Statistics

==Statistical data==
{{main|Statistical data}}

=== Data collection ===

====Sampling====
When full census data cannot be collected, statisticians collect sample data by developing specific [[design of experiments|experiment designs]] and [[survey sampling|survey samples]]. Statistics itself also provides tools for prediction and forecasting through [[statistical model]]s.

To use a sample as a guide to an entire population, it is important that it truly represents the overall population. Representative [[sampling (statistics)|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 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 [[mathematics|mathematical discipline]] of [[probability theory]]. Probability is used in [[statistical theory|mathematical statistics]] to study the [[sampling distribution]]s of [[sample statistic]]s and, more generally, the properties of [[statistical decision theory|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 [[deductive reasoning|deduce]] probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—[[inductive reasoning|inductively 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 [[Dependent and independent variables|independent variables on dependent variables]]. There are two major types of causal statistical studies: [[Experiment|experimental studies]] and [[Observational study|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 [[Level of measurement|measurements with different levels]] using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve [[Scientific control|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 controlled trial|randomized studies]], they are also applied to other kinds of data—like [[natural experiment]]s and [[Observational study|observational studies]]<ref>[[David A. Freedman (statistician)|Freedman, D.A.]] (2005) ''Statistical Models: Theory and Practice'', Cambridge University Press. {{isbn|978-0-521-67105-7}}</ref>—for which a statistician would use a modified, more structured estimation method (e.g., [[Difference in differences|difference in differences estimation]] and [[instrumental variable]]s, among many others) that produce [[consistent estimator]]s.

=====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 [[Average treatment effect|treatment effects]], [[alternative hypothesis|alternative hypotheses]], and the estimated [[experimental error|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 (statistics)|blocking]] to reduce the influence of [[confounding variable]]s, and [[randomized assignment]] of treatments to subjects to allow [[bias of an estimator|unbiased estimates]] of treatment effects and experimental error. At this stage, the experimenters and statisticians write the ''[[protocol (natural sciences)|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 [[Protocol (natural sciences)|experimental protocol]] and [[analysis of variance|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 [[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.<ref name="pmid17608932">{{cite journal |vauthors=McCarney R, Warner J, Iliffe S, van Haselen R, Griffin M, Fisher P |title=The Hawthorne Effect: a randomised, controlled trial |journal=BMC Med Res Methodol |volume=7|pages=30 |year=2007 |pmid=17608932 |pmc=1936999 |doi=10.1186/1471-2288-7-30 |issue=1 |doi-access=free }}</ref>

=====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 non-smokers, perhaps through a [[cohort study]], and then look for the number of cases of lung cancer in each group.<ref>{{cite book|editor1-last=Rothman|editor1-first=Kenneth J|editor2-last=Greenland|editor2-first=Sander|editor3-last=Lash|editor3-first=Timothy|title=Modern Epidemiology|url=https://archive.org/details/modernepidemiolo00roth|url-access=limited|date=2008|publisher=Lippincott Williams & Wilkins|page=[https://archive.org/details/modernepidemiolo00roth/page/n100 100]|edition=3rd|language=en|chapter=7|isbn=978-0781755641}}</ref> A [[case-control study]] is another type of observational study in which people with and without the outcome of interest (e.g. lung cancer) are invited to participate and their exposure histories are collected.

===Types of data===
{{main|Statistical data type||Levels of measurement}}

Various attempts have been made to produce a taxonomy of [[level of measurement|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 one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving 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 variable]]s, whereas ratio and interval measurements are grouped together as [[Variable (mathematics)#Applied statistics|quantitative variables]], which can be either [[Probability distribution#Discrete probability distribution|discrete]] or [[Probability distribution#Continuous probability distribution|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 [[integer]]s in the [[integer (computer science)|integral data type]], and continuous variables with the [[real data type]] involving [[floating-point arithmetic]]. 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)<ref>{{cite book | last1 = Mosteller | first1 = F. | author-link1 = Frederick Mosteller | last2 = Tukey | first2 = J.W | author-link2 = John Tukey | year = 1977 | title = Data analysis and regression | location = Boston | publisher = Addison-Wesley}}</ref> distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)<ref>[[John Nelder|Nelder, J.A.]] (1990). The knowledge needed to computerise the analysis and interpretation of statistical information. In ''Expert systems and artificial intelligence: the need for information about data''. Library Association Report, London, March, 23–27.</ref> described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),<ref>{{cite journal | last1 = Chrisman | first1 = Nicholas R | year = 1998 | title = Rethinking Levels of Measurement for Cartography | journal = Cartography and Geographic Information Science | volume = 25 | issue = 4| pages = 231–242 | doi=10.1559/152304098782383043| bibcode = 1998CGISy..25..231C }}</ref> van den Berg (1991).<ref>van den Berg, G. (1991). ''Choosing an analysis method''. Leiden: DSWO Press</ref>)

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."<ref>Hand, D.J. (2004). ''Measurement theory and practice: The world through quantification.'' London: Arnold.</ref>{{rp|82}}