Inferential statistics..

In statistics, statistical inference is the process of drawing conclusions from data that is subject to random variation, for example, observational errors or sampling variation. More substantially, the terms statistical inference, statistical induction and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from datasets arising from systems affected by random variation, such as observational errors, random sampling, or random experimentation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations.

The outcome of statistical inference may be an answer to the question "what should be done next?", where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy.

The use of inferential statistics is a cornerstone of research on populations and events, because it is difficult and sometimes impossible to survey every member of a population or to observe every event. Instead, researchers attempt to get a representative sample and use that as a basis for their claims. This differs from descriptive statistics, which describe only the data itself in statistical terms.

More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:

Ø a statistical model of the random process that is supposed to generate the data, which is known when randomization has been used, and

Ø a particular realization of these random process; i.e., a set of data.

The conclusion of a statistical inference is a statistical proposition. Some common forms of statistical proposition are:

Ø an estimate; i.e., a particular value that best approximates some parameter of interest,

Ø a confidence interval or set estimate; i.e., an interval constructed using a dataset drawn from a population so that, under repeated sampling of such datasets, such intervals would contain the true parameter value with the probability at the stated confidence level,

Ø a credible interval; i.e., a set of values containing, for example, 95% of posterior belief,

Ø rejection of a hypothesis

Ø clustering or classification of data points into groups

Statistical inference is generally distinguished from descriptive statistics. In simple terms, descriptive statistics can be thought of as being just a straightforward presentation of facts, in which modeling decisions made by a data analyst have had minimal influence.

Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called ill-defined, extremely limited in applicability, and even fallacious. However this argument is the same as that which shows that a so-called confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments.

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