Himani

PREDICTION OF STUDENTS’ PERFORMANCE IN FUTURE
TEST USING FUZZY APPLICATIONS
MASTER OF SCIENCE
IN
APPLIED MATHEMATICS
UNDER SUPERVISION OF:
Internal guide: External guide:
Dr. Rashmi Singh Amitabh Shukla
By
Himani Jain
Enrollment No: A4451014015

INTRODUCTION
 What is predictive model….??
A predictive model is made up of a number of predictors, which are
variable factors that are likely to influence future behavior or results.
 Why do we need it….??
Predicting student’s academic performance plays an important role in
academics. Classifying students using conventional techniques cannot
give the desired level of accuracy, while doing it with the use of soft
computing techniques may prove to be beneficial. The higher education
has to be improved if the research and skill among the students has to be
developed in universities. Many education researchers and instructors
have made extensive efforts in constructing effective models to predict
student academic performance in a class. This is to reduce the numbers
of failures and dropouts of the school. Data mining is one of the major
areas which help us to predictions and classification.

 How is it important for the institutions and colleges….??
Students and Educational Institutions are dependent on each other.
They have a certain expectations from one another. Institution is known
by the performance of its students. Thus they should adopt every
possible step to enhance the quality of their education which in return
helps the students to boost up their career and confidence.
.

SCOPE AND METHOD USED
This study is focused on developing and validating fuzzy mathematical
model to predict student academic performance in the university. The
predicted results are then validated with the actual results of students to
find the accuracy of the model. The Academics performance of a student is
affected by number of factors like previous knowledge, interest, family
background, motivation etc. In this paper we have considered the factors
like the student’s previous year results and their test attendance, their
score, percentile, accuracy, exam aptitude, attempt ratio to predict the final
year result of a degree course. We will also use Item Response Theory
(IRT) to make our prediction model more accurate.

In actual practice, there are many factors which affect a student’s academic
performance. It is very complicated to include all of them. So, some of the
important measures through which we can analyze the performance of the
student in n-tests are:
1) Percentile: Percentile scores for individual test takers represent
how an individual test taker's score compares to the scores of other test
takers within a particular comparison group. Percentile scores range from
the 1st through 99th percentile, indicating the percentage of scores in the
comparison group which are lower than the test taker's score. For example,
if your scores report says that a test taker with a CCTST Reasoning Skills
Overall score of 19 is in the 67th percentile, this means that this test taker
has tested better than 66% of the test takers compared to an aggregated
sample of test takers like themselves. As the test administrator, you choose
the comparison group when you make a test assignment for your students.

2) Accuracy: The degree to which a given quantity is correct and free
from error. It is the ratio of Correct attempt by the student to his/her total
attempt. It is very important to know how accurate does the student give
answer to any item. If we know the student’s accuracy in previous n-
tests, then we can predict his future performance on its basis.
3) Exam Aptitude: Here we have defined exam aptitude as per
Yoctel’s definition for analyzing data:
Exam aptitude tells us about the student that where he/she stands in
his/her batch.
𝐸𝑥𝑎𝑚 𝐴𝑝𝑡𝑖𝑡𝑢𝑑𝑒 =
𝑆𝑡𝑢𝑑𝑒𝑛𝑡′
𝑠 𝑚𝑎𝑟𝑘𝑠
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑚𝑎𝑟𝑘𝑠
×
𝑠 𝑎𝑡𝑡𝑒𝑚𝑝𝑡
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑎𝑡𝑡𝑒𝑚𝑝𝑡
× 100

4) Attempt Ratio: Attempt ratio is the ratio of the number of
questions attempted by the student to the total number of questions. If we
analyze the attempt ratio of the student then we can predict that how many
questions he is going to attempt in future test.
5) Test Attendance: Attendance also has a great impact on the
student’s performance. If 10 students are attempting an exam and only
one student is attempting this kind of exam for the first time while others
have appeared in all previous exams, then there is high chance that the
student does not perform well in comparison to other students as other
students have got the idea of attempting that question paper. So with the
help of students’ attendance, we can predict their performance in future
tests.

6) Item Response Theory: This procedure often involves an
analysis of the individual items on the test. One primary goal of item
analysis is to help improve the test by revising or discarding ineffective
items. Another important function is to ascertain what test takers do and
do not know. With classroom achievement tests however, there is usually
no external criterion against which items can be validated. Instead,
another procedure is used that involves determining the percentage of test
takers who passed each item and the correlation of each item with some
criterion. In this case, though, the criterion consists of total scores on the
test itself. Two statistics can help us to evaluate the usefulness of each
test item.

Item-difficulty Index (p) or p-value
• Item difficulty is used to choose items of a suitable difficulty level.
• Tests are usually arranged with items in order of difficulty, beginning with
easier items.
• p-value ranges from 0 to 1.0
• A zero means that no one got the answer correct so its very difficult item
and 1 means that everyone got the answer correct so its very easy item.
• The closer an item gets to 0 or 1, the less information it contributes about
the test takers.

Item-discrimination index (di)
Item discrimination refers to whether an item can distinguish between
the people who scored high or scored low on a test.
For this calculation, we divide the test takers into three groups
according to their scores on the test as a whole: an upper group
consisting of the 25% who make the highest scores, a lower group
consisting of the 25% who make the lowest scores, and a middle group
consisting of the remaining 50%. (Note that some text books use 27%,
not 25%.)

Let
𝑇 = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑔𝑖𝑣𝑖𝑛𝑔 𝑡𝑒𝑠𝑡
𝑈 = 𝑁𝑜. 𝑜𝑓 𝑇𝑜𝑝 25% 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑔𝑖𝑣𝑖𝑛𝑔 𝑡𝑒𝑠𝑡
𝐿 = 𝑁𝑜. 𝑜𝑓 𝐵𝑜𝑡𝑡𝑜𝑚 25% 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑔𝑖𝑣𝑖𝑛𝑔 𝑡𝑒𝑠𝑡
𝑈𝑐 = 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑖𝑛 𝑈 𝑤ℎ𝑜𝑠𝑒 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
𝐿𝑐 = 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑖𝑛 𝐿 𝑤ℎ𝑜𝑠𝑒 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
𝑝 = 𝑝 − 𝑣𝑎𝑙𝑢𝑒 / 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡𝑦 𝑖𝑛𝑑𝑒𝑥 , 𝑝 ∈ [0,1]
𝑑𝑖 = 𝐷𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑑𝑒𝑥, 𝑑𝑖 ∈ [−1,1]
Hence,
𝑝 =
𝑈 𝐶 + 𝐿 𝐶
𝑈 + 𝐿
, &
𝑑𝑖 =
𝑈 𝐶 − 𝐿 𝐶
𝑈

ASSUMPTIONS IN OUR PREDICTION MODEL
• We will consider n-tests for same set of students.
•Our prediction model will be based on these n tests.
•In 1 test, there are m students.
•We will select 2 students from this set and apply our predictions on them
simultaneously and test whether our prediction is true or not.
•We will analyze two students’ performance in n tests and then predict their
performance in the (n+1)th test.
•At the end we can verify our method by calculating the error in our
prediction.

 Suppose we have n-tests:- 𝑇1, 𝑇2, 𝑇3,…, 𝑇𝑛
 Xisa set containing 100000 students.
 𝑥𝑖 ∈ 𝑋, 1 ≤ 𝑖 ≤ 𝑚 where xi isith student.
𝑋 = {𝑥1, 𝑥2, 𝑥3,… , 𝑥 𝑚 }
 Let P be the set of studentspresent in n-tests.
 A be the set of studentsabsent in n-tests.
 With respect to n tests, there are Pj and Aj where Pj isthe set
of students present in jth test and Aj is the set of students
absent in jth test where, 1 ≤ 𝑗 ≤ 𝑛.
 So, Aj ∪ Pj = 𝑋 & 𝐴𝑗 ∩ 𝑃𝑗 = ∅.

Now define,
F1 = A1,
F2 = A1 ∩ A2,
F3 = A1 ∩ A2 ∩ A3,
………………………….
Fn = A1 ∩ A2 ∩ A3 ∩ … … … ∩ An.
i.e.,
Fi = An
i
r=1
This is clear that Fn ⊆ Fn−1 ⊆. … … … ⊆ F2 ⊆ F1. If a student
xi ∈ Fn then this means that xi ∈ F1 also, i.e. if a student was
absent in all n teststhen he wasabsent in 1st
test also.

We will only consider the studentswho belong till the set F[
n
2
], after
that we will exclude those studentsfrom our prediction base who
belong to the set F[
n
2
]+1, F[
n
2
]+2,…… , Fn. We are doing thisbecause
if the student isbelonging to any of these sets, say Fn−1 , then this
meansthat he hasskipped n-1testsso there isno use of including
him in our prediction. We will form our prediction base for the
students who have attendance more than 50% and will exclude
those having attendance below this. So now we are left
with F1,F2,F3,… , F[
n
2
].

So, now we will first see that to which set he belongs.
Let it be F3 which meansthat the student hasleft 3 testsout
of n.
Now we already know that out of n tests, 4 were of algebra
(say) and he missed 3 classesout of these 4, then thisisgoing
to affect hisperformance in (n+1)th
test ashe haslessidea as
compared to othersof how to attempt (n+1)th
test.
 But if he did not miss any test of algebra then this will not
have any effect on hisfuture performance.
So we conclude that, more the no. of testsof same subject missed,
greater will be the negative impact on future performance of that
subject.

If we talk about aptitude, we calculate the exam aptitude of a
student by the following formulae (provided by Yoctel company
for data analysis):
𝐸𝑥𝑎𝑚 𝐴𝑝𝑡𝑖𝑡𝑢𝑑𝑒 =
𝑠 𝑚𝑎𝑟𝑘𝑠
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑚𝑎𝑟𝑘𝑠
×
𝑠 𝑎𝑡𝑡𝑒𝑚𝑝𝑡
𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑎𝑡𝑡𝑒𝑚𝑝𝑡
× 100
i.e.
𝐸𝐴𝑝 =
𝑀𝑆
𝑀 𝑀𝐴𝑋
×
𝐴𝑡𝑆
𝐴𝑡 𝑀𝐴𝑋
× 100
So, exam aptitude depends on other students also. If the student
scoreshighest mark aswell ashe attemptsthe highest items, then
hisaptitude is100%.

Okay..!!! But do you really think so that the student
who attempts maximum items, is meant to score
highest….???
•No, this is practically not a valid statement.
•A student may attempt many questions just by fluke also.
•So, if the student with highest attempt ratio scores less than average of
overall score then we will go to second highest and so on.
•So the attempt of only that student will be considered as highest, whose
score is more than average of overall score also.
•Here, we see that exam aptitude depends on score as well as attempt
ratio.
• As the score and attempt ratio increase, student’s exam aptitude also
increases.
•As soon as the aptitude increases, this also means that the student’s
performance is increasing in his batch.

•We have assumed that all the students have been trained in a uniform
manner without any visible partiality.
•Also, each and every student is of same eligible age for the course.
•We will exclude those items from our prediction base model, whose
discrimination index is below 0.30 because these items are not able to
discriminate between the below average and above average students.
WHY….???
•For example, if we give a question like ‘Find x if 20+x-3=7’ to the students of
class 12th then this is quite obvious that every student will respond same and
this item is not able to judge the weak students.
•Similarly if we give a question like ‘Define a maximal ideal in a ring.’ To the
school student, then no one will be able to answer it correctly. So this kind of
item should also be excluded from prediction model.

We will try our best to make an accurate prediction model with given
possible data but many times, students just answer the question
by fluke and in daily life situations, there are many other factors
which could be possible for his/her unpredictable result. Because
of this, our prediction model may show some noise.

PROCEDURE OF THE METHOD
Now, we will discuss how the whole analytics was done.
Population and Sample
In this work all student-teachers who are studying in India
comprise the population of the study. Random sampling was
adopted for this research work and 3608 students were taken.
The sample constituted both male and female student-teachers.

Instrument
Six tests were used for data collection.
Test No. of students No. of items
A 929 40
B 1068 50
C 328 65
D 148 50
E 140 50
F 150 25
TOTAL 3608 280

Data Organization and Analysis
1) First of all the data was filtered correctly.
2) Total attempts, correct attempts, unattempts, score, exam aptitude,
accuracy and percentile of the students were entered in Microsoft Excel
sheet and it was arranged in descending order of scores.
3) Then upper group, middle group and lower group were highlighted with
different colors.
4) The formulae for difficulty levels and discriminating index discussed
above were used for analysis. The item in a test should neither be too
easy nor too difficult; hence a balance between these two must be
maintained.
5) 10 ranges were created according to the p values. The items having p
value between [0,0.1] will lie in 0 range, (0.1,0.2] in 1 range,……,
(0.9,1] in 9 range.
6) The items with discrimination index less than 0.30 are then excluded
from our prediction. Now all the items from 6 tests are compiled
together, keeping in mind that many items are repetitive.

The 7th test which we are provided consists of 845 students and
40 items whose p-values and discrimination indices are given to
us. First of all, we have to exclude the items with discrimination
index less than 0.30 as they are not in our prediction system. So,
after this 22 items are left for prediction.
We will now lookup for these 845 students’ previous test
attendance. If it is less than 50%, then we will exclude those
students from our prediction model. So, we are left with 360
students. So now we have to predict the score of 360 students for
22 items. Each item weights 1 mark and there is no negative
marking. So, maximum score will be 22 only.

As from previous 6 tests we the data that which student correctly
attempts how many no. of items in particular range. So we also
have their percentage index in particular range which iscalculated
asfollows:
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑛 𝑖𝑡ℎ 𝑟𝑎𝑛𝑔𝑒
=
𝑠𝑐𝑜𝑟𝑒 𝑖𝑛 𝑖𝑡ℎ 𝑟𝑎𝑛𝑔𝑒
𝑡𝑜𝑡𝑎𝑙 𝑛𝑜. 𝑜𝑓 𝑖𝑡𝑒𝑚 𝑖𝑛 𝑖𝑡ℎ 𝑟𝑎𝑛𝑔𝑒
× 100%
For eg, if a student attempts 3 items correctly out of 5 items in
range 4, then his percentage index will be
3
5
× 100% = 60%. We
will denote it by ‘Pi’.
So, if there are 7 items in range 4 of 7th
test, that student will
probably attempt 7 × 60% = 4.2 ≈ 4 items correctly. We will call
thisno. asour first assumption.

Similarly, we calculated first assumption for all the students giving 7th test.
Following is the snapshot of our first assumption compared with actual
score:

In next slides we will see that how the students’ test attendance
is effecting their performance:

When we compared the actual scores with our first assumption, some were
very close but rest had noise as many other factors also need to be
considered. Now we will make a fuzzy model.
Fuzzy Logic
•Fuzzy Logic can be applied to any thing that is not fixed and exact.
•Its variables take the values between zero and one.
•It is used in many fields from control theory to artificial intelligence.
•It has helped in designing many intelligent systems.
•The two major concepts of fuzzy logic which has critical role in its
applications are of a linguistic variable and of fuzzy if-then rule.
•Fuzzy logic thus helps to sum up the data and focus on decision related
information.
•As in case of students who have marks zero and 28 both are failed. Here we
use fuzzy logic for differentiating the students.

Fuzzy System
Fuzzy system has four major parts:
1. The transformation of input crisp values into fuzzy values i.e.
membership grades for linguistic variables of fuzzy sets is known as
fuzzification. This is done by a membership function which assigns a
grade to each linguistic word.
2. Knowledge base contains important definitions which are used in
control rules and manipulating data. It also defines the control scheme
and objectives by means of linguistic control rules.
3. Decision making logic directs the human decision using fuzzy
concepts and fuzzy rules.
4. Thus the fuzzy set obtained after composing the rules is converted
into a crisp value. This is called defuzzification.

Linguistic Variables
The variables whose states are fuzzy numbers signifying linguistic
concepts such as very big, big, average, small etc. as taken in a particular
situation are called linguistic variables.
Mathematical variables generally take numerical values but in applications
of fuzzy logic non-numeric are used to indicate the rules and facts.
Now we take two linguistic variables as input and one as output. The
inputs are ‘First assumption and ‘Attendance in test’. And on the basis of
these two inputs we will get an output in the form of ‘Second assumption’.

We take the following values in our case:
First assumption takes the values as
‘Extremely bad’, ‘very bad’, ‘bad’, ‘fair’, ‘good’, ‘very good’, ‘extremely
good’
Or ‘EB’, ‘VB’, ‘B’, ‘F’, ‘G’, ‘VG’, ‘EG’
Test attendance takes the values as
‘Very very low’, ‘very low’, ‘low’, ‘high’, ‘very high’, ‘very very high’
Or ‘VVL’, ‘VL’, ‘L’, ‘H’, ‘VH’, ‘VVH’
Second assumption takes the values as
‘Extremely bad’, ‘very bad’, ‘bad’, ‘fair’, ‘good’, ‘very good’, ‘extremely
good’
Or ‘EB’, ‘VB’, ‘B’, ‘F’, ‘G’, ‘VG’, ‘EG’

Membership Function
A Membership function on any set A is a function from A to the real interval
[0, 1]. Membership functions are fuzzy subsets of X. It is denoted by a
symbol for any x ∈ X is called the membership degree of x in the fuzzy set
A. The value 0 denotes x is not a member of the fuzzy set A, the value 1
denotes it is the member of A and the value between 0 and 1 denotes it
belongs to the set partially.
We form the following three membership functions for each variable:1) First assumption of score
EB [0 2.634]
VB [35.84 7]
B [6 8.4310]
F [9 11.1113]
G [12 13.89 16]
VG [15 16.0719]
EG [1820 20]

2) Test attendance in %
VVL [0 16.6725]
VL [20 33.33 40]
L [35 50 60]
H [4766.6780]
VH [75 83.33 90]
VVH [85 100 100]
3) Second assumption
EB [0 3.14]
VB [4 79]
B [8 10 13]
F [1112.5 14]
G [13 14.716]
VG [15 1720]
EG [1722 25]

Now we define fuzzy if-then rule. It is a conditional statement of the form :
IF ‘a’ is B, THEN ‘b’ is A. Here a & b are linguistic variables and A & B are
their linguistic values.
Fuzzy Rules Formulation
As we know that our first assumption made on the basis of p values and
test attendance both constitute in predicting the students’ result. Thus we
formulate the following rules with two input variables and one output
variable.
Let if A and B then C where,

First Assumption (A) Test Attendance (B) Second Assumption (C)
VB L B
B L F
F L F
G L G
VG L VG
EG L EG
VB H B
B H F
F H F
G H F
VG H VG
EG H G
B VH B
F VH F
G VH F
VG VH G
EG VH EG
B VVH B
F VVH B
G VVH F
VG VVH VG
EG VVH EG

Making second assumption of students’ performance
using MATLAB
Apply the above discussed method for analyzing result.
Figure (1): System with two inputs and one output

Figure (2): Membership function for ‘Test attendance’

Figure (3): Membership function for ‘First assumption’

Figure (4): Membership function for ‘second assumption’

Figure (5): Rules formulation for two inputs and one output

Figure (6): Analysis of result using rule viewer.

Figure (7): 3-D surface viewer
Now, after fuzzification, using rules and defuzzification process using
MATLAB we get the second assumption for students score in 7th test. Figure
(6) shows the second assumption for the student whose attendance is 16.7
and first assumption 3.43.

RESULTS AND DISCUSSION
This whole analysis is based over data provided by organization and
considering that our assumptions are true.
But conclusions may be wrong depending on the noise status in the data. For
example, since lots of questions had to be removed because of di, we were left
with only 22 items out of 40. This means the quality of paper which we were
provided was very low.
Otherwise an ideal question paper must have more than at least 90% of items
left even after removing the items with low di.
Secondly, we also observed that most of the students were having less score,
so even little variation in the score was giving high change in the percentage.
My observation says that if someone has attended first three tests, then this is
visible from the data that his score is increasing. So, attendance in the test has
a role in the prediction.

Once a student has taken few tests and the next test is from the same
subject he/she has taken, then the student has some tendency to discuss the
question at home and may practice the similar type of questions immediately.
So prediction will always be in positive sense for the student who has taken
the previous tests and in negative sense for the students who have skipped
the tests.
So, given the class attendance, test attendance, p value of each item and
response of each student on particular p-value of the item, we can make our
prediction model better.
But due to lack of appropriate data, my first assumption is showing noise
that’s why I fuzzified my model and introduced the test attendance also in it.
The second assumption, which we got after defuzzification is our final
prediction based on the data provided.

FUTURE SCOPES
 My future work will be studying the system of grading using IRT and fuzzy
logic. And what measures we can take to improve it.
 A continuous evaluation is important for knowing or keeping a track on the
learner.
 There are various ways to check the performance of students or any other
learner like exams class test mid-term exams and so on.
 In schools and colleges, marks are just given on the basis of students’
score. But they should also consider many other factors.
 Item Response Theory enables us to examine each student’s response on
an item.
 The student who gives mostly correct answers to the items of p value 2
(say) is given same score as the student who attempts the items of p value 9.
But this is wrong way of evaluation as there is a lot of difference these two
students.
 Evaluation should be done in a different and new way.
 My aim will be to improve the evaluation method using the research-based
information.

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