Independent and dependent variables

In mathematical modeling, statistical modeling and experimental sciences, the values of dependent variables depend on the values of independent variables. The dependent variables represent the output or outcome whose variation is being studied. The independent variables represent inputs or causes, i.e., potential reasons for variation or, in the experimental setting, the variable controlled by the experimenter. Models and experiments test or determine the effects that the independent variables have on the dependent variables. Sometimes, independent variables may be included for other reasons, such as for their potential confounding effect, without a wish to test their effect directly.

In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written ${\displaystyle y=f(x)}$.

It is possible to have multiple independent variables and/or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form ${\displaystyle z=f(x,y)}$, where z is a dependent variable and x and y are independent variables. Functions with multiple outputs are often referred to as vector-valued functions.

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