Additional-Mathematics.pdf
-
Upload
wylenda-winterfield -
Category
Documents
-
view
21 -
download
0
description
Transcript of Additional-Mathematics.pdf
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
SPECIFICATION
PART 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.
PART 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.
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.
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.
PART 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.
∑
Or
∑
∑
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.
(
)
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.
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.
(
) (
)
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
√(∑
∑ )
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
∑
∑
MARKS MIDPOINT, x FREQUENCY, f Fx
1-10 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, =
= 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:
(
)
Where
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
(
)
= 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
(
)
Q1 class = 31 × ¼
=7.75
= 8th
value
= 21 – 30
L = 20.5 fm = 3 N = 31
F = 5 C = 10
(
( )
)
= 29.6667
(
)
Q3 class = 31 × ¾
= 23.25
= 24th
value
= 61 – 70
L = 60.5 fm = 7 N = 31
F = 20 C = 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
√(∑
∑ )
MARKS MIDPOINT, x FREQUENCY, f fx f
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
=
= 47.7581
√
= 21.3586
Method 2
√∑ ( )
∑
MARKS MIDPOINT,
x
FREQUENCY,
f
( x - ) ( ) f( )
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
√
= 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:
MARKS LOWER
BOUNDARY
MIDPOINT,
x
FREQUENCY,f CUMULATIVE
FREQUENCY
fx
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 1,630.5 104,997.75
Mean
∑
∑
=
= 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
*
+
= 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
(
( )
)
= 29.6667
Q3 class = 31 × ¾
= 23.25
= 24th
value
= 71 – 80
L = 70.5 fm = 6 N = 31
F = 21 C = 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
√(
)
= 24.9119
b. The new student scored 97%
MARKS LOWER
BOUNDARY
MIDPOINT,
x
FREQUENCY,f CUMULATIVE
FREQUENCY
fx
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
=
= 53.9375
Standard deviation
√(
)
= 25.6307
Mode, median, and interquartile range are not affected by the adding for new marks.
FURTHER
EXPLORATION
A 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