Statistics
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Statistics is the science of the collection, organization, and interpretation of data.^{[1]}^{[2]} It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.^{[1]}
A statistician is someone who is particularly well versed in the ways of thinking necessary for the successful application of statistical analysis. Such people have often gained this experience through working in any of a wide number of fields. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject.
The word statistics, when referring to the scientific discipline, is singular, as in "Statistics is an art."^{[3]} This should not be confused with the word statistic, referring to a quantity (such as mean or median) calculated from a set of data,^{[4]} whose plural is statistics ("this statistic seems wrong" or "these statistics are misleading").
Scope
Some consider statistics to be a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data,^{[5]} while others consider it a branch of mathematics^{[6]} concerned with collecting and interpreting data. 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 and survey sampling. Statistics also provides tools for prediction and forecasting using data and statistical models. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical consultants are available to provide help for organizations and companies without direct access to expertise relevant to their particular problems.
Statistical methods can be used to summarize or describe a collection of data; this is called descriptive statistics. This is useful in research, when communicating the results of experiments. In addition, patterns in the data may be modeled in a way that accounts for randomness and uncertainty in the observations, and are then used to draw inferences about the process or population being studied; this is called inferential statistics. Inference is a vital element of scientific advance, since it provides a prediction (based in data) for where a theory logically leads.
To prove the guiding theory further, these predictions are tested as well, as part of the scientific method. 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]}
Statistics is closely related to probability theory, with which it is often grouped; the difference is roughly that in probability theory, one starts from the given parameters of a total population to deduce probabilities pertaining to samples, but statistical inference moves in the opposite direction, inductive inference from samples to the parameters of a larger or total population.
History
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.^{[10]} 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 the natural and social sciences.
Its mathematical foundations were laid in the 17th century with the development of probability theory by Blaise Pascal and Pierre de Fermat. Probability theory arose from the study of games of chance. The method of least squares was first described by Carl Friedrich Gauss around 1794. The use of modern computers has expedited largescale 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 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.
For practical reasons, a chosen subset of the population called a sample is studied â€” as opposed to compiling data about the entire group (an operation called census). Once a sample that is representative of the population is determined, data are collected for the sample members in an observational or experimental setting. These data can then be subjected to statistical analysis, serving two related purposes: description and inference.
 Descriptive statistics summarize the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include mean and standard deviation for continuous data types (like heights or weights), while frequency and percentage are more useful in terms of describing categorical data (like race).
 Inferential statistics 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.
â€œ... 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.â€
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.)
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 assures that the 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 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. Statisticians^{[cn]} describe stronger methods as more "robust".
Randomness is studied using the mathematical discipline of probability theory. Probability is used in "mathematical statistics" (alternatively, "statistical theory") 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.
Misuse of statistics can produce subtle, but serious errors in description and interpretation â€” 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. Further discussion will occur later.
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 â€” which measures the extent to which a trend could be caused by random variation in the sample â€” 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.
Experiments
The basic steps of a statistical experiment are:
 Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability. Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects.
 Design of experiments, using blocking to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the experimental protocol that shall guide the performance of the experiment and that specifies the primary analysis of the experimental data.
 Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
 Further examining the data set in secondary analyses, to suggest new hypotheses for future study.
 Documenting and presenting the results of the study.
Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was changed but because they were being observed.^{[cn]}
Observational study
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 nonsmokers, perhaps through a casecontrol study, and then look for the number of cases of lung cancer in each group.
Levels of measurement
There are four main levels of measurement used in statistics: nominal, ordinal, interval, and ratio. Each of these have different degrees of usefulness in statistical research. Ratio measurements have both a meaningful zero value and the distances between different measurements defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data.^{[cn]} 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). 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.
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 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, H_{0}, asserts that the defendant is innocent, whereas the alternative hypothesis, H_{1}, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H_{0} (status quo) stands in opposition to H_{1} and is maintained unless H_{1} is supported by evidenceâ€œbeyond a reasonable doubtâ€. However,â€œfailure to reject H_{0}â€ in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not necessarily accept H_{0} but fails to reject H_{0}. While one can notâ€œproveâ€ a null hypothesis one can test how close it is to being true with a power test, which tests for type II errors.
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.
Error also refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean. Many statistical methods seek to minimize the meansquared error, and these are called "methods of least squares."
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 important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important.
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 a range where, if the sampling and analysis were repeated under the same conditions, the interval would include the true (population) value 95% of the time. 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.) One quantity that is in fact a probability for an estimated value is the credible interval 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 pvalue).
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 will be unlikely to help the patient in a noticeable way.
Examples
Some wellknown statistical tests and procedures are:
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:
 Actuarial science
 Applied information economics
 Biostatistics
 Business statistics
 Chemometrics (for analysis of data from chemistry)
 Data mining (applying statistics and pattern recognition to discover knowledge from data)
 Demography
 Econometrics
 Energy statistics
 Engineering statistics
 Epidemiology
 Geography and Geographic Information Systems, specifically in Spatial analysis
 Image processing
 Psychological statistics
 Reliability engineering
 Social statistics
In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:
 Bootstrap & Jackknife Resampling
 Multivariate statistics
 Statistical classification
 Statistical surveys
 Structured data analysis (statistics)
 Survival analysis
 Statistics in various sports, particularly baseball and cricket
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make datadriven decisions. In these roles, it is a key tool, and perhaps the only reliable tool.
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
There is a general perception that statistical knowledge is alltoofrequently intentionally misused by finding ways to interpret only the data that are favorable to the presenter.^{[11]} The famous saying, "There are three kinds of lies: lies, damned lies, and statistics".^{[12]} which was popularized in the USA by Mark Twain and incorrectly attributed by him to Disraeli (1804–1881)^{[cn]}, has come to represent the general mistrust [and misunderstanding] of statistical science. 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."^{[cn]}
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 nonexpert. (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 pvalue than simply reporting whether a hypothesis is rejected at the given level of significance. The pvalue, 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 pvalues, 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 semistandardized methodology that was "required learning" in most sciences. This has changed with use of statistics in noninferential contexts. What was once considered a dry subject, taken in many fields as a degreerequirement, 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, methods of statistics were applied to formalize such distribution driven natural processes, in order 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 performancespecific. Though this type of artistry does not always come out as expected, it does behave in ways that are predictable and tuneable using statistics.
See also
 Glossary of probability and statistics
 Notation in probability and statistics
 List of statistics articles
 List of academic statistical associations
 List of national and international statistical services
 List of important publications in statistics
 List of university statistical consulting centers
 List of statistical packages (software)
Notes
 ↑ ^{1.0} ^{1.1} Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0199206139
 ↑ The Free Online Dictionary
 ↑ "Statistics". MerriamWebster Online Dictionary. http://www.merriamwebster.com/dictionary/statistics.
 ↑ "Statistic". MerriamWebster Online Dictionary. http://www.merriamwebster.com/dictionary/statistic.
 ↑ Moses, Lincoln E. Think and Explain with statistics, pp. 1–3. AddisonWesley, 1986.
 ↑ Hays, William Lee, Statistics for the social sciences, Holt, Rinehart and Winston, 1973, p.xii, ISBN 9780030779459
 ↑ Moore, David (1992). "Teaching Statistics as a Respectable Subject". Statistics for the TwentyFirst Century. Washington, DC: The Mathematical Association of America. pp. 14â€“25.
 ↑ Chance, Beth L.; Rossman, Allan J. (2005). "Preface". Investigating Statistical Concepts, Applications, and Methods. Duxbury Press. ISBN 9780495050643. http://www.rossmanchance.com/iscam/preface.pdf.
 ↑ Anderson, , D.R.; Sweeney, D.J.; Williams, T.A.. Statistics: Concepts and Applications, pp. 5–9. West Publishing Company, 1986.
 ↑ Willcox, Walter (1938) The Founder of Statistics. Review of the International Statistical Institute 5(4):321–328.
 ↑ Darrell Huff. How to Lie With Statistics 1954. WW Norton & Company, Inc. New York, NY. ISBN 0393310728
 ↑ Leonard H.Courtney (1832–1918) in a speech at Saratoga Springsâ€™, New York, August 1895, in which this sentence appeared: â€˜After all, facts are facts, and although we may quote one to another with a chuckle the words of the Wise Statesman, â€œLies – damned lies – and statistics,â€ still there are some easy figures the simplest must understand, and the astutest cannot wriggle out of.â€™, earliest documented use of exact phrase.
References
 Best, Joel (2001). Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists. University of California Press. ISBN 0520219783.
 DesrosiÃ¨res, Alain (2004). The Politics of Large Numbers: A History of Statistical Reasoning. Trans. Camille Naish. Harvard University Press. ISBN 0674689321.
 Hacking, Ian (1990). The Taming of Chance. Cambridge University Press. ISBN 0521388848.
 Lindley, D.V. (1985). Making Decisions (2nd ed.). John Wiley & Sons. ISBN 0471908088.
 Tijms, Henk (2004). Understanding Probability: Chance Rules in Everyday life. Cambridge University Press. ISBN 0521833299.
External links
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Online noncommercial textbooks
 "A New View of Statistics", by Will G. Hopkins, AUT University
 "NIST/SEMATECH eHandbook of Statistical Methods", by U.S. National Institute of Standards and Technology and SEMATECH
 "Online Statistics: An Interactive Multimedia Course of Study", by David Lane, Joan Lu, Camille Peres, Emily Zitek, et al.
 "The Little Handbook of Statistical Practice", by Gerard E. Dallal, Tufts University
 "StatSoft Electronic Textbook", by StatSoft
Other noncommercial resources
 Statistics (OECD)
 Probability Web (Carleton College)
 Free online statistics course with interactive practice exercises (Carnegie Mellon University)
 Resources for Teaching and Learning about Probability and Statistics (ERIC)
 Rice Virtual Lab in Statistics (Rice University)
 Statistical Science Web (University of Melbourne)
 Applied statistics applets
 Statlib: data and software archives
 StatProb – peerreviewed encyclopedia sponsored by statistics and probability societies


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