Statistical parameter: Difference between revisions

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{{Short description|Quantity that indexes a parametrized family of probability distributions}}

{{one source|date=December 2014}}

{{Other uses|Parameter (disambiguation)}}

A '''statistical parameter''' is a parameter that indexes a [[Indexed family|family]] of [[probability distribution]]s. It can be regarded as a numerical characteristic of a [[Statistical population|population]] or a [[statistical model]].<ref>Everitt, B. S.; Skrondal, A. (2010), ''The Cambridge Dictionary of Statistics'', [[Cambridge University Press]].</ref>

{{redirect|True value|the company|True Value|the logical value|Truth value}}In [[statistics]], as opposed to its general [[parameter|use in mathematics]], a '''parameter''' is any quantity of a [[statistical population]] that summarizes or describes an aspect of the population, such as a [[mean]] or a [[standard deviation]]. If a population exactly follows a known and defined distribution, for example the [[normal distribution]], then a small set of parameters can be measured which provide a comprehensive description of the population and can be considered to define a [[probability distribution]] for the purposes of extracting [[Sample (statistics)|sample]]s from this population.

A "parameter" is to a [[statistical population|population]] as a "[[statistic]]" is to a [[statistical sample|sample]]; that is to say, a parameter describes the '''true value''' calculated from the full population (such as the [[population mean]]), whereas a statistic is an estimated measurement of the parameter based on a sample (such as the [[sample mean]], which is the mean of gathered data per sampling, called sample). Thus a "statistical parameter" can be more specifically referred to as a '''population parameter'''.<ref name="ESS06">{{citation| title= Parameter | encyclopedia= [[Encyclopedia of Statistical Sciences]] | editor1-first= S. | editor1-last= Kotz | editor1-link= Samuel Kotz |display-editors=etal | year= 2006 | publisher= [[Wiley (publisher)|Wiley]]}}.</ref><ref>Everitt, B. S.; Skrondal, A. (2010), ''The Cambridge Dictionary of Statistics'', [[Cambridge University Press]].</ref>

==Definition==

Among [[parametric family|parameterized families]] of distributions are the [[normal distribution]]s, the [[Poisson distribution]]s, the [[binomial distribution]]s, and the [[exponential distribution]]s. The family of [[normal distribution]]s has two parameters, the [[mean]] and the [[variance]]: if these are specified, the distribution is known exactly. The family of [[chi-squared distribution]]s, on the other hand, has only one parameter, the number of [[degrees of freedom (statistics)|degrees of freedom]].

==Discussion==

In [[statistical inference]], parameters are sometimes taken to be unobservable, and in this case the statistician's task is to infer what they can about the parameter based on observations of [[random variables]] distributed according to the probability distribution in question, or, more concretely stated, based on a [[random sample]] taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a [[Pearson's chi-squared test]]).

===Parameterised distributions===

Suppose that we have an [[indexed family]] of distributions. If the index is also a parameter of the members of the family, then the family is a [[parameterized family]]. Among [[parametric family|parameterized families]] of distributions are the [[normal distribution]]s, the [[Poisson distribution]]s, the [[binomial distribution]]s, and the [[exponential distributionfamily|exponential family of distributions]]s. For Theexample, the family of [[normal distribution]]s has two parameters, the [[mean]] and the [[variance]]: if thesethose are specified, the distribution is known exactly. The family of [[chi-squared distribution]]s, oncan thebe otherindexed hand, has only one parameter,by the number of [[degrees of freedom (statistics)|degrees of freedom]]: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.

===Measurement of parameters===

Even if a family of distributions is not specified, quantities such as the [[mean]] and [[variance]] can still be regarded as parameters of the distribution of the population from which a sample is drawn. Statistical procedures can still attempt to make inferences about such population parameters. Parameters of this type are given names appropriate to their roles, including:

In [[statistical inference]], parameters are sometimes taken to be unobservable, and in this case the statistician's task is to estimate or infer what they can about the parameter based on observations ofa [[random variablessample]] distributedof accordingobservations totaken from the probabilityfull distributionpopulation. inEstimators question,of or,a moreset concretelyof stated,parameters basedof ona specific distribution are often measured for a [[randompopulation, sample]]under takenthe fromassumption that the population ofis interest.(at least approximately) distributed according to that specific probability distribution. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a [[Pearson's chi-squared test]]). Even if a family of distributions is not specified, quantities such as the [[mean]] and [[variance]] can generally still be regarded as statistical parameters of the population, and statistical procedures can still attempt to make inferences about such population parameters.

:*[[___location parameter]]

:*[[Statistical dispersion|dispersion]] parameter or [[scale parameter]]

:*[[shape parameter]]

Where a probability distribution has a ___domain over a set of objects that are themselves probability distributions, the term [[concentration parameter]] is used for quantities that index how variable the outcomes would be.

===Types of parameters===

Quantities such as [[regression coefficient]]s, are statistical parameters in the above sense, since they index the family of [[conditional probability distribution]]s that describe how the [[dependent and independent variables|dependent variables]] are related to the independent variables.

Parameters are given names appropriate to their roles, including the following:

:*[[___location parameter]]

:*[[Statistical dispersion|dispersion]] parameter]] or [[scale parameter]]

:*[[shape parameter]]

Where a probability distribution has a ___domain over a set of objects that are themselves probability distributions, the term ''[[concentration parameter]]'' is used for quantities that index how variable the outcomes would be.

Quantities such as [[regression coefficient]]s, are statistical parameters in the above sense, sincebecause they index the family of [[conditional probability distribution]]s that describe how the [[dependent and independent variables|dependent variables]] are related to the independent variables.

==Examples==

During an election, there may be specific percentages of voters in a country who would vote for each particular candidate – these percentages would be statistical parameters. It is impractical to ask every voter before an election occurs what their candidate preferences are, so a sample of voters will be polled, and a statistic (also called an [[estimator]]) – that is, the percentage of the sample of polled voters – will be measured instead. The statistic, along with an estimation of its accuracy (known as its [[sampling error]]), is then used to make inferences about the true statistical parameters (the percentages of all voters).

A parameter is to a [[statistical population|population]] as a [[statistic]] is to a [[statistical sample|sample]]. At a particular time, there may be some parameter for the percentage of all voters in a whole country who prefer a particular electoral candidate. But it is impractical to ask every voter before an election occurs what their candidate preferences are, so a sample of voters will be polled, and a statistic, the percentage of the polled voters who preferred each candidate, will be counted. The statistic is then used to make inferences about the parameter, the preferences of all voters. Similarly, in some forms of testing of manufactured products, rather than destructively testing all products, only a sample of products are tested,. Such totests gather statistics supporting an inference that all the products meet product design parametersspecifications.

==See alsoReferences ==

*[[Precision (statistics)]], another parameter not specific to any one distribution

*[[Parametrization]] (i.e., [[coordinate system]])

*[[Parsimony]] (with regards to the trade-off of many or few parameters in data fitting)

==References==

{{Reflist}}

{{Statistics|inference}}

[[Category:Statistical theory]]

[[Category:Statistical terminology]]

[[Category:TheoryStatistical ofparameters| probability distributions]]