a questionnaire used to measure the characteristics and/or attitudes of people. The purpose of sampling is to reduce the cost and/or the amount of work that it would take to survey the entire target population. A. Statistics also provides tools for prediction and forecasting using data and statistical models A statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions. If the model has only one equation it is called a single-equation model, whereas if it has more than one equation, it is known as a multiple-equation model. Statistics is applicable to a wide variety of academic disciplines, including natural Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general. It ranges in scale from the subatomic to the cosmic and social sciences The social sciences are the fields of academic scholarship which explore aspects of human society. Social sciences may draw upon empirical methods and attempt to emulate the standards of conventional scientific practice. By contrast, other social scientists employ critical analysis or hermeneutic methods to study objects of enquiry they regard as, government, and business.

Statistical methods can be used to summarize or describe a collection of data; this is called descriptive statistics Descriptive statistics are used to describe the main features of a collection of data in quantitative terms. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to quantitatively summarize a data set, rather than being used to support inferential statements about the population that the data. This is useful in research, when communicating the results of experiments. In addition, patterns in the data may be modeled A mathematical model uses mathematical language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most in a way that accounts for randomness Randomness is a concept with somewhat disparate meanings in several fields. It also has common meanings which may have loose connections with some of those more definite meanings. The Oxford English Dictionary defines "random" thus: and uncertainty in the observations, and are then used to draw inferences about the process or population being studied; this is called inferential statistics Statistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population. It is distinguished from descriptive statistics. Inference is a vital element of scientific advance, since it provides a prediction (based in data) for where a theory logically leads. To further prove the guiding theory, these predictions are tested as well, as part of the scientific method Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning. A scientific method consists of. If the inference holds true, then the descriptive statistics of the new data increase the soundness of that hypothesis. Descriptive statistics and inferential statistics (a.k.a., predictive statistics) together comprise applied statistics.[9]

History

Main article: History of statistics Statistics arose, no later than the 18th century, from the need of states to collect data on their people and economies, in order to administer them. Its meaning broadened in the early 19th century to include the collection and analysis of data in general. Today statistics is widely employed in government, business, and the natural and social

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 John Graunt was one of the first demographers, though by profession he was a haberdasher. Born in London, Graunt, along with William Petty, developed early human statistical and census methods that later provided a framework for modern demography. He is credited with producing the first life table, giving probabilities of survival to each age.[10] Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology Statistics arose, no later than the 18th century, from the need of states to collect data on their people and economies, in order to administer them. Its meaning broadened in the early 19th century to include the collection and analysis of data in general. Today statistics is widely employed in government, business, and the natural and social. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and the natural and social sciences.

Its mathematical foundations were laid in the 17th century with the development of probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random by Blaise Pascal Blaise Pascal , (June 19, 1623, Clermont-Ferrand – August 19, 1662, Paris) was a French mathematician, physicist, and religious philosopher. He was a child prodigy who was educated by his father, a civil servant. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the construction of mechanical and Pierre de Fermat Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the. Probability theory arose from the study of games of chance. The method of least squares The method of least squares is used to approximately solve overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis was first described by Carl Friedrich Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy around 1794. The use of modern computers A computer is a programmable machine that receives input, stores and manipulates data, and provides output in a useful format has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually.

The American Statistical Association has ranked Deming William Edwards Deming was an American statistician, professor, author, lecturer, and consultant. Deming is widely credited with improving production in the United States during the Cold War, although he is perhaps best known for his work in Japan. There, from 1950 onward he taught top management how to improve design (and thus service), product, Fisher Sir Ronald Aylmer Fisher, FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science," and Richard Dawkins described him as "the greatest of Darwin's successors", and Rao Calyampudi Radhakrishna Rao FRS (born September 10, 1920) is a world-renowned Indian born statistician, National Medal of Science Awardee, and currently Professor emeritus at Penn State University and Research Professor at the University at Buffalo. He is widely regarded as one of the greatest statisticians of all time along with Deming and Fisher as the greatest statisticians of all time.[citation needed]

Overview

In applying statistics to a scientific, industrial, or societal problem, it is necessary to begin with a population In statistics, a statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest. Notice that if we choose a population like or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal". A population can also be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of "population" constitutes what is called a time series In statistics, signal processing and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan. Time series analysis comprises methods for.

For practical reasons, a chosen subset of the population called a sample Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference is studied — as opposed to compiling data about the entire group (an operation called census A census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official count of a particular population. The term is used mostly in connection with national population and door-to-door censuses , agriculture, and business censuses. The term itself comes from). Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental In scientific research, an experiment is a method of investigating causal relationships among variables, or to test a hypothesis. An experiment is a cornerstone of the empirical approach to acquiring data about the world and is used in both natural sciences and social sciences. An experiment can be used to help solve practical problems and to setting. This data can then be subjected to statistical analysis, serving two related purposes: description and inference.

“... it is only the manipulation of uncertainty that interests us. We are not concerned with the matter that is uncertain. Thus we do not study the mechanism of rain; only whether it will rain.” —Dennis Lindley, "The Philosophy of Statistics", The Statistician (2000).

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 are 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.)

For a sample to be used as a guide to an entire population, it is important that it is truly a representative of that overall population. Representative sampling assured, inferences and conclusions can be safely extended from the sample to the population as a whole. A major problem lies in determining the extent to which the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures. There are also methods for designing experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Statisticians describe stronger methods as more "robust".(See experimental design.)

The fundamental mathematical concept employed in understanding potential randomness is probability. Mathematical statistics (also called statistical theory) is the branch of applied mathematics that uses probability theory and analysis to examine the theoretical basis of statistics. The use of any statistical method is valid only when the system or population under consideration satisfies the basic mathematical assumptions of the method.

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 statistics 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.

Statistical methods

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 or response. 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.

An example of an experimental study is the famous Hawthorne study, which attempted to test 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.[citation needed]

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

The basic steps of an experiment are:

  1. Planning the research, including determining information sources, research subject selection, and ethical considerations for the proposed research and method.
  2. Design of experiments, concentrating on the system model and the interaction of independent and dependent variables.
  3. Summarizing a collection of observations to feature their commonality by suppressing details. (Descriptive statistics)
  4. Reaching consensus about what the observations tell about the world being observed. (Statistical inference)
  5. Documenting / presenting the results of the study.

Levels of measurement

Main article: levels of measurement

There are four types of measurements or levels of measurement or measurement scales used in statistics:

They have different degrees of usefulness in statistical research. Ratio measurements have both a zero value defined and the distances between different measurements defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data. Interval measurements have meaningful distances between measurements defined, but have no meaningful zero value defined (as in the case with IQ measurements or with temperature measurements in Fahrenheit). Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values. Nominal measurements have no meaningful rank order among values.

Since variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are called together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative or continuous variables due to their numerical nature.

Key terms used in statistics

Null hypothesis

Interpretation of statistical information can often involve the development of a null hypothesis in that the assumption is that whatever is proposed as a cause has no effect on the variable being measured.

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

Error

Working from a null hypothesis two basic forms of error are recognised:

Confidence intervals

Most studies will only sample part of a population and then the result is used to interpret the null hypothesis in the context of the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval of a procedure is any range such that the interval covers the true population value 95% of the time given repeated sampling under the same conditions. If these intervals span a value (such as zero) where the null hypothesis would be confirmed then this can indicate that any observed value has been seen by chance. For example a drug that gives a mean increase in heart rate of 2 beats per minute but has 95% confidence intervals of -5 to 9 for its increase may well have no effect whatsoever.

The 95% confidence interval is often misinterpreted as the probability that the true value lies between the upper and lower limits given the observed sample. However this quantity is more a credible interval available only from Bayesian statistics.

Significance

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

When interpreting an academic paper reference to the significance of a result when referring to the statistical significance does not necessarily mean that the overall result means anything in real world terms. (For example in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect such that the drug will be unlikely to help anyone given it in a noticeable way.)

Examples

Some well-known statistical tests and procedures are:

Specialized disciplines

Main article: List of fields of application of statistics

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.

Statistical computing

gretl, an example of an open source statistical package Main article: 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.

Misuse

Main article: Misuse of statistics

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. A famous saying attributed to Benjamin Disraeli is, "There are three kinds of lies: lies, damned lies, and statistics." Harvard President Lawrence Lowell wrote in 1909 that statistics, "...like veal pies, are good if you know the person that made them, and are sure of the ingredients."

If various studies appear to contradict one another, then the public may come to distrust such studies. For example, one study may suggest that a given diet or activity raises blood pressure, while another may suggest that it lowers blood pressure. The discrepancy can arise from subtle variations in experimental design, such as differences in the patient groups or research protocols, which are not easily understood by the non-expert. (Media reports usually omit this vital contextual information entirely, because of its complexity.)

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

Deeper criticisms come from the fact that the hypothesis testing approach, widely used and in many cases required by law or regulation, forces one hypothesis (the null hypothesis) to be "favored," and can also seem to exaggerate the importance of minor differences in large studies. A difference that is highly statistically significant can still be of no practical significance. (See criticism of hypothesis testing and controversy over the null hypothesis.)

One response is by giving a greater emphasis on the p-value than simply reporting whether a hypothesis is rejected at the given level of significance. The p-value, however, does not indicate the size of the effect. Another increasingly common approach is to report confidence 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.

Statistics applied to mathematics or the arts

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

See also

Main article: Outline of statistics

Related disciplines

Notes

  1. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0199206139
  2. ^ "Statistics". Merriam-Webster Online Dictionary. http://www.merriam-webster.com/dictionary/statistics.
  3. ^ "Statistic". Merriam-Webster Online Dictionary. http://www.merriam-webster.com/dictionary/statistic.
  4. ^ Moses, Lincoln E. Think and Explain with statistics, pp. 1 - 3. Addison-Wesley, 1986.
  5. ^ Hays, William Lee, Statistics for the social sciences, Holt, Rinehart and Winston, 1973, p.xii, ISBN 978-0030779459
  6. ^ Statistics at Encyclopedia of Mathematics
  7. ^ Moore, David (1992). "Teaching Statistics as a Respectable Subject". Statistics for the Twenty-First Century. Washington, DC: The Mathematical Association of America. pp. 14–25.
  8. ^ Chance, Beth L.; Rossman, Allan J. (2005). "Preface". Investigating Statistical Concepts, Applications, and Methods. Duxbury Press. ISBN 978-0495050643. http://www.rossmanchance.com/iscam/preface.pdf.
  9. ^ Anderson, , D.R.; Sweeney, D.J.; Williams, T.A.. Statistics: Concepts and Applications, pp. 5 - 9. West Publishing Company, 1986.
  10. ^ Willcox, Walter (1938) The Founder of Statistics. Review of the International Statistical Institute 5(4):321-328.
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References

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Continuous data
Location Mean (Arithmetic, Geometric, Harmonic) · Median · Mode
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Categorical data Frequency · Contingency table
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Inference Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis
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