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parameter

(pə-răm'ĭ-tər) pronunciation

Mathematics.
1. A constant in an equation that varies in other equations of the same general form, especially such a constant in the equation of a curve or surface that can be varied to represent a family of curves or surfaces.
2. One of a set of independent variables that express the coordinates of a point.
1. One of a set of measurable factors, such as temperature and pressure, that define a system and determine its behavior and are varied in an experiment.
2. Usage Problem. A factor that restricts what is possible or what results: “all the parameters of shelter—where people will live, what mode of housing they will choose, and how they will pay for it” (New York).
3. A factor that determines a range of variations; a boundary: an experimental school that keeps expanding the parameters of its curriculum.
Statistics. A quantity, such as a mean, that is calculated from data and describes a population.
Usage Problem. A distinguishing characteristic or feature.

[New Latin parametrum, a line through the focus and parallel to the directrix of a conic : Greek para-, beside; see para–¹ + Greek metron, measure; see –meter.]

parametric par'a·met'ric (păr'ə-mĕt'rĭk) or par'a·met'ri·cal adj.
parametrically par'a·met'ri·cal·ly adv.

USAGE NOTE The term parameter, which originates in mathematics, has a number of specific meanings in fields such as astronomy, electricity, crystallography, and statistics. Perhaps because of its ring of technical authority, it has been used more generally in recent years to refer to any factor that determines a range of variations and especially to a factor that restricts what can result from a process or policy. In this use it often comes close to meaning “a limit or boundary.” Some of these new uses have a clear connection to the technical senses of the word. For example, the provisions of a zoning ordinance that limit the height or density of new construction can be reasonably likened to mathematical parameters that establish the limits of other variables. Therefore one can say The zoning commission announced new planning parameters for the historic Lamping district of the city. But other uses go one step further and treat parameter as a high-toned synonym for characteristic. Eighty percent of Panelists reject this use of parameter in the example The Judeo-Christian ethic is one of the important parameters of Western culture. • Some of the difficulties with the nontechnical use of parameter appear to arise from its resemblance to the word perimeter, with which it shares the sense “limit,” though the precise meanings of the two words differ. This confusion probably explains the use of parameter in a sentence such as U.S. forces report that the parameters of the mine area in the Gulf are fairly well established, where the word perimeter would have expressed the intended sense more exactly. This example of a use of parameter was unacceptable to 61 percent of the Usage Panel.

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Sci-Tech Encyclopedia: Parameter

An auxiliary variable, functions of which give the coordinates of a curve or surface. The coordinates of a curve are functions of one parameter. A curve in 3-space has parametric equations (1).
1. $x = f\!(t)\quad y = g(t)\quad z=h(t)$

The coordinates of a surface are functions of two parameters, shown in Eqs. (2).
2. $x = f\!(u,v)\quad y = g(u,v)\quad z = h(u,v)$

An arbitrary constant in an equation is also called a parameter. Variations in the values of the parameter generate a system of equations which may represent a family of curves or surfaces. Such families are called one-parameter, two-parameter, and so on, according to the number of independent parameters. See also Parametric equation.

Computer Desktop Encyclopedia: parameter

(1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. Parameters may be required as in parameter-driven software (see below) or they may be optional. Parameters are often entered as a series of values following the program name when the program is loaded.

A DOS switch is a parameter. For example, in the DOS Dir command dir /p the DOS switch /p (pause after every screenful) is a parameter.

(2) In programming, a value passed to a subroutine or function for processing. Programming today's graphical applications with languages such as C, C++ and Pascal requires knowledge of hundreds, if not thousands, of parameters.

In the following C function, which creates the text window for the Windows version of this database, there are 11 parameters passed to the CreateWindow routine. Some of them call yet other functions for necessary information. In order to call this routine in a program, the programmer must decide what the values are for every parameter.

 hWndText = CreateWindow    (
      "TextWClass",
       NULL,
       WS_CHILD|WS_BORDER|WS_VSCROLL|WS_TABSTOP,
       xChar*23+GetSystemMetrics(SM_CXVSCROLL)+8,
       yChar*4,
       Rect.right-Rect.left+1-xChar*23
          -2*GetSystemMetrics(SM_CXVSCROLL)+5,
       yChar*(Lines+1)+2,
       hWnd,
       IDC_TEXTLIST,
       (HANDLE)hInstance,
       NULL                 ) ;

Marketing Dictionary: parameter

Variable value; in a computer program, the parameters are changed each time the program is run. List rental selection parameters describe the criteria for selection applicable to each list rental. For example, the parameters may be set to select hotlines with male names and incomes over $50,000, or the parameters may be set for an nth-name selection.

Accounting Dictionary: Parameter

1. Constant or coefficient of a variable in an equation or a system of equations. For example, in a Cost-Volume Formula of the form y = a + bx, the constant a and the slope b are parameters. The total fixed costs, the unit variable cost, and the unit selling price are examples of parameters.

2. Numerical characteristic of a population computed using every element in the population. For example, the mean and the mode are parameters of a population.

Dental Dictionary: parameter

(pər-am′ə-tur)
n

Values that refer to a population; characteristics of a population. Because a parameter is a value of a hypothetical, infinite, unknown population, it is always an estimate.

Geography Dictionary: parameter

A numerical, characteristic of a complete data such as a mean, median, standard deviation, or variance, set, as opposed to a sample. See parametric statistics.

Sports Science and Medicine: parameter

1. An arbitrary constant or variable in a mathematical expression which gives rise to various cases of a particular phenomenon.

2. A quantity which is constant under a particular set of conditions, but which differs with changing conditions.

Science Dictionary: parameter

(puh-ram-uh-tuhr)

A quantity or number on which some other quantity or number depends. An informal example is, “Depending on the traffic, it takes me between twenty minutes and an hour to drive to work”; here, “traffic” is the parameter that determines the time it takes to get to work. In statistics, a parameter is an unknown characteristic of a population — for example, the number of women in a particular precinct who will vote Democratic.

The term is often mistakenly used to refer to the limits of possible values a variable can have because of confusion with the word perimeter.

Veterinary Dictionary: parametric

1. situated near the uterus; parametrial.
2. pertaining to or defined in terms of a parameter.

p. method — a method of testing a hypothesis which requires the user to assume a particular model for the distribution of data, e.g. Poisson, normal.

Wikipedia: parameter

For usage in computer science and programming, see Parameter (computer science).

For the journal of the U.S. Army War College, see Parameters (journal).

Parameters, in the plural form, has recently become popular with non-technical users to mean limits, but this should not be confused with the word's technical meaning.

In mathematics, statistics, and the mathematical sciences, parameters (L: auxiliary measure) are quantities that define certain relatively constant characteristics of systems or functions. Often represented by θ in general form, other symbols carry standard, specific meanings. When evaluating the function over a domain or determining the response of the system over a period of time, the independent variables are varied, while the parameters are held constant. The function or system may then be reevaluated or reprocessed with different parameters, to give a function or system with different behavior.

Loosely speaking, the term parameter is used for an argument which is intermediate in status between a variable and a constant.

Example

In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:

“

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"

”

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.

If asked to imagine the graph of the relationship $y = a x 2$ , one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different graphical appearance. The a can therefore be considered to be a parameter: less variable than the variable x, but less constant than the constant 2.

Parameters in various contexts in math and science

Mathematical functions

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:

$f(x_1, x_2, \dots; a_1, a_2, \dots) = \cdots\,$

The symbols before the semicolon in the function's definition, in this example the $x$ 's, denote variables, while those after it, in this example the $a$ 's, 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.

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

x 2 + y 2 = 1

parametric form

(x, y) = (cos t,sin t)

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

$F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx.$

In this formula, t is on the left-hand side the argument of the function F, and it is on the right-hand side the parameter that the integral depends on. 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).

Probability theory

These traces all represent Poisson distributions, but with different values for the parameter λ

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:

$f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}.$

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 k₁ 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 ten-minute 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.

Statistics and econometrics

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 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 can be used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean ( $\overline X$ ) can be used as an estimate of the mean parameter (μ) of the population from which the sample was drawn.

Other fields

Other fields use the term "parameter" as well, but with a different meaning.

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.

"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." John D. Trimmer, 1950, Response of Physical Systems (New York: Wiley), p. 13.

The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.

Computer science

Main article: Parameter (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 or just the value of f(3),

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 values passed to a function subroutine are more normally called arguments.

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