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Additional Mathematics School Based Assessment done for CSEC examinations
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Name: Jeremy Ammon
School: Fatima College
School Code:
Subject: CSEC Additional Mathematics
Project Title: Determining the types of changes in the class averages of an English class within the last three terms
Problem Statement:
An English B teacher, Ms.Habib would like to know if the class average of Form 5 Group 1 has increased over the last three terms and how much variation from the class average exists within each term
In order to analyze the marks and determine the trend between class averages, the following must be determined:
1-Individual End of Term marks for all three terms (including calculating averages for term 2 which is coursework)
2-What End of Term mark appears most frequently in each term, The averages: mean, median, mode.
3-Range of marks for each term
4-Maximum of each 25% range of marks within each term
Data was collected by:
1-The teacher’s folder that contains records of exam results for all English B students of Form 5 Group 1 for terms 1, 2 and 3
Data Representation (See Appendix) :
Matthew
Malachi
Nikolai
Robert
JeremyKyle
StephenJave
dMarc Erik
Theon
Risaan
Isaac
Giovanni
Jarrett
Jabari
Joshua
DanielJusti
n
Sebastien
Jonathan
Jordan
Donovan
Willi
am0
10
20
30
40
50
60
70
80
90
100
End of Term Marks for Form 5 Group 1 English B
Term 1 Term 2 Term 3
The following Stem and Leaf diagram is used to represent the marks of Form 5 Group 1 for Term 1:
Stem Leaf
4 3, 6, 8, 9
5 2, 8, 9
6 0, 3, 4, 4, 5, 6, 7, 8, 9
7 0, 1, 2, 5, 6, 7, 8
8 2
Key: 4│3 represents 43
Maximum of each 25% range is also known as quartiles
Q1-First 25%
Q2-25 to 50% (median)
Q3-50 to 75%
Sum of scores= ∑f
No. of students =∑fx or n
∑f= 1542
∑fx=24
Q1= ¼ (n+1)th
= ¼ (24+1)th
= ¼ (25)th
= 6.25th
= 58
Q2(median)= ½ (n+1)th
= ½ (24+1)th
= ½ (25)th
= 12.5th
=65
Q3= ¾ (n+1)th
= ¾ (24+1)th
= ¾ (25)th
= 18.75th
= 72
Class Average= ∑f/∑fx
= 1542/24
=64
Modal score (most frequent)= 64
Range= 43-82
The following table is used to calculate Standard Deviation for Term 1:
X x- (x- )2 f(x- )2
43 -21 441 441
46 -18 324 324
48 -16 256 256
49 -15 225 225
52 -12 144 144
58 -6 36 36
59 -5 25 25
60 -4 16 16
63 -1 1 1
64 0 0 0
65 1 1 1
66 2 4 4
67 3 9 9
68 4 16 16
69 5 25 25
70 6 36 36
71 7 49 49
72 8 64 64
75 11 121 121
76 12 144 144
77 13 169 169
78 14 196 196
82 18 324 324
2626
Establish that: x= score
= class average
f= number of students with that mark
Variance (S2 ) = ∑f(x- )2 /n
=2626/24
= 109
Standard Deviation= √S2
= √109
= 10.44
The following table is used to solve for quartiles and class average for term 2 (See Appendix):
Mark Frequency Cumulative Frequency
57 1 1
58 2 3
59 1 4
61 1 5
63 1 6
65 2 8
66 1 9
67 1 10
70 1 11
71 1 12
72 3 15
73 1 16
74 4 20
76 2 22
77 1 23
83 1 24
∑f= 1657
Range= 57-83
Modal Score= 74
n= no. of students
‘th’ refers to the place in the Cumulative Frequency column with respect to that mark
The following Stem and Leaf diagram represents ALL term 2 coursework marks (See Appendix):
Stem Leaf
4 8,8,9
5 0,1,1,2,4,5,8,8,8,9,9
6 0,1,1,2,2,3,3,4,4,5,5,6,6,6,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9
7 0,0,0,0,0,0,0,1,2,2,2,2,2,2,4,4,4,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8,
8,9,9,9,9
8 0,0,1,1,2,3,3,4
Key: 4│8 represents 48
Q1= ¼ (n+1)th
= ¼ (24+1)th
= ¼ (25)th
= 6.25th
= 63
Q2 (median) = ½ (n+1)th
= ½ (24+1)th
= ½ (25)th
= 12.5th
= 71
Q3= ¾ (n+1)th
= ¾ (24+1)th
= ¾ (25)th
= 18.75th
= 74
Class Average= ∑f/∑fx
= 1657/24
= 69
The following table is used to calculate Standard Deviation for Term 2:
X x- (x- )2 f(x- )2
57 -12 144 144
58 -11 121 242
59 -10 100 100
61 -8 64 64
63 -6 36 36
65 -4 16 32
66 -3 9 9
67 -2 4 4
70 1 1 1
71 2 4 4
72 3 9 27
73 4 16 16
74 5 25 100
76 7 49 98
77 8 64 64
83 14 196 196
1137
Variance (S2 ) = ∑f(x- )2/n
= 1137/24
= 47
Standard Deviation= √S2
= √47
= 6.86
The following table is used to solve for quartiles and class average for Term 3:
Mark Frequency Cumulative Frequency
57 1 1
58 1 2
59 1 3
68 1 4
70 1 5
72 1 6
73 2 8
74 1 9
77 2 1
78 3 14
79 1 15
80 1 16
81 1 17
82 1 18
83 1 19
84 1 20
87 1 21
88 1 22
89 1 23
92 1 24
∑f= 1837
Modal score= 78
Range= 57- 92
Q1= ¼ (n+1)th
= ¼ (24+1)th
= ¼ (25)th
= 6.25th
= 72
Q2 (median) = ½ (n+1)th
= ½ (25)th
= 12.5th
= 78
Q3= ¾ (n+1)th
= ¾ (24+1)th
= ¾ (25)th
= 18.75th
= 83
Class Average= ∑f/∑fx
= 1837/24
= 77
*n= number of students
The following table is used to calculate the Standard Deviation for Term 3:
Establish that: x=score
=class average
f= no. of students with that mark
x x- (x- )2 f(x- )2
57 -20 400 400
58 -19 361 361
59 -18 324 324
68 -9 81 81
70 -7 49 49
72 -5 25 25
73 -4 16 32
74 -3 9 9
77 0 0 0
78 1 1 3
79 2 4 4
80 3 9 9
81 4 16 16
82 5 25 25
84 7 49 49
87 10 100 100
88 11 121 121
89 12 144 144
92 15 225 225
2013
Variance (S2 ) = ∑f(x- )2 /n
= 2013/24
= 84
Standard Deviation= √S2
=√84
= 9.17
It is concluded that the class average of Form 5 Group 1 for English B has increased each term and is demonstrated by the following graph:
Term 1 Term 2 Term 30
10
20
30
40
50
60
70
80
90
Class Average
Class Average
Term 1 Term 2 Term 30
10
20
30
40
50
60
70
80
90
Comparing the Mean, Median and Modal scores of Form 5 Group 1 for all three terms
Mean Median Mode
We can see from the graph that the Mean, Median and Mode all increase from the first to third term, which is an indicator that students are progressively performing better in the exams each term.
In the future, marks will be kept in order so that analysis may become easier and quicker and ranges and boundaries will be used in analyzing the class marks so that an average may be of a faster result and the modal range can be clearly seen.
Appendix:(raw data)
Term 1 scores:
Matthew-64
Malachi-43
Nikolai-52
Robert-46
Jeremy-67
Kyle-65
Stephen-72
Javed-66
Marc-58
Erik-64
Theon-69
Risaan-49
Isaac-59
Giovanni-63
Jarrett-60
Jabari-71
Joshua-76
Daniel-78
Justin-77
Sebastien-70
Jonathan-68
Jordan-48
Donovan-75
William-82
Term 2 (coursework):
Matthew- 61, 69, 79, 76 -Avg mark: 71
Malachi- 48, 62, 54, 68 -Avg mark: 58
Nikolai- 51, 49, 67, 67 -Avg mark: 59
Robert- 50, 55, 59, 64 -Avg mark: 57
Jeremy- 72, 68, 77, 70 -Avg mark: 72
Kyle-68, 78, 70, 72 -Avg mark: 72
Stephen-75, 76, 70, 81 -Avg mark: 76
Javed-66, 69, 77, 74 -Avg mark: 72
Marc-62, 59, 68, 69 -Avg mark: 65
Erik-74, 68, 79, 75 -Avg mark: 74
Theon-48, abs, 67, 75 -Avg mark: 63
Risaan-52, 58, abs, 65 -Avg mark: 58
Isaac-abs, 60, 66, 70 -Avg mark: 65
Giovanni-61, 64, 72, abs -Avg mark: 66
Jarett-66, 71, 69, 74 -Avg mark: 70
Jabari-74, 58, 65, 70 -Avg mark: 67
Joshua-81, 76, 77, 68 -Avg mark: 76
Daniel-70, 63, 79, 80 -Avg mark: 73
Justin-83, abs, 72, 67 -Avg mark: 74
Sebastien-72, 74, 76, abs -Avg mark: 74
Jonathan-74, 68, 77, 78 -Avg mark: 74
Jordan-51, 63, 58, 70 -Avg mark: 61
Donovan-76, 79, 72, 80 -Avg mark: 70
William-83, 81, 82, 84 -Avg mark: 83
Rounded off as: <0.5=0 & ≥0.5=1
Avg- average
abs- absent
Term 3 marks(End of term):
Matthew- 78
Malachi- 59
Nikolai- 58
Robert- 57
Jeremy- 82
Kyle- 77
Stephen- 84
Javed- 73
Marc- 74
Erik- 79
Theon- 78
Risaan- 68
Isaac- 77
Giovanni- 70
Jarrett- 72
Jabari- 73
Joshua- 80
Daniel- 83
Justin- 88
Sebastien- 89
Jonathan- 87
Jordan- 78
Donovan- 81
William- 92
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