Statistical parameter

In statistics, as opposed to its general use in 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 considered to define a 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 sample. Thus a "statistical parameter" can be more specifically referred to as a population parameter. 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 estimate or infer what they can about the parameter based on a random 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.