theory has the task of considering the types of questions that data analysts might want to ask about the problems they are studying and of providing data analytic techniques for answering them. Some of these tasks are: Summarising populations in the form of a fitted distribution or probability density function Summarising the relationship between variables using some type of regression analysis Providing ways of predicting the outcome of a random quantity given other related variables Examining the possibility of reducing the number of variables being considered within a problem (the task of Dimension reduction) When a statistical procedure has been specified in the study protocol, then statistical theory provides well-defined probability statements for the method when applied to all populations that could have arisen from the randomization used to generate the data. This provides an objective way of estimating parameters, estimating confidence intervals, testing hypotheses, and selecting the best. Even for observational data, statistical theory provides a way of calculating a value that can be used to interpret a sample of data from a population, it can provide a means of indicating how well that value is determined by the sample, and thus a means of saying corresponding values derived for different populations are as different as they might seem; however, the reliability of inferences from post-hoc observational data is often worse than for planned randomized generation of data. Applied statistical inference Statistical theory provides the basis for a number of data-analytic approaches that are common across scientific and social research. Interpreting data is done with one of the following approaches: Estimating parameters Providing a range of values instead of a point estimate Testing statistical hypotheses Many of the standard methods for those approaches rely on certain statistical assumptions (made in the derivation of the methodology) actually holding in practice. Statistical theory studies the consequences of departures from these assumptions. In addition it provides a range of robust statistical techniques that are less dependent on assumptions, and it provides methods checking whether particular assumptions are reasonable for a given data set. See also List of statistical topics Foundations of statistics References Citations Sources Pre-publication chapters are available on-line. Cox, D.R., Hinkley, D.V. (1974) Theoretical Statistics, Chapman & Hall. Kish, L. (1965), Survey Sampling, John Wiley & Sons. . Further reading Peirce, C. S. (1876), "Note on the Theory of the Economy of Research" in Coast Survey Report, pp. 197–201 (Appendix