statistics for other uses see statistics disambiguation statistics outline statisticians glossary notation journals lists of topics articles portal category vte more probability density is found as one gets closer to the expected mean value in a normal distribution statistics used in standardized testing assessment are shown the scales include standard deviations cumulative percentages zscores and tscores scatter plots are used in descriptive statistics to show the observed relationships between different variables statistics is a branch of mathematics dealing with data collection organization analysis interpretation and presentation see glossary of probability and statistics when census data cannot be collected statisticians collect data by developing specific experiment designs and survey samples representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole 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 two main statistical methods are used in data analysis descriptive statistics which summarize data from a sample using indexes such as the mean or standard deviation and inferential statistics which draw conclusions from data that are subject to random variation eg observational errors sampling variation descriptive statistics are most often concerned with two sets of properties of a distribution sample or population central tendency or location seeks to characterize the distributions central or typical value while dispersion or variability characterizes the extent to which members of the distribution depart from its center and each other inferences on mathematical statistics are made under the framework of probability theory which deals with the analysis of random phenomena a standard statistical procedure involves the test of the relationship between two statistical data sets or a data set and synthetic data drawn from an idealized model a hypothesis is proposed for the statistical relationship between the two data sets and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false given the data that are used in the test working from a null hypothesis two basic forms of error are recognized type i errors null hypothesis is falsely rejected giving a false positive and type ii errors null hypothesis fails to be rejected and an actual difference between populations is missed giving a false negative 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 types of errors eg blunder such as when an analyst reports incorrect units can also be important the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems statistics can be said to have begun in ancient civilization going back at least to the 5th century bc but it was not until the 18th century that it started to draw more heavily from calculus and probability theory in more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis scope some definitions are merriamwebster dictionary defines statistics as a branch of mathematics dealing with the collection analysis interpretation and presentation of masses of numerical data statistician arthur lyon bowley defines statistics as numerical statements of facts in any department of inquiry placed in relation to each other statistics is a mathematical body of science that pertains to the collection analysis interpretation or explanation and presentation of data mathematical statistics mathematical statistics mathematical statistics is the application of mathematics to statistics mathematical techniques used for this include mathematical analysis linear algebra stochastic analysis differential equations and measuretheoretic probability theory overview in applying statistics to a problem it is common practice to start 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 ideally statisticians compile data about the entire population an operation called census this may be organized by governmental statistical institutes descriptive statistics can be used to summarize the population data numerical descriptors include mean and standard deviation for continuous data types like income while frequency and percentage are more useful in terms of describing categorical data like race when a census is not feasible a chosen subset of the population called a sample is studied once a sample that is representative of the population is determined data is collected for the sample members in an observational or experimental setting again descriptive statistics can be used to summarize the sample data however the drawing of the sample has been subject to an element of randomness hence the established numerical descriptors from the sample are also due to uncertainty to still draw meaningful conclusions about the entire population inferential statistics is needed it uses patterns in the sample data to draw inferences about the population represented accounting for randomness these inferences may take the form of answering yesno 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 data collection sampling when full census data cannot be collected statisticians collect sample data by developing specific experiment designs and survey samples statistics itself also provides tools for prediction and forecasting through statistical models the idea of making inferences based on sampled data began around the mid1600s in connection with estimating populations and developing precursors of life insurance to use a sample as a guide to an entire population it is important that it truly represents the overall population representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole a major problem lies in determining the extent that the sample chosen is actually representative statistics offers methods to estimate and correct for any bias 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 sampling theory is part of the mathematical discipline of probability theory probability is used in mathematical statistics 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 the difference in point of view between classic probability theory and sampling theory is roughly that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples statistical inference however moves in the opposite directioninductively inferring from samples to the parameters of a larger or total population 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 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 while the tools of data analysis work best on data from randomized studies they are also applied to other kinds of datalike natural experiments and observational studiesfor which a statistician would use a modified more structured estimation method eg difference in differences estimation and instrumental variables among many others that produce consistent estimators 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 will guide the performance of the experiment and which 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 observational study an example of an observational study is one that explores the association 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 cohort study and then look for the number of cases of lung cancer in each group a casecontrol study is another type of observational study in which people with and without the outcome of interest eg lung cancer are invited to participate and their exposure histories are collected types of data statistical data type and levels of measurement various attempts have been made to produce a taxonomy of levels of measurement the psychophysicist stanley smith stevens defined nominal ordinal interval and ratio scales nominal measurements do not have meaningful rank order among values and permit any onetoone transformation ordinal measurements have imprecise differences between consecutive values but have a meaningful order to those values and permit any orderpreserving transformation 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 and permit any linear transformation ratio measurements have both a meaningful zero value and the distances between different measurements defined and permit any rescaling transformation 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 variables which can be either discrete or continuous due to their numerical nature such distinctions can often be loosely correlated with data type in computer science in that dichotomous categorical variables may be represented with the boolean data type polytomous categorical variables with arbitrarily assigned integers in the integral data type and continuous variables with the real data type involving floating point computation but the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented other categorizations have been proposed for example mosteller and tukey 1977 the issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions the relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations whether or not a transformation is sensible to contemplate depends on the question one is trying to answer hand 2004 p82 terminology and theory of inferential statistics statistics estimators and pivotal quantities consider independent identically distributed iid random variables with a given probability distribution standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these iid variables the population being examined is described by a probability distribution that may have unknown parameters a statistic is a random variable that is a function of the random sample but not a function of unknown parameters the probability distribution of the statistic though may have unknown parameters consider now a function of the unknown parameter an estimator is a statistic used to estimate such function commonly used estimators include sample mean unbiased sample variance and sample covariance a random variable that is a function of the random sample and of the unknown parameter but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or pivot widely used pivots include the zscore the chi square statistic and students tvalue between two estimators of a given parameter the one with lower mean squared error is said to be more efficient furthermore an estimator is said to be unbiased if its expected value is equal to the true value of the unknown parameter being estimated and asymptotically unbiased if its expected value converges at the limit to the true value of such parameter other desirable properties for estimators include umvue estimators that have the lowest variance for all possible values of the parameter to be estimated this is usually an easier property to verify than efficiency and consistent estimators which converges in probability to the true value of such parameter this still leaves the question of how to obtain estimators in a given situation and carry the computation several methods have been proposed the method of moments the maximum likelihood method the least squares method and the more recent method of estimating equations null hypothesis and alternative hypothesis interpretation of statistical information can often involve the development of a null hypothesis which is usually but not necessarily that no relationship exists among variables or that no change occurred over time the best illustration for a novice is the predicament encountered by a criminal trial the null hypothesis h0 asserts that the defendant is innocent whereas the alternative hypothesis h1 asserts that the defendant is guilty the indictment comes because of suspicion of the guilt the h0 status quo stands in opposition to h1 and is maintained unless h1 is supported by evidence beyond a reasonable doubt however failure to reject h0 in this case does not imply innocence but merely that the evidence was insufficient to convict so the jury does not necessarily accept h0 but fails to reject h0 while one can not prove a null hypothesis one can test how close it is to being true with a power test which tests for type ii errors what statisticians call an alternative hypothesis is simply a hypothesis that contradicts the null hypothesis 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 giving a false negative standard deviation refers to the extent to which individual observations in a sample differ from a central value such as the sample or population mean while standard error refers to an estimate of difference between sample mean and population mean a statistical error is the amount by which an observation differs from its expected value a residual is the amount an observation differs from the value the estimator of the expected value assumes on a given sample also called prediction mean squared error is used for obtaining efficient estimators a widely used class of estimators root mean square error is simply the square root of mean squared error a least squares fit in red the points to be fitted in blue the fitted line many statistical methods seek to minimize the residual sum of squares and these are called methods of least squares in contrast to least absolute deviations the latter gives equal weight to small and big errors while the former gives more weight to large errors residual sum of squares is also differentiable which provides a handy property for doing regression least squares applied to linear regression is called ordinary least squares method and least squares applied to nonlinear regression is called nonlinear least squares also in a linear regression model the non deterministic part of the model is called error term disturbance or more simply noise both linear regression and nonlinear regression are addressed in polynomial least squares which also describes the variance in a prediction of the dependent variable y axis as a function of the independent variable x axis and the deviations errors noise disturbances from the estimated fitted curve 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 types of errors eg blunder such as when an analyst reports incorrect units can also be important the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems interval estimation interval estimation confidence intervals the red line is true value for the mean in this example the blue lines are random confidence intervals for 100 realizations most studies only sample part of a population so results dont fully represent 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 for a value is a range where if the sampling and analysis were repeated under the same conditions yielding a different dataset the interval would include the true population value in 95 of all possible cases 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 however it is true that before any data are sampled and given a plan for how to construct the confidence interval the probability is 95 that the yettobecalculated interval will cover the true value at this point the limits of the interval are yettobeobserved random variables one approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from bayesian statistics this approach depends on a different way of interpreting what is meant by probability that is as a bayesian probability in principle confidence intervals can be symmetrical or asymmetrical an interval can be asymmetrical because it works as lower or upper bound for a parameter leftsided interval or right sided interval but it can also be asymmetrical because the two sided interval is built violating symmetry around the estimate sometimes the bounds for a confidence interval are reached asymptotically and these are used to approximate the true bounds significance statistical significance statistics rarely give a simple yesno type answer to the question under analysis interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of a value accurately rejecting the null hypothesis sometimes referred to as the pvalue in this graph the black line is probability distribution for the test statistic the critical region is the set of values to the right of the observed data point observed value of the test statistic and the pvalue is represented by the green area the standard approach is to test a null hypothesis against an alternative hypothesis a critical region is the set of values of the estimator that leads to refuting the null hypothesis the probability of type i error is therefore the probability that the estimator belongs to the critical region given that null hypothesis is true statistical significance and the probability of type ii error is the probability that the estimator doesnt belong to the critical region given that the alternative hypothesis is true the statistical power of a test is the probability that it correctly rejects the null hypothesis when the null hypothesis is false 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 is unlikely to help the patient noticeably although in principle the acceptable level of statistical significance may be subject to debate the pvalue is the smallest significance level that allows the test to reject the null hypothesis this test is logically equivalent to saying that the pvalue is the probability assuming the null hypothesis is true of observing a result at least as extreme as the test statistic therefore the smaller the pvalue the lower the probability of committing type i error some problems are usually associated with this framework see criticism of hypothesis testing a difference that is highly statistically significant can still be of no practical significance but it is possible to properly formulate tests to account for this one response involves going beyond reporting only the significance level to include the pvalue when reporting whether a hypothesis is rejected or accepted the pvalue however does not indicate the size or importance of the observed effect and can also seem to exaggerate the importance of minor differences in large studies a better and 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 fallacy of the transposed conditional aka prosecutors fallacy criticisms arise because the hypothesis testing approach forces one hypothesis the null hypothesis to be favored since what is being evaluated is the probability of the observed result given the null hypothesis and not probability of the null hypothesis given the observed result an alternative to this approach is offered by bayesian inference although it requires establishing a prior probability rejecting the null hypothesis does not automatically prove the alternative hypothesis as everything in inferential statistics it relies on sample size and therefore under fat tails pvalues may be seriously miscomputed examples some wellknown statistical tests and procedures are analysis of variance anova chisquared test correlation factor analysis mannwhitney u mean square weighted deviation mswd pearson productmoment correlation coefficient regression analysis spearmans rank correlation coefficient students ttest time series analysis conjoint analysis misuse misuse of statistics misuse of statistics can produce subtle but serious errors in description and interpretationsubtle in the sense that even experienced professionals make such errors and serious in the sense that they can lead to devastating decision errors for instance social policy medical practice and the reliability of structures like bridges all rely on the proper use of statistics even when statistical techniques are correctly applied the results can be difficult to interpret for those lacking expertise the statistical significance of a trend in the datawhich measures the extent to which a trend could be caused by random variation in the samplemay 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 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 ways to avoid misuse of statistics include using proper diagrams and avoiding bias to assist in the understanding of statistics huff proposed a series of questions to be asked in each case who says so does heshe have an axe to grind how does heshe know does heshe have the resources to know the facts whats missing does heshe give us a complete picture did someone change the subject does heshe offer us the right answer to the wrong problem does it make sense is hisher conclusion logical and consistent with what we already know the confounding variable problem x and y may be correlated not because there is causal relationship between them but because both depend on a third variable z z is called a confounding factor misinterpretation correlation 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 history of statistical science gerolamo cardano the earliest pioneer on the mathematics of probability history of statistics and founders of statistics 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 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 natural and social sciences its mathematical foundations were laid in the 17th century with the development of the probability theory by gerolamo cardano blaise pascal and pierre de fermat mathematical probability theory arose from the study of games of chance although the concept of probability was already examined in medieval law and by philosophers such as juan caramuel the method of least squares was first described by adrienmarie legendre in 1805 karl pearson a founder of mathematical statistics the modern field of statistics emerged in the late 19th and early 20th century in three stages ronald fisher coined the term null hypothesis during the lady tasting tea experiment which is never proved or established but is possibly disproved in the course of experimentation the second wave of the 1910s and 20s was initiated by william gosset and reached its culmination in the insights of ronald fisher who wrote the textbooks that were to define the academic discipline in universities around the world fishers most important publications were his 1918 seminal paper the correlation between relatives on the supposition of mendelian inheritance which was the first to use the statistical term variance his classic 1925 work statistical methods for research workers and his 1935 the design of experiments a concept in sexual selection about a positive feedback runaway affect found in evolution the final wave which mainly saw the refinement and expansion of earlier developments emerged from the collaborative work between egon pearson and jerzy neyman in the 1930s they introduced the concepts of type ii error power of a test and confidence intervals jerzy neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive quota sampling today statistical methods are applied in all fields that involve decision making for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology the use of modern computers has expedited largescale statistical computations and has also made possible new methods that are impractical to perform manually statistics continues to be an area of active research for example on the problem of how to analyze big data applications applied statistics theoretical statistics and mathematical statistics applied statistics comprises descriptive statistics and the application of inferential statistics theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference as well as encompassing mathematical statistics mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference but also various aspects of computational statistics and the design of experiments machine learning and data mining there are two applications for machine learning and data mining data management and data analysis statistics tools are necessary for the data analysis statistics in society statistics is applicable to a wide variety of academic disciplines including natural and social sciences government and business statistical consultants can help organizations and companies that dont have inhouse expertise relevant to their particular questions statistical computing gretl an example of an open source statistical package computational statistics 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 examples of available software capable of complex statistical computation include programs such as mathematica sas spss and r 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 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 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 tunable using statistics specialized disciplines list of fields of application of statistics 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 assesses risk in the insurance and finance industries applied information economics astrostatistics statistical evaluation of astronomical data biostatistics business statistics chemometrics for analysis of data from chemistry data mining applying statistics and pattern recognition to discover knowledge from data data science demography statistical study of populations econometrics statistical analysis of economic data energy statistics engineering statistics epidemiology statistical analysis of disease geography and geographic information systems specifically in spatial analysis image processing medical statistics political science psychological statistics reliability engineering social statistics statistical mechanics 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 structured data analysis statistics structural equation modelling survey methodology survival analysis statistics in various sports particularly baseball known as sabermetrics 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 library resources about statistics resources in your library outline of statistics abundance estimation data science glossary of probability and statistics list of academic statistical associations list of important publications in statistics list of national and international statistical services list of statistical packages software list of statistics articles list of university statistical consulting centers notation in probability and statistics foundations and major areas of statistics foundations of statistics list of statisticians official statistics multivariate analysis of variance barbara illowsky susan dean 2014 introductory statistics openstax cnx isbn9781938168208 david w stockburger introductory statistics concepts models and applications 3rd web ed missouri state university openintro statistics 3rd edition by diez barr and cetinkayarundel stephen jones 2010 statistics in psychology explanations without equations palgrave macmillan isbn9781137282392 cohen j 1990 things i have learned so far american psychologist 45 13041312 gigerenzer g 2004 mindless statistics journal of socioeconomics 33 587606 doi101016jsocec200409033 ioannidis jpa 2005 why most published research findings are false plos medicine 2 696701 doi101371journalpmed0040168 statisticsat wikipedias sister projects definitions from wiktionary media from wikimedia commons news from wikinews quotations from wikiquote texts from wikisource textbooks from wikibooks resources from wikiversity electronic version statsoft inc 2013 electronic statistics textbook tulsa ok statsoft online statistics education an interactive multimedia course of study developed by rice university lead developer university of houston clear lake tufts university and national science foundation ucla statistical computing resources philosophy of statistics from the stanford encyclopedia of philosophy vtestatistics outline index descriptive statisticscontinuous datacenter mean arithmetic geometric harmonic median mode dispersion variance standard deviation coefficient of variation percentile range interquartile range shape central limit theorem moments skewness kurtosis lmoments count data index of dispersion summary tables grouped data frequency distribution contingency table dependence pearson productmoment correlation rank correlation spearmans rho kendalls tau partial correlation scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart qq plot run chart scatter plot stemandleaf display radar chart data collectionstudy design population statistic effect size statistical power sample size determination missing data survey methodology sampling stratified cluster standard error opinion poll questionnaire controlled experiments design control optimal controlled trial randomized random assignment replication blocking interaction factorial experiment uncontrolled studies observational study natural experiment quasiexperiment statistical inferencestatistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model lp space parameter location scale shape parametric family likelihoodmonotone locationscale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness frequentist inferencepoint estimation estimating equations maximum likelihood method of moments mestimator minimum distance unbiased estimators meanunbiased minimumvariance raoblackwellization lehmannscheff theorem median unbiased plugin interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihoodratio wald score specific tests ztest normal students ttest ftest goodness of fit chisquared gtest kolmogorovsmirnov andersondarling lilliefors jarquebera normality shapirowilk likelihoodratio test model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodgeslehmann estimator rank sum mannwhitney nonparametric anova 1way kruskalwallis 2way friedman ordered alternative jonckheereterpstra bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlationregression analysiscorrelation pearson productmoment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression model validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars linear regression simple linear regression ordinary least squares general linear model bayesian regression nonstandard predictors nonlinear regression nonparametric semiparametric isotonic robust heteroscedasticity homoscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova anova analysis of covariance multivariate anova degrees of freedom categorical multivariate timeseries survival analysiscategorical cohens kappa contingency table graphical model loglinear model mcnemars test multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal timeseriesgeneral decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickeyfuller johansen qstatistic ljungbox durbinwatson breuschgodfrey time domain autocorrelation acf partial pacf crosscorrelation xcf arma model arima model boxjenkins autoregressive conditional heteroskedasticity arch vector autoregression var frequency domain spectral density estimation fourier analysis wavelet whittle likelihood survivalsurvival function kaplanmeier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelsonaalen estimator test logrank test applicationsbiostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics chemometrics methods engineering probabilistic design process quality control reliability system identification social statistics actuarial science census crime statistics demography econometrics national accounts official statistics population statistics 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