Additional Mathematics Project Work 2013 - Statistics

SMK. ST.FRANCIS CONVENT (M), 88000 KOTA KINABALU, SABAH ADDITIONAL MATHEMATICS PROJECT WORK 2013 STATISTICS OF STUDENTS’ SCORES IN AN EXAMINATION NAME CLASS I/C NUMBER : Noor Shazlien bt. M. Jamal : 5 Pure Science 1 : EXAM REGISTRATION NO: CONTENTS            Acknowledgement......................................................................................... 2 Objectives..................................................................................................... 3 Introduction................................................................................................... 4 Task Specification......................................................................................... 8 Part 1............................................................................................................. 9 Part 2............................................................................................................. 17 Part 3............................................................................................................. 25 Further Exploration........................................................................................ 26 Conclusion.................................................................................................... 28 Reflection...................................................................................................... 29 Bibliography.................................................................................................. 30 1 ACKNOWLEDGEMENT I wish to express gratitude to God for His guidance, the strength and health He had given me to do this project work. Peace and blessings be upon to our Prophet Muhammad SAW and the family, next to his companions and tabi’in, the gentlemen scholar until the servants of God who followed in their footsteps. To my parents, who always make sure that I‟m well educated, literate and nurtured to this day; I sincerely thank, you. I appreciate the things you‟ve helped and contributed in completing this project work, such as money, encouragements and facilities like Internet and reference books. To my Additional Mathematics teacher, Mr. Saw and tuition teacher, Mr. Carette; thank you for guiding me throughout this project when I had difficulties in doing a task. Thank you so much for the lessons you‟ve taught me since last year. I‟m highly grateful for my group members; Nor Nadia Natasha, Nur Syahira Sofyani and Rowena Diane, for your freewill to did this project with me. Even though this project is presented individually, we cooperated in discussing and sharing ideas to ensure our task is done correctly and finished on time. I did not pray for a lighter load, but for a stronger back. I wouldn‟t have been motivated to complete this project work without the blessings and support from all of you. Thank you. 2 OBJECTIVES Upon completion of the Additional Mathematic project work, I am able to gain valuable experiences based on the objectives:       Apply and adapt a variety of problem solving strategies to solve routine and nonroutine problems. Experiences classroom environment which are challenging, interesting and meaningful and hence, improve thinking skills. Experiences classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems. Experiences classroom environments where expressing one‟s mathematical thinking, reasoning and communication are highly encouraged and expected. Experiences classroom environments that stimulates and enhances effective learning. Acquire effective mathematical communication through oral and writing and to use the language of mathematics to express mathematical ideas of correctly and precisely. Enhance exquisite mathematical knowledge and skills through problem solving in ways that increase interest and confidence. Prepare ourselves for the demand of our future understandings and our work place. Train ourselves to be independent learners but also to collaborate, to cooperate and to share knowledge in an engaging a healthy environment. Use technology especially the ICT appropriately and effectively. Realize the importance and the beauty of mathematics.      We are expected to submit the project work within two weeks from the first day the task is being administered to us. Failure to submit the written report will result us not to receive the certificate. 3 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" is 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. DEFINITION OF STATISTICS Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. 4 BRIEF HISTORY OF STATISTICS Statistical methods date back at least to the 5th century BC. The earliest known writing on statistics appears in a 9th-century book entitled Manuscript on Deciphering Cryptographic Messages, written by AlKindi. In this book, Al-Kindi provides a detailed description of how to use statistics and frequency analysis to decipher encrypted messages. This was the birth of both statistics and cryptanalysis, [10][11] according to the Saudi engineer Ibrahim Al-Kadi. The Nuova Cronica, a 14th-century history of Florence by the Florentine banker and official Giovanni Villani, includes much statistical information on population, ordinances, commerce, education, and religious facilities, and has been described as the first introduction of statistics as a positive element in [12] history. Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural and Political Observations [13] upon the Bills of Mortality by John Graunt. Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology. The scope of the discipline of statistics broadened in the early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences. Its mathematical foundations were laid in the 17th century with the development of the probability theory by Blaise Pascal and Pierre de Fermat (left). Probability theory arose from the study of games of chance. The method of least squares was first described by Carl Friedrich Gauss around 1794. The use of modern computers has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually. 5 History of statistics can be said to start around 1749 although, over time, there have been changes to the interpretation of the word statistics. In early times, the meaning was restricted to information about states. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as in national accounts and temperature records, and analytical work which require statistical inference. Statistical activities are often associated with models expressed using probabilities, and require probability theory for them to be put on a firm theoretical basis. A number of statistical concepts have had an important impact on a wide range of sciences. These include the design of experiments and approaches to statistical inference such as Bayesian inference, each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics. STATISTICS TODAY 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, econometry, etc.) necessitated substantial advances in statistical practices. Today the used of statistic has broadened far beyond its origin. Individuals and organizations 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. MATHEMATICS AS AN AESTHETIC DISCIPLINE This brief piece of article offers a defense of the study of mathematics for those people who are convinced either that mathematics is not worth studying or that mathematics is “just not for them.” We study mathematics for the same reasons we study poetry or music or painting or literature: for aesthetic reasons. Simply put, we study mathematics because it is one of the loveliest disciplines known to man. "The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit." – A.N. Whitehead [Wh]. 6 One of the most compelling aesthetic features of mathematics is its refined austerity. Its unadorned gracefulness is unique among the arts. In fact, part of the very essence of mathematics is its precision. People are referring to this quality when they suggest that mathematics teaches "clear thinking." Mathematics' precision does not lie in any claims of universal truth. But rather this precision, and hence power, lie in the acknowledgement of exactly the points at which mathematics consciously and deliberately abandons claims of universal truth. Mathematics is the only discipline that I am aware of that does this. And this precision and austerity allow for an elegant economy, an economy that comes from the elimination of the cluttering mire of imprecision. "Mathematical knowledge adds vigor to the mind, frees it from prejudice, credulity, and superstition." – John Arbuthnot [Mo]. The common defense is not, however, supplanted by the new defense, but rather it is subsumed by it. This assumption takes the unexpected form of an appreciation for the utility of mathematics. By this I mean that to most students of mathematics, the utility of mathematics should be presented in something like the same fashion as music is presented to students of music history, namely as a marvel to be appreciated, not an instrument to be operated. Those students interested in actually creating music (i.e. in becoming musicians or composers) are advised to study performance or composition. Similarly, those students interested in actually harnessing the utilitarian powers of mathematics (i.e. in becoming engineers and scientists and mathematicians) are advised to study engineering and applied mathematics. But for the vast majority of mathematics students, a simple, honest appreciation of the remarkable utility of mathematics should be seen as the ultimate "real world" goal. In short, the sense of agency developed in most students regarding the utility of mathematics should be of an appreciative nature, not an instrumental nature. Since "appreciation" is an aesthetic term and not a scientific term for most students, the traditional defense of the study of mathematics as a tool is subsumed by the aesthetic perspective of the new defense. Written by J.D. Phillips, Department of Mathematical Sciences 7 TASK SPECIFICATION All the form 5 students who took the Additional Mathematics subject were required to do this project. The main objective of the Additional Mathematic Work 2013 was to analyze the marks for any subject to determine the average grade of the subject in my class. As for me, I had chosen to analyze the class marks of Additional Mathematics subject in the latest examination. In the first part of the question, I had to list the importance of data analysis in daily life. I also had to specify the three types of measure of central tendency and at least two types of measure of dispersion. I had to give examples of their uses in our daily life as well. For part two, I had to calculate the mean, median, mode and standard deviation for the ungrouped data of the subject I had chosen which is Additional Mathematics by using the mark sheet of my class from the latest examination. Then, I had to construct a frequency distribution table which contains at least five class intervals of equal size. By using Mathematical method, I was able to draw histogram and ogive. From the table, graph and formula, I could find the mean, mode, median, standard deviation and interquartile range for the grouped data. Last but not least, by using Mathematics, I was able to draw an ogive to calculate the lowest mark for the Additional Mathematics subject in my class. I was also able to compare the achievement in my class with Mr. Ma‟s class (further exploration). 8 PART ONE Question 1 The importance of data analysis in daily life: Data analysis is the most important in businesses. No business can survive without analyzing available data. Visualize the following situations:    A pharma company is performing trials on number of patients to test its new drug to fight cancer. The number of patients under the trial is well over 500. A company wants to launch new variant of its existing line of fruit juice. It wants to carry out the survey analysis and arrive at some meaningful conclusion. Sales director of a company knows that there is something wrong with one of its successful products however hasn't yet carried out any market research data analysis. How and what does he conclude? These situations are indicative enough to conclude that data analysis is the lifeline of any business. Whether one wants to arrive at some marketing decisions or fine-tune new product launch strategy, data analysis is the key to all the problems. What is the importance of data analysis - instead, one should say what is not important about data analysis. Merely analyzing data isn't sufficient from the point of view of making a decision. How does one interpret from the analyzed data is more important. Thus, data analysis is not a decision making system, but decision supporting system. Data analysis can offer the following benefits:      Structuring the findings from survey research or other means of data collection. Break a macro picture into a micro one. Acquiring meaningful insights from the dataset. Basing critical decisions from the findings. Ruling out human bias through proper statistical treatment. 9 Question 2 MEASURE OF CENTRAL TENDENCY Mode, median and mean are the three measures of central tendency which indicates the central values around which the data seems to cluster. However, the values of these measures may differ greatly. Thus, it is vital to choose one that reflects the central value of data. When an extreme value exists in the data, the mean does not reflect the central value of the data. The median and the mode are not affected by the existence of the extreme value. Nevertheless, the mode is confusing when the data has more than one mode. Mean The mean of a set of ungrouped data can be obtained by adding all the values of the data and dividing the sum by the total number of values of data; ̅ total number of values of data. If the set of ungrouped data is given in a frequency distribution table, then the mean is calculated as follows; ̅ ∑ ∑ ∑ where N is the where f is the frequency. When the values of a set of data are grouped into classes in a frequency table, the value that is used to represent all the values of the data in a class is the midpoint of the class; ̅ Mode The mode of a set of ungrouped data can be determined by identifying the value which occurs most frequently. When the data is given in a frequency distribution table, then the mode is the value with the highest frequency. Example: 15, 11, 12, 14, 11, 16, 17, 11, 19, 11 The mode is 11 since 11 occurs most frequently. ∑ ∑ , where x is the class midpoint and f is the class frequency. 10 In a set of a grouped data, the mode can be found in a histogram. A histogram is a graphic representation of a frequency distribution table. It is constructed by marking the boundaries of each class along the horizontal axis and the frequencies along the vertical axis. The class with the tallest bar is the modal class and based on the diagram below, P is the mode. Median The median of a set of ungrouped data is the value in the middle position of the set when the values of the data are arranged in ascending order. Example: 11, 13, 14, 16, 17, 18, 19, 21, 23 The median is 17 since it is in the middle. 11 If the total number of values of the data is even, there will be two values in the middle. Thus, the median is the mean of the two middle values. Example: 21, 22, 24, 26, 27, 29, 31, 32 Median = = 26. 2 In a set of grouped data, the median can be estimated from an ogive or calculated from a cumulative frequency table by using the following formula: ( ) L = lower boundary of the median class N = total frequency F = cumulative frequency up to the lower boundary of the median class fm = frequency of the median class C = size of the class interval 12 The ogive is constructed by plotting the cumulative frequency against the corresponding upper boundary of each class of a set of data. MEASURE OF DISPERSION Range The simplest measure of dispersion of a set of data is the range. The range of a set of ungrouped data is the difference between the largest value and the smallest value in the data, range = largest value – smallest value. The range of a set of grouped data is the difference between the midpoint of the highest class and the midpoint of the lowest class, range = midpoint of highest class – midpoint of lowest class. Variance ∑ ̅ for a set of ungrouped data and ∑ ∑ ̅ for a set of grouped data, where x is the class midpoint. 13 Standard deviation The square root of the variance is called the standard deviation, has the same unit as each value of the data. √ which Question 3 Examples of the uses of each type of measure of central tendency in daily life: Mode The mode appears the most often out of a given set of numbers. A data set can have more than one mode or no mode at all if all of the numbers appear with equal frequency. The concept of a mode can be easily connected to many tangible, real-life situations. A bakery that sells twice as many red velvet cupcakes as chocolate brownies will need to produce more cupcakes to satisfy their customers. In this bakery, the number of red velvet cupcakes sold is the mode. Mean and average The terms mean and average are used interchangeably in mathematics. When the entirety of a data set is added and divided by the total number of data points, the resulting number is referred to as the mean or the average. Averages are used quite frequently in everyday statistics. Ask the students to find the average age of classmates in the room with or without the age of the instructor included, and in their own family. Median Median is another simple measurement used commonly in basic statistical analysis. When a data set is organized by the size of numbers, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values. Medians can be used when the data set features extremely large or small values that could significantly affect the average. Median is likely to be the best measurement of age when a given group features one person aged 80, and everyone else between 18 and 20. 14 1. In real life, suppose a company is considering to expand into an area and is studying the size of containers that competitors are offering. Would the company be more interested in the mean, the median, or the mode of their containers? Answer: The mode, because they want to know what size tends to sell most often. 2. An ad agency is planning an ad campaign for a city. Would they be more interested in the median or mean family income for the city? Answer: The median, because a few very large incomes could drastically alter the value of the mean as the "average" income in the city. 3. An economist is comparing interest rates on 90-day CDs in 8 major cities. Should he compare the mean, the median, or the mode? Answer: The mean, because these don't usually have outliers with drastically different percentage rates, so the mean is reasonable. 4. A company wants to know if their customers rate their service as excellent, good, average, or poor. Would the company be more interested in the mean, the median, or the mode? Answer: Either the median or the mode. The mode would indicate the most common answer. If numbers were assigned as 4 =excellent, 3=good, 2=average, and 1=poor, the median as the center value would provide useful information. 5. Houses sold at $203000, $214000, $220000, and $4,257,000. Which would be more helpful to a potential buyer, the mean, the median or the mode? Answer: There is no mode that occurs more often. The mean would be skewed by the outlier to indicate a mean price over 1 million dollars. The median as the average of 214000 and 220000 would be far more representative of the center of the prices for recently purchased homes. 6. Scores on a test were 99, 99, 97, 89, 88, 88, and 0. Which would be more helpful to the teacher, the mean, the median or the mode? Answer: The mean would not be helpful due to the obvious outlier. The 2 modes indicate 2 higher positions which are both helpful, but split the effect of either number. The median gives the best overall picture of the data. 15 7. SAT scores in a class are 360, 430, 450, 480,480, 500, 510, 510, 520, 520, 530, 540, 600, 620, 710. Which would be more helpful to the teacher, the mean, the median or the mode? Answer: The mode does little to provide insight. As the data is roughly bellshaped, either the mean or median would be a reasonable predictor. 16 PART TWO Question 1 Class marks of Additional Mathematics subject: Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Marks, 47 59 55 39 48 26 71 51 58 61 57 48 35 39 63 55 71 50 58 42 43 32 52 74 47 61 56 17 Question 2 a. Mean, ̅ = = = 51. 78 b. Median = 52 26, 32, 35, 39, 39, 42, 43, 47, 47, 48, 48, 50, 51, 52, 55, 55, 56, 57, 58, 58, 59, 61, 61, 63, 71, 71, 74 c. Mode = 39, 47, 48, 55, 58, 61, 71 √ ̅ d. Standard deviation, Variance, ∑ = ∑ = 76024 = = 134.5353037 =√ = 11.59893546 = 11.60 18 Question 3 a) Class Interval (Marks) 21 – 30 31 – 40 41 – 50 51 – 60 61 – 70 71 – 80 ∑ ∑ ( ) ( ) ( ) ( ) ( ) ( ) Frequency, 1 4 7 9 3 3 Midpoint, 25.5 35.5 45.5 55.5 65.5 75.5 Upper boundary 30.5 40.5 50.5 60.5 70.5 80.5 Mean, ̅ = = = = 52.17 Mode = based on histogram graph = 50.5 + 2.5 = 53 Median (Method 1) = based on ogive graph = 52 Class Interval (Marks) 11 – 20 21 – 30 31 – 40 41 – 50 51 – 60 61 – 70 71 – 80 Upper boundary 20.5 30.5 40.5 50.5 60.5 70.5 80.5 Frequency, 0 1 4 7 9 3 3 Cumulative frequency 0 1 5 12 21 24 27 Q2 = = 13.5 10.0 Mode 9.0 8.0 7.0 6.0 Fraquency 5.0 4.0 3.0 2.0 1.0 0.0 20.5-30.5 30.5-40.5 40.5-50.5 Marks 50.5-60.5 60.5-70.5 70.5-80.5 20 Ogive 28.0 26.0 24.0 22.0 20.0 18.0 Cumulative Fraquency 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 20.5 30.5 40.5 50.5 Marks 60.5 70.5 80.5 90.5 21 Median (Method 2), = ( ) = 50.5 + ( = 50.5 + (1.67) = 52.17 Standard deviation, Class Interval 21 – 30 31 – 40 41 – 50 51 – 60 61 – 70 71 – 80 Mean, ̅ = 52.17 Midpoint, 25.5 35.5 45.5 55.5 65.6 75.5 ) 10 Frequency, 1 4 7 9 3 3 ∑ = 27 25.5 142.0 318.5 499.5 196.5 226.5 = 1408.5 650.25 5041.00 14491.75 27722.25 12870.75 17100.75 ∑ = ( ̅) 711.29 1111.56 311.42 99.80 533.07 1632.87 ∑ ( ̅) = ∑ 77876.75 4400.01 Method 1 = = = 162.6151741 =√ = 12.75 ∑ ∑ Method 2 ̅ = = = 162.9633333 =√ = 12.77 ∑ ( ∑ ̅) 22 Interquartile range Method 1 Interquartile range = Q3 − Q1 = 59.5 – 43 = 16.5 = 50.5 + ( Method 2 Interquartile range = Q3 − Q1 = 59.67 – 43 = 16.67 = 50.5 + (9.167) = 59.67 = 40.5 + (2.5) = 43 ( ) Q3 = ( ) Q1 = ( ( ) ) ) 10 = 40.5 + ( ) 10 Question 3 b) Based on my answers above, the most appropriate measure of central tendency that reflect the performance of my class is the mean. The mean is able to show the average mark obtained by the class, reflecting the average performance of all the students in the class. Question 3 c) Merits of standard deviation       It is based on all the items of the distribution. It is a mean-able to algebraic treatment since actual + or – signs deviations are taken into consideration. It is least affected by fluctuations of sampling. It facilitates the calculation of combined standard deviation and coefficient of variation, which is used to compared the variability of two or more distributions. It facilitates the other statistical calculations like skewness and correlation. It provides a unit of measurement for the normal distribution. 23 Limitations of quartile deviation     It is not suited to algebraic treatment. It is very much affected by sampling fluctuations. The method of dispersion is not based on the items of the series. It ignores the 50% of the distribution. Question 4 a. Grouped data is more accurate. Ungrouped data are data that are not organized, or if arranged could only be from highest to lowest. Grouped data are data that are organized and arranged into different classes or categories. b. Data in statistics can be classified into grouped data and ungrouped data. A row of data such as 1, 2, 6, 4, 6, 3, 7, is called an ungrouped data. Ungrouped data is any list of numbers that you had gathered. Besides, this data can also be summarized neatly in a frequency distribution table as shown below: Number Frequency 1 3 2 2 3 1 4 2 Ungrouped data is usually used when there are lesser numbers to count or small numbers with only one possible answer. Example: The ages of 200 people going to a park on a Saturday afternoon. The ages are: 27, 8, 10, 49, ... On the contrary, grouped data is a data that has been organized into groups known as classes. Each of these classes is of a certain width and this is referred to as the class interval or class size. Example: Age (years) 0–9 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 Frequency 5 6 7 3 4 5 24 Grouped data is the opposite of ungrouped data which is used when you have a big amount of numbers or large numbers of possible outcomes. Example: The ages of 200 people going to a park on a Saturday afternoon. The ages have been grouped into the classes 0-9, 10-19, 20-29, ... PART THREE Question 1 a. New mean = 52.17 + 3 = 55.17 = 53 + 3 = 56 = 52 + 3 (from Method 1) = 55 = 52.17 + 3 (from Method 2) = 55.17 = 16.67 or 16.5 = 12.77 or 12.75 b. New mode c. New median d. New interquartile range e. New standard deviation Question 2 New mean, ̅ = = = 53.71 New standard deviation, =√ =√ =√ = 14.91 ∑ ∑ ∑ ∑ ̅ ( ) FURTHER EXPLORATION Question 1 = 5.4 27.0 – 5.4 = 21.6 Based on the ogive graph, the lowest mark for the top 20% of the students is 62.5. Question 2 Mr. Ma‟s class has a higher mean when compared to my class, which is 76.79 against 53.71. This shows that Mr. Ma„s class scored higher in the examination. Mr. Ma‟s class has a lower standard deviation compared to my class too, which is 10.36 against 14.91. A low standard deviation means there is a little gap between the marks of the students. Thus, this shows that Mr. Ma„s students scored nearer to their mean score of 76.79. 26 Ogive 28.0 26.0 24.0 22.0 20.0 18.0 Cumulative Fraquency 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 20.5 30.5 40.5 50.5 Marks 60.5 70.5 80.5 90.5 27 CONCLUSION In conclusion, we can conclude using mathematical method, to determine the performance of the school by analyzing the marks for every subject obtained by students to determine the average school grade. Other than that, we can also use statistic to determine the subject that can be scored highest among the students by drawing histogram and creating pie chart. This concept can also be used to find out not only the mean of the marks obtained by students in examinations but also the standard deviation. Last but not least, we can also use interquartile range to determine how far the values of data in a set of data are spread out from its average value. From this we can also compare the advantages of using standard deviation and interquartile range to be used as a better measure of dispersion. 28 REFLECTION While conducting this project, I have learnt how important data analysis is in our daily life. Apart from that, this project encourages students to work together and share their knowledge. It is also encourages student to gather information from the internet, improve their thinking skills and promote effective mathematical communication. Based on my findings, I found that the mean mark and the average grade of my class Additional Mathematics are off average. Suggestions to improve my class performance, the students must: Do more exercises on weak topics Revise on previous SPM papers Better results in Additional Mathematics examinations Complete sets of SPM model papers Pay full attention in class Not only that, I had also learned some moral values that I practiced. This project had taught me how to be responsible on the work that was given to me to be completed. This project also made me gained more confidence to do work and not to give up 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 week to complete this project and pass up to my teacher just in time. I proposed for this project work to be continued as it brings a lot of moral values to the students and also tests the students in understanding Additional Mathematics. 29 BIBLIOGRAPHY 1. Majalang, Michelle C. (2013) Additional Mathematics Project Work 2/2013. SMK Lok Yuk, Kota Kinabalu, Sabah. 2. Tze Hin, Chang and other authors. (2012) Nexus SPM A+ Additional Mathematics. Selangor Darul Ehsan, Selangor: Sasbadi Sdn. Bhd. 3. Abdul Karim, Aini Akmalia. (2011) Additional Mathematics Project Work 2/2011. Sekolah Agama Menengah Bandar Baru Salak Tinggi, 43900 Sepang, Johor. 4. Selvaraj, Nur Syafiqah bt Muhammad Faizal. (2012) Additional Mathematics Project Work 2/2012. Sekolah Berasrama Penuh Integrasi Gopeng. 5. Wikipedia. (2013) Statistics. http://en.wikipedia.org/wiki/Statistics 6. Phillips, J.D. Mathematics as an Aesthetic Discipline. Department of Mathematical Sciences, Saint Mary‟s College, Moraga, California 94575. http://euclid.nmu.edu/~jophilli/essays/math-aesth.html 30