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