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Explaining Independent and Dependent Variables

Date: 10/25/2002 at 13:01:00
From: Tony Restrepo
Subject: Independent and dependent variable

Dr. Math,

I want the definition of independent and dependent variables. 

The definitions I have are:
   independent: the value that we can change
     dependent: the value that is determined by the independent 

Can you elaborate in detail to help the students understand this a 
little more clearly, especially with real world problems? 

Thank you,
Tony Restrepo

Date: 10/25/2002 at 15:53:54
From: Doctor Achilles
Subject: Re: Independent and dependent variable

Hi Tony,

Thanks for writing to Dr. Math.

I think of independent and dependent variables in terms of functions.  
I'm not sure if your students have been taught functions yet; if not, 
you may have to modify this elaboration of independent/dependent 

A function, the way I think of it, is a little mathematical box. You 
put in numbers, variables, constants, or anything else that you can do 
math on, and it will put out an answer.

Let's say I tell you that I have a function in mind. I'm not going to 
tell you what the function is, but I am going to let you give me any 
numbers you like, and I will tell you what the output of the numbers 

So you start guessing numbers and I start telling you the function 
outputs for each guess.

  guess  output
  1      4
  2      7
  3      12
  4      19
  5      28

By now, you may have some guesses as to what the function is. Of 
course, there are many (actually an infinite number of) functions that 
COULD give rise to this behavior. The simplest one I can think of (and 
the one that I had in mind) was:

  f(x) = x^2 + 3

[x-squared plus 3]

It may seem as if this is a fairly contrived and artificial example, 
but it is actually a pretty good model of what scientific 
investigation is all about.

Science can be thought of as the process of investigating what 
mathematical functions nature uses for everything it [nature] does.

Let's say I want to find out how deep in the soil a penny goes as 
a function of the height from which I drop it. How would I do that?  
Well, I'd drop it from different heights, and measure how deep it 
goes. I can then come up with an equation that relates the height I 
dropped the penny from, x, with the depth it went into the ground, 

For another example, let's say that I'm a medical researcher testing a 
new treatment for insomnia. I know that a given drug helps, but I want 
to know how much is needed. If I have 5 doses I want to test, and 500 
people participating in the study, then I can give each dose to 100 
people. I then find out the next day what percentage of each group had 
a good night's sleep. I can then relate the dose of drug x to the 
probability of a good night's sleep, f(x), using a mathematical 

What does the height I drop the penny from in the first example have 
to do with the dose of drug in the second example?  Well, they are 
both things I control. In both cases, I want to find an equation that 
relates two variables. It is impossible for me to make the penny go a 
certain depth and then ask what height it fell from; the only thing I 
can do is adjust the height and then measure the depth. Similarly, it 
is impossible for me to make the probability of a good night's sleep 
whatever number I want and then ask what dose of drug I gave to do 
that; the only thing I can do is adjust the dose of drug and then 
measure the probability of a good night's sleep.

The height from which I drop and the dose of drug are therefore 
variables that I have control over. You can think of them as "free" 
variables or "controllable" variables. These are perfectly acceptable 
terms for input variables, and they are actually used occasionally in 
science; there's nothing wrong with these terms at all.

Once I pick a height to drop from, or a dose of drug, the depth the 
penny will go and the probability of a good night's sleep are set by 
whatever function nature uses (I don't know yet what the function 
is, but whatever it is, I don't have control over the output). You can 
think of depth and probability (in these examples) as "determined" or 
"uncontrollable" variables.

Even though there is no problem with the terms "free" and 
"controllable," there is a problem with the terms "determined" and 
"uncontrollable." The problem is that the output is not COMPLETELY 
pre-determined or completely uncontrollable; rather, it is simply 
determined once you pick the value of the input variable. So a better 
way to think of the output is that it is DEPENDENT on the input 

In addition to being called "free," the input variable can be thought 
of as "independent" because you can freely and independently change it 
to whatever you want.

While I personally agree that independent is an acceptable way to 
think of the input variable, to be honest, I think that "free" is a 
much better way to describe it. But, because "determined" is NOT an 
acceptable way to describe the output variable and people like the way 
"independent and dependent" sounds more than they like the sound of 
"free and dependent," it is most common to call the input 

Science would be trivial if we knew beforehand what functions nature 
uses, but we have to just figure them out by trying a bunch of 
different inputs (independent variables) and seeing what outputs 
(dependent variables) we get for each.

I hope this helps. If you have other questions about this or you'd 
like to talk about this some more, please write back.

- Doctor Achilles, The Math Forum 

Date: 10/25/2002 at 18:05:18
From: Tony Restrepo
Subject: Thank you (Independent and dependent variable)

Thank you for your prompt and helpful response. I really liked the 
term "free" as the variables that I have control over.
Associated Topics:
High School Definitions
High School Functions
High School Statistics
Middle School Definitions
Middle School Statistics

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