# Understanding dependent/independent variables in physics

How does one determine the independent and dependent variables?

What do the terms mean?

Can they be derived from a formula?

For example I saw in a textbook $$F = k\Delta l$$, Hooke's Law, that $$F$$ is the independent variable. Is this because $$\mathbf {F}$$ is the subject, therefore it is independent?

In an equation there is no inherent distinction between dependent and independent variables. There are to my knowledge only two contexts where the distinction makes sense.

Experimental: In an experimental context the independent variable is the one that the experimenter is controlling in the experiment. It is the treatment. For example, if the experimenter is determining the resistance of a resistor using a series of a few different voltages, then the independent variable would be the voltage and the dependent variable would be the current.

Statistical: In a statistical context the independent variable is the one that is known with no error. That is usually an approximation, so instead the independent variable is the one with negligible error. If none of the variables of interest have negligible error then unusual statistical methods must be used.

• Thank you for your answer! In your example about the voltages and resistor, is the dependent variable the resistance? In other words, the quantity we want to experiment on? – E C Dec 29 '20 at 14:50
• @EC I was thinking of the current as being the dependent variable, I will clarify in the answer – Dale Dec 29 '20 at 15:11
• Why is the dependent variable the current, aren't we experimenting for resistance? – E C Dec 29 '20 at 15:30
• @EC in this experiment the resistance is constant. It is the ratio between the voltage and current. The current varies as you change the voltage. – Dale Dec 29 '20 at 16:03

How does one determine the independent and dependent variables?

It's totally relative. In $$F = k\Delta l$$ all three variables can be considered dependent or independent, depending on your purpose.

E.g. in $$\Delta l=\frac{F}{k}$$, $$\Delta l$$ would now be considered the dependent variable.

Now suppose you studied a set of different springs, so that:

$$k=\frac{F}{\Delta l}$$

$$k$$ is 'normally' the proportionality constant (or factor) but in that study it would be the dependent variable.

• Do you have an explanation for "What do the terms mean" in my question? – E C Dec 29 '20 at 14:51
• It depends on what you're studying. The dependent variable's value depends on the independent variable's value(s). The independent variable's value can be chosen freely ('independently'). At least that's the idea... – Gert Dec 29 '20 at 14:55
• Just to double check, are you suggesting that the subject of the equation is the dependent variable? – E C Dec 29 '20 at 20:16
• It's not what I'm saying. But it probably works out that way, mostly. – Gert Dec 29 '20 at 20:57