Dependent variables are, as you might guess, variables that depend on certain conditions. An independent variable is a variable whose value can be freely changed. A variable is dependent if its value is adjusted as a direct consequence of a change in the independent variable.
For example, consider a square with side length <em>x</em>. Then its perimeter is 4 times the side length,
<em>P</em> = 4<em>x</em>
and its area is the square of the side length,
<em>A</em> = <em>x</em> ²
If <em>x</em> = 1, then <em>P</em> = 4 and <em>A</em> = 1. If <em>x</em> = 2, then <em>P</em> = 8 and <em>A</em> = 4. And so on. If we treat <em>x</em> as the independent variable, then <em>P</em> and <em>A</em> are dependent variables that depend on the value of <em>x</em>.
But you can also go the other way and express <em>x</em> as <u>functions</u> of <em>P</em> or <em>A</em>, making <em>x</em> a dependent variable that depends on the values of <em>P</em> or <em>A</em>. For example, solving for <em>x</em> in terms of <em>P</em> gives
<em>P</em> = 4<em>x</em> ===> <em>x</em> = <em>P</em>/4
Then if <em>P</em> = 4, we have <em>x</em> = 8. If <em>P</em> = 16, then <em>x</em> = 1. And so on. You can think of <em>A</em> in the same way.
You can even go one step further and express the area <em>A</em> as a function of the perimeter <em>P</em> :
<em>P</em> = 4<em>x</em> ===> <em>x</em> = <em>P</em>/4
Then
<em>A</em> = <em>x</em> ² = (<em>P</em>/4)² = <em>P</em> ²/16
So in this case, <em>P</em> is the independent variable upon which the dependent variable <em>A</em> depends. If <em>P</em> = 4, then <em>A</em> = 1. If <em>P</em> = 8, then <em>A</em> = 4. etc
In a scientific study, "independent variable" is associated with some quantity or measurement that you control, and the "dependent variable" is the quantity or measurement that you take in response to a change in the independent variable. The meaning of either is the same, but actual studies typically don't involve simple equations like the ones I use as examples here.
