# Relative impacts of different grade-scales on student success in
introductory physics

David J. Webb  Department of Physics, University of California - Davis
Cassandra A. Paul  Department of Physics & Astronomy - Science Education
Program - San Jose State University  Mary K. Chessey  Department of
Physics,University of Maryland - College Park

###### Abstract

In deciding on a student’s grade in a class, an instructor generally needs to
combine many individual grading judgments into one overall judgment. Two
relatively common numerical scales used to specify individual grades are the
4-point scale (where each whole number 0-4 corresponds to a letter grade) and
the percent scale (where letter grades A through D are uniformly distributed
in the top 40% of the scale). This paper uses grading data from a single
series of courses offered over a period of 10 years to show that the grade
distributions emerging from these two grade scales differed in many ways from
each other. Evidence suggests that the differences are due more to the grade
scale than to either the students or the instructors. One major difference is
that the fraction of students given grades less than C$-$ was over 5 times
larger when instructors used the percent scale. The fact that each instructor
who used both grade scales gave more than 4 times as many of these low grades
under percent scale grading suggests that the effect is due to the grade scale
rather than the instructor. When the percent scale was first introduced in
these courses in 2006, one of the authors of this paper had confidently
predicted that any changes in course grading would be negligible. They were
not negligible, even for this instructor.

††preprint: APS/123-QED

##  I Introduction

In higher education, the role of grades is paramount. Students enroll in
courses, and the grade that they earn in each course indicates the degree to
which the student was successful in that course. Good grades indicate that
students are successful and they are encouraged to continue in that course and
on to more advanced courses in the same topical area. Poor grades on early
assessments in a course may cause students to drop a course. Enough poor
grades can cause a student to fail a course, which can in turn affect the
student’s time-to-degree, their retention in a major, or even in their
retention in college itself. Furthermore, college grade point averages (GPAs)
are important as graduate schools and professional schools have minimum
requirements for applicants, and some employers request GPA and/or transcript
information so course grades influence student career opportunities and
pathways even after graduation. Because the stakes are high, it’s important
that educators take care to construct the meaning behind their grades and also
to understand any implications of chosen grading techniques or philosophies.

Instructors administering grades on many individual assignments and exams must
aggregate these data in assigning a single course grade that best describes
each student’s understanding and/or skill. While instructors can and do use
different factors and criteria when determining a grade [1, 2], two things are
important to achieve in this regard, 1) that the student’s grade is a
meaningful representation of their achievements in the class, and 2) that the
grading process is consistent such that students taking the same course can
expect similar grades for similar successes and failures. Choosing how to
award grades usually involves selecting a grade-scale, or numerical method,
for evaluating students work. Perhaps the most commonly used grade scale is
the percent-based grading scale (hereafter referred to as “percent scale”). A
student being graded using the percent scale earns a 90 for example if 90% of
the work being graded is correct.

Recently, the percent scale has come under criticism [3, 4, 5, 6] and
alternative methods of grading have been introduced. Some of these alternative
grade scales and methods have been shown to be quite successful [7], even when
directly compared [8] to the percent scale. However, despite the wide range of
criticisms [3, 4, 5, 6], there has yet to be significant study of the impact,
on students, of the problematic issues associated with the percent scale. In
this paper, we address this gap in the literature by examining some of the
effects of the percent scale compared to an alternative grade scale.
Specifically, we address common criticisms of the percent scale, and determine
the validity of these criticisms in one particular context as we explore the
similarities and differences in student’s course grades resulting from these
two grade scales.

##  II Percent Scale

The percent scale (shown in Table 1) is perhaps the most prevalent way of
awarding numerical grades to students both in higher education and in the K-12
context in the United States. On the percent scale, 100 is the highest grade
students can earn, zero is the lowest, the numbers represent the percent
correct, and a value around 65% is the boundary between passing and failing
(see Table 1). In general, the basic concept is that the number of points
awarded indicates the percent of the assignment that is ‘correct.’ So a grade
of 85 would mean that 85% of the assignment is evaluated as correct. There is
also a somewhat standard way to interpret this numerical grade as a letter
grade. For example, above 90 is generally a grade of ‘A’, between 80 and 90 is
usually considered a ‘B’ etc. Any grade below 60 or 65 on the percent scale is
typically considered failing, and a zero is often awarded for missing work.
There are slight variations of this scale depending on the instructor and/or
institutional rules, but any scale where a grade is based on percent and
students need to earn more than 50% of the points to earn a passing grade can
be considered percent scale grading. Despite it’s popularity, there are some
notable criticisms of this scale.

Table 1: Comparison between grade and score different grading schemes. The Letter Grade to Percent Scale and 4.0 Scale conversions are copied from the College Board website. CLASP10 is the specific version of the percent scale whose results are discussed in the paper and CLASP4 is the specific 4-point scale used. Letter | Generic Scales | Specific Scales
---|---|---
Grade | Percent Scale | 4.0 Scale | CLASP10 | CLASP4
A+ | 97-100 | 4.0 | 9.67-10 | 4.17-4.5
A | 93-96 | 4.0 | 9.33-9.67 | 3.83-4.17
A$-$ | 90-92 | 3.7 | 9.0-9.33 | 3.5-3.83
B+ | 87-89 | 3.3 | 8.67-9.0 | 3.17-3.5
B | 83-86 | 3.0 | 8.33-8.67 | 2.83-3.17
B$-$ | 80-82 | 2.7 | 8.0-8.33 | 2.5-2.83
C+ | 77-79 | 2.3 | 7.67-8.0 | 2.17-2.5
C | 73-76 | 2.0 | 7.33-7.67 | 1.83-2.17
C$-$ | 70-72 | 1.7 | 7.0-7.33 | 1.5-1.83
D+ | 67-69 | 1.3 | 6.67-7.0 | 1.17-1.5
D | 65-66 | 1.0 | 6.33-6.67 | 0.83-1.17
D$-$ |  |  | 6.0-6.33 | 0.5-0.83
E/F | 0-65 | 0.0 | 0-6.0 | 0-0.5

##  III Criticisms of the Percent Scale

One criticism of the percent scale is the portion of the scale devoted to
failing grades. In a 2013 article, Guskey [3] points out that a larger portion
of the scale is devoted to failure (60%) than success (40%). This means that
failure (grades equivalent to ‘F’) can be measured in 60 different degrees,
while each other letter grade (A, B, C and D) is limited to only ten degrees.
Another way of stating this is that the grade space devoted to ‘F’ is 6 times
larger than that of any other letter grade. That the majority of the scale is
devoted to F is a potentially a philosophical problem; in fact Guskey [3]
asks, “What message does that communicate to students?” But the problems are
mathematical as well because, as discussed by Connor and Wormeli [5] any of
the F-grades below 50 tend to skew an averaging procedure [4, 6].

The amount of the scale devoted to ‘F’ grades is particularly important [3, 5,
4, 6] when considering awarding the lowest grade, a grade of 0%. Zero grades
are often given to students who skip an assignment. While there are
conflicting viewpoints on whether missed assignments should be included in
aggregate grades at all (see Chap’s 14-17 of Ref. [6]), if we take the
viewpoint that there are some instances where a earning a 0% on the percent
scale is warranted, critics point out that it is a very difficult thing for a
student to balance out the effect of even a single zero. Table 2 shows us that
it takes a total of 19 perfect scores to fully erase (i.e. eventually receive
an ‘A’) the impact of a single zero, or a minimum of 6 perfect scores to earn
a B. In previous work [9], we have shown that zeros are more likely to be
awarded to first generation students and to students identifying as members of
racial/ethnic groups that are underrepresented in physics than to other
students. Furthermore, we provided evidence that these zeros are not clearly
indicative of lack of understanding and we noted that use of the percent scale
makes this situation more of a problem for these students than another grade
scale (CLASP4 in Table 1). Research [6, 10, 11] has shown that while good
grades are motivators of good work, poor grades may not always motivate
students to work harder.

Table 2: After receiving score of zero, how many perfect scores (either 100% or 4.5) are needed to achieve the desired average grade? N% is number needed when using a percent scale and N4 is the number needed when using the CLASP4 scale from Table 1). Desired Grade | N% | N4
---|---|---
D | 2 |  $<$ 1
C | 3 | 1
B | 6 | 2
A | 19 | 8

Another criticism of the percent scale is that because there are so many
levels (100) it is prone to inaccuracy and inconsistency. Studies of teachers
suggest that different teachers grades assigned to the same student work tend
to be distributed over a range of width of order 10 when using the percent
scale. This suggests that using fewer than 100 specific grades may lead to
less teacher-to-teacher grade variability.

Finally there are a number of criticisms of the scale that are not
mathematical in nature, for example the fact that low failing grades (grades
below 50% for example) negatively motivate students, or reduce their self-
efficacy [12, 13]. These are also important arguments to consider, but our
quantitative data does not address these issues and so in this paper we are
primarily concerning ourselves with the mathematical implications of grade
scale.

##  IV Alternative Grade Scales and Practices

There are a few different common alternatives to using the percent scale. For
example, in order to mitigate the percent scale issue of devastating zeros,
the concept of “minimum grading” was conceived [5]. Minimum grading is the
practice of raising all very low grades to some ‘minimum’ grade (usually 50%)
so that students are able to recover from missing work, or just really poor
assignment performances. Critics of minimum grading suggest that it may result
in students passing even if they have not learned the material. They argue
that minimum grading promotes student entitlement (they get something for
nothing) and leads to social promotion. Some instructors feel strongly that
it’s unfair to give students 50% if they have completed less than 50% of the
work. There are also concerns that minimum grading contributes to well-
documented grade inflation [14] (a phenomenon that may include a higher
average grade, more students being given ‘A’s, or both). Research on minimum
grading [8] in one school district shows that neither of these things
happened. Carey and Carifio [8] present an analysis of seven years of grading
data collected from a school that implemented minimum grading. They used
standardized test results to show that students who earned at least one
minimum grade actually outperformed students who did not receive any minimum
grades on standardized testing. This was true even though students receiving
minimum grades on average had lower classroom grades. This indicates that
minimum grading does not cause grade inflation and also that minimum grading
still does not entirely make up for the fact that a percent grading scheme
potentially under-measures the performance of the struggling students for whom
the practice of minimum grading was designed to help. Unfortunately, even
though this seems to be a reasonable way to address some concerns of the
percent scale, many instructors don’t like it [4], because they do not agree
with the concept of giving students scores that they don’t believe they have
earned.

Another way to address the percent scale issues is to use the concept of
Standards-Based grading (discussed in Brookhart, et al review [7] and
references therein). This method asks students to demonstrate proficiency in
certain areas, with the instructor providing ordinal grades such as well below
proficiency, approaching proficiency, proficient, and excellent, that describe
the path to proficiency [15]. This approach has been used in college physics
classes previously [16], but, because the method requires students to be able
to have multiple opportunities to attempt the same proficiency, this is
difficult to accomplish with the large class sizes that are typical of
introductory science courses.

A final alternative to the percent scale is the college 4.0 scale which is
typically used to calculate GPA. Each integer in the scale (4, 3, 2, 1, 0)
corresponds to a letter grade (A, B, C, D, F). While many college level
instructors may use the percent scale and then convert to a letter grade
(which has a numeric value tied to the 4.0 scale), others may use a 4.0 scale
from the beginning of the course and then report the numbers in the form of
letter grades. These numbers can be averaged into a single grade. Because the
4.0 scale allocates the same ‘space’ to each letter grade, the scale avoids
most of the issues associated with the percent scale. For example, in Table 2
we see that after a student earns a zero, it only takes 2 perfect scores for
them to earn a grade of “B”, and a total of 8 for them to erase the zero
completed. This is much more doable than the 6 and 19 perfect scores required
by the percent scale to achieve the same goal. This means that a grade of
‘zero’ is less disastrous for students. This scale is mathematically similar
to the practice of minimum grading, and instructors using standards-based
grading also sometimes use the integers on a 4.0 scale to mean different
levels of proficiency [15, 16], so aspects of this scale exist in both of the
other alternatives we describe. Because the 4.0 scale mitigates many of the
criticisms of the percent-scale, and does not have the any new drawbacks like
minimum grading and standards-based grading do, many suggest it as an
alternative to percent scale grading.

##  V Research Questions

The active learning introductory physics course CLASP at UC Davis, has been in
existence since 1995 [17]. Originally, all courses used the same grade scale
described as “CLASP4” in Table 1. This grade scale was based on the standard
college 4.0 scale. After many years, instructors began to move away from the
CLASP4 grade scale, and began to utilize a percent scale instead. Because the
course materials over the years were extremely similar, and some instructors
used both types of grade scale in different sections of the same course, the
circumstances provide for an ideal opportunity to compare the usage of
examples of the two scales. In this paper we consider CLASP10 to be an example
usage of the percent scale, and CLASP4 to be an example of a 4-point scale. We
use this data to compare use of the percent scale, CLASP10 to the 4-point
scale, CLASP4, to examine some common critiques of the percent scale and to
further explore similarities and differences between these two grade scales.

Specifically, regarding the controversies over percent scale grading, we ask:

1) Does the 4-point scale lead to course grades that are “inflated” compared
to percent scale grading?

2) How does the distribution of course grades differ between the percent scale
and 4-point grading?

3) How does the distribution of exam-item grades differ between the percent
scale and 4-point grading?

4) How variable are the course grades for each grade scale by instructor and
by class?

5) How does averaging grades under these different numerical scales compare
with aggregating grades using the median instead of the average?

When considering these questions it’s important to emphasize that our aim here
is to uncover how different grade scales can impact student outcomes. We are
not evaluating the philosophy behind either scale, nor making claims about
what constitutes an “A” or an “F.” Finally we will make no claims about the
connection between assigned grade and student understanding.

In this article we examine 10 years of student grades in a large enrollment
introductory physics college course. Instructors in this course sometimes used
a percent scale (CLASP10), and sometimes used an alternative scale (CLASP4)
based on the standard collegiate 4.0 scale. What we will show is that the
fraction of students failing the course is much larger when instructors use
the percent scale and that instructors assigned more F grades to their
student’s work when using the percent scale. We also find that the increase in
students failing is associated mostly with the grade scale used when
aggregating grades by averaging and not with the increase in individual
assigned F’s. These conclusions seem to be determined by the grade scale
rather than the instructor. Furthermore, we find that that using the percent
scale leads to more class-to-class variability in grade distribution. Finally,
we illustrate how using the arithmetic mean to determine course grades is
similar to using the median under the 4-point grade scale but less similar
under the percent scale.

##  VI Methods

###  VI.1 Setting and Context

When calculating a GPA, most colleges use a 4.0 scale which equates each
integer (4, 3, 2, 1, 0) to a corresponding letter grade (A, B, C, D, F). The
designers of the Collaborative Learning Through Active Sense-making in Physics
(CLASP) [17] curriculum wanted a grading system that was both transparent and
non-competitive, and so they directly linked every single graded item (be it
an exam question, or the exam itself) to a slightly modified version of this
4-point scale so that students could understand how their performance on a
given question related to the expectations of the course instructors. The
resulting CLASP4 grade scale, shown in Table 1, is therefore a version of the
4.0 scale. CLASP4 is a continuous grade scale from 0 to 4.5 with each letter
grade region, except for the F region, centered on the appropriate integer.
Generally, a grade of 4.5 (the highest A+) was earned when the
description/calculation correctly and completely applied an appropriate
physical model to the situation described in the exam.
Descriptions/calculations that were not correct or were not complete were
assigned lower grades with each grade depending on the instructor’s judgment
of the quality of the answer. An answer judged to be excellent but not perfect
was given a score in the A$-$ to A$+$ range between 3.5 and 4.5, an answer in
the B$-$ to B$+$ range was assigned between 2.5 and 3.5, in the C$-$ to C$+$
range between 1.5 to 2.5, etc. Under this grade scale a zero was almost
universally reserved for students who did not answer the question at all, the
top of the F range of grades was 0.5, and grades between 0 and 0.5 were used
for students who gave an answer but whose answer showed almost no familiarity
with or understanding of the subject. Multi-part exam problems often had a
grade for each part (sub-exam level grades) and these were averaged, with the
weight per part determined by the instructor, to determine the exam grade.
These exam grades were then averaged, with weight per exam determined by the
instructor, to give the course grade on the same 4-point scale. From 1995
until 2005 essentially all CLASP instructors used this same basic grading
method for quizzes, exams, and the class grades111Homework was generally not
graded and there are no homework grades in these databases..

In 2006 several instructors began experimenting with a 10-point grade scale,
CLASP10, that was just a re-scaled version of the standard percent scale. With
the CLASP10 grade scale, answers in the A$-$ to A$+$ range were given grades 9
to 10 (instead of the 3.5 to 4.5 of the 4-point scale), B$-$ to B$+$ range
were given grades 8 to 9, etc. for C and D grade ranges. Again the zero of
this grade scale was reserved for students who did not answer the question but
now the highest F grade given is 6.0. The result is that, relative to CLASP4
scale, CLASP10 had a much larger grading measure available for F’s (0-6) even
though the other grades have the same measure on each scale. For these reasons
we will generally refer to CLASP10 as a “percent scale”. The grades on a
single exam problem were then, as under CLASP4, averaged to give the exam
grade and the exams were similarly averaged to give a course score which then
determined the course letter grade.

###  VI.2 Data Set

Data were collected from the first two quarters of the three quarter CLASP
series from course archives spanning the ten years 2003-2012. (The “CLASP A”
content primarily covers energy and thermodynamics, while the “CLASP B”
content focuses on mechanics.) During those years the structure and content of
these classes was relatively constant. Over 75% of a student’s time in one of
these classes was spent working in discussion/laboratory sections (referred to
as DLs in the CLASP curriculum) on activities that only changed slowly over
those years. The platform for entering grades in this large enrollment class
was centralized using a separate database file for every separate course
offering. These course databases include exam scores for each student along
with the individual grades that were given as well as the calculations that
led to these exam scores. In addition, these databases sometimes included the
calculations that led to the actual course grades. Over these ten years there
were 133 of these classes, and we have found databases for 96 of them that are
identifiably graded as described above using either CLASP4 grading or percent
scale grading. This identification was determined by examining the maximum
grades given to individual student answers. If the maximum was always 10 then
we considered the class to have had percent grading and if the maximum grades
were always 4.5 then we considered the class to have had CLASP4 scale. The
resulting database contains 773,667 sub-exam level grades given to 15,757
individual students on each part of each exam. Fifty-seven of the included
classes are CLASP4 grade scale (including 478,617 grades on individual
answers) and the remaining 39 are percent grade scale (including 295,050
grades on individual answers). We also have access, from UC Davis
administration, to the recorded course grades from all of these classes as
recorded by the Registrar. Course grades were all recorded on the standard
4-point grade scale with letter grades A through D, each of which may include
a $+$ or $-$, and F. In each database that included the calculation of course
(letter) grades these letter grades were determined using the cutoffs
approximately as shown in Table 1. Although we will group together the data
from the first two courses (CLASP A and CLASP B) in this series of courses
because the differences between the grade scales show up in both courses, we
will specifically note each situation where the data from one course differ
substantially from that of the other.

Each of our research questions requires a different set of comparisons to
make. So rather than providing a list of justifications for the comparability
of each set in this section, we instead share this information when it can be
considered alongside the research question and resulting comparison.

##  VII Analysis & Results

Throughout our analysis we compare the percent scale, CLASP10, to the 4-point
scale, CLASP4, used by the instructors of CLASP. In addition to course grades
we will also be reporting on individual scores that instructors gave to
student answers on exams. These scores will always be referenced by the grade
range in which they are contained. For instance, a 3.8 given to a student
answer under CLASP4 grading will be considered to be in the A region just the
same as a 9.3 given to a student answer under CLASP10 grading. We treat
borderline grades (i.e. a 3.5 under CLASP4 grading or the equivalent 9.0 under
CLASP10 grading) as being half in each of the bordering letter grade ranges.
Most counts, averages, standard errors, etc. were calculated in Excel and most
are double-checked with STATA software. We used STATA for the standard
statistical tests, calculations, and regressions. The error estimates we give
will be standard error of the mean or propagated from standard errors unless
otherwise noted.

###  VII.1 Percent scale fails more students

When one separates the classes taught according to grade scale, either 4-point
or percent scale, a trend in student fail-rates is undeniably present. Figure
1 shows the fraction of course grades given that are less than C$-$ as a
function of the year. We choose a cutoff of C$-$ to measure because UC Davis
allows students receiving less than a C$-$ to repeat a course whereas those
with C$-$ or higher cannot. A similar way of noting that C$-$ is an important
cutoff is that courses graded Pass/Not-Pass give the Pass grade only to
students who would have received a grade of C$-$ or higher. Therefore, the
figure shows the fraction of students who have a grade considered low enough
to warrant repeating the course. From the averages shown on the figure we see
that instructors using the percent grading scale gave $5.3\pm 0.4$ times as
many grades lower than C$-$ than instructors using a 4-point scale. This
fraction was $4.5\pm 0.5$ in the first course in the series (CLASP A) and
$6.9\pm 0.9$ in the second course (CLASP B) so the two courses both showed
this grade scale effect. A chi-square test for the entire data set shows this
difference between grade scales is significant,
$\chi^{2}(2,N=22,865)=513.4,P<0.001$. Of course, it’s conceivable that this
difference is primarily an issue of student academic performance rather than
of grade scale used. To explore this possibility we will use i) the variable
$EnterGPA$, a student’s GPA upon entering the course, as a predictor of that
student’s academic performance and ii) the variable $GrScale$ (the grade
scale) as a categorical variable in logistic regression models predicting the
odds of a student receiving a grade less than C$-$. The logistic regression
model including both of these variables is

$ln[odds(CourseGrade<C-)]\\\ =b_{0}+b_{1}EnterGPA+b_{2}GrScale$ (1)

where $e^{b}$ is the appropriate odds ratio (note that the odds of receiving a
grade less than C$-$ is equal to the probability of receiving a grade less
than C$-$ divided by the probability of receiving a grade of C$-$ or higher).
First, leaving out $GrScale$ and including only $EnterGPA$ as a predictor of
the odds of receiving $<$ C$-$, we find an odds ratio for $EnterGPA$ of
$=0.095\pm 0.009$ $(z=-24.9,N=20,950,P<0.001,PseudoR^{2}=0.12)$. This means
that an increase in entering GPA by 1 (e.g. from 2.5 to 3.5) lowered the odds
of receiving a grade $<$ C$-$ by over 90%. Now we include the categorical
grade scale variable $GrScale$ along with $EnterGPA$. The odds ratio for
$EnterGPA$ changes to $0.079\pm 0.008$ and we find that a student graded under
a percent scale had $6.5\pm 0.6$ times higher odds of receiving less than C$-$
than the same student under 4-point grading ($z=-19.9,N=20,950,P<0.001$ for
the variable $GrScale$ in this model) and $PseudoR^{2}=0.195$ for this model.
We conclude that student academic performance does not explain the large
fraction of students with these low grades under percent grading. Finally, we
should point out that the student withdrawal/drop rates are essentially
independent of grade scale ($0.76\%\pm 0.08\%$ under 4-point grading and
$0.74\%\pm 0.09\%$ under percent grading) in this set of classes. What isn’t
shown in these data is why more students were failed when the percent scale
was used. In the remainder of this paper we analyze factors that contribute to
this phenomenon.

Figure 1: Fraction of students receiving a course grade less than C$-$ as a
function of year they took the course. The two main grade scales used in the
course are shown separately. The error bars are $\pm$ standard error.

###  VII.2 Is Grade Inflation Happening?

Because instructors select either the percent scale or the 4-point scale, it’s
important to consider the instructors also in this analysis. One possibility
is that instructors using the 4-point scale give generally higher grades to
their students’ answers than those given by instructors using a percent scale,
a situation commonly referred to as “Grade Inflation”. If 4-point scale
instructors were simply inflating grades, we would expect to find fewer low
grades but we would also expect to find more high grades under 4-point grading
as well as a higher average grade.

Figure 2 shows the complete course grade distribution for both grade scales.
One possible sign of grade inflation in 4-point classes, more high grades, is
easily seen to be missing. Instead, we find that students in percent scale
classes are the ones receiving more A’s (about 20% more than their peers in
4-point classes).

To look for a shift in the average grade, we computed the average course grade
given by each grade scale using the UC Davis method for calculating GPA
(A=4.0, A$-$=3.7, B+=3.3, B=3.0, etc.) except that we use A+=4.3 rather than
the UC Davis A+=4.0. This amounts to choosing, for each course grade, a value
roughly in the middle of the relevant CLASP4 range shown in Table 1. We find
the average grade given to a student graded under the percent scale was 2.846
(SD = 0.89) and under the 4-point scale was 2.915 (SD = 0.67) for a grade
shift of 0.069 (0.01). These average grades (both between a B and a B$-$) are
shown in Figure 2. The effect size of this grade shift is about 0.088 so it is
a small effect. The difference of 0.07 GPA units is less than half of the
class-to-class variation for either grade scale (standard deviation, over the
individual classes, of average class grade is 0.19 for 4-point classes and
0.34 for percent scale classes). A t-test of the two distributions shows that
this small difference in the two average grades is statistically significant
($t=6.6,df=22863,P<0.001$) and including student GPA as a covariate does not
change this conclusion. At this point we should note that the two courses that
we have grouped together here showed different results for this comparison
when considered individually. The first-quarter course (CLASP A) had
essentially the same average grades for the two grade scales ($2.858\pm 0.008$
for 4-point grading and $2.874\pm 0.012$ for percent grading). The second
quarter course (CLASP B) had a lower average under percent grading ($2.935\pm
0.009$ for 4-point grading and $2.757\pm 0.014$ for percent grading). So there
is conflicting evidence for any simply defined “grade inflation.” We find that
students graded using the percent scale are more likely than students graded
using the 4-point scale to earn “A” grades, but that they are also more likely
to fail the class, and these two differences combine in a way that the average
grade in courses graded using the 4-point scale is statistically significantly
higher but that the difference is small. When we examine the distribution of
grades for both courses in figure 2, we see evidence that suggests fewer
students fail under the 4-point scale not because the distribution of course
grades under that grade scale is shifted uniformly toward higher grades but
that the course grade distribution under 4-point grading was narrower than it
was for percent scale grading.

Figure 2: Fraction of course grades given between 2003 and 2012 for each of
the two major grade scales. Error bars are $\pm$ standard error. The average
grade is between B$-$ and B for each grade scale but the distribution for the
4-scale classes is notably narrower.

###  VII.3 Instructor Use of Grade Space

Since very low grades given under percent scale grading have a much larger
effect on course grade than the lowest grades given under 4-point scale
grading (Table 2), we might expect that this is the main difference between
the two grade scales and accounts for the difference in the student fail-
rates. However, there is another clear difference in our data that can also
lead to more failing grades. We find differences in how instructors allocate
grades on the individual answers given by their students on exams that differ
between the two scales.

First, we note that students who leave a problem blank (or mostly blank) on an
exam receive a zero, a grade that involves no instructor judgment of
understanding or skill. Figure 3 shows the fractions of scores given on
individual exam items for all 96 courses. We see that there is very little
difference between the number of zeros earned by students under each scale.
However, the fraction of F-grades that are not zero is distinctly dependent on
the grade scale. Figure 3 shows that instructors using a percent scale are
considerably more likely to judge individual student solutions on exam items
as non-zero F’s than those instructors using the 4-point scale. This amounts
to shifting about 14% of the entire exam item grade weight from higher grades
under 4-point grading down to non-zero F’s under percent grading. Notice that
none of this shifted grade weight comes from the A’s, which were actually more
common under percent grading and likely little of it from B’s because the
total fraction of (A’s + B’s) is about the same for the two scales, 56.4% for
percent grading and 56.9% for 4-point grading. In order to analyze how these
grade shifts affected individual students and whether they might be due to
student academic performance, we compute the fraction of non-zero F’s for each
student. Averaging this student-level number over all students gives
$0.0364\pm 0.0004$ under 4-point grading and $0.1636\pm 0.0014$ under percent
grading. Cohen’s d for this difference is 1.3 so this is, not surprisingly, a
large effect. The student-level distribution of non-zero F’s is,
unfortunately, both non-normal and heteroskedastic but we can still use
student GPA, $EnterGPA$, in a linear regression to see if it affects the extra
fraction of non-zero F’s seen for percent grading. The regression model we use
to model a student’s fraction of non-zero F’s is

$FractionOfNonZeroFs\\\ =b_{0}+b_{1}EnterGPA+b_{2}GrScale$ (2)

First, leaving out $GrScale$ and using only $EnterGPA$ to predict the exam
item fraction of F’s with $N=20,837$ gives, coefficient for
$EnterGPA=-0.056,t=-35,P<0.001,R^{2}=0.056$. Including $GrScale$ with
$EnterGPA$ in our model we find that the percent scales had $12.6\%\pm 0.1\%$
of extra grade weight in non-zero F’s compared to 4-point scales,
$t=99,P<0.001$ for the variable $GrScale$, and $N=20,837$ and $R^{2}=0.36$ for
this model. Controlling for student GPA did not change the fraction of shifted
grade weight very much so we conclude that the extra non-zero F’s under
percent scale grading are not due to differences in the students in these
courses. Finally, we note that the two courses that we have grouped together
here had different amounts of grade weight shifted into non-zero F’s. In the
first course of the series (CLASP A) $8.5\%\pm 0.2\%$ of grade weight was
shifted into nonzero F’s under percent grading and in the second course of the
series (CLASP B) that number was $18.6\%\pm 0.3\%$.

Figure 3: Fractions of grades given to student answers on individual exam
problems under each of the two grading regimes, 4-point scale and percent
scale. The standard errors are quite small (range from $\pm$ 0.03% to $\pm$
0.12%) and so are not shown. Nevertheless, the choice of how to distribute the
grades assigned at the boundaries does affect the totals. The largest effect
is for non-zero F’s where percent scales have an extra weight (above 4-point
grading) of between 13.3% (border grades counted in higher category) and 14.4%
(border grades counted in lower category) .

###  VII.4 Individual Instructors’ Results

One possible explanation for the difference between the failing rates of the
two grade scales is that different instructors choose the scale that serves
their interest better. Therefore, it is useful to address any selection
effects the choice of grade scale may have. Of the 60 instructors involved in
these courses over the 10 years in our data set, seven instructors used both
the 4-point scale and the percent scale at various times. This gives us seven
comparisons between the two grade scales where an instructor222Each non-summer
course has two instructors, both of whom are responsible for the grades given.
is held constant. Table 3 shows that these seven instructors gave between 4
and 12 times more course grades less than C$-$ under percent grading than
under 4-point grading with an overall average of $5.4\pm 0.5$ times as many
grades less than C$-$, a number very similar to the overall result described
previously. In addition, each of these instructors had between 4% and 20%
extra exam-item grade weight in the non-zero F region under percent grading
than that particular instructor had under 4-point grading. These numbers are
modified only very slightly if we attempt to account for class-to-class
differences in the student academic performance by using their incoming GPA as
a covariate. Specifically, the largest shift, caused by controlling for
student GPA, in Extra F Weight is from 0.12 to 0.11 for Instructor 4 and,
overall, we still find, 4% and 20% more sub-exam-level grade weight in the
non-zero F region under percent grading. These data are evidence against
either instructor or students as causes of the effects shown in figures 1, 2,
and 3.

Table 3: Seven instructors used both grade scales at various times. N’s are the numbers of students taught under the particular grading regime. The Fail Ratio is the fraction of that instructor’s students with course grades less than C$-$ under percent grading divided by that fraction under 4-point grading. Extra F Weight is the extra fractional weight of nonzero F’s that the instructor gave to individual student exam answers under percent grading above what they gave under 4-point. Standard errors are in parentheses.  Instructor | N4 | N% | Fail Ratio | Extra F Weight
---|---|---|---|---
1 | 2824 | 1700 | 7.5 (2.0) | 0.099 (0.005)
2 | 1472 | 952 | 7.7 (1.7) | 0.158 (0.004)
3 | 1920 | 267 | 4.1 (1.1) | 0.043 (0.004)
4 | 591 | 509 | 4.2 (1.2) | 0.122 (0.006)
5 | 3558 | 754 | 4.5 (0.9) | 0.163 (0.004)
6 | 662 | 297 | 6.7 (5.4) | 0.076 (0.004)
7 | 1273 | 258 | 12 (4) | 0.196 (0.008)

###  VII.5 Variability of the Percent Scale

Figure 1 shows that the fail-rates for the instructors using the percent scale
are more variable by year than the 4-point scale. We can quantify this on a
course-to-course basis by examining how the fail rate (i.e. fraction of
students with grade $<$ C$-$) varies over the courses offered. Figure 4 has a
kernel density plot of course failure rate for each grade scale. One finds not
only that the failure rate is larger for the percent scale courses but that it
is also much more variable over courses. It is possible that the students
themselves were more variable in percent scale courses and we can check this
using the student’s incoming GPA’s to find the distribution over classes, of
the class-average GPA, for each grade scale. For 4-point classes we find a
class-GPA averaged over classes of 2.97 with a standard deviation over classes
of 0.08 and for the percent scale classes we find a class-GPA average of 2.99
with a standard deviation of 0.08. A t-test of these class-average GPA’s gives
$t=1.1,df=93,P=0.26$ so the students under the two grade scales are, in terms
of average GPA, indistinguishable from each other. We can also examine whether
the fraction of students failing in a class varies with the class-average GPA.
We find the linear regression coefficient is not significant for either grade
scale (P = 0.74 for 4-point scales and P = 0.16 for percent scales). So the
variability in failing fraction is not obviously associated with an underlying
variability in the students’ academic performance. Finally, since both of the
distributions in Figure 4 include classes at the lower bound of zero students
failing, we might worry that being pushed against this lower bound has
artificially narrowed the distribution for the 4-point scale, a floor-effect.
We check this by using a higher grade cutoff. A cutoff of C still had classes
from each scale with zero students below cutoff, so we use a cutoff of C+ and
find that percent scale classes had an average fraction of 0.22 students with
lower than C+ with a standard deviation over classes of 0.12 and 4-point
graded classes had an average fraction of 0.12 student with lower than C+ and
a standard deviation of 0.07. The percent scale classes are more variable
under this measure also.

If the variability in fail-rates is not associated with the students then
maybe it is an inter-instructor effect where each instructor tends to fail
about the same number of students in each percent course but that different
instructors fail very different numbers of students. We can turn, again, to
instructors who used both 4-point and percent grading at various times during
these 10 years. Figure 5 shows the fail rates for two of these instructors.
Both of these instructors have considerable variation of the fail-rates in
their classes. The fail-rates in their 4-point courses have much smaller
variation than the fail-rates in their percent courses. Unfortunately, only
four instructors taught at least two courses under each grade scale but each
of them had a larger spread in fail rates under percent grading than under
4-point grading (the smallest increase in the spread of fail rates was a
factor of 1.7 higher for percent scales and the highest was a factor of 5.5
higher). So, the variability is not obviously an instructor selection effect.

Figure 4: Kernel density distribution of course failure rates for each of the
two grade scales. The percent scale has a much more variable failure rate
including a small weight extending to failure rates over 30%. Figure 5: “Fail”
rates for each class taught by two of the instructors identified in Table 3.
The classes are time-ordered so 1 is the first class taught by this instructor
in the 2003-2012 window of time. It is interesting that each instructor had
more widely varying “fail” rates under percent grading.

###  VII.6 Separation of grade weight effects from extra F’s

From Table 2 we see that the percent scale requires 3 perfect grades of a
student wanting to raise the lowest F-grade (i.e. a zero) to a straight C.
Similarly, we note that percent scale F-grades of 25 and 50 require 2 and 1
perfect grades respectively to average to a C. The highest percent scale F
grades average in much the same way that F’s do with a 4-point scale. Since
the lower F-grades require more perfect grades to cancel them out, we can
think of them as F’s that carry more effective “weight” than any F from a
4-point scale. These “weighty” F’s lead to the “skewing” of average grades
discussed by Connor and Wormeli [5].

As noted earlier, some of the class database files that we have access to not
only have all grades recorded but also include all of the calculations that
led to the course grades. Courses offered in 2008 and later years do not have
all of these calculations but eight of the thirteen percent graded classes
offered 2006-2007 do have complete sets of grades and grade calculations.
These eight classes gave 1839 course grades to 1272 students (567 students are
in two of these particular classes). The eight classes give us a way to
separate the effects of the heavy weighing of low F’s from the effects of just
giving more F’s on the individual exam answers. Without changing the number of
F’s (or any other grade) that were given on exams in these eight classes we
map the CLASP10 percent scale grades onto the CLASP4 scale as follows: i) for
all grades larger than 6.0 we subtract 5.5 from the grade and ii) in order to
be very conservative in our treatment of the F’s we set all original CLASP10
grades less than or equal to 6.0 to a 0 in our 4-point re-scaling (that is,
all original F’s are set to 0). We then do all the original weighted averages
of individual grades into exam grades and exam grades into course grades where
point cutoffs determining the letter grades are gotten by subtracting 5.5 from
all of the original percent scale grade cutoffs. The original percent graded
courses had a fail rate of 8.3% $\pm$ 0.6% and the 4-point re-scaled courses
had a fail rate of 1.3% $\pm$ 0.3%. The 4-point rescaled fail rate of 1.3% is
consistent with the rest of the 4-point graded classes (see Figures 1 and 4)
and the increase of a factor of 6.5 $\pm$ 1.4 in the fail rate under percent
grading is consistent with both the overall ratio of 5.3 and the individual
instructor ratios given in Table 3. These overall consistencies from a simple
rescaling onto a 4-point scale suggests that the heavy effective weight that
the low-F grades carry is the main factor increasing the failure rates and
that the extra F’s assigned by instructors using the percent scale for
individual exam answers aren’t the main difference between the two course
grade distributions.

###  VII.7 Aggregating Grades - Mean vs Median

An alternative to using the mean to aggregate student grades is to use the
median [18]. This is perhaps most straightforward with letter grades because
they are explicitly ordinal but could also be chosen for numerical grades that
are referenced to letter grades when they are given by an instructor whose
grading philosophy is that the grades they give are ordinal in nature. For the
CLASP courses the numerical grades are chosen from a continuous number line
and were always averaged but these individual grades are ultimately connected
to letter grades so one could argue that taking a median might be a reasonable
choice here in constructing course grades. In fact, the argument that it is
best to consider even numerical grades to be ordinal and so use the median to
aggregate grades has already been made by others [19]. For our grade data an
interesting similarity between the two grade scales is that the overall median
of all of the grades given to individual student answers under 4-point grading
was 3.0 (exactly middle B) and the overall median under CLASP10 was 8.5 (also
exactly a middle B). It may help better understand the differences between the
two grade scales to use the grade data we have discussed in this paper to
compare, at the student level, the two methods, median and average, of
determining course grades.

The actual course grades given during the 10 years of these data are always
made up of complicated weightings of the individual grading judgments. This
complicated weighting can even be student dependent because instructors would
almost always drop each student’s lowest quiz score, give the final exam more
weight if the student performed better on that exam than on exams during the
term, and adjust some student’s grades slightly depending on their performance
in their discussion/lab section. This makes it impossible to devise a unique
process of using the median function to determine a final grade that can be
directly compared to the actual final grades. Nevertheless, it may be useful
in understanding the similarities and differences between the two grade scales
to just construct a straight unweighted average of each student’s individual
grades and then compare those averages with a similarly unweighted median of
each student’s grades. We find that 49 of the courses graded under 4-scale
grading (11,708 students) and 24 courses under 10-scale grading (5,862
students) have a complete set of grades and so can be used in this way.

Figure 6: The unweighted average of each student’s set of grades is plotted
against the unweighted median of these grades for each student from the
classes for which all grade data were present. Both grade scales are shown. In
each graph the straight line represents average = median and the curved line
is a fit of all the grades to a quadratic polynomial done just to show the
data trend. The low grade region of the CLASP10 grade scale (the percent
scale) is cutoff so that the A, B, C, and D ranges of the two scales occupy
the same distance along each axis.

Figure 6 shows each student’s unweighted average grade as a function of their
unweighted median grade for both grade scales. The relationship between an
average grade and a median grade, for a set of grades that have both an upper
bound and a lower bound, is evident in the figure. Students with many grades
at the upper bound can have a median grade equal to the upper bound but any
other grades they have will lower their average grade and exactly what other
grades they have will determine how much lower their average grade is, so
average grades will tend to be lower than median near the upper bound.
Conversely, average grades will tend to be higher than median when the median
falls near the lower grade boundary. The result of those two general features
is that average grades and median grades must be similar for some region in
the middle of the grade space. For the grades given to the students in these
courses, the average approximates the median in the B- to B region of grades
under 4-point grading and at the F to D crossover region of grades under
percent grading. This means that under percent grading the region of grades
that includes 92% of student grades (median grade $\geq$ D) will tend to give
lower grades when the average is used than when the median is used.

Another effect that is obvious from Figure 6 is that, for any specific value
of the median grade, percent grading gives a much larger spread in average
grades. The standard deviation in the distribution of average grades is
roughly twice as large for percent as for 4-point grading for any particular
letter grade, determined by the median, from D$-$ to A$+$. So, in the courses
we are considering here, percent scale grading had both a larger systematic
shift, of the average from the median, than 4-point grading and a larger
random spread around that systematic shift.

##  VIII Discussion

Considering only the mathematical characteristics of each scale, it is perhaps
not surprising that more students fail under the percent scale. After all, if
lower grades are given more weight, it’s likely that more students will fail.
That said, there are several nuances in the overall grade distributions of
each scale uncovered by our analysis that are worth considering by instructors
who are considering using either scale.

The above results confirm many of the critiques of the percent scale discussed
in Section III. A complete understanding of the grade scale entails
understanding these critiques so that the percent scale can be used
consistently and effectively. In fact, many instructors are aware of some of
these critiques, and adjust percent scale grades accordingly (for example,
they might consider a lower grade than 65% passing or they might add some
number of points to all students grades to increase the class average.) The
intention of this discussion is to discuss the nuances of the use of the
percent scale in one particular context in order to bring these
characteristics attention so that they may be discussed both by the research
community and instructors considering their grading philosophy.

###  VIII.1  Considering Partial Credit

If a student does not complete a problem correctly but shows some small part
of understanding, instructors will often award an accordingly small amount of
“partial credit.” Figure 3 shows that a very large portion of grades earned on
exam items graded using the percent scale is devoted to “Non-Zero F’s.” In
fact, more students earn “Non-Zero F’s” on the percent scale than any other
individual grade besides “A”. Mathematically, there is a big difference
between averaging in a 40% and a 10%, but does the instructor see a
correspondingly meaningful difference between these two different grades for
failing? Perhaps, but is this the difference between these two failing grades
equivalent to distinguishing the differences between a grade of D (65%) and an
A (95%) which are also 30 points apart? If an instructor thinks that the
numerical grades they assign are actually interval in nature, rather than just
ordinal [20, 18], and they are averaging the individual grades to determine
some aggregate grade, then the same consideration as to whether to award a D
or an A should go into determining whether a student earns an F (10%) or an F
(40%). Specifically, it seems important that instructors avoid the mind frame
of awarding percents lower than 50% and thinking that this is giving the
student partial credit if they consider a grade of roughly 60% to be the
border between passing and failing. For example, while 30% is obviously better
than a zero it is still 30 points below failing, which is the same
mathematical difference between an A and an F. The data in our paper show the
collective effects of the very low F-grades of the percent scale. And the fact
that each instructor who used both grade scales gave more F’s suggests that
the grade scale itself might affect a teacher’s grading judgments in cases
when a student’s answer does not show much understanding.

###  VIII.2  The Meaning of a Zero

The effect of awarding a zero is greater in the percent scale than in the
4-point scale, as shown by Table 2 and therefore contribute to the grade
weighting issues discussed in section VII.6. When a student leaves an exam
problem blank for any reason, the instructor often awards them a zero for
this. The instructor’s justification for this is certainly logical in the
sense that the student has provided zero evidence of understanding, and
therefore has earned 0% of the possible points. The instructor might also be
using the zero for a motivational purpose in the sense that students need to
complete the work in order to earn points. However, many studies [11, 21, 22]
suggest that this simple view of motivation is unwarranted. In addition, as
seen in Figure 3, our data set shows that even though a zero carries much more
weight on the percent scale than on the 4-point scale, the overall fraction of
blank problems remains essentially constant. This is tentative evidence in
support of the fact that the number of blank responses is not affected by
choice of grade scale in this course and possible additional evidence against
the motivational use of a zero. Furthermore, leaving a physics problem blank
is a behavioral trait that may be more common [9] for women, students
identifying as underrepresented minorities, and first generation college
students. Therefore the practice of awarding zeros for missing work may well
contribute to achievement gaps.

As pointed out by many authors [23, 6], a zero that is earned because a
student did not complete an assignment is not a measurement, it’s actually
missing data. In previous work [9] we showed that the number of problems left
blank by a student is poorly correlated with other metrics of understanding,
so that leaving a blank is by no means predictive of that student’s overall
understanding of physics. Averaging in zeros for missing data would be a
terrible practice in one’s research and so we might consider alternatives to
this practice should this concern us in our teaching.

###  VIII.3 Valuing the Instructor’s Evaluation

In figure 3 we showed that the instructors using the percent scale gave more
non-zero “F” grades than instructors using the 4-point scale and, using Table
3, argued that these extra F’s were due to the grade scale rather than the
instructor. However later, in Section VII.6, we argued that the reason that
more students failed the course when graded using the percent scale as
compared to the 4-point scale, was due to the mathematics of averaging percent
scale grades, and NOT because those instructors actually gave more “F” grades
on individual exam items when using the percent scale. In some ways this
second point seems to temper the first point but the two effects should really
be considered together. For example, regarding the second point, an instructor
could argue that giving a very low F grade and using it in an averaging
process that gives it a large effective weight is entirely appropriate because
that is what their student “earned.” However the first point, that the percent
scale seems to have guided seven out of seven instructors into giving more of
these F grades, might give this instructor pause regarding their own ability
to provide impartial absolute grading judgments of these poor student answers.
The set of low F grades which average with higher effective weight are exactly
the grades that may be caused by the grade scale itself and so be particularly
hard to be confident about.

We further characterized the student-level skewing effects of these low F
grades in Section VII.7. In Figure 6 we used unweighted averages to show how
the percent scale might allow for students to earn average grades that are
consistently lower than their median grade and sometimes much lower. Of course
the actual instructor weightings (like dropping the lowest grade) would likely
reduce the scale of some of these effects but will also reduce similar but
smaller effects of the 4-point scale and are not likely to change our
conclusions that the low F grades available to instructors using the percent
scale i) can have very large effects and ii) may only have been given to a
student because of the grade scale used.

###  VIII.4 Considering Variation at the course level

Beyond the student-level effects discussed above, there also appear to be
instructor-level/class-level effects associated with the skewing effect of the
low F grades in a percent scale. We have already seen that the effective
weight of these low F grades leads to the increase in the fail rate so the
class-to-class fail rate variability that we saw in Figure 5 must be caused by
an underlying class-to-class variability in the number of these low F’s and/or
various instructor’s responses to the low average grades they see in front of
them as they determine course grades for their students. If there are a set of
possible grades that carry extra weight when averaging then the actual grades
given to the students in that class may well be very sensitive to exactly how
those powerful grades are used during the term with this sensitivity giving
rise to the class-to-class variability.

##  IX Conclusions

In summarizing our results we again emphasize that we are not casting judgment
on use of the percent scale in general or any grading practices in particular,
but instead argue that it is essential for instructors to consider the biases
of the percent scale when planning their course for the semester, so that they
can ensure that their teaching philosophies match their grading philosophies.

The primary purpose of these analyses is to fully understand the impact of the
percent scale as compared to another somewhat commonly used scale, the 4-point
scale. In fact, even knowing these results, one of the authors has chosen to
continue using a version of the percent scale in their small graduate courses.
In this use case, work that does not meet expectations earns no lower than
60%, but zeros for missing assignments are used to ensure it’s not possible to
pass the course without completing the assignments. This example is shared NOT
as an example of an exemplar use of the scale (we do not have such data to
support such a claim), but rather to emphasize that the intention of this
paper is not to discredit the percent scale, but rather to expose some
characteristics of the percent scale that may have previously avoided
consideration due to its widespread use.

With respect to our research questions, we draw the following conclusions.

RQ1) The 4-point scale does not appear to inflate grades in the traditional
use of the word because the average grades of the two scales are close and
students are actually over 30% less likely to earn A grades in courses using
the 4-point scale.

RQ2) Although the average course grades under the two scales are close, the
width of the distribution is much larger under percent grading. This led to
many more students receiving failing course grades when the percent scale is
used as compared to the 4-point scale. We found that the odds of failing is
over 5 times higher under percent scale grading, $P<0.001$. The overall fail
rate varies by instructor and by individual class for the percent scale with
an average fail rate of about 8%. Nevertheless, each instructor who used both
scales at various times failed, overall, at least four times as many students
under the percent scale.

RQ3) Instructors tend to give out more “F” grades on individual exam items
when using the percent scale than when using the 4-point scale ( 13% to 14% of
the entire grade weight was shifted down into the F-region under percent
grading, effect size = 1.3). We have not seen this sort of effect reported in
the literature. However, the number of extra F’s is not found to be the main
contributor to the higher course fail rate under percent scales. Rather, the
extra “effective weight” of these low F’s in the averaging process is the main
contributor. The grade scale is more important in determining the fail rate
than the instructor.

RQ4) The percent scale has a much more variable fail rate when compared to the
4-point scale. The class-to-class variation of the fail rate under percent
grading is over seven times higher than under 4-point scale grading even
though the variation in the students was negligible.

RQ5) The two grade-aggregating methods, mean and median, gave results more
consistent with each other under 4-point scales than under percent scales.
Under percent scale grading 88% of students had unweighted average grades
lower than their unweighted median grade and 12% had higher averages.
Conversely, for 4-point scales 56% of students had lower unweighted averages
than unweighted means and 44% had higher averages. In addition to the
systematically lower averages under percent grading, the distribution of
average grades for any particular median grade was about twice as large under
percent grading as under 4-point grading.

These results are derived from a data set that comes entirely from two
(sequential) courses offered over a period of ten years by one department at a
single institution. This course has a fairly low fail rate regardless of which
scale is used. Findings would likely vary across institutions. While we have
made every effort to account for student and instructor selection effects,
this is not a randomized controlled study. It is possible that there are
unseen factors contributing to higher fail-rates in the courses graded using
the percent scale. Future work will examine similar data sets at other
institutions offering CLASP and other Introductory Physics for Life Sciences
courses. Furthermore, a big question we have not addressed here is what
happens to these students after completing this course. Are students who would
have failed under the percent scale, but passed under the 4-point scale
successful in future courses? We do not address this question in this paper,
as our intent is not to prove one grading method is superior to another, but
rather to uncover characteristics of the percent and 4-point scales that are
important for instructors to consider when deciding on a grading practice for
their courses. We do plan to investigate this question in forthcoming work.

Each college or university level instructor has their own opinion about the
quality of a student’s work but this judgment should represent an unbiased
opinion of that work. Toward that end, it is useful for instructors to know
the origins of possible biases so that they can account for these in assigning
grades. Our results indicate that instructor use of the 4-point scale led to
many more students passing their introductory physics course as compared with
classes using the percent scale. This result was achieved without grade
inflation. Our findings align with previous critiques of the percent scale,
and indicate that instructors should consider the specific issues we highlight
in this paper when using the percent scale.

##  X Acknowledgements

None of these results would have been possible without the organized databases
that Wendell Potter set up in 1997 and continued through his retirement in
2006 so, in memoriam, we owe him a debt of gratitude. We would also like to
thank the education research groups at UC Davis and San Jose State for useful
comments on the research and the manuscript. We also would like to thank
Jayson Nissen for providing feedback on an earlier draft of this paper.

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