MUHAMMAD SHARIFF BIN ARIFIN960713-11-5237SABAH SCIENCE SECONDARY SCHOOL

TABLE OF CONTENT

NUM.CONTENTPAGE

1. Acknowledgement

2. Objective

3. Introduction

4. Part 1 Importance of data analysis in daily life Measure of central tendency Measure of dispersion

5. Part 2 Raw Data Ungrouped Data Analysis Mean Median Mode Standard Deviation Grouped Data Analysis Mean Mode Median Standard Deviation Interquartile Range Comparison between Ungrouped and Grouped Data

6. Part 3 Conjecture on Uniform Data Changes on Grouped Data

7. Further Exploration

8. Reflection

9. Conclusion

ACKNOWLEDGEMENT

First and foremost, praise be to Allah, God the Almighty for giving me strength and will to done this Additional Mathematics Project Work Form 5 2013.

I would like to give a lot of appreciation to the school management lead by Tuan Haji Zaini b. Zair, the Excellent Principal of Sabah Science Secondary School for giving me opportunity and support to done this task.

I would like to thank the most important person, Mr. Rasman b. Sabar, my Additional Mathematics teacher, as he gives us important guidance and commitment during this project work. He has been a very supportive figure throughout the whole project.

I also would like to thanks to all my classmates from 5 Beta for helping me and always supporting me to complete this project work. They have done a vital job, sharing fundamental information with other people including me. Without them this project would never have had its conclusion.

For their strong support, I would like to express my gratitude to my beloved parents for helping me to find the mark to complete this project. They have always been by my side and I hope they will still be there in the future.

Last but not least, I would also like to thank the entire squad of teachers and my friends for helping me collect the much needed data and statistics for this task.

Not forgotten, to all the individuals who were involved directly or indirectly towards making this project a reality.

OBJECTIVE

The aims of carrying out this project work are;

To apply and adapt a variety of problem-solving strategies to solve routine and non-routine problems.

To experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills.

To experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems.

To experience classroom environments where expressing students mathematical thinking, reasoning and communication are highly encouraged and expected.

To experience classroom environments that stimulates and enhances effective learning.

To acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely.

To enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increases interest and confidence.

To prepare students for the demands of their future undertakings and in workplace.

To realise that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.

To train students not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in an engaging and healthy environment.

To use technology especially the ICT appropriately and effectively.

To train students to appreciate the intrinsic values of mathematics and to become more creative and innovative.

To realise the importance and the beauty of mathematics.

INTRODUCTION

By the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. In the early 19th century, the meaning of "statistics" broadened, then including the discipline concerned with the collection, summary, and analysis of data. Today statistics is widely employed in government, business, and all the sciences.

Electronic computers have expedited statistical computation, and have allowed statisticians to develop "computer-intensive" methods. The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly used probability theory, whose initial results were found in the 17th and 18th centuries, particularly in the analysis of games of chance (gambling). By 1800, astronomy used probability models and statistical theories, particularly the method of least squares, which was invented by Legendre and Gauss.

Early probability theory and statistics was systematized and extended by Laplace; following Laplace, probability and statistics have been in continual development. In the 19th century, social scientists used statistical reasoning and probability models to advance the new sciences of experimental psychology and sociology; physical scientists used statistical reasoning and probability models to advance the new sciences of thermodynamics and statistical mechanics. The development of statistical reasoning was closely associated with the development of inductive logic and the scientific method.

Statistics is not a field of mathematics but an autonomous mathematical science, like computer science or operations research. Unlike mathematics, statistics had its origins in public administration and maintains a special concern with demography and economics. Being concerned with the scientific method and inductive logic, statistical theory has close association with the philosophy of science; with its emphasis on learning from data and making best predictions, statistics has great overlap with the decision science and microeconomics. With its concerns with data, statistics has overlap with information science and computer science

During the 20th century, the creation of precise instruments for agricultural research, public health concerns (epidemiology, biostatistics, etc.), industrial quality control, and economic and social purposes (unemployment rate, econometric, etc.) necessitated substantial advances in statistical practices. Today the used of statistic has broadened far beyond its origin. Individuals and organisations use statistics to understand data and make informed decisions throughout the natural and social sciences, medicines, business, and other area. Statistics are generally regarded not as the subfield of mathematics but rather as a distinct, allied, field. Many universities maintain separate mathematics and statistic departments. Statistic is also taught in department as diverse as psychology, education and public health.

PART 1Importance of Data Analysis in Daily LifeData analysis is a process of inspecting, cleaning, transforming, and modelling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. It has multiple facts and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains. Data analysis can be done in different method according to its purpose, needs and requirement. As example, if the dean of a university wants to know whether the place of living of its student affected their academic performance. In other words, does student who live in university hostel can perform better in examination compared with students who live outside the alma mater? Or are there other factors that responsible for the variance in academic performance? For proving this theory, an analysis of data will help describe students performance and explain the relationship between the performance issues and students place of living.Analysis does not involving complex statistics. Data analysis in schools involves the collection of data and using the data to improve teaching and learning. Interestingly, principals and teachers have it simple. In most cases, the collection of data has already been done. Schools regularly collect attendance data, academic performance records, discipline referrals, and a variety of other useful data. Rather than complex statistical formulas and tests, it is generally simple counts, averages, percents, and rates that educators are interested in. There are many benefits of data analysis. For example, data analysis helps in structuring the findings from different sources of data collection like survey research. It is again very helpful in breaking a macro problem into micro parts. Data analysis acts like a filter when it comes to acquiring meaningful insights out of huge data-set. Every researcher has sort out huge pile of data that he/she has collected, before reaching to a conclusion of the research question. Mere data collection is of no use to the researcher. Data analysis proves to be crucial in this process. It provides a meaningful base to critical decisions. It helps to create a complete dissertation proposal. One of the most important uses of data analysis is that it helps in keeping human bias away from research conclusion with the help of proper statistical treatment. With the help of data analysis a researcher can filter both qualitative and quantitative data for an assignment writing projects. Thus, it can be said that data analysis is of utmost importance for both the research and the researcher. Or to put it in another words data analysis is as important to a researcher as it is important for a doctor to diagnose the problem of the patient before giving him any treatment.

Measure Of Central Tendency Central tendency gets at the typical score on the variable, while dispersion gets at how much variety there is in the scores. When describing the scores on a single variable, it is customary to report on both the central tendency and the dispersion. Not all measures of central tendency and not all measures of dispersion can be used to describe the values of cases on every variable. What choices you have depend on the variables level of measurement.

a) Mean

The mean is what in everyday conversation is called the average. It is calculated by simply adding the values of all the valid cases together and dividing by the number of valid cases.

The mean is an interval/ratio measure of central tendency. Its calculation requires that the attributes of the variable represent a numeric scale. In the physical sciences, such variability may result from random measurement errors: instrument measurements are often not perfectly precise, i.e., reproducible, and there is additional inter-rater variability in interpreting and reporting the measured results. One may assume that the quantity being measured is stable, and that the variation between measurements is due to observational error.

b) ModeThe mode is the attribute of a variable that occurs most often in the data set.

For ungroup data, we can find mode by finding the modal class and draw the modal class and two classes adjacent to the modal class. Two lines from the adjacent we crossed to find the intersection. The intersection value is known as the mode.This information is used by business company to find their opportunity to make profit. For example, a shoe manufacturer can find out the most popular model and size of shoes that consumers demand by using this calculation.c) MedianThe median is a measure of central tendency. It identifies the value of the middle case when the cases have been placed in order or in line from low to high. The middle of the line is as far from being extreme as you can get.

There are as many cases in line in front of the middle case as behind the middle case. The median is the attribute used by that middle case. When you know the value of the median, you know that at least half the cases had that value or a higher value, while at least half the cases had that value or a lower value.This calculation is used to find out the average of something. For example, we can find out the average height of 5 Beta students by simply line up the students according to their height. Then, the student in the middle of the line tells his or her height. His or her height picturing the average of the class height.

Measure Of Dispersiona) RangeThe distance between the minimum and the maximum is called the range. The larger the value of the range, the more dispersed the cases are on the variable; the smaller the value of the range, the less dispersed (the more concentrated) the cases are on the variableRange = maximum value minimum valueInterquartile range (IQR) is the distance between the 75th percentile and the 25th percentile. The IQR is essentially the range of the middle 50% of the data. Because it uses the middle 50%, the IQR is not affected by outliers or extreme values.

IQR = Q3 - Q1b) Standard DeviationThe standard deviation tells you the approximate average distance of cases from the mean. This is easier to comprehend than the squared distance of cases from the mean. The standard deviation is directly related to the variance. If you know the value of the variance, you can easily figure out the value of the standard deviation. The reverse is also true. If you know the value of the standard deviation, you can easily calculate the value of the variance. The standard deviation is the square root of the variance.

PART 2

1. Mid-Semester 1 Additional Mathematics examination scores for 34 students in 5 Beta.

StudentsMarks

1 72

2 70

3 68

4 60

5 57

6 56

7 56

8 55

9 53

10 49

11 48

12 47

13 47

14 47

15 46

16 46

17 43

18 42

19 42

20 42

21 41

22 41

23 39

24 39

25 38

26 38

27 38

28 38

29 36

30 35

31 34

32 30

33 28

34 21

2. Calculation of mean, median, mode and standard deviation.StudentsMarks (x)

1 725184

2 704900

3 684624

4 603600

5 573249

6 563136

7 563136

8 553025

9 532801

10 492401

11 482304

12 472209

13 472209

14 472209

15 462116

16 462116

17 431849

18 421764

19 421764

20 421764

21 411681

22 411681

23 391521

24 391521

25 381444

26 381444

27 381444

28 381444

29 361296

30 351225

31 341156

32 30900

33 28784

34 21441

Question 2a) Mean

N = 34

b) Median

Arrange all of the marks in ascending order:

21, 28, 30, 34, 35, 36, 38, 38, 38, 38, 39, 39, 41, 41, 42, 42, 42, 43, 46, 46, 47, 47, 47, 48, 49, 53, 55, 56, 56, 57, 60, 68, 70, 72.

17th valueMedian of the marks =

= 17th value = 42 marks

c) Mode of the marks:

21, 28, 30, 34, 35, 36, 38, 38, 38, 38, 39, 39, 41, 41, 42, 42, 42, 43, 46, 46, 47, 47, 47, 48, 49, 53, 55, 56, 56, 57, 60, 68, 70, 72.

Occur frequently

Number of the marks that occur frequently = 38 marks.

d) Standard deviation

=

= = 12.50

Question 3 Frequency distributions table:MARKSFREQUENCY

21-303

31-409

41-5013

51-606

61-702

71-801

a) I) Mean The mean mark of 34 students can be found by using the formula:

MarksMidpoint,

Frequency,

21-3025.5376.5

31-4035.59319.5

41-5045.513591.5

51-6055.56333

61-7065.52131

71-8075.5175.5

From the table,f = 34fx = 1527Therefore, mean, = 44.91

II) Mode

The modal class is 41-50, this because, majority of the students got that marks.

To find the mode weight, we draw the modal class and two classes adjacent to the modal class.

From the histogram, the mode mark is 45.5

III) Median

Method 1 By using formula Median is the value of the centre of a set of data Median weight for 34 students can be obtained by using the formula:

MarksLower boundaryUpper boundaryFrequency,Cumulative frequency

21-3020.530.533

31-4030.540.5912

41-5040.550.51325

51-6050.560.5631

61-7060.570.5233

71-8070.580.5134

From the table, Median class = 34 2 = 17th value = 41-50 classL = 70.5 fm = 13 Total frequency N = 34

F = 12 C = 50.5-40.5 = 10

Method 2 by drawing an ogive

Ogive

Ogive is a graph constructed by plotting the cumulative frequency of a set of data against the corresponding upper boundary of each class.

Not only that, ogive is also the method of calculation, the median, and the interquartile range of a set of data can also be estimated from its ogive.

From the ogive,

Median = 44.5IV) Interquartile range

Method I By using formula

class = 34 x = value= 31-40

Total frequency N= 34

= 36.61

class = 34 x = value = 51-60

Therefore, the interquartile range, : = 14Method II By using ogive

From the ogiveInterquartile range = 50.5 36.5 = 14

V) The Standard deviationMethod IMarksMidpoint,Frequency,

21-3025.5376.51950.75

31-4035.59319.511342.25

41-5045.513591.526913.25

51-6055.5633318481.5

61-7065.521318580.5

71-8075.5175.55700.25

= 44.91

= 11.37Method IIMarksMidpoint,Frequency,

21-3025.53-19.41376.751130.25

31-4035.59-9.4188.55796.95

41-5045.5130.590.354.55

51-6055.5610.59112.15672.9

61-7065.5220.59423.95847.9

71-8075.5130.59935.75935.75

b) Mean, x = 44.91 kg Median, m = 44.35 Mode = 45.5

From the above measure of central tendency, mean is the best measure of central tendency because all the values in the data are taken into account when determining the mean. It also quite evenly distributed and no extreme values exist in the data, whereas mode and median does not take all the values in the data into account which decrease the accuracy of central tendency.

b) The standard deviation gives a measure of dispersion of the data about the mean. A direct analogy would be that of the interquartile range, which gives a measure of dispersion about the median. However, the standard deviation is generally more useful than the interquartile range as it includes all data in its calculation. The interquartile range is totally dependent on just two values and ignores all the other observations in the data. This reduces the accuracy it extreme value is present in the data. Since the marks does not contain any extreme value, standard deviation give a better measures compared to interquartile range.

Question 4a) Group data gives a more accurate representation as the data that been grouped is easier to look for patterns and saves more spaces than ungrouped data. Besides, group data also make the calculation much easier than ungrouped data.b) Grouped data is when there is a large number of possible outcomes, we will usually need to group the data. For example, the ages of 200 people entering a park on a Saturday afternoon. The ages have been grouped into the classes 0-9, 10-19, 20-29, and so on. Ungrouped data is the opposite of grouped data with only one possible answer. For example, the ages of 200 people entering a park on a Saturday afternoon. The ages are: 27, 8, 10, 49 and so on.

PART 3

Question 1)

a) If the teacher adds 3 marks for each student in class for completing all their assignment, the new marks for 5 Beta students as shown below

StudentsMarks

158

241

331

442

545

650

737

849

949

1045

1171

1259

1341

1439

1558

1641

1775

1845

1944

2052

2141

2263

2350

2459

2538

2651

2773

2842

2933

3044

3146

3260

3324

3450

b) New frequency distributions table:MarksLower boundaryMidpoint,Frequency,Cumulative frequency

21-3020.525.51125.5650.25

31-4030.535.556177.56301.25

41-5040.545.51521682.531053.75

51-6050.555.5930499.527722.25

61-7060.565.513165.54290.25

71-8070.575.5334226.517100.75

a) New mean,

b) New Mode,

New Mode = 46.5

c) New MedianMedian class = 41-50

Class interval remain same Class interval = 10 c) New interquartile range Q1 class = 34 x = 8.5th value = 31-40

class = 34 x = value = 51-60

Therefore the interquartile range, Q3 Q1 = 51.33-36.61 = 14.72

d) New standard deviation,

Question 2: MarksLower boundaryMidpoint

Frequency,

Cumulative frequency

21-3020.525.51125.5650.25

31-4030.535.556177.56301.25

41-5040.545.51521682.531053.75

51-6050.555.5930499.527722.25

61-7060.565.513165.54290.25

71-8070.575.5334226.517100.75

81-9080.585.503400

91-10090.595.513595.59120.25

New mean,

New standard deviation,

FURTHER EXPLORATION

2. Aspect5 BetaMr. Mas class

Mean44.7976.79

Standard Deviation12.0010.36

Mr. Mas class has gotten better marks for Additional Mathematics subject for this Mid-Year Examination. Mean shows the average scores of students. The mean mark of 5 Beta is 47.8 while Mr. Mas class is 76.79 which are higher than 5 Beta class. This case shows that the students in Mr. Mas class scored better than 5 Beta class. The standard deviation shows that dispersion between the data sets. Standard deviation of Mr. Mas class is lower than the class of 5 Beta. Standard deviation of 5 Beta is 20.28 while standard deviation of Mr. Mas class is 10.36 which are lower than 5 Beta class. This case shows that the marks have small differences. It is proven that Mr. Mas class scored very close to the average. The standard deviation for 5 Beta is greater which means the marks dispersion is larger. This case shows that the marks have greater difference. Given the mean of Mr. Mas class is greater than the class of 5 Beta showing that the average is higher than 5 Beta, concluded that Mr. Mas class has better result than 5 Beta.

REFLECTIONWhile I conducting this project, a lot of information that I found. I have learnt how statistics appear in our daily life.

Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.

Not only that, I had learned some moral values that I practice. This project had taught me to responsible on the works that are given to me to be completed. This project also had made me felt more confidence to do works and not to give easily when we could not find the solution for the question. I also learned to be more discipline on time, which I was given about a month to complete this project and pass up to my teacher just in time. I also enjoy doing this project I spend my time with friends to complete this project and it had tighten our friendship. Last but not least, I proposed this project should be continue because it brings a lot of moral value to the student and also test the students understanding in Additional Mathematics.Additional Mathematics...From The Moment I Was Born...From The Moment I Was Able To Hold a Pencil...From The Moment I Start Learning...And..From The Moment I Started to Know Your Name...I Always Thought That You Will Be My Greatest EnemyAndRival In Excelling In My Life...But After Countless Of Hours...Countless Of Days...Countless Of Seasons...After Sacrificing My Precious Time Just For You...Sacrificing My IPhone...Sacrificing My Nikon Camera...Sacrificing My Galaxy Tab...Sacrificing My Facebook...I Realized Something Really Important In You...I Really Love You...You Are My Best Friend Forever...You Are My Superheroes...You Are My Soul Mate...

ADDITIONAL MATHEMATICS PROJECT WORK | 29STATISTICS