Statistics

From Wikipedia, the free encyclopedia

This article is about the field of Statistics. = For other=20 uses, see Statist= ics=20 (disambiguation).

Statistics is the study of the collection, organization, = analysis, and=20 interpretation of data.^=
[1][2]=20 It deals with all aspects of this, including the planning of data = collection in=20 terms of the design of surveys=20 and experiments= .= [1]

A statistician is = someone who=20 is particularly well versed in the ways of thinking necessary for the = successful=20 application of statistical analysis. Such people have often gained this=20 experience through working in any of a wide=20 number of fields. There is also a discipline called mathematica= l=20 statistics that studies statistics mathematically.

The word statistics, when referring to the scientific = discipline, is=20 singular, as in "Statistics is an art."^[3]=20 This should not be confused with the word statistic, referring to = a=20 quantity (such as mean or median)=20 calculated from a set of data,^[4]=20 whose plural is statistics ("this statistic seems wrong" or = "these=20 statistics are misleading").

More probab= ility=20 density will be found the closer one gets to the expected (mean) = value in a=20 normal=20 distribution. Statistics used in standardized=20 testing assessment are shown. The scales include standard=20 deviations, cumulative percentages, percentile equivalents, = Z-scores,=20 T-scores, standard nines, and percentages in standard = nines.

[edit]=20 Scope

Some consider statistics to be a mathematical body of science = pertaining to=20 the collection, analysis, interpretation or explanation, and = presentation of data,^[5]=20 while others consider it a branch of mathematics[6]=20 concerned with collecting and interpreting data. Because of its = empirical roots=20 and its focus on applications, statistics is usually considered to be a = distinct=20 mathematical science rather than a branch of mathematics.^[7][8]=20 Much of statistics is non-mathematical: ensuring that data = collection is=20 undertaken in a way that allows valid conclusions to be drawn; coding = and=20 archiving of data so that information is retained and made useful for=20 international comparisons of official=20 statistics; reporting of results and summarised data (tables and = graphs) in=20 ways that are comprehensible to those who need to make use of them; = implementing=20 procedures that ensure the privacy=20 of census information.

Statisticians improve the quality of data with the design of=20 experiments and survey=20 sampling. Statistics also provides tools for prediction and = forecasting=20 using data and statistical=20 models. Statistics is applicable to a wide variety of academic=20 disciplines, including natural=20 and social = sciences,=20 government, and business. Statistical = consultants are available to provide help for organizations and = companies=20 without direct access to expertise relevant to their particular = problems.

Statistical methods can be used for summarizing or describing a = collection of=20 data; this is called descriptive = statistics. This is useful in research, when communicating the = results=20 of experiments. In addition, patterns in the data may be modeled in = a way that=20 accounts for randomness=20 and uncertainty in the observations, and are then used for drawing = inferences=20 about the process or population being studied; this is called inferential = statistics. Inference is a vital element of scientific advance, = since it=20 provides a means for drawing conclusions from data that are subject to = random=20 variation. To prove the propositions being investigated further, the = conclusions=20 are tested as well, as part of the scientific=20 method. Descriptive statistics and analysis of the new data tend to = provide=20 more information as to the truth of the proposition.

Descriptive statistics and the application of inferential statistics = (a.k.a.,=20 predictive statistics) together comprise applied statistics.^[9][verification=20 needed] Theoretical statistics concerns both = the=20 logical arguments underlying justification of approaches to statistical=20 inference, as well encompassing mathematica= l=20 statistics. Mathematical statistics includes not only the = manipulation=20 of probabilit= y=20 distributions necessary for deriving results related to methods of=20 estimation and inference, but also various aspects of computatio= nal=20 statistics and the design=20 of experiments.

Statistics is closely related to probability=20 theory, with which it is often grouped; the difference is roughly = that in=20 probability theory, one starts from the given parameters of a total = population=20 to deduce=20 probabilities pertaining to samples, but statistical inference moves in = the=20 opposite direction, inductive=20 inference from samples to the parameters of a larger or total=20 population.

[edit]=20 History

Main article: History of=20 statistics

The earliest writing on statistics was found in a 9th century book = entitled:=20 "Manuscript on Deciphering Cryptographic Messages", written by Al-Kindi = (801=E2=80=93873 CE). In his=20 book, Al-Kindi gave a detailed description of how to use statistics and = frequency=20 analysis to decipher encrypted messages, this was the birth of both=20 statistics and cryptanalysis, according to Ibrahim Al-Kadi.^[10][11]

Some scholars pinpoint the origin of statistics to 1663, with the = publication=20 of Natural and Political Observations upon the Bills of Mortality = by John=20 Graunt.^[12]=20 Early applications of statistical thinking revolved around the needs of = states=20 to base policy on demographic and economic data, hence its = stat-=20 etymology. The scope of the discipline of statistics broadened in = the early=20 19th century to include the collection and analysis of data in general. = Today,=20 statistics is widely employed in government, business, and the natural = and=20 social sciences.

Its mathematical foundations were laid in the 17th century with the=20 development of probability=20 theory by Blaise=20 Pascal and Pierre=20 de Fermat. Probability theory arose from the study of games of = chance. The=20 method=20 of least squares was first described by Carl = Friedrich=20 Gauss around 1794. The use of modern computers=20 has expedited large-scale statistical computation, and has also made = possible=20 new methods that are impractical to perform manually.

[edit]=20 Overview

In applying statistics to a scientific, industrial, or societal = problem, it=20 is necessary to begin with a population=20 or process to be studied. Populations can be diverse topics such as "all = persons=20 living in a country" or "every atom composing a crystal". A population = can also=20 be composed of observations of a process at various times, with the data = from=20 each observation serving as a different member of the overall group. = Data=20 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=20 studied =E2=80=94 as opposed to compiling data about the entire = group (an operation=20 called census).=20 Once a sample that is representative of the population is determined, = data are=20 collected for the sample members in an observational or experimental = setting. This=20 data can then be subjected to statistical analysis, serving two related=20 purposes: description and inference.

Descriptive = statistics summarize the population data by describing what was = observed=20 in the sample numerically or graphically. Numerical descriptors = include mean and = standard = deviation=20 for continuous=20 data types (like heights or weights), while frequency and = percentage are=20 more useful in terms of describing categorical=20 data (like race).
Inferential = statistics uses patterns in the sample data to draw inferences = about the=20 population represented, accounting for randomness. These inferences = may take=20 the form of: answering yes/no questions about the data (hypothesis=20 testing), estimating numerical characteristics of the data (estimation), = describing associatio= ns=20 within the data (correlat= ion)=20 and modeling relationships within the data (for example, using regression=20 analysis). Inference can extend to forecasting,=20 prediction and = estimation=20 of unobserved values either in or associated with the population being = studied; it can include extrapolation=20 and interpolation = of time=20 series or spatial=20 data, and can also include data=20 mining.^[13]

"... it is only the manipulation of uncertainty that interests us. We = are not=20 concerned with the matter that is uncertain. Thus we do not study the = mechanism=20 of rain; only whether it will rain."

Dennis=20 Lindley, 2000^[14]

The concept of correlation is particularly noteworthy for the = potential=20 confusion it can cause. Statistical analysis of a data=20 set often reveals that two variables (properties) of the population = under=20 consideration tend to vary together, as if they were connected. For = example, a=20 study of annual income that also looks at age of death might find that = poor=20 people tend to have shorter lives than affluent people. The two = variables are=20 said to be correlated; however, they may or may not be the cause of one = another.=20 The correlation phenomena could be caused by a third, previously = unconsidered=20 phenomenon, called a lurking variable or confounding=20 variable. For this reason, there is no way to immediately infer the=20 existence of a causal relationship between the two variables. (See Correlation=20 does not imply causation.)

For a sample to be used as a guide to an entire population, it is = important=20 that it is truly a representative of that overall population. = Representative=20 sampling assures that the inferences and conclusions can be safely = extended from=20 the sample to the population as a whole. A major problem lies in = determining the=20 extent to which the sample chosen is actually representative. Statistics = offers=20 methods to estimate and correct for any random trending within the = sample and=20 data collection procedures. There are also methods of experimental=20 design for experiments that can lessen these issues at the outset of = a=20 study, strengthening its capability to discern truths about the = population.

Randomness is studied using the mathematical=20 discipline of probability=20 theory. Probability is used in "mathematica= l=20 statistics" (alternatively, "statistical=20 theory") to study the sampling=20 distributions of sample=20 statistics and, more generally, the properties of statist= ical=20 procedures. The use of any statistical method is valid when the = system or=20 population under consideration satisfies the assumptions of the = method.

Misuse=20 of statistics can produce subtle, but serious errors in description = and=20 interpretation =E2=80=94 subtle in the sense that even experienced = professionals=20 make such errors, and serious in the sense that they can lead to = devastating=20 decision errors. For instance, social policy, medical practice, and the=20 reliability of structures like bridges all rely on the proper use of = statistics.=20 See below = for=20 further discussion.

Even when statistical techniques are correctly applied, the results = can be=20 difficult to interpret for those lacking expertise. The statistica= l=20 significance of a trend in the data =E2=80=94 which measures = the extent to=20 which a trend could be caused by random variation in the = sample =E2=80=94 may or=20 may not agree with an intuitive sense of its significance. The set of = basic=20 statistical skills (and skepticism) that people need to deal with = information in=20 their everyday lives properly is referred to as statistical=20 literacy.

[edit]=20 Statistical=20 methods

[edit]=20 Experimental and observational studies

A common goal for a statistical research project is to investigate causality, and in = particular=20 to draw a conclusion on the effect of changes in the values of = predictors or independent=20 variables on dependent=20 variables or response. There are two major types of causal = statistical=20 studies: experimental = studies and observational=20 studies. In both types of studies, the effect of differences of an=20 independent variable (or variables) on the behavior of the dependent = variable=20 are observed. The difference between the two types lies in how the study = is=20 actually conducted. Each can be very effective. An experimental study = involves=20 taking measurements of the system under study, manipulating the system, = and then=20 taking additional measurements using the same procedure to determine if = the=20 manipulation has modified the values of the measurements. In contrast, = an=20 observational study does not involve experimental manipulation. Instead, = data=20 are gathered and correlations between predictors and response are=20 investigated.

[edit]=20 Experiments

The basic steps of a statistical experiment are:

Planning the research, including finding the number of replicates = of the=20 study, using the following information: preliminary estimates = regarding the=20 size of treatment = effects, alternative = hypotheses, and the estimated experimental=20 variability. Consideration of the selection of experimental = subjects and=20 the ethics of research is necessary. Statisticians recommend that = experiments=20 compare (at least) one new treatment with a standard treatment or = control, to=20 allow an unbiased estimate of the difference in treatment = effects.
Design=20 of experiments, using blocking = to reduce the influence of confounding = variables, and randomized=20 assignment of treatments to subjects to allow unbiased=20 estimates of treatment effects and experimental error. At this = stage, the=20 experimenters and statisticians write the experim= ental=20 protocol that shall guide the performance of the experiment = and that=20 specifies the primary analysis of the experimental data.
Performing the experiment following the experim= ental=20 protocol and analyzing=20 the data following the experimental protocol.
Further examining the data set in secondary analyses, to suggest = new=20 hypotheses for future study.
Documenting and presenting the results of the study.

Experiments on human behavior have special concerns. The famous Hawthorne = study examined=20 changes to the working environment at the Hawthorne plant of the Western=20 Electric Company. The researchers were interested in determining = whether=20 increased illumination would increase the productivity of the assembly line = workers. The=20 researchers first measured the productivity in the plant, then modified = the=20 illumination in an area of the plant and checked if the changes in = illumination=20 affected productivity. It turned out that productivity indeed improved = (under=20 the experimental conditions). However, the study is heavily criticized = today for=20 errors in experimental procedures, specifically for the lack of a control=20 group and blindness.=20 The Hawthorne=20 effect refers to finding that an outcome (in this case, worker = productivity)=20 changed due to observation itself. Those in the Hawthorne study became = more=20 productive not because the lighting was changed but because they were = being=20 observed.^{[citation=20
needed]}

[edit]=20 Observational=20 study

An example of an observational study is one that explores the = correlation=20 between smoking and lung cancer. This type of study typically uses a = survey to=20 collect observations about the area of interest and then performs = statistical=20 analysis. In this case, the researchers would collect observations of = both=20 smokers and non-smokers, perhaps through a case-control=20 study, and then look for the number of cases of lung cancer in each=20 group.

[edit]=20 Levels of=20 measurement

Main article: levels=20 of measurement

There are four main levels=20 of measurement used in statistics: nominal, ordinal, interval, and = ratio.=20 Each of these have different degrees of usefulness in statistical research. Ratio = measurements=20 have both a meaningful zero value and the distances between different=20 measurements defined; they provide the greatest flexibility in = statistical=20 methods that can be used for analyzing the data.^{[citation=20
needed]} Interval measurements have meaningful = distances=20 between measurements defined, but the zero value is arbitrary (as in the = case=20 with longitude and = temperature=20 measurements in Celsius or Fahrenheit). = Ordinal=20 measurements have imprecise differences between consecutive values, but = have a=20 meaningful order to those values. Nominal measurements have no = meaningful rank=20 order among values.

Because variables conforming only to nominal or ordinal measurements = cannot=20 be reasonably measured numerically, sometimes they are grouped together = as categorical=20 variables, whereas ratio and interval measurements are grouped = together as=20 quantitative=20 variables, which can be either discrete=20 or continuous,=20 due to their numerical nature.

[edit]=20 Key = terms used in=20 statistics

[edit]=20 Null = hypothesis

Interpretation of statistical information can often involve the = development=20 of a null=20 hypothesis in that the assumption is that whatever is proposed as a = cause=20 has no effect on the variable being measured.

The best illustration for a novice is the predicament encountered by = a jury=20 trial. The null hypothesis, H₀, asserts that the defendant is = innocent, whereas the alternative hypothesis, H₁, asserts = that the=20 defendant is guilty. The indictment comes because of suspicion of the = guilt. The=20 H₀ (status quo) stands in opposition to H₁ and is=20 maintained unless H₁ is supported by evidence"beyond a = reasonable=20 doubt". However,"failure to reject H₀" in this case does not = imply=20 innocence, but merely that the evidence was insufficient to convict. So = the jury=20 does not necessarily accept H₀ but fails to = reject=20 H₀. While one can not "prove" a null hypothesis one can test = how=20 close it is to being true with a power=20 test, which tests for type II errors.

[edit]=20 Error

Working from a null=20 hypothesis two basic forms of error are recognized:

Type=20 I errors where the null hypothesis is falsely rejected giving a = "false=20 positive".
Type=20 II errors where the null hypothesis fails to be rejected and an = actual=20 difference between populations is missed giving a "false = negative".

Error also refers to the extent to which individual observations in a = sample=20 differ from a central value, such as the sample or population mean. Many = statistical methods seek to minimize the mean-squared error, and these = are=20 called "methods=20 of least squares."

Measurement processes that generate statistical data are also subject = to=20 error. Many of these errors are classified as random=20 (noise) or systematic=20 (bias), = but other=20 important types of errors (e.g., blunder, such as when an analyst = reports=20 incorrect units) can also be important.

[edit]=20 Interval=20 estimation

Main article: interval=20 estimation

Most studies will only sample part of a population and so the results = are not=20 fully representative of the whole population. Any estimates obtained = from the=20 sample only approximate the population value. Confidence=20 intervals allow statisticians to express how closely the sample = estimate=20 matches the true value in the whole population. Often they are expressed = as 95%=20 confidence intervals. Formally, a 95% confidence interval for a value is = a range=20 where, if the sampling and analysis were repeated under the same = conditions=20 (yielding a different dataset), the interval would include the true = (population)=20 value 95% of the time. This does not imply that the probability = that the=20 true value is in the confidence interval is 95%. From the frequentist=20 perspective, such a claim does not even make sense, as the true value is = not a=20 random = variable. Either=20 the true value is or is not within the given interval. However, it is = true that,=20 before any data are sampled and given a plan for how the confidence = interval=20 will be constructed, the probability is 95% that the = yet-to-be-calculated=20 interval will cover the true value: at this point, the limits of the = interval=20 are yet-to-be-observed random=20 variables. One approach that does yield an interval that can be = interpreted=20 as having a given probability of containing the true value is to use a = credible = interval from=20 Bayesian=20 statistics: this approach depends on a different way of interpr= eting=20 what is meant by "probability", that is as a Bayesian=20 probability.

[edit]=20 Significance

Main article: Statistica= l=20 significance

This section = includes a=20 list=20 of references, related reading or external=20 links, but its sources remain unclear because it lacks inline=20 citations. Please improve=20 this article by introducing more precise citations. (May = 2012)

Statistics rarely give a simple Yes/No type answer to the question = asked of=20 them. Interpretation often comes down to the level of statistical = significance=20 applied to the numbers and often refers to the probability of a value = accurately=20 rejecting the null hypothesis (sometimes referred to as the p-value).

Referring to statistical significance does not necessarily mean that = the=20 overall result is significant in real world terms. For example, in a = large study=20 of a drug it may be shown that the drug has a statistically significant = but very=20 small beneficial effect, such that the drug will be unlikely to help the = patient=20 in a noticeable way.

Criticisms arise because the hypothesis testing approach forces one=20 hypothesis (the null=20 hypothesis) to be "favored," and can also seem to exaggerate the = importance=20 of minor differences in large studies. A difference that is highly = statistically=20 significant can still be of no practical significance, but it is = possible to=20 properly formulate tests in account for this. (See also criticism=20 of hypothesis testing.)

One response involves going beyond reporting only the significance=20 level to include the p-value=20 when reporting whether a hypothesis is rejected or accepted. The = p-value,=20 however, does not indicate the size of the effect. A better and = increasingly=20 common approach is to report confidence=20 intervals. Although these are produced from the same calculations as = those=20 of hypothesis tests or p-values, they describe both the size of = the=20 effect and the uncertainty surrounding it.

[edit]=20 Examples

Some well-known statistical test= s=20 and procedures=20 are:

Analysis=20 of variance (ANOVA)
Chi-squared=20 test
Correlation
Factor=20 analysis
Mann=E2=80=93= Whitney=20 U
Mean= =20 square weighted deviation (MSWD)
Pearson=20 product-moment correlation coefficient
Regression=20 analysis
Spearman's=20 rank correlation coefficient
Student's=20 t-test
Time=20 series analysis

[edit]=20 Specialized=20 disciplines

Main article: List=20 of fields of application of statistics

Statistical techniques are used in a wide range of types of = scientific and=20 social research, including: biostatistics,=20 computational= =20 biology, computation= al=20 sociology, network=20 biology, social=20 science, sociology and social = research. Some=20 fields of inquiry use applied statistics so extensively that they have = specialized= =20 terminology. These disciplines include:

Actuarial=20 science
Appli= ed=20 information economics
Biostatistics
Business=20 statistics
Chemometrics = (for=20 analysis of data from chemistry)
Data mining = (applying=20 statistics and pattern=20 recognition to discover knowledge from data)
Demography
Econometrics
Energy=20 statistics
Engineering = statistics
Epidemiology
Geography and Geog= raphic=20 Information Systems, specifically in Spatial=20 analysis
Image=20 processing
Psychologi= cal=20 statistics
Reliability= =20 engineering
Social=20 statistics

In addition, there are particular types of statistical analysis that = have=20 also developed their own specialised terminology and methodology:

Bootstra= p=20 & Jackknife=20 Resampling
Multivariat= e=20 statistics
Statisti= cal=20 classification
Statistical=20 surveys
Structured=20 data analysis (statistics)
Struc= tural=20 equation modelling
Survival=20 analysis
Statistics in various sports, particularly baseball = and cricket

Statistics form a key basis tool in business and manufacturing as = well. It is=20 used to understand measurement systems variability, control processes = (as in statist= ical=20 process control or SPC), for summarizing data, and to make = data-driven=20 decisions. In these roles, it is a key tool, and perhaps the only = reliable=20 tool.

[edit]=20 Statistical=20 computing

gretl,=20 an example of an open=20 source statistical package

Main article: statistical=20 computing

The rapid and sustained increases in computing power starting from = the second=20 half of the 20th century have had a substantial impact on the practice = of=20 statistical science. Early statistical models were almost always from = the class=20 of linear models, = but powerful=20 computers, coupled with suitable numerical algorithms, caused = an=20 increased interest in nonlinear=20 models (such as neural=20 networks) as well as the creation of new types, such as generalize= d linear=20 models and multilevel=20 models.

Increased computing power has also led to the growing popularity of=20 computationally intensive methods based on resampling<= /A>, such=20 as permutation tests and the bootstra= p,=20 while techniques such as Gibbs=20 sampling have made use of Bayesian models more feasible. The = computer=20 revolution has implications for the future of statistics with new = emphasis on=20 "experimental" and "empirical" statistics. A large number of both = general and=20 special purpose statis= tical=20 software are now available.

[edit]=20 Misuse

Main article: Misuse of=20 statistics

There is a general perception that statistical knowledge is=20 all-too-frequently intentionally misused=20 by finding ways to interpret only the data that are favorable to the=20 presenter.^=
[15]=20 A mistrust and misunderstanding of statistics is associated with the = quotation,=20 "T= here=20 are three kinds of lies: lies, damned lies, and statistics". Misuse = of=20 statistics can be both inadvertent and intentional, and the book How=20 to Lie With Statistics^=
[15]=20 outlines a range of considerations.

If various studies appear to contradict one another, then the public = may come=20 to distrust such studies. For example, one study may suggest that a = given diet=20 or activity raises blood=20 pressure, while another may suggest that it lowers blood pressure. = The=20 discrepancy can arise from subtle variations in experimental design, = such as=20 differences in the patient groups or research protocols, which are not = easily=20 understood by the non-expert. (Media reports usually omit this vital = contextual=20 information entirely, because of its complexity.)

By choosing (or rejecting, or modifying) a certain sample, results = can be=20 manipulated. Such manipulations need not be malicious or devious; they = can arise=20 from unintentional biases of the researcher. The graphs used to = summarize data=20 can also be misleading.

[edit]=20 Statistics applied to mathematics or the = arts

Traditionally, statistics was concerned with drawing inferences using = a=20 semi-standardized methodology that was "required learning" in most = sciences.=20 This has changed with use of statistics in non-inferential contexts. = What was=20 once considered a dry subject, taken in many fields as a = degree-requirement, is=20 now viewed enthusiastically. Initially derided by some mathematical = purists, it=20 is now considered essential methodology in certain areas.

In number=20 theory, scatter=20 plots of data generated by a distribution function may be = transformed with=20 familiar tools used in statistics to reveal underlying patterns, which = may=20 then lead to hypotheses.
Methods of statistics including predictive methods in forecasting, are = combined=20 with chaos theory = and fractal=20 geometry to create video works that are considered to have great=20 beauty.
The process art of = Jackson = Pollock relied=20 on artistic experiments whereby underlying distributions in nature = were=20 artistically revealed. With the advent of computers, methods of = statistics=20 were applied to formalize such distribution driven natural processes, = in order=20 to make and analyze moving video art.
Methods of statistics may be used predicatively in performance = art, as in=20 a card trick based on a Markov=20 process that only works some of the time, the occasion of which = can be=20 predicted using statistical methodology.
Statistics can be used to predicatively create art, as in the = statistical=20 or stochastic=20 music invented by Iannis=20 Xenakis, where the music is performance-specific. Though this type = of=20 artistry does not always come out as expected, it does behave in ways = that are=20 predictable and tunable using statistics.

[edit]=20 See also

Statistics=20 portal

Find = more about=20 Statistics on Wikipedia's sister=20 projects:
	Definiti= ons=20 and translations from Wiktionary
	Imag= es=20 and media from Commons
	Learnin= g=20 resources from Wikiversity
	News=20 stories from Wikinews
	Quotation= s=20 from Wikiquote
	Source=20 texts from Wikisource
	Textbooks= =20 from Wikibooks

Main article: Outline of=20 statistics

Glossary=20 of probability and statistics
Notation=20 in probability and statistics
List=20 of statistics articles
List=20 of academic statistical associations
List=20 of national and international statistical services
List=20 of important publications in statistics
List=20 of university statistical consulting centers
List=20 of statistical packages (software)

Foundatio= ns=20 of statistics
Official=20 statistics
List=20 of statisticians

[edit]=20 References

^ ^=
a=20 ^=
b=20 Dodge, Y. (2003) The Oxford = Dictionary of=20 Statistical Terms, OUP. ISBN= =20 0-19-920613-9
^=20 The Free Online Dictionary
^=20 "Statistics". Merriam-Webster Online = Dictionary. http://www.merriam-webster.com/dictionary/statistics= .
^=20 "Statistic". Merriam-Webster Online = Dictionary. http://www.merriam-webster.com/dictionary/statistic<= /SPAN>.
^=20 Moses, Lincoln E. (1986) Think and = Explain=20 with Statistics, Addison-Wesley, I= SBN=20 978-0-201-15619-5 . pp. 1=E2=80=933
^=20 Hays, William Lee, (1973) Statistics = for the=20 Social Sciences, Holt, Rinehart and Winston, p.xii, I= SBN=20 978-0-03-077945-9
^=20 Moore, = David (1992).=20 "Teaching Statistics as a Respectable Subject". In F. Gordon and S. = Gordon.=20 Statistics for the Twenty-First Century. Washington, DC: The=20 Mathematical Association of America. pp. 14=E2=80=9325. = ISBN 978-0-88385-078-7.
^=20 Chance, = Beth L.;=20 Rossman, Allan J. (2005). "Preface". Investigating Statistical Concepts, = Applications, and Methods. Duxbury Press. = ISBN 978-0-495-05064-3. http://www.rossmanchance.com/iscam/preface.pdf.
^=20 Anderson, D.R.; Sweeney, D.J.; = Williams, T.A..=20 (1994) Introduction to Statistics: Concepts and Applications, = pp. 5=E2=80=939.=20 West Group. I= SBN=20 978-0-314-03309-3
^=20 Singh, = Simon (2000).=20 The code book : the science of secrecy from ancient Egypt to = quantum=20 cryptography (1st Anchor Books ed.). New York: Anchor Books. = ISBN 0= -385-49532-3. ^{[page needed]}
^<= /A>=20 Al-Kadi, Ibrahim A. (1992) "The origins = of=20 cryptology: The Arab contributions=E2=80=9D, Cryptologia,=20 16(2) 97=E2=80=93126. doi:<= A=20 class=3D"external text" = href=3D"http://dx.doi.org/10.1080%2F0161-119291866801"=20 rel=3D"nofollow">10.1080/0161-119291866801
^=20 Willcox, Walter (1938) "The = Founder of=20 Statistics". Review of the International=20 Statistical Institute 5(4):321=E2=80=93328. JSTOR 1400906
^=20 Leo=20 Breiman (2001). "Statistical Modelling: the two cultures", Statistical=20 Science 16 (3), pp.199-231. doi:<= A=20 class=3D"external text" = href=3D"http://dx.doi.org/10.1214%2Fss%2F1009213726" = rel=3D"nofollow">10.1214/ss/1009213726=20 MR1874152
^=20 Lindley,=20 D. (2000) "The Philosophy of Statistics", Journal=20 of the Royal Statistical Society, Series D (The Statistician), = 49 (3),=20 293-337 JSTOR 2681060 doi:<= A=20 class=3D"external text" = href=3D"http://dx.doi.org/10.1111%2F1467-9884.00238" = rel=3D"nofollow">10.1111/1467-9884.00238
^ ^=
a=20 ^=
b=20 Huff, Darrell (1954) How=20 to Lie With Statistics, WW Norton & Company, Inc. New = York, NY. ISBN= =20 0-393-31072-8

Statistics

Descriptive = statistics

Continuous=20 data

Location	Mean=20 (Arithmetic,=20 Geometric,=20 Harmonic) Median Mode

Dispersion	Range Standard=20 deviation Coefficien= t=20 of variation Percentile Interquartile=20 range

Shape=	Variance Skewness Kurtosis Moments L-moments =

C= ount=20 data

Index=20 of dispersion

Summary = tables

Grouped=20 data
Frequency=20 distribution
Contingency=20 table

Dependen= ce

Statistical=20 graphics

Bar=20 chart
Biplot
Box=20 plot
Control=20 chart
Correlogram
Forest=20 plot
Histogram
Q-Q=20 plot
Run=20 chart
Scatter=20 plot
Stemplot
Radar=20 = chart

Data=20 collection

Designing = studies	Effect=20 size Standard=20 error Statistical=20 power Sample=20 size = determination

Survey=20 methodology	Sampling<= /LI> Stratified=20 sampling Opinion=20 poll Questionnaire

C= ontrolled=20 experiment	Design=20 of experiments Randomized=20 experiment Random=20 assignment Replicatio= n Blocking<= /LI> Factorial=20 experiment Optimal=20 design

Uncontrolled=20 studies	Natural=20 experiment Quasi-experiment Observational=20 = study

Statistical=20 inference

Statistical=20 theory	Sampling=20 distribution Sufficient=20 statistic Meta-analysis

Bayesian=20 inference	Bayesian=20 probability Prior Posterior= Credible=20 interval Bayes=20 factor Bayesian=20 estimator Max= imum=20 posterior = estimator

Frequentist=20 inference	Confidence=20 interval Hypo= thesis=20 testing Likelihood-ra= tio

Specific = tests	Z-test=20 (normal) Student's=20 t-test F-test Chi-squared=20 test Wald=20 test Mann=E2=80=93= Whitney=20 U Shapiro=E2= =80=93Wilk Signed-ra= nk Kol= mogorov=E2=80=93Smirnov=20 test

General=20 estimation	Bias Robustness Efficiency<= /A> Maximum=20 likelihood Meth= od=20 of moments Minimum= =20 distance Density=20 = estimation

Correlat= ion=20 and regression=20 analysis

Correlat= ion	Pearson=20 product-moment correlation Partial=20 correlation Confounding=20 variable Coeffi= cient=20 of determination

Regression=20 analysis	= Errors=20 and residuals Regress= ion=20 model validation Mixed=20 effects models Simult= aneous=20 equations models

Linear=20 regression	Simple=20 linear regression Ordinary=20 least squares General=20 linear model Bayesian= =20 regression

Non-standard=20 predictors	Nonlinear=20 regression Nonparamet= ric Semiparam= etric Isotonic Robust

Generalize= d=20 linear model	Exponential=20 families Logistic=20 (Bernoulli) Binomial Poisson =

Partit= ion=20 of variance	Analysis=20 of variance (ANOVA) Analysis=20 of covariance M= ultivariate=20 ANOVA Deg= rees=20 of = freedom

Categorical,=20 multivariat= e,=20 time-series,=20 or survival=20 analysis

Categorical=20 data

Cohen's=20 kappa
Contingency=20 table
Graphical=20 model
Log-linear=20 model
McNemar's=20 test

Multivariat= e=20 statistics

Multivariate=20 regression
Princi= pal=20 components
Factor=20 analysis
Cluster=20 analysis
Classifi= cation
Copulas=

Time=20 series analysis

General	Decomp= osition Trend Stationarity= Seasonal=20 adjustment

Time=20 domain	ACF PA= CF XCF<= /LI> ARMA=20 model ARIMA=20 model Vector=20 = autoregression

Frequency=20 domain	Spectra= l=20 density = estimation

Survival=20 analysis

Survival=20 function
Kapl= an=E2=80=93Meier
Logrank=20 test
Failure=20 rate
Proport= ional=20 hazards models
Acce= lerated=20 failure time = model

Applications

Biostatistics	Bioinformatics Biometrics Clinical=20 trials & studies Epidemiology Medical=20 statistics

Engineering = statistics	Chemometrics Methods=20 engineering Probabilistic = design Process= =20 & Quality=20 control Reliability= System=20 identification

Social=20 statistics	Actuarial=20 science Census Crime=20 statistics Demography Econometrics National=20 accounts Official=20 statistics Population Psychometrics

Spatial=20 statistics	Cartography Environmen= tal=20 statistics Geogr= aphic=20 information system Geostatistics Kriging =

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v t e Areas=20 of mathematics

Areas	Arithmetic Algebra elementary linear multilinear= abstract Geometry discrete = algebraic differential<= /A> Calculus/Analysis<= /LI> Set=20 theory Logic Category=20 theory Number=20 theory Combinatorics Graph=20 theory Topology Lie=20 theory Differential = equations/Dynamical = systems Mathematical=20 physics Numerical=20 analysis Computation Information=20 theory Probability<= /LI> Statistics<= /A> Optimizat= ion Control=20 theory Game=20 theory

Divisions	Pure=20 mathematics Applied=20 mathematics Discrete=20 mathematics Computati= onal=20 mathematics

Cate= gory Mathematics=20 portal Outlin= e Lists =

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color: rgb(102, 51, 102); } .mw-body a.external:active { color: rgb(187, 102, 51); } img { border: currentColor; vertical-align: middle; } hr { margin: 0.2em 0px; border: 0px currentColor; height: 1px; color: = rgb(170, 170, 170); background-color: rgb(170, 170, 170); } h1 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h2 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h3 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h4 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h5 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h6 { margin: 0px; color: black; overflow: hidden; padding-top: 0.5em; = padding-bottom: 0.17em; font-weight: normal; border-bottom-color: = rgb(170, 170, 170); border-bottom-width: 1px; border-bottom-style: = solid; } h1 { font-size: 188%; } h1 .editsection { font-size: 53%; } h2 { font-size: 150%; } h2 .editsection { font-size: 67%; } h3 { font-weight: bold; border-bottom-color: currentColor; = border-bottom-width: medium; border-bottom-style: none; } h4 { font-weight: bold; border-bottom-color: currentColor; = border-bottom-width: medium; border-bottom-style: none; } h5 { font-weight: bold; border-bottom-color: currentColor; = border-bottom-width: medium; border-bottom-style: none; } h6 { font-weight: bold; border-bottom-color: currentColor; = border-bottom-width: medium; border-bottom-style: none; } h3 { font-size: 132%; } h3 .editsection { font-size: 76%; font-weight: normal; } h4 { font-size: 116%; } h4 .editsection { font-size: 86%; font-weight: normal; } h5 { font-size: 100%; } h5 .editsection { font-weight: normal; } h6 { font-size: 80%; } h6 .editsection { font-size: 125%; font-weight: normal; } h1 { margin-bottom: 0.6em; } h2 { margin-bottom: 0.6em; } h3 { margin-bottom: 0.3em; } h4 { margin-bottom: 0.3em; } h5 { margin-bottom: 0.3em; } p { margin: 0.4em 0px 0.5em; line-height: 1.5em; } p img { margin: 0px; } ul { margin: 0.3em 0px 0px 1.6em; padding: 0px; line-height: 1.5em; = list-style-type: square; } ol { margin: 0.3em 0px 0px 3.2em; padding: 0px; line-height: 1.5em; = list-style-image: none; } li { margin-bottom: 0.1em; } dt { font-weight: bold; margin-bottom: 0.1em; } dl { margin-top: 0.2em; margin-bottom: 0.5em; } dd { line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; } q { font-family: Times,"Times New Roman",serif; font-style: italic; } pre { font-family: monospace,Courier; } code { font-family: monospace,Courier; } tt { font-family: monospace,Courier; } kbd { font-family: monospace,Courier; } samp { font-family: monospace,Courier; } .mw-code { font-family: monospace,Courier; } code { background-color: rgb(249, 249, 249); } pre { padding: 1em; border: 1px dashed rgb(47, 111, 171); color: black; = background-color: rgb(249, 249, 249); } .mw-code { padding: 1em; border: 1px dashed rgb(47, 111, 171); color: black; = background-color: rgb(249, 249, 249); } table { font-size: 100%; } fieldset { margin: 1em 0px; padding: 0px 1em 1em; border: 1px solid rgb(47, 111, = 171); line-height: 1.5em; } fieldset.nested { margin: 0px 0px 0.5em; padding: 0px 0.5em 0.5em; } legend { padding: 0.5em; font-size: 95%; } form { margin: 0px; border: currentColor; } textarea { padding: 0.1em; width: 100%; } select { vertical-align: top; } .center { width: 100%; text-align: center; } *.center * { margin-right: auto; margin-left: auto; } .small { font-size: 94%; } table.small { font-size: 100%; } #toc { padding: 5px; border: 1px solid rgb(170, 170, 170); font-size: 95%; = background-color: rgb(249, 249, 249); } .toc { padding: 5px; border: 1px solid rgb(170, 170, 170); font-size: 95%; = background-color: rgb(249, 249, 249); } .mw-warning { padding: 5px; border: 1px solid rgb(170, 170, 170); font-size: 95%; = background-color: rgb(249, 249, 249); } #toc h2 { padding: 0px; border: currentColor; font-size: 100%; font-weight: bold; = display: inline; } .toc h2 { padding: 0px; border: currentColor; font-size: 100%; font-weight: bold; = display: inline; } #toc #toctitle { text-align: center; } .toc #toctitle { text-align: center; } #toc .toctitle { text-align: center; } .toc .toctitle { text-align: center; } #toc ul { padding: 0px; text-align: left; margin-left: 0px; list-style-type: = none; list-style-image: none; } .toc ul { padding: 0px; text-align: left; margin-left: 0px; list-style-type: = none; list-style-image: none; } #toc ul ul { margin: 0px 0px 0px 2em; } .toc ul ul { margin: 0px 0px 0px 2em; } #toc .toctoggle { font-size: 94%; } .toc .toctoggle { font-size: 94%; } .toccolours { padding: 5px; border: 1px solid rgb(170, 170, 170); font-size: 95%; = background-color: rgb(249, 249, 249); } .mw-warning { text-align: center; margin-right: 50px; margin-left: 50px; } div.floatright { margin: 0px 0px 0.5em 0.5em; border: 0px currentColor; } table.floatright { margin: 0px 0px 0.5em 0.5em; border: 0px currentColor; } div.floatright p { font-style: italic; } div.floatleft { margin: 0px 0.5em 0.5em 0px; border: 0px currentColor; } table.floatleft { margin: 0px 0.5em 0.5em 0px; border: 0px currentColor; } div.floatleft p { font-style: italic; } div.thumb { width: auto; margin-bottom: 0.5em; background-color: transparent; } div.thumbinner { padding: 3px !important; border: 1px solid rgb(204, 204, 204); = text-align: center; overflow: hidden; font-size: 94%; background-color: = rgb(249, 249, 249); } html .thumbimage { border: 1px solid rgb(204, 204, 204); } html .thumbcaption { padding: 3px !important; border: currentColor; line-height: 1.4em; = font-size: 94%; } div.magnify { border: currentColor !important; } div.magnify a { border: currentColor !important; display: block; } div.magnify img { border: currentColor !important; display: block; } div.tright { margin: 0.5em 0px 1.3em 1.4em; } div.tleft { margin: 0.5em 1.4em 1.3em 0px; } img.thumbborder { border: 1px solid rgb(221, 221, 221); } #userlogin { margin: 0.5em; padding: 1.2em; border: 1px solid rgb(204, 204, 204); = float: left; } #userloginForm { margin: 0.5em; padding: 1.2em; border: 1px solid rgb(204, 204, 204); = float: left; } .catlinks { padding: 5px; border: 1px solid rgb(170, 170, 170); clear: both; = margin-top: 1em; background-color: rgb(249, 249, 249); } .usermessage { margin: 2em 0px 1em; padding: 0.5em 1em; border: 1px solid rgb(255, = 165, 0); color: black; font-weight: bold; vertical-align: middle; = background-color: rgb(255, 206, 123); } #siteNotice { margin: 0px; text-align: center; position: relative; } #localNotice { margin-bottom: 0.9em; } .firstHeading { line-height: 1.2em; padding-bottom: 0px; margin-bottom: 0.1em; } #firstHeading { line-height: 1.2em; padding-bottom: 0px; margin-bottom: 0.1em; } #siteSub { display: none; } #jump-to-nav { margin-top: -1.4em; margin-bottom: 1.4em; } #contentSub { margin: 0px 0px 1.4em 1em; width: auto; color: rgb(125, 125, 125); = line-height: 1.2em; font-size: 84%; } #contentSub2 { margin: 0px 0px 1.4em 1em; width: auto; color: rgb(125, 125, 125); = line-height: 1.2em; font-size: 84%; } span.subpages { display: block; } html { margin: 0px; padding: 0px; height: 100%; font-family: sans-serif; = font-size: 1em; } body { margin: 0px; padding: 0px; height: 100%; font-family: sans-serif; = font-size: 1em; } body { background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAAAAAA6fptVAA= AAFElEQVR4XgXAIQ0AAACAMPoHvmYZAPgA9zBoSQ8AAAAASUVORK5CYII=3D"); = background-color: rgb(243, 243, 243); } div#content { padding: 1em; color: black; margin-left: 10em; direction: ltr; = background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAIAAACQd1PeAA= AAF0lEQVR4XgWAMQ0AAACC7F/ESGThYzsGBKACeCZUlawAAAAASUVORK5CYII=3D"); = background-repeat: repeat-y; background-color: white; } #mw-page-base { background-position: left bottom; height: 5em; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAsCAIAAAArRUU2AA= AAOklEQVR4XlWNSQ4AQAjCgP//uXMYl3gwkcZKgEiaQURM/rsJ0uV1R3soLp/x+/f6mx3zOdu= 1TnHfngdpZCM7Hvq/swAAAABJRU5ErkJggg=3D=3D"); background-repeat: = repeat-x; background-color: white; } #mw-head-base { background-position: left bottom; height: 5em; margin-top: -5em; = margin-left: 10em; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAIAAACQd1PeAA= AAF0lEQVR4XgWAMQ0AAACC7F/ESGThYzsGBKACeCZUlawAAAAASUVORK5CYII=3D"); = background-repeat: repeat-x; } div#mw-head { top: 0px; width: 100%; right: 0px; position: absolute; } div#mw-head h5 { margin: 0px; padding: 0px; } div.emptyPortlet { display: none; } #p-personal { top: 0.33em; right: 0.75em; position: absolute; } #p-personal h5 { display: none; } #p-personal ul { list-style: none; margin: 0px; padding-left: 10em; } #p-personal li { line-height: 1.12em; float: left; } #p-personal li { font-size: 0.75em; margin-top: 0.5em; margin-left: 0.75em; white-space: = nowrap; } #left-navigation { left: 10em; top: 2.5em; position: absolute; } #right-navigation { margin-top: 2.5em; float: right; } div.vectorTabs h5 { display: none; } div.vectorMenu h5 span { display: none; } div.vectorTabs { height: 2.5em; float: left; } div.vectorTabs { background-position: left bottom; padding-left: 1px; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAuCAIAAABmjeQ9AA= AARElEQVR4XmVO0QqAQAwy//+P87QjGUf0IDo3ZZREAD84eXkZ5YDBRQcFwpXv3kW1N4/GmZX= e326/xnf9ya8zzy/NtusBX91B1f55qzcAAAAASUVORK5CYII=3D"); = background-repeat: no-repeat; } div.vectorTabs ul { float: left; } div.vectorTabs ul { list-style: none; margin: 0px; padding: 0px; height: 100%; } div.vectorTabs ul li { float: left; } div.vectorTabs ul li { background-position: left bottom; margin: 0px; padding: 0px; height: = 100%; line-height: 1.12em; display: inline-block; white-space: nowrap; = background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAABkCAIAAADITs03AA= AAPElEQVR4XuWK2xEAIAzCsPvv5DSu0ahX3yv4wQVyGGCSvg2dnJvduT8sz/IwKOIfjCi2z/7= 6FhHeJcslVZQFLUC06LZ/AAAAAElFTkSuQmCC"); background-repeat: repeat-x; = background-color: rgb(243, 243, 243); } div.vectorTabs ul > li { display: block; } div.vectorTabs li.selected { background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAABkAQAAAABvV2fNAA= AAFklEQVR4Xt3BAQ0AAABAMNFF18OODwGJ5TIB2xD1DwAAAABJRU5ErkJggg=3D=3D"); } div.vectorTabs li a { height: 1.9em; color: rgb(6, 69, 173); padding-right: 0.5em; = padding-left: 0.5em; font-size: 0.8em; display: inline-block; cursor: = pointer; } div.vectorTabs li > a { display: block; } div.vectorTabs li.icon a { background-position: right bottom; background-repeat: no-repeat; } div.vectorTabs span a { padding-top: 1.25em; display: inline-block; } div.vectorTabs span > a { float: left; display: block; } div.vectorTabs span { background-position: right bottom; display: inline-block; = background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAuCAIAAABmjeQ9AA= AARElEQVR4XmVO0QqAQAwy//+P87QjGUf0IDo3ZZREAD84eXkZ5YDBRQcFwpXv3kW1N4/GmZX= e326/xnf9ya8zzy/NtusBX91B1f55qzcAAAAASUVORK5CYII=3D"); = background-repeat: no-repeat; } div.vectorTabs li.selected a { color: rgb(51, 51, 51); text-decoration: none; } div.vectorTabs li.selected a:visited { color: rgb(51, 51, 51); text-decoration: none; } div.vectorTabs li.new a { color: rgb(165, 88, 88); } div.vectorTabs li.new a:visited { color: rgb(165, 88, 88); } div.vectorMenu { background-position: 100% 60%; float: left; cursor: pointer; direction: = ltr; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAQBAMAAADgw5IVAA= AAD1BMVEXz+fuamprd3d15eXmxsbG+1Cf6AAAAAXRSTlMAQObYZgAAADFJREFUeF6tjUENADA= IxJZJ2AxsEkgd4F8UIQEMwP2ufXTNbV/gxPkgr4TjFI5TaDtqAFsCqYRIsz8AAAAASUVORK5C= YII=3D"); background-repeat: no-repeat; } div.vectorMenuFocus { background-position: 100% 60%; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAQBAMAAADgw5IVAA= AAD1BMVEWampqSkpKtra2/v7/j4+Mm0C2mAAAAAXRSTlMAQObYZgAAADNJREFUeF6tjUENADA= IxPaYgeKABAkIWOZf1EIyMAD3u/bRNbetgP1zQU6JwCkCp/B29AHbvAJZh9m+yQAAAABJRU5E= rkJggg=3D=3D"); } body.rtl div.vectorMenu { direction: rtl; } div#mw-head div.vectorMenu h5 { float: left; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAuCAIAAABmjeQ9AA= AARElEQVR4XmVO0QqAQAwy//+P87QjGUf0IDo3ZZREAD84eXkZ5YDBRQcFwpXv3kW1N4/GmZX= e326/xnf9ya8zzy/NtusBX91B1f55qzcAAAAASUVORK5CYII=3D"); = background-repeat: no-repeat; } div#mw-head div.vectorMenu h5 { background-position: left bottom; margin-left: -1px; } div#mw-head div.vectorMenu > h5 { background-image: none; } div#mw-head div.vectorMenu h4 { border: currentColor; padding-top: 1.37em; padding-left: 0.5em; = font-size: 0.8em; font-weight: normal; float: left; display: = inline-block; } div.vectorMenu h5 a { width: 24px; height: 2.5em; text-decoration: none; display: = inline-block; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAuCAIAAABmjeQ9AA= AARElEQVR4XmVO0QqAQAwy//+P87QjGUf0IDo3ZZREAD84eXkZ5YDBRQcFwpXv3kW1N4/GmZX= e326/xnf9ya8zzy/NtusBX91B1f55qzcAAAAASUVORK5CYII=3D"); = background-repeat: no-repeat; } div.vectorMenu h5 a { background-position: right bottom; } div.vectorMenu h5 > a { display: block; } div.vectorMenu div.menu { text-align: left; clear: both; display: none; position: relative; } body.rtl div.vectorMenu div.menu { margin-left: 24px; } body.rtl div.vectorMenu > div.menu { margin-left: auto; } div.vectorMenu:hover div.menu { display: block; } div.vectorMenu div.menuForceShow { display: block; } div.vectorMenu ul { border-width: 0px 1px 1px; border-style: solid; border-color: silver; = list-style: none; margin: 0px 0px 0px -1px; padding: 0px; text-align: = left; position: absolute; background-color: white; } div.vectorMenu li { margin: 0px; padding: 0px; text-align: left; line-height: 1em; } div.vectorMenu li a { padding: 0.5em; color: rgb(6, 69, 173); font-size: 0.8em; display: = inline-block; white-space: nowrap; cursor: pointer; } div.vectorMenu li > a { display: block; } div.vectorMenu li.selected a { color: rgb(51, 51, 51); text-decoration: none; } div.vectorMenu li.selected a:visited { color: rgb(51, 51, 51); text-decoration: none; } #p-search h5 { display: none; } #p-search { float: left; } #p-search { margin-right: 0.5em; margin-left: 0.5em; } #p-search form { margin: 0.4em 0px 0px; } #p-search input { margin: 0.4em 0px 0px; } div#simpleSearch { border: 1px solid rgb(170, 170, 170); width: 14em; height: 1.4em; = color: black; margin-top: 0.65em; display: block; position: relative; = min-height: 1px; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAQCAIAAABY/YLgAA= AAJklEQVR4Xl2KoREAIBCAgP3n/ez5wWDgINDMBKReXiOJ2+sI/N8Dum4DHnZbWmcAAAAASUV= ORK5CYII=3D"); background-repeat: repeat-x; background-color: white; } div#simpleSearch label { top: 0.25em; font-size: 13px; direction: ltr; } div#simpleSearch input { color: black; direction: ltr; } div#simpleSearch input:focus { =09 } div#simpleSearch input.placeholder { color: rgb(153, 153, 153); } div#simpleSearch input#searchInput { margin: 0px; padding: 0.2em 0px 0.2em 0.2em; border: currentColor; = left: 0px; top: 0px; width: 90%; font-size: 13px; position: absolute; = direction: ltr; background-color: transparent; } div#simpleSearch button#searchButton { margin: 0px; padding: 0.3em 0.4em 0.2em 0px; border: currentColor; top: = 0px; width: 10%; right: 0px; position: absolute; cursor: pointer; = background-image: none; background-color: transparent; } div#simpleSearch button#searchButton img { margin: -3px 0px 0px; padding: 0px; border: currentColor; } div#simpleSearch button#searchButton > img { margin: 0px; } div#mw-panel { left: 0px; top: 160px; width: 10em; padding-top: 1em; position: = absolute; } div#mw-panel div.portal { padding-bottom: 1.5em; direction: ltr; } div#mw-panel div.portal h5 { padding: 0px 0.25em 0.25em 1.75em; border: currentColor; color: rgb(68, = 68, 68); font-size: 0.75em; font-weight: normal; cursor: default; } div#mw-panel div.portal div.body { margin: 0px 0px 0px 1.25em; padding-top: 0.5em; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIwAAAABCAAAAAAphRnkAA= AAKUlEQVR4XrXIwQkAIBDEwI39d+v/IixoB84nELZRM9ZLp/mAAGFRN50HPt8Y9Kicg/8AAAA= ASUVORK5CYII=3D"); background-repeat: no-repeat; } div#mw-panel div.portal div.body ul { list-style: none; margin: 0px; padding: 0px; } div#mw-panel div.portal div.body ul li { margin: 0px; padding: 0px 0px 0.5em; line-height: 1.12em; overflow: = hidden; font-size: 0.75em; } div#mw-panel div.portal div.body ul li a { color: rgb(6, 69, 173); } div#mw-panel div.portal div.body ul li a:visited { color: rgb(11, 0, 128); } div#footer { padding: 0.75em; margin-top: 0px; margin-left: 10em; direction: ltr; = background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAIAAACQd1PeAA= AAF0lEQVR4XgWAMQ0AAACC7F/ESGThYzsGBKACeCZUlawAAAAASUVORK5CYII=3D"); = background-repeat: repeat-x; } div#footer ul { list-style: none; margin: 0px; padding: 0px; } div#footer ul li { margin: 0px; padding: 0.5em 0px; color: rgb(51, 51, 51); font-size: = 0.7em; } div#footer #footer-icons { float: right; } body.ltr div#footer #footer-places { float: left; } div#footer #footer-info li { line-height: 1.4em; } div#footer #footer-icons li { text-align: right; line-height: 2em; margin-left: 0.5em; float: left; } div#footer #footer-places li { line-height: 2em; margin-right: 1em; float: left; } #p-logo { left: 0px; top: -160px; width: 10em; height: 160px; position: absolute; } #p-logo a { background-position: center; width: 10em; height: 160px; = text-decoration: none; display: block; background-repeat: no-repeat; } #preftoc { background-position: left bottom; margin: 0px !important; padding: 0px = !important; width: 100%; clear: both; float: left; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAhCAMAAAAvZzGNAA= AAYFBMVEUAAAD+/v79/f38/Pzt7e36+vr4+Pj39/f19fXz8/Py8vLw8PD7+/vq6uro6Ojm5ub= l5eXj4+Ph4eHf39/d3d3b29vZ2dnY2NjW1tbU1NTT09PR0dHQ0NDPz8/Ozs7Nzc0koCh/AAAA= 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color: rgb(6, 69, 173); font-size: 0.9em; = text-decoration: none; display: inline-block; position: relative; = background-image: none; } #preftoc a:hover { text-decoration: underline; } #preftoc a:focus { text-decoration: underline; } #preftoc li.selected a { background-position: bottom; color: rgb(51, 51, 51); text-decoration: = none; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAAhCAQAAACysAk0AA= AAIElEQVR4Xm3CxwkAAAwCQPcfOc08RRAOBEi5ZN24TkoeTRJAcK98h4wAAAAASUVORK5CYII= =3D"); background-repeat: repeat-x; } #preferences { margin: -2px 0px 0px; border: 1px solid rgb(204, 204, 204); width: = 100%; clear: both; float: left; background-image: = url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAAAAAA6fptVAA= AAFElEQVR4XgXAIQ0AAACAMPpHvmIZAPwA+4zbLQMAAAAASUVORK5CYII=3D"); = background-color: rgb(249, 249, 249); } #preferences fieldset { border-width: 1px medium medium; border-style: solid none none; = border-color: rgb(204, 204, 204) currentColor 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Statistics

[edit]=20 Scope

[edit]=20 History

[edit]=20 Overview

[edit]=20 Statistical=20 methods

[edit]=20 Experimental and observational studies

[edit]=20 Experiments

[edit]=20 Observational=20 study

[edit]=20 Levels of=20 measurement

[edit]=20 Key = terms used in=20 statistics

[edit]=20 Null = hypothesis

[edit]=20 Error

[edit]=20 Interval=20 estimation

[edit]=20 Significance

Contents

[edit]=20 Examples

[edit]=20 Specialized=20 disciplines

[edit]=20 Statistical=20 computing

[edit]=20 Misuse

[edit]=20 Statistics applied to mathematics or the = arts

[edit]=20 See also

[edit]=20 References

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