Subject: Re: EPR Approach to Intro Stat: Relationships Between
   Date: Sun, 28 Jul 1996 15:35:21 -0400 (EDT)

   From: "Donald Macnaughton" <>



This post contains a medium-length body, followed by a long appendix, 
followed by a list of four references.  If you have trouble viewing 
the entire post, it is also available at

On July 19, 1996 Bob Frick wrote

>      ( snip )
> Macnaughton defined a variable as the property of an entity.  

Actually, I said (in reference 1) that 
    a variable is *equivalent to* a property of an entity.  
Equivalence is different from equality.

[Note added on December 1, 1996:  I now believe a more precise 
definition is:  A "variable" is a formal representation of a
property of entities.]

I view variables as being different from properties, although there 
is obviously a close kinship.  Properties come first in human thought 
as a way of characterizing the various entities that constitute real-
ity.  Variables come later as a formal invention of mathematically-
minded scientists.  Variables come later in the development of human 
thought in individual people, and variables also come later in the 
development of human thought down through the ages of civilization.  
Variables enable systematic mathematical treatment of properties.  
That is, they allow us to model properties.  

In mathematical discussion, mathematicians usually dispense complete-
ly with the entities that lie behind properties, and instead use var-
iables to model properties in a general abstract sense.  This is use-
ful because the resulting mathematical systems are guaranteed to work 
properly with any properties of any type of entity provided that the 
properties and entities obey the axioms.  (Of course, mathematicians 
often lay down axioms that reflect some aspect of the real world, be-
cause then the resulting system will be immediately useful.)  

As soon as the variables of abstract mathematics are brought into an 
empirical (real world) situation then, invariably, one or more sets 
(types) of entities appear in the discussion, and each variable be-
comes associated with a property of one type of entity.

I invite readers to try to propose situations in empirical research 
in which a variable is present, but the variable is not equivalent to 
a property of some type of entity.

In the spirit of the preceding paragraph, Bob describes a situation 
in which my definition of "variable" appears to be violated.  Consid-
er a two-group between-subjects medical experiment to study the re-
lief of headache pain with aspirin.  Bob writes

>      ( snip )
> the properties of the entities [i.e., the properties of the
> patients in the experiment] are going to be whether or not they
> took aspirin and how much pain they are suffering [at] a given
> later time.  Because the difference between taking aspirin and not
> taking aspirin is not a property of any entity in this experiment,
> it is not a variable according to Macnaughton's definition.

Bob is suggesting that the predictor variable in this experiment does 
not satisfy my definition of "variable".  However, Bob's suggestion 
is incorrect.

We can view the example as follows:  The response variable is "re-
ported amount of headache pain at t minutes after taking the pill" 
(or something like that), where t has some specific value (e.g., 20 
minutes).  The predictor variable is "amount of aspirin taken".  This 
variable has a value of zero for patients in the control group and 
some specific non-zero value (e.g., 500 milligrams) for the patients 
in the treated group.  This variable reflects a property of the pa-
tients, and therefore my definition of "variable" is satisfied in 
this example.

Note that the property of the patients called "amount of aspirin 
taken" comes into existence only after the patients have taken the 
pills (i.e., placebo or aspirin).  This may at first be unacceptable 
to those readers who are used to viewing properties as having more 
permanence, or being more firmly attached to their entities.  But the 
"property" that I've called "amount of aspirin taken" is a valid fea-
ture, attribute, or aspect of the patients.  Thus we should not aban-
don our fundamental conceptual tool of "property" because this "prop-
erty" is less permanent than the property of say, "height", which is 
with a person for life (although the value changes over time).

In his next sentence, Bob writes

> This suggests, interestingly, that Macnaughton's definition and the
> mathematical definition are different.  

Bob doesn't specify what "the mathematical definition" of "variable" 
is.  I suggest that we should have a specific mathematical definition 
on the table before we discuss whether my definition is the same as 
or different from it.  We have the following choices:
- "Kolmogorov's informal definition" quoted in Bill Oliver's post of 
  June 29
- Rubin's definition in his post of June 30
- Rubin's metaphorical "definition" in the same post
- six other definitions or characterizations of variables or random 
  variables given in the appendix of reference 1
- other definitions of the term "variable" in the statistical and 
  mathematical literature.
Without specifying which of the above definitions to focus on, we 
cannot easily decide whether my definition and "the mathematical 
definition" are different.

> It seemed to me that this [i.e., the possibility that Macnaughton's
> definition and the mathematical definition are different] was al-
> ready suggested by several previous posts, who criticized
> Macnaughton's definition from a mathematical perspective.  

I'm aware of six posts that discussed (or came close to discussing) 
my definition of "variable".  These posts were in the thread 'Defin-
ing "variable"' that appeared between June 28 and July 5 in this 
newsgroup/mail-list.  The authors of the posts are Frick (2 posts), 
Oliver (1 post), and Rubin (3 posts).  I summarize the six posts in 
the appendix.  Although the posts all discuss definitions of "vari-
able", I have been unable to find any criticisms of my definition in 
them from a mathematical perspective, or from any other perspective.  

Thus Bob's speculation that the previous posts criticized my defini-
tion would appear to be incorrect.

Regardless of whether the previous posts *criticized* my definition, 
it is important to determine whether my definition is consistent with 
mathematical definitions.  Thus in reading the six posts, I have also 
looked for inconsistencies between my definition and the definitions 
given in the posts.  (I describe the details of my search for incon-
sistencies in the appendix.)  Although the concept of "entity" in my 
definition generally fades into the background in the other defini-
tions, I found no significant inconsistencies between my definition 
and the definitions in the six posts.

Consider three premises:
1. The main role of the field of statistics is to support empirical
2. Entities always lie behind variables in empirical research.
3. The entities in an empirical research project are often the most
   tangible aspect of the research.

The premises imply that it is helpful to emphasize the concept of 
"entity" in the definition of "variable" in the introductory statis-
tics course.  

The above points are part of a broader discussion of an approach to 
the introductory statistics course available at

Donald B. Macnaughton      MatStat Research Consulting Inc.         Toronto, Canada

This appendix discusses the six posts in the thread 'Defining 
"Variable"' that was begun by Robert Frick on June 28, 1996.  (Bob 
began the thread in response to my post of June 25/26, 1996--refer-
ence 1).  The full text of the six posts is available in reference 3.  

The purpose of discussing the six posts is to address two questions:
- what are the criticisms of my definition of "variable" in the six 
- are there inconsistencies between my definition of "variable" and 
  the definitions discussed in the six posts?

Frick's first post does not criticize my definition but instead opens 
by praising it:

> I thought Don Macnaughton's definition of a variable was
> brilliant.  

Bob is obviously not as sure now about the correctness of my defini-
tion as he was then, and I respect his public change of heart--or at 
least his expression of doubts when they arose.  Bob's questioning 
and doubting approach is the same as mine.  It is the only way to 
find truth.  (I hope I can remove Bob's doubts.)

Bob goes on in his post to argue that the concept of "variable" is a 
tough concept for students to grasp.  He concludes the post by offer-
ing a two-paragraph "paraphrase" of my discussion of entities, prop-
erties, and variables.

Thus Frick's first post neither criticizes my definition of "vari-
able", nor does it suggest any inconsistencies between my definition 
and other definitions.

To prepare for discussing Rubin's first post, I shall briefly di-
gress:  C. C. Heyde, writing in the _Encyclopedia of Statistical 
Sciences_, gives a "sample space" definition of a random variable 
(reference 4).  The definition says that a random variable is a mem-
ber of a certain class of real-valued functions of points in the sam-
ple space.  (The class of functions is described in detail in the 

We can view the points in the sample space as being equivalent to en-
tities.  Thus my definition and the "sample space" definition of 
"variable" are consistent with each other in the sense that the enti-
ties in my definition map to the points in the sample space and the 
property maps to the function.

Herman Rubin's first post is in response to Frick's first post.  Ru-
bin suggests that Frick's paraphrase of my discussion of entities, 
properties, and variables is

> still not too good for random variables, where it is more appropri-
> ate.  

By the use of the word "still" in the phrase "still not too good" Ru-
bin may mean that Frick's paraphrase of my definition may be better 
than my definition, but is still unsatisfactory.  Thus Rubin may be 
indirectly disapproving of my definition.  However, since Rubin nei-
ther makes a direct reference to my definition, nor states any spe-
cific criticism of the definition, I shall assume that this quotation 
is not criticizing my definition.

I am unsure what Rubin means by the phrase "where it is more appro-
priate", but he may mean that since we are in a statistical context, 
it is more appropriate to define the term "random variable" instead 
of the term "variable".  If that is what he means, I suggest that it 
is reasonable to try to define the general term "variable" in a way 
that subsumes "random" variables.

Rubin continues:

> I would define a random variable as something which can be computed
> knowing the real world situation; it need not be a number.

I believe that Rubin's definition and my definition are consistent 
with each other in the sense that whenever we compute something 
"knowing the real world situation" we will find ourselves computing 
the value of a property of some type of entity.  Admittedly, it can 
occasionally be difficult to determine what the entity is, especially 
if the entity is an "event", or a "trial" in an experiment, or some-
thing else not very tangible.  However, although I have looked rather 
carefully, I have been unable to find a situation in which we have 
the value of a variable in the real world, but that variable is not 
associated with some property of some type of entity.

Rubin continues by stating that the "sample-space" approach to defin-
ing a random variable (presumably including Heyde's definition above) 
leads to problems:

> 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.

Rubin concludes his first post with a second "definition" of 

> 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.

By putting the word "definition" in quotation marks Rubin appears to 
be indicating that he isn't really offering a definition but is in-
stead offering a metaphor.  That is, he is saying that a variable is 
a place holder for a set of different values, just as a pronoun is a 
place holder for the names of individual people or things.  

In the case of a pronoun, the person or thing being referenced by the 
pronoun--the referent--is determined from a common or proper noun 
that appears earlier in the discussion.  Thus when a pronoun is used 
in proper grammar the name of the referent is always known, and the 
pronoun is merely a one-syllable abbreviation for that name.  In con-
trast, in the case of a variable, the referent (i.e., the *value* of 
the variable) is usually unknown and it is this unknown value that we 
are modeling in general terms.

There is a parallelism between Rubin's pronoun characterization of a 
variable and my definition in the sense that (like a pronoun) a prop-
erty can be viewed as a place holder for a group of different values 
(i.e., a property can "take" different values in different entities 
or take different values in the same entity across time.)  In the 
case of a variable, the "referent" is a specific value of the prop-

However, there does not seem to be a clear mapping between the enti-
ties that I say are associated with a variable and the "entities" of 
the pronoun metaphor (i.e., pronouns and nouns, which are, to make 
things more complicated, themselves names for entities).  However, I 
don't view this lack of a mapping as an inconsistency between the two 
definitions.  Instead, I view it as (1) a weakness of the metaphor 
and (2) a ramification of the fact (discussed at the beginning of the 
present post) that mathematical treatments of variables usually dis-
pense with the concept of "entity" to allow a very general discus-
sion.  But as soon as the mathematical discussion is linked to the 
real world, the entities (which were always implicit but invisible 
behind the variables) will reappear.

Thus Rubin's first post does not (directly) criticize my definition 
of "variable".  Furthermore, (except for the matter discussed in the 
preceding paragraph) my definition and Rubin's two definitions appear 
to be consistent with each other.

William Oliver, in his post of June 29, states that he hasn't seen my 
definition of "variable", but has only seen Frick's first post.  
Oliver presents "Kolmogorov's informal definition" and wonders if 
there is anything wrong with it.  He states the definition as

> "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..."

Note that the Kolmogorov definition states that a random variable is 
equivalent to a "quantity" with certain restrictions placed on it.  A 
"quantity" is a property of an entity (although, as I've noted, the 
entities can fade into the background in abstract mathematical dis-
cussions).  Thus the Kolmogorov definition and my definition appear 
to be consistent with each other.

Thus Oliver's post neither criticizes my definition of "variable", 
nor does it suggest any inconsistencies between my definition and 
other definitions.

In his second post (June 30), Rubin responds to Oliver's post.  In 
the first paragraph Rubin criticizes the Kolmogorov definition for 
its dependence on the concept of a probability distribution and notes 
that the concept of probability is a representation (model) in the 
mathematical world of "something" in the real world.  In the second 
paragraph he makes the important point that we must distinguish be-
tween the thing being modeled and the model.  In the third paragraph 
he notes that this distinction implies that a "unique" sample space 
for dealing with a problem does not exist, although different 
"compatible" sample spaces will yield the same "answers".

In the fourth and final paragraph Rubin writes

> 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.

Rubin makes the point here that if one wishes to define "random vari-
able" in terms of a function on the sample space, then a problem 
arises because no explicit conceptual machinery is present to account 
for the random element.  That is, both functions (mappings) and 
sample spaces are quite fixed, not random.  Rubin proposes to solve 
this problem by adding a second mapping to the situation.  This 
mapping is between the real world and the abstract sample space.  

My definition of "variable" and Rubin's dual mapping approach are 
consistent with each other in the sense that the property in my defi-
nition corresponds to the union of the instances of the two mappings 
in Rubin's approach.  The entities in my definition correspond to the 
points in the sample space and their referents in the real world. 

Thus Rubin's second post neither criticizes my definition of 
"variable", nor does it suggest any inconsistencies between my defi-
nition and other definitions.

Frick's second post (July 2) is in response to Oliver's post.  Frick 
suggests that "Kolmogorov's informal definition" of "variable" (as 
given by Oliver) might not be clear to students.  He reinforces this 
by noting how he had discovered that he and his students were unknow-
ingly using certain terms in different senses.  He goes on to de-
scribe two different misinterpretations that students could easily 
make of the Kolmogorov definition.  He then refers (at the end of his 
last paragraph) to my definition as follows:

> 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.

Thus Frick seems to be beginning to wonder whether my definition and 
the "algebra" or mathematical definition are consistent.  As I sug-
gest at the beginning of the present post, I believe that the two 
definitions are consistent, although (for the sake of generality) the  
entities usually fade into the background in mathematical treatments 
of variables (to reappear whenever the variables are used in the real 

Thus although Frick's second post speculates that my definition of 
"variable" may be different from the standard definition of 
"variable" in the context of algebra, it does not *criticize* my 
definition of "variable".  

The post suggests the possibility of an inconsistency between my 
definition and an algebraic definition although the post does not 
touch on the nature of the possible inconsistency.  Because the post 
does not specify the nature of any inconsistency, I conclude that the 
post does not suggest any significant inconsistencies between my 
definition of "variable" and other definitions.

In his third post (July 5), Rubin responds to Frick's second post.  
Rubin begins by quoting "Kolmogorov's informal definition" (as posted 
by Oliver) and then quotes Frick's comments about discovering that he 
(Frick) and his students were unknowingly using certain terms in dif-
ferent senses, and how he (Frick) wonders whether students might mis-
interpret the Kolmogorov definition.  Rubin then writes

> This is one of the places where attempts to make things too sim-
> plistic 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 start out and have an adequate language.  But the
> adequate language here is mathematics, and the idea of represent-
> ation of a "real world" situation by a mathematical model.

Although he doesn't use the word "variable" in the preceding para-
graph, Rubin seems (since the title of the thread is 'Defining 
"Variable"') to be saying here that we should not try to *define* the 
term "variable", either with a "mathematical" definition or with my 
definition.  Instead, we should only try to *characterize* the term.

Rubin next quotes (1) Frick's description of the two misinterpreta-
tions that students could make of the Kolmogorov's definition of 
"variable" and (2) Frick's comment that there might be a difference 
between the definition of a variable in the context of an experiment 
and the definition of a variable in the context of algebra.

Rubin next affirms Frick's last point, distinguishing the standard 
variables of mathematics from random variables as follows:

> There are too [sic] 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 pre-
> cise 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.

It is unclear what Rubin means in his last sentence above by the 
phrase "the above intuition" although presumably it applies to some-
thing in Frick's post as opposed to something in the same paragraph 
by Rubin.  However, (recalling the questions being addressed in this 
appendix) Rubin does not point out any specific inconsistencies be-
tween my definition of "variable" and other definitions of "variable" 
or "random variable".

Rubin next writes:

> After the restrictions got worked out, the linguistic use of
> variable is essentially that of pronoun.  Mathematics is a gram-
> matically 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.

The preceding paragraph reiterates Rubin's characterization of a 
variable as a pronoun, and notes that the majority of items in the 
language of mathemetics are general names for "objects".  (Are these 
"objects" equivalent to what I call entities or properties?)  The 
paragraph ends by characterizing constants.

> 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.

The discussion is linguistic, aimed at building a powerful general 
terminology for handling applied problems.  

> 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.

The above paragraph notes that real world problems require statisti-
cal interpretation and advice for action.  No single optimal way of 
modeling a particular real world situation exists, but different 
"correct" models lead to the same results.  The sample space is an 
abstract set representing the real world "outcomes" being modeled.  A 
dictionary (function?) maps between each outcome in the real world 
and a point in the sample space.  (The sample space may have more 
points than the real world has outcomes.)

> 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.

Rubin begins the above paragraph, his final paragraph, by echoing his 
earlier definition of "random variable".  He next notes that a random 
variable can be "represented" by a function on a sample space, and 
thereby makes the point that a random variable is, in his view, not 
identical to a function.  He reiterates his point that a function and 
a sample space are both fixed.  The random element of a random vari-
able occurs in the many-to-one mapping (representation) between ob-
jects in the real world and the sample space.

On the basis of his three posts, I summarize Rubin's description of a 
random variable as follows:
   A random variable is something that can be computed from the state 
   of the real world.  It can be characterized by a fixed function on 
   the fixed sample space.  The values of the function are the values 
   of the variable, which need not be numbers.  The random element 
   comes through a second many-to-one mapping between the phenomena 
   in the real world and the points in the sample space.

As I noted above, my definition of "variable" and Rubin's description 
of a random variable are consistent with each other in the sense that 
the property in my definition corresponds to the union of the instan-
ces of the two mappings in Rubin's description.  The entities in my 
definition correspond to the phenomena (entities?, outcomes?) in the 
real world and the linked elements in the sample space in Rubin's de-

Thus Rubin's third post neither criticizes my definition of "vari-
able", nor does it suggest any inconsistencies between my definition 
and other definitions.

As I noted at the beginning of this appendix, the first purpose of 
the appendix is to identify criticisms of my definition of "variab-
le".  Apart from a possible oblique disapproval of my definition in 
Rubin's first post, I found no criticisms in any of the six posts. 

The second purpose of the appendix is to determine if inconsistencies 
exist between my definition of "variable" and other definitions.  Al-
though my definition of "variable" is clearly different from the 
other definitions, I found no specific inconsistencies between my 
definition and other definitions in any of the six posts.  

Although there are no inconsistencies between my definition and the 
other definitions, the entities in my definition may fade completely 
into the background in other definitions when these definitions are 
used in abstract mathematical situations.  The entities reappear, 
however, when the variables (random or non-random) are used to model 
a real world (i.e., empirical) situation.  

Like Herman Rubin, I believe that an important goal of statistics is 
to help scientists model the real world.  Therefore, I believe that 
the entities that appear in real-world situations should be carefully 
taken account of in statistical discussions--especially in the defi-
nition of the concept of "variable".

1. Macnaughton, D. B. (1996), "EPR Approach to Intro Stat: Entities 
   and Properties." Posted to the UseNet newsgroup (= 
   EdStat-L e-mail list) on June 25/26, 1996.  Available at
2. Macnaughton, D. B. (1996), "The Introductory Statistics Course:  A 
   New Approach."  This 8000-word draft paper is available at
3. Frick, R., Rubin, H., and Oliver, W. (1996), "(six separately 
   authored posts to the 'Defining "Variable"' thread in the UseNet newsgroup [= EdStat-L e-mail list] between June 
   28 and July 5, 1996)".  The text of these posts may be available 
   from the Journal of Statistical Education archive at
   or from the authors.  The text of the six posts is also available 
4. Heyde, C. C. (1986), "Probability Theory (Outline)" in 
   _Encyclopedia of Statistical Sciences_ (Vol. 7), ed. S. Kotz and 
   N. L. Johnson, New York:  John Wiley, pp. 248-252.

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