Score Normalization as a Fair Grading Practice.
by Winters, R. Scott
Course instructors want to evaluate students in a manner that is fair
and based upon the student's representative performance.Discussions of
fair grading practice tend to focus on: grading methodology and individual
assignments (i.e., Glenn, 1998), the determination of an appropriate metric
and clearly articulating expectations to students (i.e., Davis, 1993).
Few guidelines address practical considerations for integrating multiple
assignments (e.g., determining final grades based upon multiple exams written
by different instructors) and the prerequisite statistical methodologies(but
see Cross, 1995). This Digest outlines an appropriate means to handle these
situations in a fair and equitable manner. Included is a detailed example,
based upon real class data, which illustrates the disparity in grade assignment
with and without proper normalization.
ALL SCORES ARE NOT EQUAL
While fair grading is easily understood when discussing a single assignment
(such as an exam or paper) it becomes a more difficult issue when multiple
assignments are considered. For instance, if a student gets a 50 on an
exam that is very hard (hence the 50 is the highest grade among all students),
and a 60 on a second exam that is very easy (hence the lowest grade among
all students), are these exams equitable? If a student is given the option
of dropping the "lowest grade" of the two, does it make sense to drop the
exam that, a) reflects the lowest numerical score (the 50), or b) reflects
poorer performance (the 60)? If we set our evaluation criterion as a performance
measure, then the score reflecting poor performance should be dropped.
However, in order to make such an evaluation, the exams need to be converted
into a common currency; specifically, they need to be placed upon a standard
scale for comparison. Therefore, using raw scores to calculate final grades
may not accurately capture a student's true performance within a class.
As variation in performance evaluation increases, so does the impact on
the student's final ranking.
Ideally, we would like the distribution of individual student performance
for all exams to be equal, despite differences in time, instructor, teaching
assistant, and other factors. Only then can evaluations be considered comparable.
Without this common currency or scale, errors in grade assignment will
result. Fortunately, the methodology for placing diverse assignments on
an equitable scale is straightforward. Appropriate normalization requires
nothing more than adjusting the exams' means to be equal as well as their
variances. If different teaching assistants instruct different subsets
of the class, then these subsets also need to be standardized for equal
means and variances across teaching assistants.
The need for normalization is intuitive to most: an exam with a mean
of 40 is not equitable to an exam with a mean of 70. The obvious correction
is to readjust the scores such that the means are equal; this is a good
first step, but alone, it is insufficient. Equally important is the need
to correct for differences in the variances. A template for making such
calculations is introduced below.
THE NORMALIZATION PROCESS
We begin by converting an individual score into a context-free evaluation
of relative performance. Next, we will transpose this context-free evaluation
into a performance measure (a normalized score) based upon a distribution
that we define (that is, we will dictate what the mean and variance are
to be). In this manner, scores from different evaluations (exams, instructors,
laboratory sections, etc.) can be transposed onto a common scale. When
all of the course's evaluations are based upon the same distribution, they
can reasonably be compared.
The context-free evaluation we will work with is a z-score. A z-score
captures an individual performance relative to the population's mean and
variance. Z=(X-M)/S where: z refers to the z-score, M is the estimate of
the population's mean, S is the estimate of the population's standard deviation,
and X is an individual score within the distribution having mean M and
Since z-scores give us a relative performance measure, then the same
z-score can be derived from significantly different distributions. Thus,
any score from one distribution can be converted into a score for a second
distribution, while maintaining that same relative performance (the same
For any assignment in a class, we know the absolute score for every
student and can estimate the mean and the standard deviation for that assignment
based upon all students' scores. Therefore, we can convert each student's
absolute score into a z-score. With z-score in hand, we can calculate a
new absolute score for any distribution we define. That is, we can declare
a mean and standard deviation we wish the new distribution to have and
then solve for the absolute numerical value that the z-score would take.
This is called the T-score or transformation score. T=m + (s)(z) where:
T refers to the transformed score on the new distribution, m is the target
mean, s is the target standard deviation, and z is the z-score.
WORKING THROUGH AN EXAMPLE-ONE STUDENT
Let us take a specific example of one student's performance on three
separate exams where we intend to drop the "lowest" exam score. The vernacular
of "lowest exam score" is misleading since our true intention is to drop
the grade representing the student's worst performance on any of the three
exams. Table 1 gives the student's grades along with the average and standard
deviation for the performance of all students on each exam.
See Table at End of Digest
Normalization begins by choosing an arbitrary average and standard deviation
for the distribution we wish to set as our baseline. In this example, an
average of 70 and a standard deviation of 15 are selected. In order to
normalize the student's performance on exam 1, we simply fill in those
values that we have. Thus, for Exam 1, the student's z-score is z = (69-58)/
22 = .5 and T = 70+(15)(.5) = 77.5
While the numerical value may have changed, the student's relative performance
(the z-score) has not. A grade of 77.5 within a distribution having an
average of 70 and standard deviation of 15 represents the same relative
performance as a grade of 69 within a distribution having an average of
58 and a standard deviation of 22.
If we were normalizing the grades of an entire class, then we would
use the same equation and change the values for the original grades for
each student in order to obtain each student's normalized grade (T-score).
Performing similar calculations for Exam 2 and Exam 3 generates normalized
scores of 77.1 and 86.67, respectively. Therefore, Exam 2 should be dropped
since the student's performance is the lowest.
WORKING THROUGH AN EXAMPLE-AN ENTIRE CLASS
This example illustrates how final scores for individual students can
change dramatically depending on whether normalization procedures are adopted.
The example is derived from real data for an introductory biology course
taught at a large university and is based upon scores for 205 students.
For each student, there are five grades: three exams, a final, and a laboratory
score. It is the policy of the department that grades be calculated according
to the following criteria:
A. the "lowest" of the three exam scores is to be dropped,
B. each of the two remaining exams is worth the same as the final, and
C. the laboratory score is worth one and one half times any exam (which
represents one third of the course evaluation). Complicating the matter
is the fact that students are pseudo-randomly assigned to one of seven
laboratory instructors. Laboratory instructors vary tremendously in their
knowledge, experience, and difficulty. Finally, two instructors co-lectured
the course and exams were written independently (with the exception of
For simplicity, let us assume that grades are based upon the following
schema: the top 5% will receive an A+, the next 5% an A, the next 15% a
B, the next 50% a C, the next 15% a D, and the last 10% an F. In reality,
a far more complicated method is and should be used that bases an individual's
grade on an absolute score rather than a relative measure such as intra-class
Differences in grade assignment between pre-normalization (raw) and
post-normalization are profound. Approximately 27% of the class (56 out
of 205 students) would have been assigned the wrong grade had the instructors
not normalized the scores. In fact, the grades for 52 students changed
by one letter grade, and 4 students changed by two letter grades. Looking
at one superficial aspect of these dynamics, we note that 37% of students
have a different exam score dropped post-normalization. The effects of
such changes influence the top, more competitive, tiers. Without normalization,
40% of A+ grades are incorrectly assigned and the ranking of the top three
students is incorrect. In fact, the student who performed the best in class
would have been wrongly assigned a B without normalization. More dramatically,
prior to normalization, another student would have incorrectly been considered
average, C, when in fact their work merited an A relative to his or her
The author would like to thank Heather Trobert and Michael J. Balsai
for helpful comments on an earlier draft of this manuscript.
Cross, L. 1995. Grading Students. Practical Assessment, Research &
Evaluation, 4 (8). Available at: http://ericae.net/pare/getvn.asp?v=4&n=8
Davis, B.G. (1993). Tools for Teaching. San Francisco: Jossey-Bass Publishers.
Glenn, B.J. (1998). The golden rule of grading: Being fair. The American
Political Science Association Online, 31 (4). Available at: http://www.apsanet.org/PS/dec98/glenn.cfm
Exam 1 Exam 2 Exam 3
Student's Performance 69 75 72
Class Average 58 66 62
Class Standard Deviation 22 19 9