Subject: Re: Eight Features of an Ideal Intro Stat Course
         (Response to comments by Dennis Roberts)

     To: EdStat-L (July 23, 1998) (July 28, 1998)

   From: Donald B. Macnaughton <>

In a July 17 post I recommend that teachers emphasize the concept 
of a relationship between variables and I recommend

    a de-emphasis of less important topics such as univariate 
    distributions ...

Referring to this text, Dennis Roberts writes (on July 20)

>    ( snip )
> i disagree ....  i find it hard to think that you will be able
> to cogently discuss relationships between variables without
> first having studied such mundane issues as central tendency,
> or variability, or position measures (i.e., z scores for exam-
> ple) ... on single variables ...
> these form the foundation of higher level things. 

Dennis' comments reflect traditional thinking about the introduc-
tory statistics course.  This thinking holds that students must 
understand univariate distributions before they can understand 
the main ("higher level") ideas of statistics.  While I fully 
agree that the concept of a univariate distribution is necessary 
to understand the main *mathematical* ideas of statistics, I do 
not believe the concept is necessary to understand the main ideas 
of statistics *as they are used in empirical research*.

As I discuss in a paper (1998), I believe the concept of a rela-
tionship between variables is a candidate for the most important 
concept in both empirical research and statistics.  Thus I be-
lieve it is important to emphasize this concept in an introduc-
tory statistics course.  

It is clear that students can understand the concept of a rela-
tionship between variables without understanding the concept of a 
univariate distribution.  We can see this by noting that students 
learn and understand many examples of relationships between vari-
ables in high school *before* they know much about univariate 

For example, students in high school physics courses learn about 
the relationship between acceleration (a) and force (f) with the 
model equation

                            f = ma

where m is the mass of the body being accelerated.  Students seem 
quite capable of understanding this relationship without knowl-
edge of univariate distributions.  Similarly, we can easily talk 
with students about the relationship in people between the vari-
ables EDUCATION and INCOME without reference to the concept of a 
univariate distribution.  Similarly, although they are generally 
not specifically characterized as such, many high school mathe-
matics problems are problems of relationships between variables 
-- these problems make no reference to univariate distributions.

Thus we can discuss relationships between variables in the intro-
ductory statistics course with almost no reference to the concept 
of a univariate distribution.  (I demonstrate one approach in a 
paper for students [1996].)  

In section 9.1 of the 1998 paper I note that univariate distribu-
- play only a *peripheral conceptual role* in real empirical re-
- are *not necessary* for initial study and understanding of the 
  concept of a relationship between variables
- are *boring* for students because students see no practical use 
  of univariate distributions
- are *less accurate* for making predictions than predictions 
  based on relationship between variables.

I believe we alienate students by burdening them with concepts of 
univariate distributions that are boring and have no obvious 
practical use.  (I discuss two examples of univariate distribu-
tions a teacher submitted that are clearly not boring in appendix 
G of the 1998 paper.  I suggest that both examples are better 
viewed as examples of relationships between variables.)

Studying univariate distributions certainly *was* necessary when 
students had to do all the mathematical computations of statis-
tics themselves because many of the underlying mathematical ideas 
rely on the concept of a univariate distribution.  But nowadays a 
computer can do all the standard computations, so it is no longer 
necessary to understand the underlying mathematical ideas right 
from the start.

Finally, as I say in the paper, an understanding of univariate 
distributions is *mandatory* for full understanding of the field 
of statistics.  Therefore, I am not suggesting that the topic of 
univariate distributions be removed from the curriculum -- I am 
only suggesting that it be moved later, *after* students have a 
good understanding of the more interesting and more important 
concept of a relationships between variables.

Because this approach is more closely tied to the use of statis-
tics in actual empirical research, I suggest that the approach is 
more likely to give students a lasting appreciation of the vital 
role of our field.

Donald B. Macnaughton   MatStat Research Consulting Inc      Toronto, Canada


Macnaughton, D. B. 1996. The entity-property-relationship ap-
   proach to statistics:  An introduction for students.  Avail-
   able at

Macnaughton, D. B. 1998. Eight features of an ideal introductory 
   statistics course.  Available at

Home page for Donald Macnaughton's papers about introductory statistics