Statistical parameter

In statistics rather than in general mathematics, a parameter is any measured quantity of a statistical population that summarises 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 completely describes the population, and can be used as probability distribution for the purposes of extracting samples from this population.

A parameter is to a population as a statistic is to a sample; that is to say, a parameter describes the true value calculated from the full population, whereas a statistic is an estimated measurement of the parameter based on a subsample. Thus a "statistical parameter" can be more specifically referred to as a population parameter.

Discussion

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 parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential family of distributions. For example, the family of normal distributions has two parameters, the mean and the variance: if those are specified, the distribution is known exactly. The family of chi-squared distributions can be indexed by the number of degrees of freedom: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.

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
• 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 coefficients are statistical parameters in the above sense because they index the family of conditional probability distributions that describe how the dependent variables are related to the independent variables.

Examples

A parameter is to a population as a statistic is to a sample. At a particular time, there may be some parameters 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 tests gather statistics supporting an inference that the products meet specifications.

See also

• Parameter