An independent variable is a type of variable that is used in mathematics, statistics, and the experimental sciences. It is the variable that is manipulated in order to determine whether it has an effect on the dependent variable.
Real world examples of independent variables include things like fertilizer given to plants, where the dependent variable may be plant height; medication, where one group gets a placebo and the other gets the medication, and the dependent variable may be their health outcomes; the amount of caffeine a person drinks, where the dependent variable may be the number of hours they sleep.
Independent variables in algebra
In algebra, independent variables are usually discussed in the context of equations and functions. Most commonly, the independent variable is "x," (though others, such as t for time, are used as well) as in the equation
y = x + 5
or in function notation:
f(x) = x + 5
In the above, x is the independent variable because it is the variable that we control. Depending on what value of x is plugged into the function, f(x) (or y) changes. As such, it is common to characterize the independent variable as the input of a function, while the dependent variable is the output.
Referencing the above example, if the independent variable, x, is equal to 5, we can write this in function notation as f(5), and can compute the dependent variable as follows:
f(x) = x + 5
f(5) = 5 + 5 = 10
In this function, f(x) is always 5 more than x.
In graphs, independent variables are graphed along the x-axis, and dependent variables are graphed along the y-axis:
It is possible for a function to have multiple independent and dependent variables, though this is more common in higher mathematics, not algebra.
Independent variables in experiments
In the context of statistics and experiments, the independent variable is the control. It is the known variable that is manipulated, while the dependent variable is the variable that is expected to change as a result of manipulating the independent variable. In an experiment, the goal is typically to determine whether the independent variable has any effect on the dependent variable, and if so, how it affects the dependent variable. It follows that an independent variable may also be referred to as the explanatory variable, manipulated variable, and predictor variable, among other things. Similarly, a dependent variable may be referred to as the explained variable, response variable, predicted variable, and so on.
As an example, in an experiment that measures the growth of a group of plants that are given varying amounts of fertilizer, the independent variable is the amount of fertilizer administered, and the dependent variable is the growth of the plant. Adding more fertilizer might increase (or decrease) the growth of the plant. However, the growth of the plant will not directly affect the amount of fertilizer added.