Instatistics, as opposed to its generaluse in mathematics, aparameter is any quantity of astatistical population that summarizes or describes an aspect of the population, such as amean or astandard deviation. If a population exactly follows a known and defined distribution, for example thenormal distribution, then a small set of parameters can be measured which provide a comprehensive description of the population and can be considered to define aprobability distribution for the purposes of extractingsamples from this population.
A "parameter" is to apopulation as a "statistic" is to asample; that is to say, a parameter describes thetrue value calculated from the full population (such as thepopulation mean), whereas a statistic is an estimated measurement of the parameter based on a sample (such as thesample mean, which is the mean of gathered data per sampling, called sample). Thus a "statistical parameter" can be more specifically referred to as apopulation parameter.[1][2]
Suppose that we have anindexed family of distributions. If the index is also a parameter of the members of the family, then the family is aparameterized family. Amongparameterized families of distributions are thenormal distributions, thePoisson distributions, thebinomial distributions, and theexponential family of distributions. For example, the family ofnormal distributions has two parameters, themean and thevariance: if those are specified, the distribution is known exactly. The family ofchi-squared distributions can be indexed by the number ofdegrees of freedom: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.
Instatistical 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 arandom sample of observations taken from the full population. Estimators of a set of parameters of a specific distribution are often measured for a population, under the assumption that the population is (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 aPearson's chi-squared test). Even if a family of distributions is not specified, quantities such as themean andvariance can generally still be regarded as statistical parameters of the population, and statistical procedures can still attempt to make inferences about such population parameters.
Parameters are given names appropriate to their roles, including the following:
Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the termconcentration parameter is used for quantities that index how variable the outcomes would be.Quantities such asregression coefficients are statistical parameters in the above sense because they index the family ofconditional probability distributions that describe how thedependent variables are related to the independent variables.
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 anestimator) – 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 itssampling error), is then used to make inferences about the true statistical parameters (the percentages 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 tests gather statistics supporting an inference that the products meet specifications.
