Normalization of student grades by class performance
Shoba Ittyipe and Peter Zizler
Abstract
We normalize student GPA grades by their relative performance in the
classes they have taken.

1

Introduction.

Students at universities take many courses to graduate. At some universities,
most often larger institutions, the student course grade performance is a function
of their classmates. The course grades are awarded ”on a curve” adjusted to
reflect an expected mean and standard deviation of the performance in the given
class. Therefore, in such a case, the student’s grade very much depends on their
classmate’s performance. Smaller institutions, Mount Royal University, might
shy away from such grading on a curve and award student grades independent
of the performance of their peers.
Whatever the scenario for the grade assignment is, it is conceivable that
students might be awarded harsher grades due to the high performance of their
classmates. If the particular class had higher scores on average, and another
class had lower scores on average, this process would normalize these scores
based on how an individual student scored relative to the mean value of that
class.
It is of interest to generate a marker that would detect such aspects of
grading. Such a marker could be used by an institution to detect to what extent,
if any, the student grade awards are affected by the student peer performance.
Once such adjusted grades are produced, the student might assess to what extent
their overall GPA was affected by their peers. If the adjusted GPA is higher
than the actual one, the student’s grade performance might have been assessed
more harshly due to having high-performing classmates. On the other hand, if
the adjusted GPA was lower than the actual one, then the student’s performance
might have been overestimated due to lower-performing classmates.
We propose a method to produce adjusted student grades that would reflect
their class performance and as a consequence, we could generate adjusted GPA
based on the class performance of the student peers. These should not, of course,
replace the actual grades, but should be only informative.

1

Suppose a student has the following 8 grades
TERM
201501
201501
201501
201501
201504
201504
201504
201601

CRN
19249
18950
15057
19514
45407
48039
51200
16332

SUBJ
COMP
COMP
GNED
GNED
CHIN
MATH
MGMT
COMP

CRSE
1103
1615
1101
1103
2217
1505
2130
1501

GRADE
B
C+
B
B
BB+
C
A+

SCORE
3.0
2.3
3.0
3.0
2.7
3.3
2.0
4.0

from the 8 courses they have taken. We have chosen a sample of only 8 courses
to illustrate the procedure, understanding that a student would have needed to
have 40 grades from all the courses they have taken in order to graduate. The
z scores of the class performance of each course grade is found to be
Grade
z score

B
0.5214

C+
0.1000

B
0.2980

B
-0.0479

B-0.7271

B+
0.5072

C
-0.7528

A+/A
0.5318

The grade of A+ and A has the same score. The course grades the student
obtained and the associated z scores are unrelated. For example, the first grade
of B+ (3.0) was obtained in COMP 1103. The course grade value of 3.0 has
some corresponding z score in that particular class and that z score happens to
be z = 0.5214. In particular, suppose the class COMP 1103 had 40 students
there (including our student). The course grades (as values) of all 40 students
happened to be
40

{ti }i=1
Then the z score value of 0.5 is obtained as
0.5 =

3.0 − t
σt

where

t=

40
1 X

40 i=1

s
ti ; σ t =

P40

i=1

ti − t
39


2

If it happens that all the students in the class (including our student) obtained identical grade scores, for example pass and fail grade and all passed,
then we set the z score value for our student in that class to be z = 0.
We want to adjust all the 8 grades of our student, based on the z scores from
all the 8 classes. The second grade of C+ had the associated z score from the
class (COMP 1615) z = 0.1000. The student was above average and the grade
seems harsh, we expect a bump up. On the other hand, the 5th grade is B- but
the associated z score from the class (CHIN 2217) is z = −0.7271, not too big.
We expect a mild drop as a result of the adjusted grade. We now present the
method to generated these adjusted scores.
2

2

Method.

A student has the following GPA grades in their n classes they have taken
{Xi }ni=1
Each grade Xi was taken in a class and the corresponding z score from the class
performance is given by zi . Thus each GPA grade Xi now has the associated z
score zi . Define
Xmax = max{Xi }ni=1 ; Xmin = min{Xi }ni=1
zmax = max{zi }ni=1 ; zmin = min{zi }ni=1

z=

n
1X

n i=1

s

P

zi ; σ z =

(zi − z)
n−1

2

Each student has z scores {zi }ni=1 of their class performance attached to the
corresponding GPA grade {Xi }ni=1 . It is desirable to generate adjusted student
GPA scores {Yi }ni=1 that would reflect the z scores obtained in the respective
classes. In particular we would like to have
Yi − Y
zi − z
=
σY
σz
where

Y =

n
1X

n i=1

s
Yi ; σY =

Pn

Yi − Y
n−1


2

i=1

while keeping the property
Ymax = Xmax ; Ymin = Xmin
This guarantees the range of grades is not changed. To that end we define
normalized GPA grades by class performance as


Xmin + Xmax
Xmax − Xmin
zmax + zmin
Yi =
+
zi −
2
zmax − zmin
2
Result 1.1 The z score of Yi in the set {Yi }ni=1 is equal to the z score of zi in
the set {zi }ni=1 . In particular, for each i ∈ {1, . . . , n}, we have
zi − z
Yi − Y
=
σY
σz
Proof. Let

3

Yi = a + b (zi − c)
where
a=

Xmin + Xmax
Xmax − Xmin
zmax + zmin
;b=
;c=
2
zmax − zmin
2

We have
Y = a + b (z − c)
thus
Yi − Y = b (zi − z)
Now
n
n

2
1X
1 X
2
(zi − z)
Y i − Y = b2
n i=1
n i=1

thus
σY = bσz
We have
b (zi − z)
(zi − z)
Yi − Y
=
=
σY
bσzi
σz i
♣
Result 1.2 We have


Xmin + Xmax
Xmax − Xmin
zmax + zmin
Y =
+
z−
2
zmax − zmin
2
Furthermore suppose for some a, b ≥ 0 with a + b = 1 we have
z = azmax + bzmin
then
Y = aXmax + bXmin
Proof
Y

=
=
=



Xmin + Xmax
Xmax − Xmin
zmax + zmin
+
z−
2
zmax − zmin
2


Xmin + Xmax
Xmax − Xmin
zmax + zmin
+
azmax + bzmin −
2
zmax − zmin
2
aXmax + bXmin

♣
Note that we can not determine, based on the above, whether Y > X or Y < X.
4

3

Example.

We continue with the data presented in the introduction. We illustrate the
grade adjustment procedure on this example. We recall the lowest grade is C
and the highest grade is A+. The new adjusted grades will keep this grade
range the same. The vector X of the actual grades is given by
X

=

3.3

2.3

3.0

3.0

2.7

3.3

2.0

4.0

and the z score vector for the original grades obtained from respective class
performance, independent of these grades is given by
z

=

0.5214

0.1000

0.2980

-0.0479

-0.7271

0.5072

-0.7528

0.5318

2.7
2.0402

3.3
3.9616

2.0
2.0000

4.0
4.0000

BC

C
C

We apply the procedure above and obtain
X
Y

=
=

3.3
3.9838

2.3
3.3277

3.0
3.6359

3.0
3.0975

C+
B+

B
B

yielding
Original
Adjusted

B
A-

B
B+

B+
A-

A+/A
A+/A

A statistic of interest is the mean GPA score from all the 8 classes for our
student, both actual and adjusted.
actual
adjusted

2.9500
3.2558

Our student has a higher adjusted mean GPA score than the actual (nonadjusted) mean GPA score from all the 8 classes. As a result, the student is
informed that their overall grade performance during their studies was possibly
assessed more harshly due to the higher performance of their classmates during
their studies.
We have collected blind data by stripping student identification information
altogether. We have used the data from students who graduated from the BCIS
program in 2022 and 2023. For each student, all 40 course grades applied for
graduation are used. Subsequently, for each grade that the student received
in each section, all student grades from that class are considered to calculate
the z-score for that class. The z-score generates values that can be negative or
positive. A z-score value of zero means the student had an average score in that
class. An above zero or below zero indicates above average or below average
grades, respectively. For example the z-score value of z = 0.5 would indicate
that a student class performance was half a standard deviation above the mean.
Similarly, the z-score value of z = −1.00 would indicate that a student class
performance was one standard deviation below the mean.
5

Also, it is noteworthy to indicate that the adjusted grade is computed using
the maximum and minimum of the grades for the entire set. The final normalized
grade for each student in each class is converted back to the original scale, by
using the same range. Hence, the original range is preserved, although individual
grades may change within that range.

6