Dependent and independent variables

A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. 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, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question. Rather, they are controlled by the experimenter.

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 or set of numbers). 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 y = f(x).

It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and y are independent variables. Functions with multiple outputs are often referred to as vector-valued functions.

In mathematical modeling, the relationship between the set of dependent variables and set of independent variables is studied.^{[citation needed]}

In the simple stochastic linear model y_i = a + bx_i + e_i the term y_i is the ith value of the dependent variable and x_i is the ith value of the independent variable. The term e_i is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.^{[citation needed]}

With multiple independent variables, the model is y_i = a + bx_i,1 + bx_i,2 + ... + bx_i,n + e_i, where n is the number of independent variables.^{[citation needed]}

In statistics, more specifically in linear regression, a scatter plot of data is generated with X as the independent variable and Y as the dependent variable. This is also called a bivariate dataset, (x₁, y₁)(x₂, y₂) ...(x_i, y_i). The simple linear regression model takes the form of Y_i = a + Bx_i + U_i, for i = 1, 2, ... , n. In this case, U_i, ... ,U_n are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence, the independence of U_i implies independence of Y_i, even though each Y_i has a different expectation value. Each U_i has an expectation value of 0 and a variance of σ². Expectation of Y_i Proof:

$${\displaystyle \operatorname {E} [Y_{i}]=\operatorname {E} [\alpha +\beta x_{i}+U_{i}]=\alpha +\beta x_{i}+\operatorname {E} [U_{i}]=\alpha +\beta x_{i}.}$$