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Statistics

[[File:The Normal Distribution.svg|thumb|350px|right|More Probability density function|probability density will be found the closer one gets to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include ''standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines,'' and ''percentages in standard nines.'']]
Statistics is the science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of Survey sampling|surveys and experimental design|experiments.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0199206139 A statistician is someone who is particularly versed in the ways of thinking necessary for the successful application of statistical analysis. Often such people have gained this experience after starting work in any of a list of fields of application of statistics|number of fields. There is also a discipline called ''mathematical statistics'', which is concerned with the theoretical basis of the subject. The word ''statistics'' can either be singular or plural. In its singular form, a ''statistic'' is a quantity (such as a mean) calculated from a set of data, whereas ''statistics'' is the mathematical science discussed in this article.

Scope

Statistics is considered by some to be a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data,Moses, Lincoln E. ''Think and Explain with statistics'', pp. 1 - 3. Addison-Wesley, 1986. while others consider it to be a branch of mathematicsHays, William Lee, ''Statistics for the social sciences'', Holt, Rinehart and Winston, 1973, p.xii, ISBN 978-0030779459 concerned with collecting and interpreting data.Statistics at Encyclopedia of Mathematics Because of its empirical roots and its focus on applications, statistics is usually considered to be a distinct mathematical science rather than a branch of mathematics. Statisticians improve the quality of data with the design of experiments and survey sampling. Statistics also provides tools for prediction and forecasting using data and statistical models. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical methods can be used to summarize or describe a collection of data; this is called ''descriptive statistics''. This is useful in research, when communicating the results of experiments. In addition, patterns in the data may be mathematical model|modeled in a way that accounts for randomness and uncertainty in the observations, and are then used to draw inferences about the process or population being studied; this is called ''inferential statistics''. Inference is a vital element of scientific advance, since it provides a prediction (based in data) for where a theory logically leads. To further prove the guiding theory, these predictions are tested as well, as part of the scientific method. If the inference holds true, then the descriptive statistics of the new data increase the soundness of that hypothesis. Descriptive statistics and inferential statistics (a.k.a., predictive statistics) together comprise ''applied statistics''.Anderson, , D.R.; Sweeney, D.J.; Williams, T.A.. ''Statistics: Concepts and Applications'', pp. 5 - 9. West Publishing Company, 1986.

History

Some scholars pinpoint the origin of statistics to 1663, with the publication of ''Natural and Political Observations upon the Bills of Mortality'' by John Graunt.Willcox, Walter (1938) ''The Founder of Statistics''. Review of the International Statistical Institute 5(4):321-328. Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its History of statistics#Etymology|''stat-'' etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and the natural and social sciences. Its mathematical foundations were laid in the 17th century with the development of probability theory by Blaise Pascal and Pierre de Fermat. Probability theory arose from the study of games of chance. The method of least squares was first described by Carl Friedrich Gauss around 1794. The use of modern computers has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually. The American Statistical Association has ranked W. Edwards Deming|Deming, Ronald Fisher|Fisher, and CR Rao|Rao as the greatest statisticians of all time.

Overview

In applying statistics to a scientific, industrial, or societal problem, it is necessary to begin with a statistical population|population or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal". A population can also be composed of observations of a process at various times, with the data from each observation serving as a different member of the overall group. Data collected about this kind of "population" constitutes what is called a time series. For practical reasons, a chosen subset of the population called a sampling (statistics)|sample is studied — as opposed to compiling data about the entire group (an operation called census). Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental setting. This data can then be subjected to statistical analysis, serving two related purposes: description and inference.
- Descriptive statistics summarize the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include mean and standard deviation for Continuous probability distribution|continuous data types (like heights or weights), while frequency and percentage are more useful in terms of describing categorical data (like race).
- Inferential statistics uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. These inferences may take the form of: answering yes/no questions about the data (hypothesis testing) estimating numerical characteristics of the data (estimation), describing associations within the data (correlation), modeling relationships within the data (regression analysis|regression), extrapolation, interpolation, or other mathematical model|modeling techniques like ANOVA, time series, and data mining. 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 are connected. For example, a study of annual income that also looks at age of death might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable or confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the two variables. (See Correlation does not imply causation.) For a sample to be used as a guide to an entire population, it is important that it is truly a representative of that overall population. Representative sampling assured, inferences and conclusions can be safely extended from the sample to the population as a whole. A major problem lies in determining the extent to which the sample chosen is actually representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures. There are also methods for designing experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Statisticians describe stronger methods as more "robust".(See experimental design.) The fundamental mathematical concept employed in understanding potential randomness is probability. Mathematical statistics (also called statistical theory) is the branch of applied mathematics that uses probability theory and mathematical analysis|analysis to examine the theoretical basis of statistics. The use of any statistical method is valid only when the system or population under consideration satisfies the basic mathematical assumptions of the method. Misuse of statistics can produce subtle, but serious errors in description and interpretation — subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics. Even when statistics are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data — which measures the extent to which a trend could be caused by random variation in the sample — may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy.

Statistical methods

Experimental and observational studies

A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables or response. There are two major types of causal statistical studies: experimental studies and observational studies. In both types of studies, the effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between predictors and response are investigated. An example of an experimental study is the famous Hawthorne study, which attempted to test changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved (under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and double-blind|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. An example of an observational study is one that explores the correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers, perhaps through a case-control study, and then look for the number of cases of lung cancer in each group. The basic steps of an experiment are:
- Planning the research, including determining information sources, research subject selection, and ethics|ethical considerations for the proposed research and method.
- Design of experiments, concentrating on the system model and the interaction of independent and dependent variables.
- summary statistics|Summarizing a collection of observations to feature their commonality by suppressing details. (Descriptive statistics)
- Reaching consensus about what statistical inference|the observations tell about the world being observed. (Statistical inference)
- Documenting / presenting the results of the study.

Levels of measurement

There are four types of measurements or level of measurement|levels of measurement or measurement scales used in statistics:
- nominal,
- ordinal,
- interval, and
- ratio. They have different degrees of usefulness in statistical research. Ratio measurements have both a zero value defined and the distances between different measurements defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data. Interval measurements have meaningful distances between measurements defined, but have no meaningful zero value defined (as in the case with IQ measurements or with temperature measurements in Fahrenheit). Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values. Nominal measurements have no meaningful rank order among values. Since variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are called together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative or continuous variables due to their numerical nature.

Key terms used in statistics

Null hypothesis

Interpretation of statistical information can often involve the development of a null hypothesis in that the assumption is that whatever is proposed as a cause has no effect on the variable being measured. The best illustration for a novice is the predicament encountered by a jury trial. The null hypothesis, 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.

Error

Working from a null hypothesis two basic forms of error are recognised:
- Type I and type II errors|Type I errors where the null hypothesis is falsely rejected giving a "false positive".
- Type I and type II errors|Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed.

Confidence intervals

Most studies will only sample part of a population and then the result is used to interpret the null hypothesis in the context of the whole population. Any estimates obtained from the sample only approximate the population value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals. Formally, a 95% confidence interval of a procedure is any range such that the interval covers the true population value 95% of the time given repeated sampling under the same conditions. If these intervals span a value (such as zero) where the null hypothesis would be confirmed then this can indicate that any observed value has been seen by chance. For example a drug that gives a mean increase in heart rate of 2 beats per minute but has 95% confidence intervals of -5 to 9 for its increase may well have no effect whatsoever. The 95% confidence interval is often misinterpreted as the probability that the true value lies between the upper and lower limits given the observed sample. However this quantity is more a credible interval available only from Bayesian statistics.

Significance

Statistics rarely give a simple Yes/No type answer to the question asked of them. Interpretation often comes down to the level of statistical significance applied to the numbers and often refer to the probability of a value accurately rejecting the null hypothesis (sometimes referred to as the p-value). When interpreting an academic paper reference to the significance of a result when referring to the statistical significance does not necessarily mean that the overall result means anything in real world terms. (For example in a large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect such that the drug will be unlikely to help anyone given it in a noticeable way.)

Examples

Some well-known statistical Statistical hypothesis testing|tests and procedures are:

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

Specialized disciplines

Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:

- Actuarial science
- Applied information economics
- Biostatistics
- Business statistics
- Chemometrics (for analysis of data from chemistry)
- Data mining (applying statistics and pattern recognition to discover knowledge from data)
- Demography
- Economic statistics (Econometrics)
- Energy statistics
- Engineering statistics
- Epidemiology
- Geography and Geographic Information Systems, specifically in Spatial analysis
- Image processing
- Psychological statistics
- Reliability engineering
- Social statistics In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology:
- Bootstrapping (statistics)|Bootstrap & Resampling (statistics)|Jackknife Resampling
- Statistical classification
- Statistical surveys
- Structured data analysis (statistics)
- Survival analysis
- Statistics in various sports, particularly Baseball statistics|baseball and Cricket statistics|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 data-driven decisions. In these roles, it is a key tool, and perhaps the only reliable tool.

Statistical computing

[[File:Gretl screenshot.png|thumb|300px|gretl, an example of an open source statistical package]] 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 regression|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 (statistics)|resampling, such as permutation tests and the bootstrapping (statistics)|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 List of statistical packages|statistical software are now available.

Misuse

There is a general perception that statistical knowledge is all-too-frequently intentionally Misuse of statistics|misused by finding ways to interpret only the data that are favorable to the presenter. A famous saying attributed to Benjamin Disraeli is, "Lies, damned lies, and statistics|There are three kinds of lies: lies, damned lies, and statistics." Harvard President Lawrence Lowell wrote in 1909 that statistics, ''"...like veal pies, are good if you know the person that made them, and are sure of the ingredients."'' If various studies appear to contradict one another, then the public may come to distrust such studies. For example, one study may suggest that a given diet or activity raises blood pressure, while another may suggest that it lowers blood pressure. The discrepancy can arise from subtle variations in experimental design, such as differences in the patient groups or research protocols, which are not easily understood by the non-expert. (Media reports usually omit this vital contextual information entirely, because of its complexity.) By choosing (or rejecting, or modifying) a certain sample, results can be manipulated. Such manipulations need not be malicious or devious; they can arise from unintentional biases of the researcher. The graphs used to summarize data can also be misleading. Deeper criticisms come from the fact that the hypothesis testing approach, widely used and in many cases required by law or regulation, forces one hypothesis (the null hypothesis) to be "favored," and can also seem to exaggerate the importance of minor differences in large studies. A difference that is highly statistically significant can still be of no practical significance. (See Hypothesis test#Criticism|criticism of hypothesis testing and Null hypothesis#Controversy|controversy over the null hypothesis.) One response is by giving a greater emphasis on the p-value|''p''-value than simply reporting whether a hypothesis is rejected at the given level of significance. The ''p''-value, however, does not indicate the size of the effect. Another increasingly common approach is to report confidence intervals. Although these are produced from the same calculations as those of hypothesis tests or ''p''-values, they describe both the size of the effect and the uncertainty surrounding it.

Statistics applied to mathematics or the arts

Traditionally, statistics was concerned with drawing inferences using a semi-standardized methodology that was "required learning" in most sciences. This has changed with use of statistics in non-inferential contexts. What was once considered a dry subject, taken in many fields as a degree-requirement, is now viewed enthusiastically. Initially derided by some mathematical purists, it is now considered essential methodology in certain areas.
- In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses.
- Methods of statistics including predictive methods in forecasting, are combined with chaos theory and fractal geometry to create video works that are considered to have great beauty.
- The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed. With the advent of computers, methods of statistics were applied to formalize such distribution driven natural processes, in order to make and analyze moving video art.
- Methods of statistics may be used predicatively in performance art, as in a card trick based on a Markov process that only works some of the time, the occasion of which can be predicted using statistical methodology.
- Statistics is used to predicatively create art, as in applications of statistical mechanics with the statistical or stochastic music invented by Iannis Xenakis, where the music is performance-specific. Though this type of artistry does not always come out as expected, it does behave within a range predictable using statistics.