Independent and dependent variables

Independent and dependent variables are types of variables that are used in mathematics, statistics, and the experimental studies. Generally, the dependent variable is the variable in a function or experiment whose value depends on the independent variable. The independent variable is the known variable that is manipulated in order to determine its effect (if any) on the dependent variable

Independent variable vs dependent variable

Another way to think of independent variables, particularly in the context of functions, is that the independent variable is the input value of a function, commonly denoted as x. They are sometimes called the argument of the function. The independent variable is not affected by any other variable, hence its name.

On the other hand, the value of a dependent variable is determined by some input, or independent variable. Dependent variables therefore represent the output value of a function, and are commonly denoted as y, or f(x). They are sometimes also referred to as the value of the function. Below is an example of a basic function.

y = 2x + 1

or

f(x) = 2x + 1

In the above function, y or f(x) is the dependent variable, and x is the independent variable. We can see from this relationship that f(x) is dependent on the value of x. Whatever the value of x, the value of f(x) is twice x, plus 1. For example:

f(5) = 2(5) + 1 = 11

The independent variable, x, is some value we choose, or manipulate, to determine the value of the dependent variable. There is no way for f(x) to affect x, but any change in x affects f(x). This is the relationship between dependent and independent variables.

On a graph, the dependent variable is typically plotted on the y-axis and the independent variable is plotted on the x-axis:

Independent and dependent variables are commonly used in statistics and experimentation when experimenters want to determine if one variable has an effect on another, and whether and how the effect can be manipulated or controlled. One real-world example is the testing of new medications. It is common to give the control group a placebo, which is some substance that is designed to have no therapeutic value. It should, in theory, have no effect on the patient.

While one group gets the placebo, the other group gets the medication that is intended to have therapeutic value. Ideally, the medication should help patients with whatever it is intended to treat. However, this is not necessarily the case, hence the experiment.

In this case, the independent variable is what the experimenters give each group: the placebo or the medication. This is the controlled variable of the experiment, and the dependent variable is the effect that the placebo or medication have on the patient. The purpose of the experiment is therefore to determine how each affects a patient, and whether any measured differences between the placebo and the medication are desired, or significant enough to conclude that the medication has the intended benefit over the placebo.