Dependent and independent variables

In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function. The dependent variable depends on the independent variables; hence the names.

When there is only one independent variable and its values and those of the dependent variable are real numbers, then one conventionally draws the graph of the function with values of the independent variable on the horizontal axis&mdash;the x-axis&mdash;and the values of the dependent variable on the vertical axis&mdash;the y-axis (see Cartesian coordinates).

In calculus, the identification of the independent and dependent variable is significant, since the rate of change, or derivative, of the dependent variable is calculated with respect to the independent variable.

Experiments
In the design of experiments, independent variables, also called predictor variables, regressors, controlled variables, manipulated variables,or explanatory variables, are those whose values are controlled or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). In such an experiment, an attempt is made to find evidence that the values of the independent variable determine the values of the dependent variable (that which is being measured, also known as response variable, responding variable, explained variable, or regressand). The independent variable can be changed as required, and its values do not represent a problem requiring explanation in an analysis, but are taken simply as given. The dependent variable on the other hand, usually cannot be directly controlled.

More generally, the independent variable is the thing whose value one actively controls and can change, while the dependent variable is the thing whose value then changes as a result.

Examples

 * In a study of how different doses of a drug affect the severity of symptoms, a researcher could compare the frequency and intensity of symptoms (the dependent variables) when different doses (the independent variable) are administered, and attempt to draw a conclusion.


 * In measuring the acceleration of a vehicle, time is usually the independent variable and speed is the dependent variable. This is because when taking measurements, times are usually predetermined, and the resulting speed of the vehicle is recorded at those times. As far as the experiment is concerned, the speed is dependent on the time. Since the decision is made to measure the speed at certain times, time is the independent variable.


 * In measuring the amount of colour removed from beetroot samples at different temperatures, the dependent variable would be the amount of pigment removed, because it is depending on the temperature (which is the independent variable).