Independent variables

In mathematical modelling and statistical modelling, there are dependent and independent variables. The models investigate how the former depend on the latter. 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. Models test or explain 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 (usually number or set of numbers) and providing an output (which is also usually 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 written as vector-valued functions.

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