Statistical parameter: Difference between revisions

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A '''statistical parameter''' or '''population parameter''' is a quantity thatentering indexesinto the [[probability distribution]] of a [[Indexedstatistic]] familyor a [[random variable]].<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> Thus, it indexes an [[indexed family]] of distributions, thereby forming a [[probabilityparameterized distributionfamily]]s. It can be regarded as a numerical characteristic of a [[Statisticalstatistical population|population]] or a [[statistical model]].<ref>Everitt, B. S.; Skrondal, A. (2010), ''The Cambridge Dictionary of Statistics'', [[Cambridge University Press]].</ref>

==DefinitionDiscussion==

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

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]] (approximately) 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]]).

Even if a family of distributions is not specified, quantities such as the [[mean]] and [[variance]] can generally 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 the following.

:*[[___location parameter]]

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

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==Examples==

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 also==

*[[Model selection]]

*[[Parameter]]

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

==References==