Dependent and Independent Variables

Every problem that one decides to solve requires knowledge of dependent and independent variables of the system to which it is related. In this article, I try to elucidate the meaning of 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 variable and analyzing their relation is a major step while modeling the behavior of any kind of system.

What is a Variable?

Prior to understanding the attributes of a variable being dependent or independent, let us clarify what one means by a 'Variable'. In simple words, a variable 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' variables. Also needed is a relation or more precisely an equation that connects the known variable to 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 in nature has many variables that decide what will happen next! The proliferation of variables that control a system make it difficult to model its behavior and have some predictability about future developments. Sifting out the dependent from independent variables, makes things easier.

What are Dependent and Independent Variables?

An independent variable is independent of change from any other parameters that control a system. It is any variable 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 other variables. These other variables which change according to changes in the independent variable are (not surprisingly) 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.

What are Controlled and Extraneous Variables?

As I mentioned previously, 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 rest of the variables 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 study in an experiment or a problem.

Use in Mathematics

In pure mathematics, the complexity of a problem rises with the number of unknown independent variables that it contains. For a solution, every independent variable 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' variable, 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 value of independent variable on X-axis while the values of dependent variable on Y axis. Thus the variability of a function can be visualized with the help of a graph.

Scientific Application

Every field of science tries to explore nature and guess the fundamental laws that govern the nature of 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. Understanding the entire phenomenon sometimes 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 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 variable.

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