Statistical parameters

A statistical parameter or population parameter is a quantity that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a statistical model.

Among parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential family of distributions. The family of normal distributions has two parameters, the mean and the variance: if these are specified, the distribution is known exactly. The family of chi-squared distributions, on the other hand, has only one parameter, the number of 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 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 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:

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