What are Independent and Dependent Variables?

Every problem, that one decides to solve, requires knowledge of dependent and independent variables of the system, to which they are related. Take a glance at these terms and their importance in analyzing and solving problems in mathematics, statistics, and science in general.

Knowing these concepts is vitally important, as selecting dependent and independent variables and analyzing their relation is a major step, while modeling the behavior of any kind of system.

In simple words, it is any quantity that varies. Of course, there are 'constant variables' that can choose not to vary.

Solving any problem mathematically, requires the statement of the 'known' variables, to find the values of 'unknown' ones. Also needed, is a relation or more precisely an equation that connects the known variable to the unknown ones, to find the value of unknowns.

Variables are all those things that can change in any system that you are observing. The real world is dynamic in nature and therefore, every single phenomenon has many variables that decide what will happen next.

The proliferation of variables that control a system, makes it difficult to model its behavior and have some predictability about future developments. Sifting out the dependent ones, from independent variables, makes things easier.

An independent variable is independent of change, from any other parameters that control a system. It is any one, whose values do not change, according to changes in any of the other variables. However, changes in the independent variable can affect the values of others.

These other variables which change, according to changes in the independent variable, are called dependent variables. For example, the height of Mercury rise in a thermometer, is a dependent variable, whose value is controlled by the independent variable of room temperature.

As mentioned previously, the real world is quite complex and the number of independent variables controlling a system, is large. To determine the dependence relation between two variables, one needs to make sure that the rest of them, are kept constant. This enables one to isolate the relation between a pair of independent and dependent variables.

These variables, that are kept constant in an experiment, designed to probe the relation between two specific variables, are called the controlled variables. Extraneous variables are ones that do not affect the relationship between independent and dependent variables under consideration or under study, in an experiment or a problem.

In pure mathematics, the complexity of a problem rises, with the number of unknown independent variables that it contains. For a solution, every independent one, needs to be isolated. In mathematical language, a dependent variable is a function of the independent variable(s).

For example, consider the following equation:

y = x + 4

Here, 'x' and 'y' are variables. A change in the value of 'x', will change the value of 'y'. So 'y' is dependent on 'x' or it is a function of 'x'. In mathematical language, this is written as:

y = f(x) = x + 4

A variable's value could be dependent upon more than one independent variables. Variation of dependent and independent variables on a graph, is generally plotted with the values of the independent variable on X-axis, while the values of dependent variable are on Y-axis. Thus, the variability of a function can be visualized, with the help of a graph.

Every field of science tries to explore nature and guess the fundamental laws that govern various phenomena. To be able to predict what will happen in a given situation, science needs to understand the relations between dependent and independent variables.

Sometimes, understanding the entire phenomenon is difficult. That's why, science breaks the whole thing into small manageable pieces, whose governing independent variables are then investigated. A law is derived that states how a change in one independent variable, changes the other dependent variables.

An example is the Newton's law of force:

F (Force) = Mass x Acceleration

So, force is a function of acceleration. Rather, force as a variable is dependent on the independent variable of acceleration, while mass is a constant.

Knowing the independent and dependent variables of a system helps mathematicians and scientists to formulate theoretical models, that exactly describe the behavior of that system.