# 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 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. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.

## Contents

## Types of parameter

### Mathematical

In mathematics the difference in meaning between a *parameter* and an *argument* of a function is that the parameters are the symbols that are part of the function's *definition*, while arguments refer to the symbols that are supplied to the function when it is used. The value or objects assigned to the *parameters* by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during the evaluation or processing of that function or system. The value of arguments can change outside of the function and between function usages. This distinction, the parameter's constancy, is a key part of the meaning of a parameter in any situation, often in usage beyond just mathematics.

In some informal situations people regard it as a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters.

### Computer science

When the terms **formal parameter** and **actual parameter** are used, they generally correspond with the definitions used in computer science. In the definition of a function such as

*f*(*x*) =*x*+ 2,

*x* is a formal parameter. When the function is used as in

*y*=*f*(3) + 5,

3 is the actual parameter value that is used to solve the equation. These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic.

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

### Logic

In logic, the parameters passed to (or operated on by) an *open predicate* are called *parameters* by some authors (e.g. Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called *variables*. This extra distinction 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*.

## 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2=1}**

- "parametric" form

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)=(\cos t,\sin t)}**- where
*t*is the "parameter".

A somewhat more detailed description can be found here.

## Mathematical analysis

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx}**

In this formula, *t* is the *argument* of the function *F* on the left-hand side, and the *parameter* that the integral depends on, on the right-hand side. The quantity *x* is a *dummy variable* or *variable (or parameter) of integration*. Now, if we performed the substitution *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.

It is possible to make statistical inferences without assuming a particular *parametric family* of probability distributions. In that case, one speaks of *non-parametric statistics* as opposed to the *parametric statistics* described in the previous paragraph. For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.

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* (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline X}**
) is an estimate of the *mean* parameter (μ) of the population from which the sample was drawn.

## Computer

On computers, parameters are used to differentiate behavior or pass input data to computer programs or their subprograms. See **parameter (computer science)** for details.

## Climate model parameters

## See also

Parametrization (i.e. coordinate system).