Statistics is a mathematical science Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns.They formulate new conjectures and establish truth by rigorous deduction from appropriately chosen axioms and definitions pertaining to the collection, analysis, interpretation or explanation, and presentation of data Data are pieces 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 knowledge are.[1] 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. Often the experimenter is interested in the effect of some process or intervention (the "treatment") on some objects (the "experimental units") 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 charateristics 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 comprise academic disciplines concerned with the study of the social life of human groups, animals and individuals including anthropology, archeology, communication studies, cultural studies, demography, economics, human geography, history, linguistics, media studies, political science, psychology, social work, and sociology, 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 inductive statistics in that they aim to quantitatively summarize a data set, rather than being used to support statements about the population that the data are thought to represent. Even when. 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 Informally, it is typically used to denote a lack of order, or purpose, or cause[citation needed]. In addition more closely connected with the concept of entropy, there is the sense of lack of predictability 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 bodies 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.[2]

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 "statistical theory" but also embraces modelling for actuarial science and non-, which is concerned with the theoretical basis of the subject.

The word statistics can either be singular or plural.[3] In its singular form, statistics refers to the mathematical science discussed in this article. In its plural form, statistics is the plural of the word statistic A statistic is the result of applying a function (statistical algorithm) to a set of data, which refers to a quantity (such as a mean It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a) calculated from a set of data.[4]

Contents

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 1662, 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.[5] 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.

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.[6][7] 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, in Clermont-Ferrand, France – August 19, 1662) 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 and Pierre de Fermat Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and a 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 then. 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, electrostatics, astronomy and optics around 1794. The use of modern computers A computer is a machine that manipulates data according to a set of instructions has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually.

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 many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying context of the data points (Where did they come from?.

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. Each observation measures one or more properties of an observable entity enumerated to distinguish objects or individuals. Survey is studied — as opposed to compiling data about the entire group. 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 inquiry, an experiment is a method of investigating causal relationships among variables. 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 support or negate theoretical 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 Dennis Victor Lindley is a British statistician, decision theorist and leading advocate of Bayesian statistics, "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 A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. It lists values for each of the variables, such as height and weight of an object or values of random numbers. Each value is known as a datum. The data set may 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 In statistics, a confounding variable is an extraneous variable in a statistical model that correlates (positively or negatively) with both the dependent variable and the independent variable. The methodologies of scientific studies therefore need to control for these factors to avoid a type 1 error; an erroneous 'false positive' conclusion that or confounding variable In statistics, a confounding variable is an extraneous variable in a statistical model that correlates (positively or negatively) with both the dependent variable and the independent variable. The methodologies of scientific studies therefore need to control for these factors to avoid a type 1 error; an erroneous 'false positive' conclusion that. 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 "Correlation does not imply causation" is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other . The phrase's opposite, correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-.)

For a sample to be used as a guide to an entire population, it is important the is be truly 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 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 Probability, or chance, is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the. 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&nbsp- which measures the extent to which a trend could be caused by random variation in the sample&nbsp- 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.

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:

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

Notes

  1. ^ Moses, Lincoln E. Think and Explain with statistics, pp. 1 - 3. Addison-Wesley, 1986.
  2. ^ Anderson, , D.R.; Sweeney, D.J.; Williams, T.A.. Statistics: Concepts and Applications, pp. 5 - 9. West Publishing Company, 1986.
  3. ^ "Statistics". Merriam-Webster Online Dictionary. http://www.merriam-webster.com/dictionary/statistics.
  4. ^ "Statistic". Merriam-Webster Online Dictionary. http://www.merriam-webster.com/dictionary/statistic.
  5. ^ Willcox, Walter (1938) The Founder of Statistics. Review of the International Statistical Institute 5(4):321-328.
  6. ^ 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.
  7. ^ 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.
This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be and removed. (September 2008)

References

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