A dependent variable is a type of variable that is used in mathematics, statistics, and the experimental sciences. It is the variable whose value depends on how the independent variable is manipulated, hence its name.
Real world examples of a dependent variable include: the height of a plant as a function of the amount that it is watered, where the amount it is watered is the independent variable; the score on a test a student achieves as a function of the amount that they studied, where the independent variable is the amount of time that they studied; the amount of money you earn as a function of the amount of time worked, where the amount of time worked is the independent variable.
Dependent variables in algebra
In algebra, dependent variables are usually discussed in the context of equations and functions. Most commonly, the dependent variable is denoted as "y" or "f(x), though other symbols are also used:
y = x - 7
In function notation:
f(x) = x - 7
In the above equations, y or f(x) are the dependent variable, and x is the independent variable. Function notation is more commonly used in this context in part because it more clearly shows the dependence of the function on the variable x. Because of this dependence, it is also common to characterize the dependent variable as the output of a function, where the independent variable is the input.
Referencing the above example, if the independent variable, x, is equal to 12, we can write this in function notation as f(12), and can compute the dependent variable as follows:
f(x) = x - 7
f(12) = 12 - 7 = 5
In this function, f(x) is always 7 less than x.
In graphs, dependent variables are graphed along the y-axis, and independent variables are graphed along the x-axis:
Functions may have multiple dependent and independent variables, but this is more common in higher mathematics, not algebra.
Dependent variables in experiments
In the context of statistics and experiments, the dependent variable is the variable that is being measured or tested. In such an experiment, the goal is usually to determine whether the independent variable, or control, has an effect on the dependent variable, and if so, how. Thus, in an experiment, the independent variable is the known variable that is manipulated to determine its effect (if any) on the dependent variable.
The independent variable is also referred to as the manipulated variable, predictor variable, and explanatory variable, among other things. It follows that the dependent variable may also be referred to as the response variable, predicted variable, explained variable, and so on.
As an example, if a student's grade on a test increases by 10 points for every hour they study, then their grade is dependent on the number of hours they study. In this situation, the student's grade is the dependent variable.
No matter what grade any student gets on their test, the number of hours they study will not change. Thus, the number of hours studied is the independent variable, also referred to as the control, since it is what the experimenter manipulates in order to determine its effect on the dependent variable.
There can also be other dependent variables, but in an experiment, it is important to test these separately to increase the likelihood that any observed effect is actually a result of the manipulation of the independent variable. For example, the number of hours a student studies may depend on some other factor (e.g. motivation, time spent playing video games, etc.), but in this scenario we are only examining the relationship between the received grade and study time. An experiment would also study a group of students, not just one, but would follow the same procedure for each.