Parameters are a special type of mathematical variable. A parametric equation contains one or more parametric variables that have multiple possible values. The value of each parameter is held constant when the function is used. In the statistical branches of mathematics, a parameter is an estimated numerical value for a population characteristic.
The quadratic equation is a familiar example that can be written as a parametric equation. In the form a*x^2 + b*x + c = 0, a, b, and c are parameters. If the parametric variables are assigned values — such as a = 1, b = 2, c = 3 — the equation is no longer parametric. x^2 + 2x + 3 is one distinct member of the family of quadratic functions.
Another familiar example is the equation for a straight line drawn on a Cartesian coordinate system. The most general form of the equation is y = m*x + b. Variables m and b are usually called the slope and the intercept, respectively. By varying m and b, an infinite number of distinct straight lines can be produced. The equation can never produce a parabola or a circle, however, no matter what combination of m and b is used. The equation is said to produce a family of functions because each function produces the same result, a straight line.
A parameter may also be used to describe a system of equations. If a ball is thrown and its trajectory is plotted on a Cartesian coordinate system, for example, both the x and y components of the trajectory depend on the time after the ball was thrown and the ball's initial velocity. The equations may look something like x = v*t and y = v*t - 5*t^2. Velocity and time are parameters in this case.
A more advanced application of parameters is the variation of parameters method, which is used to solve differential equations. In this method, the parameters are actually functions that replace unknown constants in the solution to a differential equation. By solving for these parametric functions, the unknown constants can be determined and the general and particular solutions for a differential equation can be found.
In statistics, a parameter is an estimate of a given population. Common statistical parameters include the mean and median. These estimates are used in the equations to calculate the test statistic for various statistical tests. For example, the test statistic for a student's t-test is calculated using Z = X*√n/σ, where X is the mean parameter and sigma is the standard deviation parameter.