From: hayden@OZ.PLYMOUTH.EDU (Dr. Robert Hayden)
Newsgroups: bit.listserv.edstat-l
Subject: computing stuff
Date: 16 Oct 92 23:52:15 GMT
Reply-To: "Dr. Robert Hayden" <hayden@OZ.PLYMOUTH.EDU>
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU
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There has been some discussion of the role of computers,
computational formulae, and hand computation on EDSTAT-L.  Below
is an unpublished paper I wrote on these issues some years ago.
If you think it is too long or you are not interested, please
procede to your next message.  It runs about 4 pages.  I
guarantee that it does not contain any theories of the universe
named after myself.

*****************************************************************

	   USING COMPUTERS TO IMPROVE STUDENT LEARNING

			  OF STATISTICS

			Robert W. Hayden




In an earlier paper, with a very similar title, I argued that

     ...writing becomes not just another subject to teach,
     nor even a tool for achieving traditional goals, but
     rather a necessary path to developing higher-level
     quantitative skills.

I would make a similar argument for using computers to teach
statistics.

To one who reads older introductory statistics textbooks without
having ever done much statistics, it is not always clear how much
the content of these books is compromised by the state of
technology at the time they were written. Most statisticians see
statistics as the art and science of drawing conclusions from
numerical data. Thus, a statistics book without data is like a
calculus book without real numbers or an algebra book without
variables. Even so, the older introductory statistics textbooks
often contained no real data whatever. (One of the oldest,
Snedecor and Cochran, was a welcome exception.) The rationale for
excluding data was that real data sets were too large and too
messy for hand calculation. This is a valid point when we are
trying to teach students how to do statistical calculations by
hand. It becomes a dangerous point of view when it reduces
statistics to learning to do the computations, rather than
learning to think critically about our choice of technique or the
meaning of our data.

What can we do to keep arithmetic from driving statistics out of
a beginning course? To begin with, I suggest that everything we
do in an introductory statistics course should be evaluated on
the basis of whether or not it helps students to draw conclusions
from (real) data. (Even if one does not use real data in class, I
presume that the object of a statistics course is to help
students deal with real data outside of class.) In particular,
any computational techniques we teach our students must be
evaluated by this standard. With The Student Version of Minitab
available for about $40, no one who has any quantity of real data
to analyze is going to do the analysis by hand. So, to begin
with, I think our starting premise should be that students will
be taught to do statistical computations the way they are
actually done in practice, with real data on a computer. It is
not using real data or the computer in statistics class that
requires special justification, but rather using fictitious data
or any other means of computation.

So, let us turn now to the real question:

     Is there any place for hand calculation in statistics
     class?

I believe that there is a definite but limited role for such
calculation. The main value of hand computation is to help the
student to understand what the result means after it is computed.
Before giving examples of how I think this should or should not

be done, there are a couple of underlying issues I wish to deal
with.

First, I think that mathematicians are often overly optimistic
about how much our students learn from seeing how something is
calculated. A mathematician looking at the formula for sample
variance can see that this is some sort of measure of the degree
of deviation from the mean occurring throughout the sample. Few
of our students would see this in the formula, and I suspect that
many fail to see it even when exposed to numerous example
calculations of variances. Thus, if we show students this
computation, we need to concentrate on helping them to see what
the steps mean, rather than, say, concentrating on teaching them
to reproduce the steps on homework problems or on an exam. Hand
computation must be seen as a teaching technique, not as an end
in itself.

Second, there often seems to be an assumption that hand
computation automatically instills understanding in a way that
machine computation does not. In my experience, most students
learn hand computation as rote procedures and know nothing about
the meaning of the steps anyway. For example, most of our
students have had far more experience doing long division than
they have had computing variances. You might select a random
sample of these students and determine the proportion who have
any idea that the algorithm represents a shortened repeated
subtraction. Unless there is real evidence that a particular hand
computation leads to greater understanding, students might better
do it on a computer. Indeed, doing it on a computer leaves time
for them to think about what the result means. (Of course, they
only think about this if so doing, and writing down their
thoughts, is part of the assignment.)

There also seems to be a common assumption that hand computation
somehow demonstrates understanding. If you want to see if a
student understands what a variance is, a much more direct
approach than asking the student to compute one is to set a
situation in which the variance might be the appropriate thing to
compute, and see if the student notices this, or show the student
an already completed variance calculation and ask for an
explanation of what the various numbers, including the variance
itself, mean.

Here are some places where I think hand calculations are useful.

   1.  Computing medians and other order statistics. The
       preparatory step of sorting the data helps students to
       understand what we mean by order statistics. A stem-and-
       leaf diagram can help with sorting, and provides a
       valuable summary of the data.

   2.  Computing means, variances, and standard deviations. The
       residuals should be given special emphasis, as this
       concept recurs throughout statistics. It is hard to
       communicate this idea unless the students see the
       residuals, and they are rarely exhibited by statistical
       software. An important interpretation of the standard
       deviation (here and throughout statistics) is that it is a
       typical value for the residuals.

   3.  Computing ANOVA by hand in the case where all groups have
       the same size. This should not be done using any of the
       formulae commonly appearing in textbooks. Instead, means
       and variances should be computed for each group. Then the
       variance of the group means and the mean of the group
       variances should be computed. The latter provides an
       estimate of the (common) population variance directly,
       while the Central Limit Theorem assures us that the former
       estimates that variance divided by the group size. This
       provides a good review of means, variances, and the
       Central Limit Theorem at the end of the course, as well as
       giving some idea of what the F-ratio measures.

   4.  Exploring the meaning of the least squares criterion for
       goodness of fit in regression. Here we can explore
       different criteria for "best" fit, including fits by eye,
       and sum the squared residuals as one measure of goodness.
       This helps to make it clear what least squares means, even
       though we can not explain why the formulae for the
       coefficients of the least squares line take the form they
       do.


Here are some places where I would view hand calculations as
counterproductive.

   1.  So called "computational formulae" do generally speed up
       hand calculations, especially for the kind of fictional
       data sets commonly used in textbooks. However, they also
       generally obscure the meaning of the computation, and so
       should not be used in an educational context. If your
       students are doing so much hand calculation they feel a
       need for shortcuts, they should be doing those
       calculations on a computer. This will save time that can
       be used for more important matters.

   2.  Correlation coefficients and regression equations. Even in
       the unlikely event that the students have any idea what
       the computations mean, the arithmetic is so messy that it
       diverts large amounts of the students' time and energy
       away from much more important issues, such as an
       examination of regression residuals. This makes possible a
       deeper coverage of regression in less time.

   3.  Analysis of variance with unequal group sizes. This is
       another topic that can be covered quite quickly with
       computer support, but takes lots of time and energy if
       done manually.


Here are some things that the computer allows us to do that we
could not have done, or could not have done so easily, in the
past.

   1.  Examine real data sets and answer real questions about
       them. In the older statistics books, just computing the
       mean may have been the whole point of an exercise. In
       reality, the mean is only computed if it tells us
       something interesting about the data. With the computer
       doing the work, students can get the mean, mode, and
       median of several sets of real data, learn the strengths
       and weaknesses of each, and see what they tell us about
       the data. This does not mean that there is no place for
       the carefully crafted educational example that makes a
       single pedagogical point without the distractions inherent
       in real data. However, such examples are only a
       preparation for dealing with real data, not a substitute.

   2.  Examine large data sets. While big is not necessarily
       beautiful, real data sets are almost always larger than
       textbook fictions, and many important aspects of data are
       invisible when n=5. (See below.)
   3.  Examine a large number of data sets. Statistics is an art
       as much as a science. Students need to see lots of
       examples of real data appropriately analyzed in order to
       learn to do so themselves. Using the computer to handle
       the calculations, you can cover several data sets in a
       lecture that would allow time for only one if the
       calculations were carried out by hand. A similar savings
       applies to students' homework.

   4.  Check assumptions. It is impossible to pay more than lip
       service to checking the hypotheses of a statistical
       theorem without some means of carrying out those checks.
       Any time a technique assumes that the data come from a
       normally distributed population, students should be taught
       to plot the data and see if this is plausible. (You really
       can't tell much unless the data set is much larger than a
       textbook fiction.) Regression residuals (as well as the
       data and the regression line) should always be plotted to
       check the regression assumptions. Indeed, the regression
       assumptions are so complicated and technical, I can not
       imagine how students could possibly understand them except
       in terms of what they say about the residuals. Again, this
       requires realistically large data sets to be worthwhile.
       An analysis of variance should always be preceded by
       boxplots of the groups, all to the same scale. These
       checks would be impractically time consuming to perform by
       hand, and students could only examine a few small and
       probably unrealistic data sets.

   5.  Transformations can be regularly used, as they are in
       practice. This provides students with a partial answer to
       the question of what to do if the data are not normally
       distributed. Students can easily explore the effects of a
       variety of transformations on the same data set.

   6.  More generally, students can try out a variety of analyses
       of the same data set. This enables them to explore and get
       some sense of how data are actually analyzed, and provides
       the statistical equivalent of manipulative materials in
       arithmetic. For common, artificial textbook data sets, the
       mode of analysis is given by the text. Real data never
       arrive with a label telling us what method of analysis to
       use. Exploring real data for which the method of analysis
       is not given helps students to understand the methods
       tried as well as the data at hand, and gives them hands-on
       training in one of the most crucial of all statistical
       skills: selecting an appropriate technique. No one without
       this skill should be allowed to do statistics, by hand or
       by machine.

--


      _
     | |          Robert W. Hayden
     | |          Department of Mathematics
    /  |          Plymouth State College
   |   |          Plymouth, New Hampshire 03264  USA
   | * |
  /    |          (603) 536-2497
 |     )          (603) 535-2489
 L_____/          hayden@oz.plymouth.edu
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU