Statistical Analysis of Test and Test Scores

n
= Number of items of the test

M
= Arithmetic mean of the scores

s =
Standard deviation

A.M.=
Assumed Mean

f
= Frequency

d
= Deviation from the class Interval containing Assumed Mean

Σfd
= Summation of the products of f and d

N
= Total frequency

i
= Interval length

L
= Lower point of the median class

Cfu
= Cumulative frequency of the class below the median class

fm
= Frequency of the median class

i
=  Interval length

N
= Total frequency

Q
= Quartile deviation

Q₃
= 75^th percentile

Q₁
= 25^th percentile

L₁
= Lower point of the 25^th percentile class interval

F₁
= Cumulative frequency of the class below the 25^th Percentile
class 

fq₁
= frequency of the class interval of 25^th percentile

i
=  Length of the class interval

N = Total frequency

L₃
= Lower point of the 75^th percentile class interval

F₃
= Cumulative frequency of the class below the 75^th Percentile
class 

fq₃
= frequency of the class interval of 75^th percentile

i
=  Length of the class interval

N = Total frequency

D
=  Distance between the midpoint of the
class and the mean

f
= Frequency of that class

N
=  Sum of the frequencies

D
= Deviation

F
= frequency

i
= Length of the class interval

N = Total frequency

f_c = Cumulative frequency of the classes
below the class of that particular class

X = The score of which the P.R has to be determined

L = Lower limit of the class which contains that
particular score

i = Length of the class interval

q = The frequency of that class

N = Total frequency

X = Score of the student

M = Mean score of the class

s2.11.3 Standard score (T – Score)

A T – Score is nothing more than a Z
– Score expressed in a different form.

2.12 Measures of relationship

Degree of relationship is expressed as a correlation
coefficient, which is computed, based on the two sets of scores. If the change
in one set of score causes the change of the other set of score, then this
phenomenon is called the correlation. The most appropriate measure of
correlation is :

N
SSSY

¡ =

ÖSX²
– ( SX )² }{ NSSY )²
}

X = Score of test 1 Y = Score of test 2 N = Total frequency

2.13 Measures of relationship

Degree of relationship is expressed
as a correlation coefficient, which is computed, based on the two sets of
scores. If the change in one set of score causes the change of the other set of
score, then this phenomenon is called the correlation. The most appropriate
measure of correlation is :

N
SSSY

¡ =

ÖSX² – ( SX )² }{ NSY² – ( SY )² }

X = Score of test 1 Y = Score of test 2 N = Total frequency

2.14 Normal Distribution and Normal
Probability Curve

If a variable is normally
distributed, that is, does for a normal curve, and then several things are
true:

1. 50% of the scores are above the mean and 50% are below
   the mean.
2. The mean, the median, and the mode are the same.
3. Most scores are near the mean and the further from the
   mean a score is, the fewer the number of subject who attained that score.

·
·
·
·

· Mean – 3.0 S.D = Approximately the
0.1 Percentile

· Mean – 2.0 S.D = Approximately the 2^nd
Percentile

· Mean – 1.0 S.D = Approximately the 16^th
Percentile

· Mean = Approximately the 50^th
Percentile

· Mean + 1.0 S.D = Approximately the 84^th
Percentile

· Mean + 2.0 S.D = Approximately the 98^th
Percentile

· Mean + 3.0 S.D = Approximately the 99⁺
Percentile

0.13%

2.15%

13.59%

34.13%

34.13%

13.59%

2.15%

0.13%

sss
Mean 1 sss

Fig: The normal probability curve

2.15 Abnormal Distribution
2.15.1 Skewness

# Skewness = 3 ( mean – median ) /
S.D

2.15.2 Kurtosis

Kurtosis indicates to what degree a
distribution is pointed or flat in comparison to normal probability curve. It
can be of three types:

2.15.2.1 Leptokurtic

It is more pointed than the normal
curve.

2.15.2.2
Platykurtic

It is more flat than a normal curve.

2.15.2.3 Mesokurtic

If the curve is neither too much
pointed, nor too much flat, then the curve is mesokurtic or normal.

2.16 Grading System

The grading system is based on
integration of the traditional absolute marking system and grading student
performance assuming normal distribution. The system is a two tier evaluation
system, which will ensure gradation of the group as a whole in a course, and
that of individuals which the group in comparison with one another.

3.0 General considerations

In preparing the essay type test,
understated matters have been tried to maintain:

 
i.
1.
It is a 15-mark test. Total item no. is 6.

ii.
The
duration of the test is 30 minutes. Time per mark is 2 minutes.

iii.
The
test has been constructed mainly with short essay type questions rather than
broad essays in order cover more areas of the content.

iv.
Short
essays enabled the increase of items in number in order to cover more areas of
content.

v.
Overlapping
of content between the essay type and MCQ type questions has been avoided.

vi.
Effort
is made to use proper language so that the pupils can easily understand the
requirements of the test. Simple to complex sequence was tried to maintain.

 
vii.
Mix
up of other type of questions with the essay type test has been avoided.

viii.
Optional
questions are not given.

ix.
Item
and sub-item wise number distribution was shown in the test and key words were
underlined so that the students can prepare their answers according to the
number distribution.

3.0.1Test and Examination –
Explanation of the situation

1. To fulfill my practicum requirement, I had to execute a
   50-mark essay type test and a 50 mark MCQ test. But school authority did
   not allow me to take a two-hour or of more duration test. So I had to take
   the 50-mark essay type test in 4 parts:

Essay Type Test

QUESTION
-1

4.1.1 Analysis of Test-1

4.1.1.2 Subject wise
Distribution

Integrated Social Science has been
designed including major six subjects:

1. Sociology
2. History
3. Geography
4. Political Science
5. Economics
6. Population Education

But
Political Science and Population Education were not included for 2^nd
semester. So, I had to teach and take tests without those subjects.

The essay type test has included all
other subjects. Subject wise number distribution is shown below:

Comment

Distribution of marks differed
because of the difference of ranges among the subjects. This table clearly
shows that an overall equivalency of number distribution according to the
subject range has been tried to maintain in sociology, geography, history and
economics Number per page is 1.61 (approx). According to page range if 100%
equivalency of number distribution is to be maintained, then for sociology
(7×1.61)=11.27, for history (8×1.61)=12.88, for geography (9×1.61)=14.49,for
economics (7×1.61)=11.27, number is to be required to distribute. This figure
is not too far from the real distributed mark.

4.1.1.3 Item
Specification

The item specification according
to the domains of objectives of the essay type test is shown below:

Comment

A good test covers all the domains of
learning such as knowledge, comprehension, application, analysis, synthesis,
and evaluation. This test fulfills this requirement covering all the domains of
learning. A test should not be too easy or too difficult. 32% number has been
given from knowledge sub-domain; items have also been taken from higher
sub-domains such synthesis, evaluation, etc

Measure of difficulty level and
discrimination power is appropriate for MCQ or objective type tests; not for
essay type test. So these two parameters is not measured here for the essay
type test.

A good test requires validity,
reliability and objectivity. Objectivity can’t be statistically determined. I
don’t have sufficient data to measure the validity of the test. Reliability can
be measured following the ‘Kuder and Richardson Method’. The reliability of the
test is determined below:

4.1.1.4 Reliability

#
Reliability,  r₁ =
n/(n-1)[1-{M(n-M)/ns²}]

s² } ]
= 1.13 (Apprx.)

5.1.2.2 The Quartile Deviation

# Q₁ = L₁ + {(N/4-F₁)/fq₁}I

Q₁ = L₁ +
{(N/4-F₁)/fq₁} i

= 15.5
+{(14.25 – 22)/16} 5

= 15.5 + {(- 7.25)/23} 5

= 15.5 + (-0.32) 5

= 15.5 – 1.6

=
13.9

Q₁= 13.9  (Apprx.)

# Q₃ = L₃ + {(3N/4 – F₃)/fq₃}
i

Quartile Deviation, Q = ( Q₃  –
Q₁ )/2

= (22.72 – 13.9)/2

= 8.82/2

= 4.41

5.1.2.3
Mean Deviation

S||
/N

S||
= 286.4

S||
/N

=
286.4/58

= 4.94

5.1.2.4 Standard Deviation

s
= ÖSfd²/N
– (Sfd/N)²}I

∑fd = – 1
∑fd² =89

sÖSSfd/N)²}i

• 29
  scores (50%) stand below the mean and 29 scores (50%%) stand above the
  mean. So the distribution is supposed to be normal.
• The
  mean, median and mode are very near to the same. But Mean>Median>Mode.
• Most
  scores are near the mean and the further from the mean a score is, the
  fewer the number of subjects who attained that score.
• Mean
  ± 1.0 S.D., that is, 11.7 – 24.1 contains a total of 39 scores (67.24%).
  So the distribution is supposed to be normal

• These findings show that
  although the distributions of Mean ± 1.0 S.D but the distribution of Mean ± 2.0
  S.D. and Mean ± 3.0 S.D is too far from the normal curve. The distribution is
  said to be normal and kurtosis.

  The distribution is very near to
  normal but very little positively skewed

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  
  

  Fig:
  Positive Skewness

  Skewness of the distribution = 3 (
  mean – median ) / S.D = 0.1 (Apprx).

  The value is very low which indicates
  the distribution is very near to the normal distribution.

  The distribution is very near to normal but is very little
  platykurtic

  
  
  
  
  

  
  

  Leptokurtic – where more
  scores than normal distribution stand in mean ±2.0  & 3.0S.D.

  5.1.3
  Measures of Relative Positions

  Major measures of relative
  positions are percentile ranks, Z-score and T-score. I shall determine only
  Z-score and T-score here as they are more reliable than percentile ranks.

  # Standard score (Z – Score), Z
  =  (X – M) / s

  # T = 50 + 10Z

  Student’s Roll No.	Z –score (Z = (X-M)/ s)	T-score (T = 50 + 10Z)
  01	(30 – 17.9) / 6.2 = 1.95	69.5
  05	(22 – 17.9) / 6.2 = 0.66	56.6
  08	(16.5 – 17.9) / 6.2 = – 0.23	47.7
  09	(19 – 17.9) / 6.2 = 0.18	51.8
  10	(11 – 17.9) / 6.2 = – 1.11	49.9
  11	(16 – 17.9) / 6.2 = – 0.30	47
  12	(14 – 17.9) / 6.2 = – 0.63	43.7
  13	(16 – 17.9) / 6.2 = – 0.30	47
  14	(14 – 17.9) / 6.2 = – 0.63	43.7
  15	(21 – 17.9) / 6.2 = – 0.50	45
  16	(9 – 17.9) / 6.2 = – 1.44	35.6
  17	(19 – 17.9) / 6.2 = 0.18	51.8
  18	(31.5 – 17.9) / 6.2 = 2.19	71.9
  19	(24 – 17.9) / 6.2 = 0.98	59.8
  20	(11 – 17.9) / 6.2 = – 1.11	48.9
  21	(26.5 – 17.9) / 6.2 = 1.39	63.9
  22	(12.5 – 17.9) / 6.2 = – 0.87	41.3
  23	(22 – 17.9) / 6.2 = 0.66	56.6
  25	(9 – 17.9) / 6.2 = – 1.44	35.6
  26	43.4
  27	(15 – 17.9) / 6.2 = – 0.47	45.3
  28	(26 – 17.9) / 6.2 = 1.30	63
  29	(12.5 – 17.9) / 6.2 = –0.87	41.3
  32	(17.5 – 17.9) / 6.2 = – 0.06	49.4
  33	(8 – 17.9) / 6.2 = – 1.60	34
  34	(11.5 – 17.9) / 6.2 = – 1.03	39.7
  36	(10 – 17.9) / 6.2 = -1.27	37.3
  37	(19.5 – 17.9) / 6.2 = 0.26	52.6
  38	(30 – 17.9) / 6.2 = 1.95	69.5
  39	(18 – 17.9) / 6.2 = 0.02	50.2
  40	(13 – 17.9) / 6.2 = – -0.80	42
  41	(20 – 17.9) / 6.2 = 0.34	53.4
  42	(24.5 – 17.9) / 6.2 = 1.66	66.6
  43	(19 – 17.9) / 6.2 = 0.18	51.8
  44	(12 – 17.9) / 6.2 = – 0.95	40.5
  45	(12 – 17.9) / 6.2 = – 0.95	40.5

  Comment

  From this table relative
  position of any student in comparison to other or whole as a group can easily
  be made. The T-score shown above it is a comprehensive presentation of relative
  position. Scores above 50 are more than average; scores below 50 are less than
  average.

  5.1.4
  Grading of Test-1 Results

  Norm
  of Grading

  Norm	Number Range (Apprx)	Grade	GPA
  Mean + 2.5 S.D. & above	33 and above	A	4.00
  Mean +1.5 S.D. – up to Mean + 2.5 S.D.	27 -32	B	3.50
  Mean + 0.5 S.D. – up to Mean + 1.5 S.D.	21 – 26	C	3.00
  Mean – 0.5 S.D.-up to Mean + 1.5 S.D.	14 – 19	D	2.50
  Below Mean – 0.5 S.D.	0 – 13	F	——

  Grading
  of students

  Roll No.	Obtained Mark	Grade	GPA
  01	30	B	3.50
  02	22	C	3.00
  03	16.5	D	2.50
  04	19	D	2.50
  05	11	F	——
  06	16	D	2.50
  07	14	D	2.50
  08	16	D	2.50
  09	14	D	2.50
  10	21	C	3.00
  11	9	F	——
  12	19	D	2.50
  13	31.5	B	3.50
  14	24	C	3.00
  15	11	F	——
  16	26.5	B	3.50
  17	12.5	F	——
  18	22	C	3.00
  19	9	F	——
  20	22	C	3.00
  21	15	D	2.50
  22	26	C	3.00
  23	12.5	F	——
  24	17.5	D	2.50
  25	18	D	2.50
  26	11.5	F	——
  27	10	F	——
  28	19.5	D	2.50
  29	30	B	3.50
  30	18	D	2.50
  31	13	F	——
  32	20	D	2.50
  33	24.5	C	3.00
  34	19	D	2.50
  35	12	F	——
  36	12	F	——
  37	18	D	2.50
  38	18	D	2.50
  39	15	D	2.50
  40	23	C	3.00
  41	19	D	2.50
  42	23.5	C	3.00
  43	11.5	F	——
  44	22	C	3.00
  45	28.5	B	3.50

  It
  is to be noted that this grading was not executed in the school’s test result;
  it is done only to fulfill the purpose of statistical analysis report. Combined
  score of MCQ and Essay type test was executed for grading of students and that
  result was submitted to school. Then the score was converted into 15 and was
  added as class test number

  Percentage of pass and
  Fail

  P/F	No. of students	Percentage
  Pass	40	69%
  Fail	18	31%

  Percentage of grading

  Grade	No. of students	Percentage
  A	0	0%
  B	6	10.34%
  C	14	24.14%
  D	20	34.48%
  F	18	31.04%

  5.1.5 Graphic presentation of Test-1
  result analysis

  
  
  
  

  
  
  
  

  
  
  
  

  
  
  
  

  
  
  

  
  
  
  

  
  
  
  

  5.2 Analysis of Test-2 results (MCQ
  test)

  Student’s score in Test-2

  Roll No.	Name	Obtained Mark (out of 50)
  01	Parvin Akter Sumi	26
  05	Tania Akter	29
  08	Rawnok Sultana	29
  09	Farhana Akter	22
  10	Afroza Akter Anika	22
  11	Farzana Abedin Chadni	18
  12	Shahana Akter Runa	22
  13	Mahmuda Akter Asha	23
  14	Ivy Moni	21
  15	Aesha Akter Lucky	22
  16	Moni Akter	20
  17	Sabina Yasmin Boishakhi	23
  18	Shompa Akter	27
  19	Shakil Hossain	25
  20	Soleman Hossain	20
  21	Aesha Akter Liza	27
  22	Surma Akter	20
  23	Md. Rubel	23
  25	Kohinur Akter Munni	19
  26	Eti Rani Das Brishti	24
  27	Shamsun Nahar	24
  28	Kamrul Hasan Meraj	26
  29	Nur Fatema	19
  32	Tania Akter	16
  33	Bithi Akter Nadia	22
  34	Jasim Uddin	21
  36	Shima Akter Shimu	21
  37	Rita Rani Roy	23
  38	Maksuda Akter Priya	28
  39	Mominur Parvez Rafil	20
  40	Shahadat Hossain	22
  41	Sazzad Hossain	18
  42	Alamin Mia	23
  43	Miraj Hossain Shuvo	22
  44	Shongita Rani Roy	27
  45	Nargis Akter	20

  Tabulation of scores

  # Number of the class = (Range/class Interval) +1

  =  3.2 + 1

  Frequency Distribution Table

  Class Interval (C.I.)	Tallies	Frequency
  11 -15	|	1
  16 – 20	|||| |||| ||||	14
  21 – 25	|||| |||| |||| |||| |||| ||||	30
  26 – 30	|||| |||| |||	13

  5.2.1 Measures of Central Tendency

  5.2.1.1 The Mean

  # True Mean = A.M.+(Σfd/N)i

  C.I.	Mid point	Frequency (f)	Deviation (d)	Product (fd)
  11 -15	13	1	-2	– 2
  16 – 20	18	14	– 1	– 14
  21 – 25	23	30	0	0
  26 – 30	28	13	+ 1	+ 13

  N = 58   Σfd = -3

  Here – A.M.= 23 N = 58 Σfd = – 3 i = 5

  True Mean =
  A.M.+(Σfd/N)i

  23 + (- 3/58) 5

C.I.

Lower and upper limit of C.I.

Frequency (f)

Cumulative Frequency (cfu)

11 -15

10.5 – 15.5

1

1

16 – 20

15.5 – 20.5

14

15

21 – 25

20.5 – 25.5

30

45

26 – 30

25.5 –30.5

13

58

Here-

L = 20.5

Cfu = 15

fm = 30

i = 5

N = 58

Median = 22.9

Mode = 23.2

C.I.

Mid point (X)

Mean

f

d = x – M

fd

11 -15

13

22.75

1

– 9.75

– 9.75

16 – 20

18

22.75

14

– 4.75

+ 66.5

21 – 25

23

22.75

30

+ 0.75

26 – 30

28

22.75

13

+ 5.25

+ 68.25

Here-

S|fd| = 145.25

N = 58

Mean Deviation = 2.50 (Apprx.)

C.I.

f

Deviation (d)

fd

fd²

11 -15

1

-2

– 2

+ 4

16 – 20

14

– 1

– 14

+14

21 – 25

30

0

0

0

26 – 30

13

+ 1

+ 13

13