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Topic: EPR Approach to Intro Stat: Entities and Properties
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Donald Macnaughton

Posts: 8
Registered: 12/6/04
EPR Approach to Intro Stat: Entities and Properties
Posted: Jun 28, 1996 10:42 PM
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[I've posted this message to the UseNet newsgroup
and to four mail lists where there may be interest. Apologies to
recipients who receive multiple copies.]

Consider three questions:

1. Is the concept of "entity" the most fundamental concept of
human reality?

2. Are the concepts of "entity" and "property of an entity" ap-
propriate concepts to teach at the beginning of an introduc-
tory statistics course?

3. What is a variable?

Consider a condensed version of how the concepts of entities,
properties, and variables could be presented to students:

If you stop and observe your train of thought at this moment, you
will probably agree that you think about "things". For example,
during the course of a minute or so, you might think about, among
other things, a friend, an appointment, today’s weather, and an
idea. Each of these things is an example of an "entity".

Many different types of entities exist, for example,
- physical objects
- processes
- organisms
- events
- ideas
- societal entities (e.g., educational institutions)
- symbols
- forces
- waves
- mathematical entities (e.g., sets, numbers, vectors).

People usually view entities as existing in two different places:
in the external world and in their minds. We use entities in our
minds mainly to stand for entities in the external world, much as
we use a map to stand for its territory.

Most people begin to use the concept of "entity" when they are
very young. Most of us use the concept automatically as a way of
organizing the multitude of stimuli that enter our minds minute
by minute when we are awake.

Since everything (every thing) can be usefully viewed as being an
entity, the concept of "entity" may be the most fundamental con-
cept of human reality.

Because people use the concept of "entity" almost entirely at an
unconscious level, some people have difficulty grasping the fun-
damental role that the concept plays in their thought.

The concept of "entity" is further concealed because it does not
often appear directly in discussion in either (1) everyday life,
(2) empirical research, or (3) statistics. Direct discussion is
usually omitted because, when dealing with specific issues, it is
usually not necessary to drill down all the way to the founda-
tional concept and discuss "things" at such a basic level. In-
stead, discussions (of specific issues) usually concern one or
more particular *types* of entities, which are best referred to
by their type names. For example, medical researchers often
study a type of entity called "human beings".

However, as I argue below, the concept of "entity" serves as a
foundation for most other concepts in statistics and empirical
research. Therefore, discussion of the concept of "entity" at
the beginning of an introductory statistics course is invaluable.

Every entity has associated with it a set of attributes or
"properties". For example, all human beings have thousands of
different properties, two of which are "height" and "blood

For any particular entity, each of its properties has a "value".
We usually report the value of a property with words, with sym-
bols, or with numbers. For example, your height might be 5 feet
9 inches.

If we need to *know* the value of a property of an entity, we can
apply an appropriate measuring instrument *to* the entity. If
the instrument is measuring properly, it will return a measure-
ment to us that is an estimate of the value of the property in
the entity at the time of the measurement. For example, if we
need to know the (value of the) height (property) of a person, we
can apply a height-measuring instrument (e.g., a tape measure) to
the person, and the instrument will give us a number that is an
estimate of the person’s height.

Empirical researchers and statisticians usually refer to proper-
ties of entities as *variables*. That is, when researchers or
statisticians refer to a variable, they are usually referring
(either specifically or generally) to some property of some type
of entity.

Thus the important statistical concept of "variable" can be de-
fined in terms of the three more fundamental concepts of "enti-
ty", "property of an entity", and "value of a property of an en-
tity". A simple version of the definition is
A "variable" is equivalent to a property of an entity.

I discuss other definitions of the concept of "variable" in the

I have argued that we can use the concepts of "entity" and "prop-
erty of an entity" and "value of a property of an entity" to de-
fine the concept of "variable". The three defining concepts are
simple, intuitive, and fundamental. Thus I maintain that it is
useful to introduce the three concepts at the beginning of an
introductory statistics course, as a way of helping students to
understand the concept of "variable".

I invite readers who disagree to present their views in the UseNet newsgroup (= EdStat-L).

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
Joint Statistical Mtgs, Session 201, Tuesday August 6, 2 PM

To help evaluate the above characterization of a variable, it is
useful to consider definitions of the concept that have been pro-
posed by others.

Kruskal and Tanur (1978) lack entries for either "variable" or
"random variable".

Kotz and Johnson (1982-1988) also do not define the term "vari-
able". Their entry for "random variable" consists of "See PROB-
ABILITY THEORY". In the entry for "probability theory" Heyde
(1986) defines a random variable as a member of a certain class
of real-valued functions of points in the sample space. Of
course, the "points" are equivalent to entities.

Marriott (1990) gives the following definition:
variable Generally any quantity which varies. More pre-
cisely, a variable in the mathematical sense, i.e. a quan-
tity which may take any one of a specified set of values.
It is convenient to apply the same word to denote non-
measurable characteristics, e.g. 'sex' is a variable in this
sense since any human individual may take one of two 'val-
ues', male or female.

Marriott defines variables in terms of the concepts of "quantity"
and "characteristic". These two somewhat abstract concepts are
equivalent to the more tightly delineated concept of "property of
an entity".

Marriott makes no reference to the general concept of "entity".
However, it is clear that entities lurk in the background of his
definition. For example, whenever a "sex" variable has a value,
an entity, a particular organism whose sex has been determined
("measured") is somewhere about. In fact, for virtually *all*
variables, it is reasonable to see entities existing behind the
variables. (The entities are whatever are associated with the
rows in a standard computer-package data table, in which the col-
umns represent variables.)

I believe that we should move the entities in statistical analy-
sis to the foreground since, from the point of view of empirical
researchers, the entities are an important and tangible aspect of
the research, and as such should not be left lurking in the back-

Vogt (1993) defines a variable as
Variable Any finding (an attribute or characteristic) that
can change, that can _vary_, or that can be expressed as
more than one value or in _various_ values or categories.
The opposite of a variable is a constant.
For example, height: 5'7", 5'8", and so on; or religion:
Catholic, Protestant, Jewish, Other; or experimental treat-
ment: Drug A, Drug B, Drug C.

Vogt uses the concepts of "finding", "attribute", and "charac-
teristic" to define variables. As with the defining concepts in
Marriott's definition, these three somewhat abstract concepts are
all equivalent to the more tightly delineated concept of "prop-
erty of an entity".

Like Marriott, Vogt makes no direct reference to the concept of
"entity" although there are entities lurking in the background in
each of his three examples.

Modern definitions of the concept of "variable" are beginning to
embrace the concept of "entity" although once referred to in a
definition, entities are still often given short shrift in the
rest of the discussion.

For example, Freedman, Pisani, Purves, and Adhikari define a var-
iable as
A _variable_ is a characteristic which changes from person
to person in a study (1991, 40).
These writers use the concept of "entity" in their definition but
they seem to assume that only people can be entities. (One sus-
pects, however, that this limitation is not their actual intent,
and is instead an editing error.)

Moore, in his exemplary introductory statistics textbook, begins
by defining "individuals" as
the objects described by a set of data. Individuals may be
people, but they may also be animals or things.
He then defines a "variable" as
any characteristic of an individual. A variable can take
different values for different individuals (1995, 10).

Moore defines the concept of "individuals" (= "entities") in
terms of the concepts of "objects", "people", "animals", and
"things". Similarly, he defines the concept of "variable" in
terms of the concept of "characteristic" (= "property").

(The choice of which *names* to use for the concepts "entity" and
"property" is of some importance, with the choice perhaps being
dictated by considerations of generality and ease of understand-
ing. However, the present discussion is not about the choice of
names for the concepts, but is about the concepts themselves, re-
gardless of what we decide to call them.)

Note that Moore defines "individuals" in terms of the concept of
a "set of data". If we assume that Moore is following the con-
vention of defining each term in a conceptual system in terms of
other more fundamental terms, his definition suggests that he
views the concept of "set of data" as being more fundamental than
the concept of "individual". Thus Moore appears to be taking a
phenomenalistic approach.

My approach to defining the concept of "variable" is similar to
Moore's except I suggest that it is useful to view the concept of
"entity" (= "individual") as being more fundamental than the con-
cept of a "set of data". In fact, I suggest we leave the concept
of "entity" as a primitive. And although we can *illustrate* the
concept of "entity" for students by discussing many examples of
entities, we should tell students that the concept itself will,
to avoid circularity, be left verbally undefined.

[As noted above, humans acquire the concept of "entity" as young
children through non-verbal linking of consistent sets of stim-
uli. Thus one could argue that the stimuli (sense data) that one
receives are the fundamental units of reality. At a preconscious
level this approach seems quite reasonable. But at the conscious
level, which is the level at which all human discussion about
statistics must operate, the concept of "entity" seems to hold
sway as the concept that is the basis of all other concepts.
(After all, even properties and sets of data are entities.) Thus
at the discussion level it makes sense to designate the concept
of "entity" as fundamental and therefore verbally undefined.]

Similarly, I believe that the concepts of "properties of enti-
ties" and "values of properties of entities" are best left as
primitives, defined solely through human experience and through
discussion of examples.

On the other hand, I agree with Moore that we can give the con-
cept "variable" a formal or informal verbal definition in terms
of the concepts of entities, properties, and values.

In their comprehensive unified view of many of the main statisti-
cal topics, Kendall, Stuart, and Ord characterize variables in a
way that is similar to the approach described in this note al-
though they use somewhat different underpinnings. In particular,
in volume 1 in the first sentence of chapter 1 they assert that
the concept of "population" is "the fundamental notion in statis-
tical theory" (1987, 1994). They then give five examples of dif-
ferent types of entities to illustrate what a population can be
made of. However, although Kendall et al populate their popula-
tions with entities, they do not recognize the concept of "enti-
ty" as being a concept in its own right, more fundamental than
the concept of "population". Thus they seem to want to start in
the ball game at second base.

In the second paragraph of chapter 1 Kendall et al prepare for
discussion of the concept of "variable" by discussing the concept
of "properties". However, they concentrate on "properties of
populations" as opposed to the more general concept of "proper-
ties of entities". (Entities are more general than populations
because all populations are also entities, but not vice versa.)
I believe that we should first introduce students to the funda-
mental concepts of "entity" and "property of an entity". Then we
can define the concepts of "population" and "variable" in terms
of those concepts. By building the discussion around what appear
to be the most fundamental concepts of human reality, I believe
we make the field of statistics substantially easier for students
to understand.

Freedman, D., Pisani, R., Purves, R., and Adhikari, A. (1991),
_Statistics_ (2nd ed.), New York: Norton.

Heyde, C. C. (1986), "Probability Theory (Outline)" in _Encyclo-
pedia of Statistical Sciences_ (Vol. 7), ed. S. Kotz and N. L.
Johnson, New York: John Wiley, pp. 248-252.

Kendall, M., Stuart, A., and Ord, J. K. (1987, 1994) _Kendall's
Advanced Theory of Statistics,_ (5th and 6th eds, 3 vols),
London: Charles Griffin, Edward Arnold.

Kotz, S. and Johnson, N. L., eds. (1982-1988), _Encyclopedia of
Statistical Sciences_ (9 vols), New York: John Wiley.

Kruskal, W. H. and Tanur, J. M., eds. (1978), _International En-
cyclopedia of Statistics_ (2 vols), New York: Free Press.

Marriott, F. H. C. (1990), _A Dictionary of Statistical Terms_
(5th ed.), Harlow, UK: Longman Scientific and Technical.

Moore, D. S. (1995), _The Basic Practice of Statistics,_ New
York: Freeman.

Vogt, W. P. (1993), _Dictionary of Statistics and Methodology,_
Newbury Park, CA: Sage.

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