TESTING MEMO 6: WHAT KIND OF GRADES SHOULD BE AVERAGED?
by
Lawrence H. Cross
Virginia Polytechnic Institute and State
University
Some instructors record letter grades for tests and
assignments, and others record numerical values, often
the percent correct on tests. Later, under either
method, the grades are averaged, often employing a
weighting process designed to make some grades count
more heavily than others. Discussion of the merits of
different approaches usually centers around the question
of whether it is better to average letter or numerical
grades or around some feature of the weighting process.
However, an important characteristic of the grades as
initially recorded is seldom questioned, namely the
variability of the scores of each test or assignment.
Indeed, it is ironic that instructors may go to
considerable trouble to weight grades according to their
perceived importance when, in fact, the result may be
quite different from what was intended, due to failure to
account for differences in score variability from one
test to another.
To see how this outcome could occur, consider a course
in which the midterm examination was much more difficult
than the instructor intended; scores ranged from 35% to 95%
with an average of 65%. Contrary to the advice of TESTING
MEMO 2, the instructor did not view this outcome as desirable
and, with the intention of being fair to the students,
included a large proportion of easy questions on the final
examination. Their presence caused a great reduction in
score variation. Final examination scores ranged only from
88% to 100% with and average of 94%. Only a small number of
harder questions kept everyone from earning very high scores
in a narrow range. The result was that differences from
one student to another in final course averages were
largely attributable to scores on the midterm. Thus, a
student's achievement in the latter part of the course was
effectively devalued, which was hardly fair or in keeping with
the presumed intention that grades reflect achievement across
the entire course.
The best approach to avoiding situations like the one
just presented is to record and average standardized test
scores. In order to calculate standardized scores it is
necessary to know the standard deviation of the scores prior
to standardization. This statistic is a measure of how
"spread out" the scores are and is explained in any
elementary statistics text. Though nearly any program
reporting test results will include this statistic, a
fair approximation for most classroom tests may be obtained
by subtracting the lowest score from the highest and dividing
by 4. In the example above, the standard deviations are about
15 and 3 percentage points respectively for the midterm and
final examinations. A standard score is then the number of
standard deviations the number-right or percentage score is
above or below average. Commonly called a z-score, its formula
is: z = (x - xbar)/s, in which x is the observed score, xbar
is the average score, and s is the standard deviation. Then
scores of 80% and 97% on the midterm and final respectively
would each yield z-scores of 1.0, because both are one
standard deviation above average. Similarly, scores of 50% and
91% would correspond to z-scores of -1.0. It may be difficult
to work with z-scores, because half of them will be negative
and all will probably lie between -3 and 3. Therefore, it
is convenient to transform the z-scores into T-scores as
follows: T = 50 + 10z. T-scores will have a mean (average) of
50 and a standard deviation of 10. Thus, a T-score of 60
represents a number-right score one standard deviation above
the average. If the distribution of scores approximates the
shape of the normal curve, about 16% of the T-scores will be
above 60 and about 10% above 63. Similarly, about 16% of
T-scores will be below 40 and about 10% below 37.
If T-scores are computed for every test, averaging them
will provide a composite score from which the influence of the
variability of the scores has been eliminated. (Strictly
speaking, if more than two scores are to be averaged,
the intercorrelations among the scores should be taken
into consideration in order to control for the degree of
"overlap." However, simple averaging of T-scores should
produce a good approximation of the more precise result.)
T-scores are typically provided for multiple-choice tests
processed by measurement services offices at universities.
Moreover, T-scores can be calculated for any numerically
evaluated non-test assignments you may wish to include
in the course composite. Like other scores, T-scores
may be weighted differentially. For example, if you wish to
weight the final exam twice as much as the midterm, multiply
the T-scores from the final by 2, add the midterm T-scores
and divide by 3.
It should be noted at this point that T-scores report
only a student's relative position in the class and not an
absolute measure of achievement. However, we contend that
the difficulty level of nearly all academic tests is arbitrary
and that, regardless of the scoring method, they provide
nothing more than ranking information. (See TESTING MEMO 2 for
a more complete discussion of this point.) The concern of
this MEMO is that the scores be averaged in a manner
consistent with the instructor's intention.
Finally, when the T-scores have been averaged, there is the
problem of assigning letter grades for the course. Until
this point, we have been able to speak with conviction,
deducing conclusions logically through arguments based
on statistical principles. However, when it is necessary
to determine the dividing line between As and Bs or Ds and
Fs, there is no such clear-cut approach available. Of course,
if student X's average is higher than student Y's, student
Y's letter grade must not be higher than student X's, but
beyond this recommendation our best advice is to inspect
the distribution of average T-scores with the following
questions in mind:
1. What is a typical letter grade distribution for a course
of this type with this kind of student?
2. Are there any circumstances which might warrant
altering this "typical" distribution, e.g., did the course
progress especially well or poorly?
3. Where in the distribution are key students whose work
you know especially well, students you believe might
deserve especially good or poor grades for reasons other than
test performance?
4. Where are naturally occurring "breaks" in the
distribution of average T-scores? (There is no "scientific"
reason for letting these points determine letter grades, but
if their use is not inconsistent with other considerations, it
will help to prevent hard feelings on the part of students
who otherwise might miss a better grade by one T-score point.)
Two ideas to be avoided or at least questioned in
determining letter grades are:
1. That the T-score spread should be the same for each
letter grade.
2. That an equal number of As and Fs, Bs and Ds,
should necessarily be awarded.
Finally, it must be remembered that assignment of letter
grades a cross a range of average scores is essentially
arbitrary and a matter of professional judgement.
For more information, contact Bob Frary at
Robert B. Frary, Director of Measurement
and Research Services
Office of Measurement and Research Services
2096 Derring Hall
Virginia Polytechnic Institute and State
University
Blacksburg, VA 24060
703/231-5413 (voice)
frary#064;vtvm1.cc.vt.edu
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