In mathematics, statistics, and the mathematical sciences, a parameter (G: auxiliary measure) is a quantity that serves to relate functions and variables using a common variable (often t) when such a relationship would be difficult to explicate with an equation. In different contexts the term may have special uses.
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“  W.M. Woods...a mathematician...writes... "...a variable is one of the many things a parameter is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.  ” 
“  [Kilpatrick quoting Woods] "Now...the engineers...change the lever arms of the linkage...the speed of the car...will still depend on the pedal position...but in a...different manner. You have changed a parameter"  ” 
Mathematical functions typically can have one or more variables and zero or more parameters. The two are often distinguished by being grouped separately in the list of arguments that the function takes:
The symbols before the semicolon denote variables, and those after it denote parameters.
Strictly speaking, parameters are denoted by the symbols that are part of the function's definition, while arguments are the values that are supplied to the function when it is used. Thus, a parameter might be something like "the ratio of the cylinder's radius to its height", while the argument would be something like "2" or "0.1".
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.
In the special case of parametric equations the independent variables are called the parameters.
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:
A somewhat more detailed description can be found at parametric equation.
In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form
In this formula, t is the argument of the function F, and on the righthand side the parameter on which the integral depends. When evaluating the integral, t is held constant, and so it considered a parameter. If we are interested in the value of F for different values of t, then, we now consider it to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration).
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". The function defining the distribution (the probability mass function) is:
This example nicely illustrates the distinction between constants, parameters, and variables. e is Euler's Number, a fundamental mathematical constant. The parameter λ is the mean number of observations of some phenomenon in question, a property characteristic of the system. k is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample. If we want to know the probability of observing k_{1} occurrences, we plug it into the function to get f(k_{1};λ). Without altering the system, we can take multiple samples, which will have a range of values of k, but the system will always be characterized by the same λ.
For instance, suppose we have a radioactive sample that emits, on average, five particles every ten minutes. We take measurements of how many particles the sample emits over tenminute periods. The measurements will exhibit different values of k, and if the sample behaves according to Poisson statistics, then each value of k will come up in a proportion given by the probability mass function above. From measurement to measurement, however, λ remains constant at 5. If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.
Another common distribution is the normal distribution, which has as parameters the mean μ and the variance σ².
It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.
In statistics and econometrics, 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 treated as random variables, and their uncertainty is described as a distribution.
It is possible to make statistical inferences without assuming a particular parametric family of probability distributions. In that case, one speaks of nonparametric statistics as opposed to the parametric statistics described in the previous paragraph. For example, Spearman is a nonparametric 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 can be used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean () can be used as an estimate of the mean parameter (μ) of the population from which the sample was drawn.
Other fields use the term "parameter" as well, but with a different meaning.
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.
In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. 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.
"Speaking generally, properties are those physical quantities which directly describe the physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal."^{[1]}
The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
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
x is a formal parameter. When the function is used as in
3 is the actual parameter value that is substituted for the formal parameter in the function definition. These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic.
In computing, parameters are often called arguments, and the two words are used interchangeably. However, some computer languages such as C define argument to mean actual parameter (i.e., the value), and parameter to mean formal parameter.
Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in a Universal Grammar within a Principles and Parameters framework.
Parameters
