This is a sample of an internal assessment for csec physical education.
SBA
sba
past paper
CXC Additional Mathematics School Based Assessment Title: Using statistical knowledge to determine whether absenteeism in school affects Form 5 students’ Mathematics marks. Name: Candidate #: School: Centre #: Class: Teacher: -
Table of Contents Item ▪ Title of Project ▪ Purpose of the Project ▪ Method of Data Collection ▪ Presentation of Data ▪ Mathematical Knowledge & Analysis of Data ▪ Discussion of Findings & Conclusion ▪ Appendix
Page#
Title of Project Using statistical knowledge to determine whether absenteeism in school affects students’ Mathematics marks.
Purpose of the Project In recent times, many students of High School have complained about Mathematics being too difficult as an excuse to receiving unsatisfactory examination marks. But other factors may have influenced their results mainly: absenteeism. The purpose of this project is to investigate whether absenteeism affects Mathematics marks of students. The independent variable is the number of times a student was absent. The dependent variable is Mathematics marks.
Method of Data Collection There are 160 students in the entire Form 5 of East High School; it was decided that data would be obtained from 50% of the students. Therefor the sample frame comprised of 80 students. In order obtain the data needed, 80 questionnaires were printed and distributed randomly among the Form 5’s. The data collected was divided in to 2 groups: Group 1 contained students who were absent 3 times or less. (≤ 3) Group 2 contained students who were absent more than 3 times. (> 3)
Copy of questionnaire: In appendix
Presentation of Data The back-to-back stem and leaf diagram below shows data obtained from students of Group 1 and Group 2.
Back-to-Back Stem and Leaf Diagram ≤ 3 times absent (Group 1)
> 3 times absent (Group 2)
1
4
2
0 2 3
3
6
9 7
4
0 0 4 4 4 4 7 7 8 8 8 8 9 9
8 5 3 0
5
0 2 5 6 6
9 8 5 5 5 5 2 2 1 0
6
1 2 2 3 3 4 5 5 6 6 8 9 9 9
7 5 5 3 2 1 0 0 0
7
2 6
8 8 8 7 5 3 1 0
8
5 5 2 0 0 0
9
0
10
Key: 5/6 means 65
Key: 5/0 means 50
Box and Whisker Diagrams
Mathematical Knowledge and Analysis of Data Data from Groups 1 and 2 were grouped.
Group 1
Marks
Frequency (𝑓)
Cumulative Frequency
Midpoint of Class interval (𝑥 )
𝑓 × 𝑥
= 𝑓𝑥
𝑥 − 𝑥ҧ =𝑥−
𝑥 − 𝑥ҧ
2
𝑓 𝑥 − 𝑥ҧ
74.25
40 – 49
2
2
44.5
89
-29.75
885.06
1770.12
50 - 59
4
6
54.5
218
-19.75
390.06
1560.24
60 – 69
10
16
64.5
645
-9.75
95.06
950.6
70 - 79
9
25
74.5
670.5
0.25
0.06
0.54
80 - 89
8
33
84.5
676
10.25
105.06
840.48
90 - 99
6
39
94.5
567
20.25
410.06
2460.36
100-109
1
40
104.5
104.5
30.25
915.06
915.06
n=σ 𝑓=40
2
σ 𝑓 𝑥= 2970
Histogram and Cumulative Frequency are displayed on proceeding pages.
σ 𝑓(x-𝑥)ҧ 2 = 8497.4
Group 2 Marks
Frequency (𝑓)
Cumulativ e Frequency
Midpoint of Class interval (𝑥 )
𝑥 − 𝑥ҧ
2
𝑓 𝑥 − 𝑥ҧ
= 𝑓𝑥
𝑥 − 𝑥ҧ =𝑥− 51.75
𝑓 × 𝑥
10 – 19
1
1
14.5
14.5
-37.25
1387.56
1387.56
20 – 29
3
4
24.5
73.5
-27.25
742.56
2227.68
30 – 39
1
5
34.5
34.5
-17.25
297.56
297.56
40 – 49
14
19
44.5
623
-7.25
52.56
735.84
50 – 59
5
24
54.5
272.5
2.75
7.56
37.8
60 – 69
14
38
64.5
903
12.75
162.56
2275.84
70 - 79
2
40
74.5
149
22.75
517.56
1035.12
n=σ 𝑓=40
σ 𝑓 𝑥= 2070
2
σ 𝑓(x-𝑥)ҧ 2 = 7997.4
Histogram and Cumulative Frequency curve are displayed on the proceeding pages.
Measures of Central Tendency ▪ Mean (𝑥)ҧ The mean is the arithmetic average. It is determined by finding the sum of all the numbers and then dividing it by the amount of numbers. Mathematical σ 𝑓𝑥 formula for mean is: 𝑥ҧ = 𝑛
Mean for Group 1:
σ 𝑓𝑥 𝑛
= 2970 ÷ 40 = 74.25 (to 2 d.p.)
Mean for Group 2:
σ 𝑓𝑥 𝑛
= 2070 ÷ 40 = 51.75 (to 2 d.p.)
▪ Mode The mode is the most frequent value. It can be found on the histogram by estimating the value(s) from the modal class or classes. The estimated mode of Group 1 is: 68 The estimated mode of Group 2 is both: 45.5 & 63.5. It is bimodal.
▪ Median (𝑄2 )
The median (𝑄2 ) of a set of data is the middle observation. The data must be arranged in numerical order. Formula for median is: 𝑄2 = (𝑛+1) . The 2 median can also be estimated from a cumulative frequency curve. From the respective cumulative frequency curves: Estimated Median of Group 1, 𝑄2 = 74.5 Estimated Median of Group 2, 𝑄2 = 51
Measures of Dispersion Measures of dispersion are used to show the spread of the data.
▪ Variance Variance uses the mean as a point of reference to measure spread of data. σ 𝑓(x−𝑥)ҧ 2 The formula for variance is: 𝑠 2 = 𝑛
Variance for Group 1 =
σ 𝑓(x−𝑥)ҧ 2 𝑛
8497.4 40 = 212.44 (to 2 d.p.)
=
Variance for Group 2 =
σ 𝑓(x−𝑥)ҧ 2 𝑛
7997.4 = 40 = 199.94 (to 2 d.p.)
▪ Standard Deviation The standard deviation is the square root of the variance. The formula is: s=
σ 𝑓(x−𝑥)ҧ 2 𝑛
Standard deviation for Group 1 =
σ 𝑓(x−𝑥)ҧ 2 𝑛
= 212.44 = 14.58 (to 2 d.p.) Standard deviation for Group 2 =
σ 𝑓(x−𝑥)ҧ 2 𝑛
= 199.94 = 14.14 (to 2 d.p.)
▪ For Group 1: Range: This is upper class boundary subtract the lower class boundary = 109.5 – 39.5 = 70
Quartiles: The lower quartile is the median for the lower half of the data obtained. It can be estimated from the cumulative frequency curve. The lower quartile, 𝑄1 is 65.5 The upper quartile is the median for the upper half of the data obtained. It can be estimated from the cumulative frequency curve. The upper quartile, 𝑄3 is 85.5 Interquartile range = 𝑄3 - 𝑄1 = 85.5 – 65.5 = 20
Semi-Interquartile range =
𝑄3
=
− 𝑄1
2 20 2
= 10
▪ For Group 2:
Range: This is upper class boundary subtract the lower class boundary = 79.5 – 9.5 = 70
Quartiles: The lower quartile is the median for the lower half of the data obtained. It can be estimated from the cumulative frequency curve. The lower quartile, 𝑄1 is 43.5 The upper quartile is the median for the upper half of the data obtained. It can be estimated from the cumulative frequency curve. The upper quartile, 𝑄3 is 62.5 Interquartile range = 𝑄3 - 𝑄1 = 62.5 – 43.5 = 19
Semi-Interquartile range =
𝑄3
− 𝑄1 2
=
19 2
= 9.5
Discussion of Findings & Conclusion A back to back stem-and-leaf diagram was used to compare the 2 groups of data obtained. The lowest mark of Group 1 was 47 while Group 2’s lowest mark was 14, indicating that a student who was absent more than 3 times had the lowest mark of the data obtained. The highest mark of Group 1 was: 100 while Group 2’s highest mark was: 76 showing therefore student from Group 1 gained the highest mark of the total. Group 1’s mean was 74.5 and Group 2’s mean was 51.75. This shows that average mark of Group 1 was higher than that of Group 2. The mode of Group 1 was 65 and Group 2 was both 45.5 & 63. (bimodal). Comparing the box and whisker diagrams, Group 1’s median value lies closer the centre than Group 2 , hence Group 1 was more symmetrical than Group 2. Group 2’s median lies closer to the left indicating it is more positively skewed than Group 1. This indicates that Group 2’s students’ marks were lower.
Since Group 2 was both bimodal and positively skewed while Group 1 was fairly symmetrically skewed and the best measure of central tendency to use would be the mean. The standard deviation of Group 1 was 14.58 and Group 2 was 14.14. This indicates that Group 1 and Group 2 had a similar spread of marks but Group 1 had a slightly greater range of marks in the end. For future analysis, it is recommended that questionnaires should be distributed to the entire Form 5 student population for more accurate results i.e. increasing the sample size. From the results of the analysed data, it can be concluded that students who were absent more than three times (Group 2) did worse the students who were absent three times or less (Group 1); hence absenteeism affects academic performance negatively.
Appendix The questionnaire that was used to gather the data for the project:
CSEC Add Maths SBA Questionnaire I am an Additional Mathematics student from Couva East Secondary School. The purpose of this questionnaire is to collect the necessary data for my School Based Assessment. Please be completely honest and truthful. Place a tick in the boxes.
1. What is your gender ? Male
Female
2. What is your percentage in Mathematics based on the previous term ? State your mark here ___________%
3. On estimate how many times were you absent in the previous term ? 3 times or less
More than 3 times
Thank You for participating
The raw data obtained from the questionnaires. For Group 1: 88
