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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