# Parameter

A**parameter**is a measurement or value on which something else depends.

For example, a *parametric equaliser* is a tone control circuit that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency and level of the peak or trough, are two of the **parameters** of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as *skew*. These **parameters** each describe some aspect of the response curve seen as a whole, over all frequencies. By way of contrast, a *graphic equaliser* provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.

In mathematics there is little difference in meaning between **parameter** and **argument of a function**. It is usually a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters. The best way to explain this is to illustrate it with examples.

In computing the parameters passed to a function subroutine are more normally called *arguments*.

In logic, by some authors (eg Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover") the parameters passed to (or operated on by) an *open predicate* are called *parameters*. Parameters locally defined within the predicate are called *variables*. This extra effort pays off when defining substitution (without this distinction special provision has to be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate *variables*, and when defining substitution have to distinguish between *free variables* and *bound variables*.

Table of contents |

2 Mathematical analysis 3 Probability theory 4 Statistics 5 Computer |

## Analytic geometry

In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parameter". A circle of radius 1 centered at the origin can be specified in more than one form:

- "implicit" form

- "parametric" form

*t*is the "parameter".

## Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form

*x*=

*g*(

*y*), it would be called a "change of variable".

## Probability theory

In probability theory, one may describe the distribution of a random variable as belonging to a *family* of probability distributions, distinguished from each other by the values of a finite number of *parameters*. For example, one talks about "a Poisson distribution with mean value λ", or "a normal distribution with mean μ and variance σ^{2}". The latter formulation and notation leaves some ambiguity whether σ or σ^{2} is the second parameter; the distinction is not always relevant.

It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.

## Statistics

In statistics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own.

Statistics are mathematical characteristics of samples which are used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the *sample mean* () is an estimate of the *mean* parameter (μ) of the population from which the sample was drawn.

## Computer

On the computer, parameters are used to differentiate behavior or pass input data to computer programs or their subprograms. See parameter for detail.

See also: Parametrization (i.e. coordinate system).