Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function), on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question. In this sense, some common independent variables are time, space, density, mass, fluid flow rate, and previous values of some observed value of interest (e.g. human population size) to predict future values (the dependent variable).
Of the two, it is always the dependent variable whose variation is being studied, by altering inputs, also known as regressors in a statistical context. In an experiment, any variable that can be attributed a value without attributing a value to any other variable is called an independent variable. Models and experiments test the effects that the independent variables have on the dependent variables. Sometimes, even if their influence is not of direct interest, independent variables may be included for other reasons, such as to account for their potential confounding effect.
In pure mathematics
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 , and the most common symbol for the output is ; the function itself is commonly written .
It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form , where is a dependent variable and and are independent variables. Functions with multiple outputs are often referred to as vector-valued functions.
In modeling and statistics
In mathematical modeling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model the term is the th value of the dependent variable and is the th value of the independent variable. The term is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.
With multiple independent variables, the model is , where is the number of independent variables.
In statistics, more specifically in linear regression, a scatter plot of data is generated with as the independent variable and as the dependent variable. This is also called a bivariate dataset, . The simple linear regression model takes the form of , for . In this case, are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence, the independence of implies independence of , even though each has a different expectation value. Each has an expectation value of 0 and a variance of . Expectation of Proof:
The line of best fit for the bivariate dataset takes the form and is called the regression line. and correspond to the intercept and slope, respectively.
In an experiment, the variable manipulated by an experimenter is something that is proven to work, called an independent variable. The dependent variable is the event expected to change when the independent variable is manipulated.
In data mining tools (for multivariate statistics and machine learning), the dependent variable is assigned a role as (or in some tools as label attribute), while an independent variable may be assigned a role as regular variable. Known values for the target variable are provided for the training data set and test data set, but should be predicted for other data. The target variable is used in supervised learning algorithms but not in unsupervised learning.
Synonyms
Depending on the context, an independent variable is sometimes called a "predictor variable", "regressor", "covariate", "manipulated variable", "explanatory variable", "exposure variable" (see reliability theory), "risk factor" (see medical statistics), "feature" (in machine learning and pattern recognition) or "input variable". In econometrics, the term "control variable" is usually used instead of "covariate". From the Economics community, the independent variables are also called "exogenous".