statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, 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 In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...

. 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

for the purposes of extracting samples from this population. A parameter is to a population as a

statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...

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..Everitt, B. S.; Skrondal, A. (2010), ''The Cambridge Dictionary of Statistics'', Cambridge University Press.

Discussion

Parameterised distributions

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 In mathematics and its applications, a parametric family or a parameterized family is a indexed family, family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. Common examples are p ...

. Among parameterized families of distributions are the normal distributions, the Poisson distributions, the

binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...

s, 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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.

Measurement of parameters

statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...

, 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. 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 Pearson's chi-squared test (\chi^2) is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests (e.g., ...

). Even if a family of distributions is not specified, quantities such as the mean and variance can generally still be regarded as statistical parameters of the population, and statistical procedures can still attempt to make inferences about such population parameters.

Types of parameters

Parameters are given names appropriate to their roles, including the following: * location parameter *

dispersion parameter Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...

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

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 an estimator) – that is, the percentage of the subsample of polled voters – will be measured instead. The statistic, along with an estimation of its accuracy (known as its sampling 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.

References