NAME:

Class:

CONTENT

Acknowledgement

Objectives

Introduction

Task Specification

Problem Solving

Further Exploration

Reflection

ACKNOWLEDGEMENT First of all, I would like to say Alhamdulillah, for giving me the strength and

health to do this project work and finish it on time.

Not forgotten to my parents for providing everything, such as money, to buy anything

that are related to this project work, their advise, which is the most needed for this project and

facilities such as internet, books, computers and all that. They also supported me and encouraged

me to complete this task so that I will not procrastinate in doing it.

Then I would like to thank to my teacher, M iss Salhalida for guiding me throughout this

project. Even I had some difficulties in doing this task, but she taught me patiently until we knew

what to do. She tried and tried to teach me until I understand what I am supposed to do with the

project work.

Besides, my friends who always supporting me. Even this project individually but we are

cooperated doing this project especially in discussion and sharing ideas to ensure our task will

finish completely.

Last but not least, any party which involved either directly or indirect in completing this

project work.

Thank you everyone.

OBJECTIVES

develop mathematical knowledge in a way which increases students’ interest and

confidence;

apply mathematics to everyday situations and to begin to understand the part that

mathematics plays in the world in which we live;

improve thinking skills and promote effective mathematical communication;

assist students to develop positive attitude and personalities, intrinsic mathematical

values such as accuracy, confidence and systematic reasoning;

stimulate learning and enhance effective learning.

INTRODUCTION

Vision 2020 aims to produce a balanced human capital in terms of physical, emotional, spiritual

and intellectual in accordance with the National Education Philosophy. In order to expand the

intellectual aspect, every individual should have the ability to analyze data.

The picture above shows students in a secondary school having their final year examination. The

School Examination secretary will collect the marks for each subjects to determine the average

grade of the subjects, the average grade school and which will give the picture of the

performance of the school.

Data representation reflects the general characteristics of data that allows us to compare

and thus predict and plan for the future.

Data analysis is a process used to transform, remodel and revise certain information (data) with a

view to reach to a certain conclusion for a given situation or problem. Data analysis can be done by

different methods as according to the needs and requirements. For example if a school principal

wants to know whether there is a relationship between students’ performance on the district writing

assessment and their socioeconomic levels. In other words, do students who come from lower

socioeconomic backgrounds perform lower, as we are led to believe? Or are there other variables

responsible for the variance in writing performance? Again, a simple correlation analysis will help

describe the students’ performance and help explain the relationship between the issues of

performance and socioeconomic level.

Analysis does not have to involve complex statistics. Data analysis in schools involves collecting

data and using that data to improve teaching and learning. Interestingly, principals and teachers have

it pretty easy. In most cases, the collection of data has already been done. Schools regularly collect

attendance data, transcript records, discipline referrals, quarterly or semester grades, norm- and

criterion-referenced test scores, 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.

TASK SPECIFICATIONP ART 1:

1. List the importance of data analysis in daily life.2. Specify the types of Measure of Central Tendency and of Measure of Dispersion.

P ART 2 :

1. Get the March Additional Mathematics test scores for your class. Attach the scores sheet.2. Construct a frequency table as in Table 1 which contains at least five class intervals.

Choose a suitable class size.

Marks Tally Frequency

Table 1

.a) From Table 1, find

(i) the mean,

(ii) the mode,

(iii) the median using two methods.

b) Based on your findings from (a) above, state the appropriate measure of central tendency to reflect the performance of your class in Additional Mathematics. Explain why.

P ART 3:

Measure of Dispersion is a measurement to determine how far the values of data in a set of data is spread out from its average value.

Based on the data from Table 1,

a) using two methods, find(i) the interquartile range

(ii) the standard deviation

b) Explain the advantages of using standard deviation compared to interquartile range to describe the data.

P ART 4:

a. If your teacher wants to make adjustments by adding 3 marks for each student in your class for their commitment and discipline shown, find the new value of mean, mode, median, class interval, interquartile range and standard deviation. Check your answers with other methods.

b. In April 2012, a new student has enrolled in your class. The student has scored 97% in the Additional Mathematics March Test in his/her former school. If the student scores were taken into account in the analysis of your school March Test, state the effect of the presence of this student to the mean, mode, median, interquartile range and standard deviation.

PROBLEM SOLVING

PART 1

1. Importance of data analysis in daily life

There are many benefits of data analysis however; the most important ones are as follows: - 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.

2. Types of Measure of Central Tendency and of Measure of

Dispersion

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 variable’s level of measurement.

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.

x=∑ x

N Or x=

∑ fx

∑ f

The mean is an interval/ratio measure of central tendency. Its calculation requires that the attributes of the variable represent a numeric scale

Mode

The 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.

Median

The 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.

m=L+( N2

−F

fm )CThere 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.

Range

The 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 variable

Range = maximum value – minimum value

Interquartile range 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.

Q 1=L+( 14

N−F

fm )C Q 3=L+( 34

N−F

fm )C

Interquartile range = Q3 - Q1

Standard Deviation

The 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

σ=√(∑ f x2

∑ f )−x2

PART 2

1. the March Additional Mathematics test scores for your class

STUDENTS MARKS

1 80

2 72

3 85

4 88

5 70

6 67

7 78

8 77

9 68

10 58

11 35

12 50

13 70

14 50

15 70

16 60

17 58

18 43

19 35

20 50

21 17

22 53

23 23

24 37

25 17

26 37

27 17

28 12

29 27

30 22

31 15

2. Frequency table

MARKS TALLY FREQUENCY

0-10 0

11-20 5

21-30 3

31-40 4

41-50 4

51-60 4

61-70 7

71-80 2

81-90 2

91-100 0

a) (1) mean

x=∑ fx

∑ f

MARKS MIDPOINT, x FREQUENCY, f Fx1-10 5.5 0 0

11-20 15.5 5 77.5

21-30 25.5 3 76.5

31-40 35.5 4 142

41-50 45.5 4 182

51-60 55.5 4 222

61-70 65.5 7 458.5

71-80 75.5 2 151

81-90 85.5 2 171

91-100 95.5 0 0

TOTAL 31 1480.5

∑f = 31

∑fx = 1408.5

Mean, x = 1480.5

31

= 47.7581

(2) mode

The modal class is 61-70, the majority of the students got that marks.

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

modal class.

(REFER TO HISTOGRAM 1)

Based on the histogram;

Mode = 64.5

(3) median

Median is the value of the centre of a set of data

Method 1 – By using formula

Median mark for 31 students can be obtained by using the formula:

m=L+( N2

−F

fm )CWhere

L = lower boundary of median class,

N = total frequency,

F = cumulative frequency before the median class,

fm = frequency of median class,

C = class interval size

MARKS LOWER

BOUNDARY

UPPER

BOUNDARY

FREQUENCY, f CUMULATIVE

FREQUENCY

1-10 0.5 10.5 0 0

11-20 10.5 20.5 5 5

21-30 20.5 30.5 3 8

31-40 30.5 40.5 4 12

41-50 40.5 50.5 4 16

51-60 50.5 60.5 4 20

61-70 60.5 70.5 7 27

71-80 70.5 80.5 2 29

81-90 80.5 90.5 2 31

91-100 90.5 100.5 0 31

Median class = 31÷ 2

=15.5

=16th value

= 41 - 50

L = 40.5 fm = 4 N = 31

F = 12 C = 50.5-40.5

=10

m=40.5+( 312

−12

4 )10

= 49.25

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.

(REFER TO OGIVE 1)

Based on the ogive;

Median = 49.5

b) Appreciate measure of central tendency

From the above measure of central tendency, mean is suitable measure of central tendency

because the minimum value of raw data is not extreme where the data seems to be clustered,

whereas mode and median does not take all the values in the data into account which decrease

the accuracy of central tendency.

PART 3

Measure of Dispersion is a measurement to determine how far the values of data in a set of data

is spread out from its average value.

a) (1) interquartile range

Method 1 – Using Formula

Q 1=L+( 14

N−F

fm )C

Q1 class = 31 × ¼

=7.75

= 8th value

= 21 – 30

L = 20.5 fm = 3 N = 31

F = 5 C = 10

Q 1=20.5+( 14

(31 )−5

3 )10

= 29.6667

Q 3=L+( 34

N−F

fm )CQ3 class = 31 × ¾

= 23.25

= 24th value

= 61 – 70

L = 60.5 fm = 7 N = 31

F = 20 C = 10

Q 3=60.5+( 34

(31 )−20

7 )10

= 65.1429

Therefore,

Interquartile range = Q3 – Q1

= 65.1429 – 29.6667

= 35.4762

Method 2 – Using ogive

(REFER TO OGIVE 2)

Q1 = 29.75

Q3 = 65.5

Interquartile range = 65.5 – 30.0

= 35.5

(2) standard deviation

Method 1

σ=√(∑ f x2

∑ f )−x2

MARKS MIDPOINT, x FREQUENCY, f fx fx2

1-10 5.5 0 0 0

11-20 15.5 5 77.5 1,201.25

21-30 25.5 3 76.5 1,950.75

31-40 35.5 4 142 5,041

41-50 45.5 4 182 8,281

51-60 55.5 4 222 12,321

61-70 65.5 7 458.5 30,031.75

71-80 75.5 2 151 11,400.5

81-90 85.5 2 171 14,620.5

91-100 95.5 0 0 0

TOTAL 31 1480.5 84,847.75

x = 1480.5

31

= 47.7581

σ=√ 84847.7531

−47.75812

= 21.3586

Method 2

σ=√∑ f (x−x )2

∑ f

MARKS MIDPOINT,

x

FREQUENCY,

f

( x - x ) (x−x )2 f(x−x )2

1-10 5.5 0 -40.4355 1635.0297 0

11-20 15.5 5 -29.9355 896.1342 4,480.671

21-30 25.5 3 -19.9355 397.4242 1,192.2726

31-40 35.5 4 -9.9355 98.7142 394.8568

41-50 45.5 4 0.0645 0.004160 0.01664

51-60 55.5 4 10.0645 101.2942 405.1768

61-70 65.5 7 20.0645 402.5842 2,818.0894

71-80 75.5 2 30.0645 903.8742 1,807.7484

81-90 85.5 2 40.0645 1,605.1642 3,210.3284

91-100 95.5 0 50.0645 2,506.4542 0

TOTAL 14,309.16

σ=√ 14,309.1631

= 21.4845

b) advantages of using standard deviation

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.

PART 4

a. The new marks for 31 students

STUDENTS MARKS

1 83

2 75

3 88

4 91

5 73

6 70

7 81

8 80

9 71

10 61

11 38

12 53

13 73

14 53

15 73

16 63

17 61

18 46

19 38

20 53

21 20

22 56

23 26

24 40

25 20

26 40

27 20

28 15

29 30

30 25

31 18

New frequency distributions table:

MARK

S

LOWER

BOUNDAR

Y

MIDPOINT

, x

FREQUENCY,

f

CUMULATIV

E

FREQUENCY

fx fx2

1-10 0.5 5.5 0 0 0 0

11-20 10.5 15.5 5 5 77.5 1,201.25

21-30 20.5 25.5 3 8 76.5 1,950.75

31-40 30.5 35.5 4 12 142 5,041

41-50 40.5 45.5 1 13 45.5 2,070.25

51-60 50.5 55.5 4 17 222 12,321

61-70 60.5 65.5 4 21 262 17,161

71-80 70.5 75.5 6 27 453 34,201.5

81-90 80.5 85.5 3 30 256.5 21,930.75

91-100 90.5 95.5 1 31 95.5 9,120.25

TOTAL 1,630.

5

104,997.7

5

Mean

x=∑ fx

∑ f

= 1630.5

31

= 52.5968

Mode

The modal class is 71-80

(REFER TO HISTOGRAM 2)

Based on the histogram;

Mode = 74.5

Median

Method 1: Formula

Median class = 31÷ 2

=15.5

=16th value

= 51 - 60

L = 50.5 fm = 4 N = 31

F = 13 C = 10

m=50.5+[ 312

−13

4 ]10

= 56.75

Method 2: Ogive

(REFER TO OGIVE 3)

Based on the ogive,

Median = 56.5

Class interval

Class interval remain same

C = 10

Interquartile range

Method 1: Formula

Q1 class = 31 × ¼

=7.75

= 8th value

= 21 – 30

L = 20.5 fm = 3 N = 31

F = 5 C = 10

Q 1=20.5+( 14(31)−5

3 )10

= 29.6667

Q3 class = 31 × ¾

= 23.25

= 24th value

= 71 – 80

L = 70.5 fm = 6 N = 31

F = 21 C = 10

Q 3=70.5+( 34(31)−21

6 )10

= 74.25

Interquartile range = Q3 – Q1

= 74.25 – 29.6667

= 44.5833

Method 2: Ogive

(REFER TO OGIVE 4)

Based on the ogive,

Interquartile range = Q3 – Q1

= 77.0 – 29.5

= 47.5

Standard deviation

σ=√( 104,997.7531 )−52.59682

= 24.9119

b. The new student scored 97%

MARKS LOWER

BOUNDARY

MIDPOINT,

x

FREQUENCY,f CUMULATIV

E

FREQUENCY

fx fx2

1-10 0.5 5.5 0 0 0 0

11-20 10.5 15.5 5 5 77.5 1,201.25

21-30 20.5 25.5 3 8 76.5 1,950.75

31-40 30.5 35.5 4 12 142 5,041

41-50 40.5 45.5 1 13 45.5 2,070.25

51-60 50.5 55.5 4 17 222 12,321

61-70 60.5 65.5 4 21 262 17,161

71-80 70.5 75.5 6 27 453 34,201.5

81-90 80.5 85.5 3 30 256.5 21,930.75

91-100 90.5 95.5 2 32 191 18,240.5

TOTAL 1,726 114,118

Mean

x= 1726

32

= 53.9375

Standard deviation

σ=√( 114,11832 )−53.93752

= 25.6307

Mode, median, and interquartile range are not affected by the adding for new marks.

FURTHER EXPLORATIONA statistic (singular) is a single measure of some attribute of a sample (e.g. its arithmetic

mean value). It is calculated by applying a function (statistical algorithm) to the values of the

items comprising the sample which are known together as a set of data.

More formally, statistical theory defines a statistic as a function of a sample where the

function itself is independent of the sample's distribution; that is, the function can be stated

before realization of the data. The term statistic is used both for the function and for the value of

the function on a given sample.

A statistic is distinct from a statistical parameter, which is not computable because often the

population is much too large to examine and measure all its items. However a statistic, when

used to estimate a population parameter, is called an estimator. For instance, the sample mean is

a statistic which estimates the population mean, which is a parameter.

Examples

In calculating the arithmetic mean of a sample, for example, the algorithm works by

summing all the data values observed in the sample then divides this sum by the number of data

items. This single measure, the mean of the sample, is called a statistic and its value is frequently

used as an estimate of the mean value of all items comprising the population from which the

sample is drawn. The population mean is also a single measure however it is not called a

statistic; instead it is called a population parameter.

Other examples of statistics include

Sample mean discussed in the example above and sample median

Sample variance and sample standard deviation

Sample quartiles besides the median, e.g., quartiles and percentiles

Test statistics, such as t statistics, chi-squared statistics, f statistics

Order statistics, including sample maximum and minimum

Sample moments and functions thereof, including kurtosis and skewness

Various functional of the empirical distribution function

Properties

Observability

A statistic is an observable random variable, which differentiates it from a parameter that

is a generally unobservable quantity describing a property of a statistical population. A parameter

can only be computed exactly if the entire population can be observed without error; for instance,

in a perfect census or for a population of standardized test takers.

Statisticians often contemplate a parameterized family of probability distributions, any member

of which could be the distribution of some measurable aspect of each member of a population,

from which a sample is drawn randomly. For example, the parameter may be the average height

of 25-year-old men in North America. The height of the members of a sample of 100 such men

are measured; the average of those 100 numbers is a statistic. The average of the heights of all

members of the population is not a statistic unless that has somehow also been ascertained (such

as by measuring every member of the population). The average height of all (in the sense of

genetically possible) 25-year-old North American men is a parameter and not a statistic.

Statistical properties

Important potential properties of statistics

include completeness, consistency, sufficiency, unbiasedness, minimum mean square error,

low variance, robustness, and computational convenience.

Information of a statistic

Information of a statistic on model parameters can be defined in several ways. The most

common one is the Fisher information which is defined on the statistic model induced by the

statistic. Kullback information measure can also be used.

REFLECTION

While 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.

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.  ~S. Gudder