## Defining "Variable"

This page contains six posts in the thread 'Defining "variable"' that were posted to the sci.stat.edu UseNet newsgroup (= e-mail list EdStat-L) between June 28 and July 5, 1996. The authors are Robert Frick (2 posts), Herman Rubin (3 posts) and William Oliver (1 post).

```Date: Fri, 28 Jun 1996 11:45:02 -0400
Message-Id: <7395BF061E@psych1.psy.sunysb.edu>
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From: ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu>
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Defining "variable"

I thought Don Macnaughton's definition of variable was brilliant.
Last semester I gave my class the problem to name a varible that
usually takes the value 1.  While this question perhaps can be
faulted, and two Ph.D. psychologists solved it slowly, nonetheless
they both came
up with good answers.  Anyway, for most of the class, their answer
was not even a variable.  Same thing for another class with the easier
question to name a variable that usually takes the value yes.

So, I suspect that variable is a tough concept for students to grasp.
Rereading the standard definitions, they don't seem to make any
sense unless you first know what a variable is.

The following is my attempt to paraphrase Macnaughton for the
research methodology textbook I am writing.

People conceptualize the world as containing "things".  These
things can be physical objects, such as people, buildings,
hammers, or cities.  They can also be processes, such as driving
to school. Each thing has associated with it a collection of
properties, and the goal in learning about a thing is to learn
its properties.  For example, my daughter is 23 pounds, has
blond hair, and is cautious; my drive to school takes 15 minutes
and has 3 stop lights.

For a *collection* of things, the property in question is
called a variable.  For example, for a collection of children,
the properties of weight, hair color, and amount of cautiousness
are variables; for a collelction of drives to work, the
properties of duration and number of stoplights are variables.
For the collection of people in an experiment, whether or not
they run and their GPA are variables.  For each thing in the
collection, the variable takes a particular value.

Bob Frick
rfrick@psych1.psy.sunysb.edu
```

```Date: Sun, 30 Jun 1996 09:10:25 -0400
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From: hrubin@b.stat.purdue.edu (Herman Rubin)
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Re: Defining "variable"

In article <7395BF061E@psych1.psy.sunysb.edu>,
ROBERT FRICK  <RFRICK@psych1.psy.sunysb.edu> wrote:
>I thought Don Macnaughton's definition of variable was brilliant.
>Last semester I gave my class the problem to name a varible that
>usually takes the value 1.  While this question perhaps can be
>faulted, and two Ph.D. psychologists solved it slowly, nonetheless
>they both came
>up with good answers.  Anyway, for most of the class, their answer
>was not even a variable.  Same thing for another class with the easier
>question to name a variable that usually takes the value yes.

>So, I suspect that variable is a tough concept for students to grasp.
> Rereading the standard definitions, they don't seem to make any
>sense unless you first know what a variable is.

>The following is my attempt to paraphrase Macnaughton for the
>research methodology textbook I am writing.

>     People conceptualize the world as containing "things".  These
>     things can be physical objects, such as people, buildings,
>     hammers, or cities.  They can also be processes, such as driving
>     to school. Each thing has associated with it a collection of
>     properties, and the goal in learning about a thing is to learn
>     its properties.  For example, my daughter is 23 pounds, has
>     blond hair, and is cautious; my drive to school takes 15 minutes
>     and has 3 stop lights.

>     For a *collection* of things, the property in question is
>     called a variable.  For example, for a collection of children,
>     the properties of weight, hair color, and amount of cautiousness
>     are variables; for a collelction of drives to work, the
>     properties of duration and number of stoplights are variables.
>     For the collection of people in an experiment, whether or not
>     they run and their GPA are variables.  For each thing in the
>     collection, the variable takes a particular value.

This is still not too good for random variables, where it is
more appropriate.  I would define a random variable as something
which can be computed knowing the real world situation; it need
not be a number.  The tie-in with the usual "definition" found
in statistics books is that, for every sample space representation,
if the representation is adequate for that random variable, the
variable is represented by a function on the sample space.  You
are using the function here as the variable, and this does lead
to problems.

But for variable in mathematics, a far better "definition" is a
pronoun.  One might say that a restricted variable is a common noun,
which in many respects is closer to pronoun than to proper noun.
It is this which should be taught in first grade.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu	 Phone: (317)494-6054	FAX: (317)494-0558

```

```Date: Sun, 30 Jun 1996 19:55:41 -0400
Originator: edstat-l@jse.stat.ncsu.edu
Sender: edstat-l@jse.stat.ncsu.edu
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Re: Defining "variable"

> In article <7395BF061E@psych1.psy.sunysb.edu>,
> ROBERT FRICK  <RFRICK@psych1.psy.sunysb.edu> wrote:
> >I thought Don Macnaughton's definition of variable was brilliant.
> >Last semester I gave my class the problem to name a varible that
> >usually takes the value 1.  While this question perhaps can be
> >faulted, and two Ph.D. psychologists solved it slowly, nonetheless
> >they both came
> >up with good answers.  Anyway, for most of the class, their answer
> >was not even a variable.  Same thing for another class with the easier
> >question to name a variable that usually takes the value yes.
>
<snip>

I've missed this thread, so perhaps I've missed out on why the following
informal definition of Kolmogorov's is lacking.

"A *random variable* is the name given to a quantity which under
conditions S may take various values with specific probabilities. For us
it is sufficient to consider random variables that may take on only a
finite number of different values. To give the *probability distribution*,
as it is called, of such a random variable, it is sufficient to state its
possible values..."

I find this definition quite clear, but perhaps it's wrong in some way.

-Bill

--

```

```Date: Sun, 30 Jun 1996 23:23:19 -0400
Message-Id: <4r56l2\$2777@b.stat.purdue.edu>
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From: hrubin@b.stat.purdue.edu (Herman Rubin)
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Re: Defining "variable"

>> In article <7395BF061E@psych1.psy.sunysb.edu>,
>> ROBERT FRICK  <RFRICK@psych1.psy.sunysb.edu> wrote:
>> >I thought Don Macnaughton's definition of variable was brilliant.
>> >Last semester I gave my class the problem to name a varible that
>> >usually takes the value 1.  While this question perhaps can be
>> >faulted, and two Ph.D. psychologists solved it slowly, nonetheless
>> >they both came
>> >up with good answers.  Anyway, for most of the class, their answer
>> >was not even a variable.  Same thing for another class with the easier
>> >question to name a variable that usually takes the value yes.

><snip>

>I've missed this thread, so perhaps I've missed out on why the following
>informal definition of Kolmogorov's is lacking.

>"A *random variable* is the name given to a quantity which under
>conditions S may take various values with specific probabilities. For us
>it is sufficient to consider random variables that may take on only a
>finite number of different values. To give the *probability distribution*,
>as it is called, of such a random variable, it is sufficient to state its
>possible values..."

>I find this definition quite clear, but perhaps it's wrong in some way.

First of all, to understand random variables, we must eliminate the
connection with probability distribution as essential.  Kolmogorov was
coming at the whole problem as a means of "defining" probability, and
it is this which causes problems by itself.  Probabilty, TO BE USED,
must be something in the real world, which is only represented in the
mathematical world.  However, it is only in the mathematical world
which we can understand the ideas of calculation.  So it is necessary
to carefully model the real world in the mathematical world.

The representation is not what is being represented.  An oversimplified
representation may, in some cases, be a reasonable approximation for the
solution of a particular type of problem, but it blocks understanding of
the general problem.  So:

There is no "unique" sample space for a situation.  Different
representations are appropriate, and one can even have representations
for aspects of a problem; one can move between representations as is
convenient.  If the various representations are compatible, any answers
will not change.

The representation of a random variable, which need not be numeric,
corresponds to a function on the sample space used, if that sample
space is adequate.  This is not a matter of measurability; in a
problem of tossing a coin more than once, the result of the first
toss is not representable on the sample space consisting of the
non-negative integers, where the representation is the number of
heads.  When one looks at a fixed function on a fixed space, there
is nothing random or variable about it, but it can represent a
random variable.  The randomness comes from the fact that there
is a mapping of the real world phenomenon into the sample space.
This mapping, which is essentially a dictionary, is what is not
stressed in teaching, but which is important in understanding.

--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu	 Phone: (317)494-6054	FAX: (317)494-0558

```

```Date: Tue, 2 Jul 1996 13:11:23 -0400
Message-Id: <D4F8874150@psych1.psy.sunysb.edu>
Originator: edstat-l@jse.stat.ncsu.edu
Sender: edstat-l@jse.stat.ncsu.edu
From: ROBERT FRICK <RFRICK@psych1.psy.sunysb.edu>
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Re: Defining "variable"

Bill Oliver asked what was lacking with the standard definition:

> "A *random variable* is the name given to a quantity which under
> conditions S may take various values with specific probabilities. For us
> it is sufficient to consider random variables that may take on only a
> finite number of different values. To give the *probability distribution*,
> as it is called, of such a random variable, it is sufficient to state its
> possible values..."
>
> I find this definition quite clear, but perhaps it's wrong in some way.

I once thought that my definition of "experiment" was perfectly
clear to students, until I discovered that I was using "manipulate"
in a technical sense and the students weren't.  (Suppose in an
experiment testing if music can reduce sadness, I first make all of
my subjects sad.  Students tend to think, quite appropriately, that I

So "Something that can take two or more values" makes perfect sense to
me, but I know what a variable is.  What would a student make of
this definition?  Speculations:

1. One good example would be the price of a car.  One price is
listed, but the salesman is willing to sell it for a second price.

Well, perhaps "price of car" is a variable, but not for the
student's reason.

2. Another good example would be a compass.  Normally a compass has no
value, but if you were lost, it would be very valuable.

I think these examples fit the definition fairly well.  I don't know
what subjects actually make of the standard definition, I just felt
that I was having trouble teaching the concept of variable using the
standard definition and that after reading Macnaughton's definition,
the standard definition felt shallow and incomplete.  Perhaps
importantly, I was trying to teach the definition of a variable in
the context of an experiment, which might be different from the
context of algebra.

Bob Frick
rfrick@psych1.psy.sunysb.edu

```

```Date: Fri, 5 Jul 1996 22:27:21 -0400
Message-Id: <4rj36c\$v3c@b.stat.purdue.edu>
Originator: edstat-l@jse.stat.ncsu.edu
Sender: edstat-l@jse.stat.ncsu.edu
From: hrubin@b.stat.purdue.edu (Herman Rubin)
To: Multiple recipients of list <edstat-l@jse.stat.ncsu.edu>
Subject: Re: Defining "variable"

In article <D4F8874150@psych1.psy.sunysb.edu>,
ROBERT FRICK  <RFRICK@psych1.psy.sunysb.edu> wrote:
>Bill Oliver asked what was lacking with the standard definition:

>> "A *random variable* is the name given to a quantity which under
>> conditions S may take various values with specific probabilities. For us
>> it is sufficient to consider random variables that may take on only a
>> finite number of different values. To give the *probability distribution*,
>> as it is called, of such a random variable, it is sufficient to state its
>> possible values..."

>> I find this definition quite clear, but perhaps it's wrong in some way.

>I once thought that my definition of "experiment" was perfectly
>clear to students, until I discovered that I was using "manipulate"
>in a technical sense and the students weren't.  (Suppose in an
>experiment testing if music can reduce sadness, I first make all of
>my subjects sad.  Students tend to think, quite appropriately, that I

>So "Something that can take two or more values" makes perfect sense to
>me, but I know what a variable is.  What would a student make of
>this definition?

This is one of the places where attempts to make things too simplistic
result in confusion.  There is too much of a tendency to try to DEFINE,
rather than to CHARACTERIZE.  Now some things must be defined, to
here is mathematics, and the idea of representation of a "real world"
situation by a mathematical model.

Speculations:
>1. One good example would be the price of a car.  One price is
>listed, but the salesman is willing to sell it for a second price.

>Well, perhaps "price of car" is a variable, but not for the
>student's reason.

>2. Another good example would be a compass.  Normally a compass has no
>value, but if you were lost, it would be very valuable.

>I think these examples fit the definition fairly well.  I don't know
>what subjects actually make of the standard definition, I just felt
>that I was having trouble teaching the concept of variable using the
>standard definition and that after reading Macnaughton's definition,
>the standard definition felt shallow and incomplete.  Perhaps
>importantly, I was trying to teach the definition of a variable in
>the context of an experiment, which might be different from the
>context of algebra.

There are too quite different uses of "variable" here.  One of them
is the use as a key part of the language of mathematics; it should be
added to the more customary languages as a means of precise communication.
The other is as "random variable".  This is very definitely NOT the same.
And in neither case is the above intuition even self-consistent.

After the restrictions got worked out, the linguistic use of variable
is essentially that of pronoun.  Mathematics is a grammatically simple
language, with a few linguistic constants, and the rest consists of
names for "objects" (this is not a good term) upon which restrictions
MAY be imposed, but are not inherent.  These names can be changed;
this is the use of substitution.  In some cases, it can be shown that
a particular name refers to exactly one mathematical object; this is
the general introduction of constants, they are variables which can
only take on one value.

I repeat that this is linguistic, but it is the most important part
of mathematics for the non-mathematician.  The applied problem,
translated into this terminology, becomes a problem on which the
full power of mathematics can be unleashed, just as the typesetter
and press operator do not need to understand the book.

Now what about random variables and related items?  There is a
real world situation, which calls for probabilistic or statistical
interpretation and advice for action.  Again, this needs to be
modeled, and there is no one way of modeling.  In fact, one may
want to use different models, with different amounts of coverage,
in the same problem.  If something can be computed in different
correct models, the results are equal.  One should start out with
a sample space, which is a purely abstract set, together with a
dictionary, such that each real world outcome corresponds to
exactly one point of the set.  Insisting that each point of the
set is attained is neither necessary nor desirable.  It may not
even be known if a point is attained, but including it does
absolutely no harm.

Now a random variable is something which can be computed from the
state of the real world.  It can be REPRESENTED by a function on
a sample space, if all real world occurrences giving rise to any
point in that sample space give the same value to the random
variable.  The function doing the representation is a fixed function
on a fixed space; the randomness in the real world is contained
entirely in the representation.  And it is even possible that the
random variable is a constant.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu	 Phone: (317)494-6054	FAX: (317)494-0558

```