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A parameter ( ''auxiliary measure'') is an argument of a Mathematical Function that is considered in some way incidental or secondary within a particular context of discussion. Thus, what is considered a principal argument and what is considered a parameter may change from one context to the next. For example, a Parametric Equaliser is an Audio Filter 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. 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 are 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 Constant s 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'' is a formal parameter. When the function is used as in y 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''. Engineering In Engineering (especially involving data acquisition) the term ''parameter'' sometimes loosely refers to an individual measured item. For example an airliner Flight Data Recorder may record 88 different items, each termed a parameter. This usage isn't consistent, as sometimes the term ''channel'' refers to an individual measured item, with ''parameter'' referring to the setup information about that channel. ANALYTIC GEOMETRY In Analytic Geometry , Curve s 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:
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: :where ''t'' is the ''parameter''. A somewhat more detailed description can be found at Parametric Equation . MATHEMATICAL ANALYSIS In Mathematical Analysis , one often considers "integrals dependent on a parameter". These are of the form : 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 Distribution s, 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 σ2". The latter formulation and notation leaves some ambiguity whether σ or σ2 is the second parameter; the distinction is not always relevant. It is possible to use the sequence of Moments (mean, mean square, ...) or Cumulant s (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. Statistic s 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. SEE ALSO
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