# Dependent and independent variables

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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. Thus if we have a function *f*(*x*), then *x* is an independent variable, and *f*(*x*) is a dependent variable. 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—the *x*-axis—and the values of the dependent variable on the vertical axis—the *y*-axis (see Cartesian coordinates).

In calculus, the identification of 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.

Depending on the context, independent variables are also known as **predictor variables**, **regressors**, **controlled variables**, **manipulated variables**, or **explanatory variables**.

The dependent variable is also known as the **response variable**, the **regressand**, the **measured variable**, the **responding variable**, the **explained variable**, or the **outcome variable**.

## Experiments

In the design of experiments, *independent 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). 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.

*Controlled variables* are also important to identify in experiments. They are the variables that are kept constant to prevent their influence on the effect of the independent variable on the dependent.

In summary:

- The independent variable answers the question "What do I change?".
- The dependent variables answer the question "What do I observe?".
- The controlled variables answer the question "What do I keep the same?".

### Examples

- If one were to measure the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used (the changing factor of the experiment). The dependent variables would be the growth in height and/or mass of the plant (the factors that are influenced in the experiment) and the controlled variables would be the type of plant, the type of fertilizer, the amount of sunlight the plant gets, the size of the pots, etc. (the factors that would otherwise influence the dependent variable if they were not controlled).

- 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).

- In sociology, in measuring the effect of education on income or wealth, the dependent variable could be a level of income or wealth measured in monetary units (United States Dollars for example), and an independent variable could be the education level of the individual(s) who compose(s) the household (i.e. academic degrees).

## See also