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# Statistics

Statistics is the science and practice of developing human knowledge through the use of empirical data. It is based soundly on statistical theory which is a branch of applied mathematics. Within statistical theory, randomness and uncertainty are modelled by probability theory. Statistical practice includes the planning, summarizing, and interpreting of uncertain observations. Because the aim of statistics is to produce the "best" information from available data, some authors make statistics a branch of decision theory.

## Origin

The word statistics comes from the modern Latin phrase statisticum collegium (lecture about state affairs), from which came the Italian word statista, which means "statesman" or "politician" (compare to status) and the German Statistik, originally designating the analysis of data about the state. It acquired the meaning of the collection and classification of data generally in the early nineteenth century. The collection of data about states and localities continues, largely through national and international statistical services; in particular, censuses provide regular information about the population.

## Statistical methods

We describe our knowledge (and ignorance) mathematically and attempt to learn more from whatever we can observe. This requires us to
1. plan our observations to control their variability (experiment design),
2. summarize a collection of observations to feature their commonality by suppressing details (descriptive statistics), and
3. reach consensus about what the observations tell us about the world we observe (statistical inference).

In some forms of descriptive statistics, notably data mining, the second and third of these steps become so prominent that the first step (planning) appears to become less important. In these disciplines, data often are collected outside the control of the person doing the analysis, and the result of the analysis may be more an operational model than a consensus report about the world.

## Probability

The probability of an event is often defined as a number between one and zero. In reality however there is virtually nothing that has a probability of 1 or 0. You could say that the sun will certainly rise in the morning, but what if an extremely unlikely event destroys the sun? What if there is a nuclear war and the sky is covered in ash and smoke?

We often round the probability of such things up or down because they are so likely or unlikely to occur, that it's easier to recognise them as a probability of one or zero.

However, this can often lead to misunderstandings and dangerous behaviour, because people are unable to distinguish between, e.g., a probability of 10-4 and a probability of 10-9, despite the very practical difference between them. If you expect to cross the road about 105 or 106 times in your life, then reducing your risk of being run over per road crossing to 10-9 will make you safe for your whole life, while a risk per road crossing of 10-4 will make it very likely that you will have an accident, despite the intuitive feeling that 0.01% is a very small risk.

Use of prior probabilities of 0 (or 1) causes problems in Bayesian statistics, since the posterior distribution is then forced to be 0 (or 1) as well. In other words, the data is not taken into account at all! As Lindley puts it, if a coherent Bayesian attaches a prior probability of zero to the hypothesis that the Moon is made of green cheese, then even whole armies of astronauts coming back bearing green cheese cannot convince him. Lindley advocates never using prior probabilities of 0 or 1. He calls it Cromwell's Rule, from a letter Oliver Cromwell wrote to the synod of the Church of Scotland on August 5th, 1650 in which he said "I beseech you, in the bowels of Christ, consider it possible that you are mistaken."

## Specialized disciplines

Some sciences 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.

Lindley, D. Making Decisions. John Wiley. Second Edition 1985.

Statistical Science Web
Hierarchical directory of resources of interest to statisticians with special attention to Australia.
http://www.statsci.org/

IRT Tutorial
Item Response Theory (IRT) tutorial from the University of Illinois, Urbana-Champaign Psychology Department.
http://work.psych.uiuc.edu/irt/tutorial.asp

Advanced statistics and optimization including Response and Yield Surface Modeling.
http://www.geocities.com/ecmaass/indexa.html

Biostatistics Resources on the World Wide Web
Links to departments, associations, data, software, teaching resources.
http://www.sph.emory.edu/bios/bioslist.html

The Interdisciplinary Inversion Group.
The focus of the research is on statistical aspects of inverse problems. Courses, publications and other resources.
http://www.math.auc.dk/~mbh/Interest/inverse.html

Statistics Every Writer Should Know
Basic statistical concepts described in plain English. Designed for journalists or other beginners.
http://www.robertniles.com/stats/

Non Symmetrical Data Analysis
Discusses methods of how to use structured information for multi-dimensional data analysis. From the University of Naples.
http://www.dms.unina.it/nsda.html

The Chart of the Week
Graphical display of social, economic, and political statistical data
http://lilt.ilstu.edu/gmklass/COW/

Analysis of Multiple Informant Data
Statistical methodology for the analysis of multiple informants/sources with applications to child psychopathology and gerontology.
http://www.biostat.harvard.edu/multinform

Glossary of Statistical Terms
http://dorakmt.tripod.com/mtd/glosstat.html

StatiBot
An online system for finding statistical tests that would best suited to a specific dataset.
http://www.statibot.com/

Effect Size Calculator
On-line calculator for Cohen's d and r, presented by Lee A. Becker.
http://www.uccs.edu/~lbecker/psy590/escalc3.htm

Virtual Laboratories in Probability and Statistics
Interactive, web-based resources for students and teachers of probability and statistics.
http://www.math.uah.edu/stat/

Statistical Data Analysis
Covers time series analysis, popular distributions, and the computer implementations.
http://obelia.jde.aca.mmu.ac.uk/resdesgn/arsham/opre330.htm

Power Analysis for ANOVA Designs
On-line demonstration which runs an SAS program that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design.
http://www.math.yorku.ca/SCS/Online/power/

Entropy in Science
Specializing in the uses of entropy in statistics and science.
http://www.math.psu.edu/gunesch/entropy.html

Latent Class Analysis
Directory of software for this statistical technique for modeling unmeasured categorical variables.
http://members.aol.com/KMarkus/lca.html

Free Statistics On The Web
An index of free statistical software, data and resources.
http://freestatistics.altervista.org/

Statistical Inference on the TI-83/86/89
Contains problems and solutions on general topics of probability distributions, confidence intervals, hypothesis testing, analysis of variance, and correlation analysis. Detailed survey sampling projects are also included.
http://www.wku.edu/~david.neal/statistics/

Directory of statistical links, organized by specialty.
http://www.statoo.info/en/resources/anthill/

Power Calculator
Returns distribution values based on on-line user input for many common statistical distributions.
http://calculators.stat.ucla.edu/powercalc/

Allstat Statistics Mailing List
A UK-based worldwide e-mail broadcast system for the statistical community, operated by CTI Statistics and maintained by Dr Stuart G Young.
http://www.ltsn.gla.ac.uk/allstat/

Statistics Resources
A collection of resources: software, FAQs, papers, books, links. Ranges from elementary to advanced. Compiled by Stephen Soldz.
http://soldzresearch.com/statisticsresources.htm

SEMNET FAQ
http://www.gsu.edu/~mkteer/semfaq.html

History of the Monte Carlo Method
The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer.

Chance Database Welcome Page
This data base contains materials designed help teach a Chance course or a more standard introductory probability or statistics course.
http://www.dartmouth.edu/~chance/

Statistical List Subscription Service
An easy-to-use, form-based way of subscribing to and unsubscribing from a variety of statistical e-mail discussion lists.
http://www.stattransfer.com/lists.html

Modern Regression and Classification
2 day course for overview of computer-based regression and classification.
http://www-stat.stanford.edu/~trevor/mrc.general.html

Describes adaptive statistical designs and their benefits, and some programs for optimizing and analyzing such designs.
http://www.eecs.umich.edu/~qstout/abs/Seattle97.html

Statistical Review
Free formulae of rules for the expectation of mean, variance, covariance, correlation coefficient, proofs available for purchase.
http://www.kaspercpa.com/statisticalreview.htm

Interactive Statistical Calculations Page
Collection of over 300 links to web pages that perform statistical calculations.
http://members.aol.com/johnp71/javastat.html

Meta-Analysis
Handbook for the understanding and practice of Meta-Analysis.
http://www.media.it/meta

Item Response Theory and Its Applications
Scientific resources for item response theory approaches to measurement.
http://www.inform.umd.edu/EdRes/Colleges/EDUC/Depts/EDMS/tutorials/frontpage.html

Common Distribution Calculators
Calculators, plotters, and random number generators for pdf and cdf of the common continuous and discrete probability distributions.
http://www.stat.ucla.edu/calculators/cdf

Spatial Aggregation
A framework for organizing computations around image-like, analogue representations of physical processes in data interpretation and control tasks.
http://www.cis.ohio-state.edu/insight/sa.html

A site about Parrando's paradox: alternating two losing games yield a winning game.
http://seneca.fis.ucm.es/parr/GAMES

Research Randomizer
A free web-based service that offers instant random sampling and random assignment for researchers. Also has a user-friendly tutorial for students.
http://www.randomizer.org/

Second Moment
Academic research and resources for industry analysts and businesses interested in applying the latest statistical and analytical tools and technology.
http://secondmoment.org/

The World-Wide Web Virtual Library: Statistics
Well-organized, comprehensive list of web Statistics resources.
http://www.stat.ufl.edu/vlib/statistics.html

Spatial Statistical Sites Source
This site provides links to many pages with primary or secondary relevance to spatial statistics.
http://www.statistical.org/

The Internet Glossary of Statistical Terms
Glossary of terms organized according to learning order for those unfamiliar with the subject, and also an alphabetical list for already familiar with statistical procedure.
http://www.animatedsoftware.com/statglos/statglos.htm

Calculation for the Sobel Test
Interactive calculator performs tests of mediation in path analysis or structural equation models.
http://quantrm2.psy.ohio-state.edu/kris/sobel/sobel.htm

Experimental Designs, DOE
A collection of experimental designs that a statistical practicioner may find useful.
http://home.hccnet.nl/kees.duineveld/

Statistical Methods for Rater Agreement
Resources for researchers who want to analyze data on agreement among raters or diagnostic methods. Includes raw agreement indices, kappa coefficient, weighted kappa, latent class and latent trait models, and tetrachoric and polychoric correlations.
http://ourworld.compuserve.com/homepages/jsuebersax/agree.htm

Information about statistics for genetic analysis.

MediaChannel.org: Numbers Game
The media's use of statistics in a culture where public policy is often based on "scientific" findings is more than a topic for journalism-school debates. Links to in-depth articles from a variety of perspectives.
http://www.mediachannel.org/atissue/sample4/

Latent Class Analysis and Latent Class Models
Resource for applied researchers and statisticians interested in the statistical method, Latent Class Analysis.
http://ourworld.compuserve.com/homepages/jsuebersax

Statistics to Use
Online statistics demos.
http://www.physics.csbsju.edu/stats/

Statistical Analysis
Principles of mixed model analysis. Includes discussion of REML estimation of variances, BLUEs and BLUPs, fixed and random effects, and inference space; with small examples.

Statistics on the Web
Clay Helberg's annotated list of statistical method resources on the web.
http://www.execpc.com/~helberg/statframes.html

Statistics Resources
Sidney Tyrrell's Resources' page, part of the National Grid for Learning.
http://www.mis.coventry.ac.uk/~styrrell/resource.htm

Information and links on WWW servers in Statistics, mailing lists and newsgroups, statistical associations and departments, statistical software, miscellaneous, statistical quotes, and a directory of professional statisticians worldwide.
http://ourworld.compuserve.com/homepages/Rainer_Wuerlaender/stathome.htm

Statistics.com
Searchable database of statistical information and data sources on the internet.
http://www.statistics.com/

Chebyshev's Inequality
Chebyshev's inequality with one-tailed and unimodal versions, putting statistical limits on the dispersion of probability distributions.
http://www.btinternet.com/~se16/hgb/cheb.htm

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