Independent variables

An independent variable is that variable presumed to affect or determine a dependent variable. It can be changed as required, and its values do not represent a problem requiring explanation in an analysis, but are taken simply as given.

More generally, the independent variable is the thing that someone actively controls/changes; while the dependent variable is the thing that changes as a result. Thus independent variables act as catalysts for dependent variables. In other words, the independent variable is the "presumed cause", while dependent variable is the "presumed effect" of the independent variable.

In experimental design, an independent variable is a random variable used to define treatment groups.

Examples
The wages of an employee depend on the time worked. Time is the independent variable that varies among employees, and the wages are calculated directly from a specific amount of time. Thus wages are dependent on time worked.

In a study of how different dosages of a drug are related to the severity of symptoms of a disease, a researcher could compare the varying symptoms (the dependent variable) for varying dosages (the independent variable) and attempt to draw a conclusion.

The independent variable is also called the predictor variable.

Mathematics usage
When graphing a set of collected data, the independent variable is graphed on the x-axis (see Cartesian coordinates).

In mathematics, in functional analysis, it was traditional to define the set of independent variables as the only set of variables in a given context which could be altered. For, even though any function defines a bilateral relation between variables, and it's even true that two variables might be connected by an implicit equation (for instance, cf. the definition of a circle, $$x^2 + y^2 = R^2$$), when computing derivatives it is nonetheless necessary to take a group of variables as "independent", or else the derivative has no meaning.