MCE673

ADVANCED MATHEMATICS WITH MODELLING AND

SIMULATION TECHNIQUES

The course covers modelling and simulation

principles as applied in complex systems in the
engineering domain.

• The first part covers the basic concepts and

approaches to solving systems of linear and non-
linear ODE’s and PDE’s with emphasis on integral
and discrete transform techniques such as the
Laplace and Fourier transforms.
• The second part covers modelling and simulation
principles such as continuous, discrete (including
the combination), and other modelling techniques
applications.
MCE673
ADVANCED MATHEMATICS WITH MODELLING AND Syllabus
SIMULATION TECHNIQUES
MCE673
ADVANCED MATHEMATICS WITH MODELLING AND Evaluation
SIMULATION TECHNIQUES

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Learning Objectives – Week 6
• Statistics
• Normal Distribution
Statistics and Normal Distribution
Data Gathering
Data analysis
Distribution of Data
Objectives
At the end of the lesson, the students are expected to
• Define basic terms and phrases used in statistics;
• Identify the importance of statistics in everyday life;
• Compare and contrast descriptive and inferential statistics; and
• Explain the concepts of methods of data collection and presentation.
 Statistics
The field of statistics deals with the collection,
presentation, analysis, and use of data to make decisions,
solve problems, and design products and processes. In
simple terms, statistics is the science of data.
 Branches of Statistics
Descriptive Statistics (DS)
• Concerned with describing the characteristics and
properties of a group of persons, places or things.
• Based on easily verifiable facts or meaningful
information.
• Does not draw inferences or conclusions about a larger
set of data.
 Descriptive Statistics
Examples
• How many passed in the recent Electrical Engineering
Licensure examination?
• In Applied Life Data Analysis (Wiley, 1982), Wayne
Nelson presents the breakdown time of an insulating
fluid between electrodes at 34 kV. The times in
minutes, are as follows: 0.19, 0.78, 0.96, 1.31, 2.78,
3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06,
31.75, 35.52, 33.91, 36.71, and 72.89.
 Branches of Statistics
Inferential Statistics (IS)
• Draws inferences about a population based on the data
gathered from the samples using the techniques of DS.
• Composed of those methods concerned with the
analysis of a smaller group of data leading to
predictions or inferences about the larger set of data.
• Statistics that deals in giving a generalization about the
whole from an analysis of the part of the group.
 Inferential Statistics
Examples
• Is there a significant correlation between the amount
spent in studying and final grade in a computer
programming course?
• Study shows that ABET accredited programs draw more
students to enrol at Mapúa Institute of Technology in
such programs.
 Population and Sample
Population
• Totality of all observations from which the dataset is
acquired
• All of the possible events should be considered.
• Variable that describes population is known as
parameter.
Example:
There are 5,786 students enrolled in MCE673.
Population: Students of MCE673
Parameter: 5,786 (population size)
 Population and Sample
Sample
• Small group taken from the population
• A group heterogeneous as possible taken from the
large group to represent the population
• Variable that describes sample is known as statistic.
Example:
Of the 5,786 students enrolled in MCE673, 3,456 are
females.
Sample: Female students in MCE673
Statistic: 3,456 (sample size)
 Variables
Variables are the parameters being studied in statistics.

Qualitative Variables
• Also known as categorical data which are commonly
answered by non numeric data usually qualitative in
form
• Examples are preferences, gender, civil status, and
location.
 Variables
Quantitative Variables
• Also known as numerical data which are information
and observations that are countable or measurable
quantities
• Examples are force, weight, height, voltage, current,
resistance, tensile strength, and grades.
 Variables
Examples: Classify as Quantitative (QN) or Qualitative
(QL).
• Weekly allowance
• Income of parents
• Religion
• Age
• Address
• Educational attainment
• Jobs
• Schools attended
Categories of Quantitative Data
Continuous Data
• Measurable quantities. Have infinite values between
intervals.
• Data that have been measured by analog devices and
have infinite values based on interpolations
• Examples are height, weight, and ratio of persons.
Categories of Quantitative Data
Discrete Data
• Countable quantities. Have finite equal intervals.
• Data that have been measured by digital measuring
device that tends to have exact values
• Examples are number of individuals and months of the
year.
Dependent vs Independent Variable
Independent Variable
• A naturally occurring phenomenon that can be altered
by increasing or decreasing its magnitude.

Dependent Variable
• A variable that is observed upon application of the
changes applied to the independent variable.

Example: The number of hours spent in studying and test

scores.
Dependent vs Independent Variable
Controlled Variable
• Kept constant to check for the external effects of the
dependent to the independent variable

Extraneous Variable
• Would have minimal effect to the result of the
dependent variable to the independent variable
 Scales of Measurement
• Nominal
- Assigning numerical to categorical data.
• Ordinal Data
- Assigning rank to the levels of data.
• Interval
- Assigning a constant difference between numeric
data.
• Ratio
- Assigning continuous range of data over a range.
 Nominal Data
• Commonly categorical data assigned to numbers.
• The applicable measurement is simply counting the
number of times a certain data would fall on the
category, like assigning 1 for males and 2 for females.
• Other examples include course, civil status, color, and
preference.
 Ordinal Data
• Quantities where the numbers are used to designate
the rank order of the data
• The correlation or the effect of the ranking of one
variable can be measured. However, the range for each
rank is not constant.
• Examples are results of a race, ranking of a beauty
pageant, and level of hardness of a material in the
Moh’s scale.
 Interval Data
• The range between the numeric values is constant.
• Addition and subtraction is applicable, but not for
multiplication and division.
• Multiplication and division can only be done in the
difference between intervals.
• Zero point is arbitrary.
• Examples include years (1994, 2004, etc.), times (00:00,
20:00, etc.) and temperature in Celsius and Fahrenheit
scales.
 Ratio Data
• Widely used data in science and engineering
• Almost all the basic mathematical operations can be
performed in this data type.
• There is a non arbitrary zero point.
• Examples include length, mass, angles, charge, and
energy.
 Sampling
Sampling is the process of taking samples from the
population.
• Probability Sampling
- This eliminates the biases against certain event that has
no chance to be selected by listing all the possible events
and taking a chance that they will be selected to be part
of the sample.
• Non-Probability Sampling
- This type of sampling technique has certain or has no
chance of an individual of being selected to be part of
the sample.
 Probability Sampling
• Simple Random Sampling
- Performed by arranging the population according to a
certain rule, each element being numbered and a
sample is taken by various randomizing principles.
- Randomizing events examples are table of random
numbers, random number generator in computers and
calculators, and lottery or fish bowl technique.
- Each event in the population has equal chance of being
selected as part of the sample.
 Probability Sampling
• Systematic Sampling
- Done by arranging the population in accordance to a
certain order and the sample will be taken by dividing
the population into equal groups and obtaining the kth
element in each group
Examples:
- Getting the temperature of the device every 4 hours
- Getting the voltage of the signal every constant interval
and converting to another signal
 Probability Sampling
• Stratified Sampling
- Done by grouping the population into strata, a
subpopulation with generally homogeneous or similar
characteristics
- After dividing the population into several strata, a
random sampling is performed in each stratum
proportional to the size of each stratum relative to the
population.
 Probability Sampling
• Stratified Sampling
Example: A survey to find out if families living in a certain
city are in favor of construction of manufacturing plant
will be conducted. To ensure all income groups
represented, respondents will be divided into:
Class A – high income
Class B – middle income
Class C – low income
 Probability Sampling
Strata Number of Families
A 1000
B 2500
C 1500
N = 5000
• Stratified Sampling
- Using a 5% margin of error, how many families should
be included in the survey? Use Slovin’s formula: 𝑛 =
𝑁
1+𝑁𝑒 2
- Using proportional allocation, how many from each
group should be taken as samples?
 Probability Sampling
• Cluster Sampling
- Done by identifying groups called clusters, a
subpopulation with elements as heterogeneous or
diverse characteristics as possible
- The clusters must be similar to each other with respect
to the parameter being examined.
- A cluster or clusters will be selected as sample.
- Preferred since it will save time and money to go to
various clusters
- Example: Selection of a certain region.
Non-Probability Sampling
• Convenience Sampling
- Based primarily on the availability of the respondents
- Used because of the convenience it offers to the
researcher
- Example: Gathering data through telephone.
• Quota Sampling
- There is a desired number of sample and the
respondents were taken as they volunteered
themselves to become part of the experiment.
- Almost similar to the stratified random sampling
- Example: Phone call survey where the first 100 callers
are taken
Non-Probability Sampling
• Purposive Sampling
- The sample is obtained based on a certain premise.
- Example: A study about pregnant women where the
male population would have zero chance of being
selected as part of the survey
 Summary
• There are two fields of Statistics: Descriptive and
Inferential Statistics.
• Population is the totality of all observations from which
the dataset is acquired. Sample is a subset of
population.
• Variables are classified as quantitative or qualitative
and independent or dependent.
• The scales of measurement are nominal, ordinal,
interval, and ratio.
• Sampling techniques are classified as probability
(random, systematic, stratified, and cluster) and non-
probability (convenience, quota, and purposive).
Data Presentation
 Objectives
At the end of the lesson, the students are expected to
• Identify and learn various ways of presenting data;
• Describe data through tables, graphs, and charts;
• Describe and interpret data presented in various
charts; and
• Practice different ways or presenting data.
Types of Data Presentation
• Textual Form
- Data presentation using sentences and paragraphs in
describing data

• Tabular Form
- Data presentation that uses tables arranged in rows
and columns for various parameters

• Graphical Form
- Pictorial representation of data
Grouped and Ungrouped Data
• Ungrouped Data
- Data points are treated individually.

• Grouped Data
- Data points are treated and grouped according to
categories.
 Data Presentation
Stem-and-Leaf Diagram
Steps to construct a stem-and-leaf diagram:
1. Divide each number xi into two parts: a stem,
consisting of one or more of the leading digits, and a
leaf, consisting of the remaining digit.
2. List the stem values in a vertical column.
3. Record the leaf for each observation beside its stem.
4. Write the units for stems and leaves on the display.
 Stem-and-Leaf Diagram
Example:
1. Express the following data as a stem-and-leaf diagram
with the tens digit as the stems and the ones digit as the
leaves.
12, 23, 12, 11, 10, 25, 29, 39, 31, 43, 42, 54,
53, 53, 56, 57, 56, 67, 54, 65, 76, 76, 75, 74
 Data Presentation
Frequency Distribution Table
Numerous data can be analyzed by grouping the data
into different classes with equal class intervals and
determining the number of observations that fall within
each class. This procedure is done to lessen work done
in treating each data individually by treating the data by
group.
Frequency Distribution Table
Class limits
- The smallest and the largest values that fall within
the class interval (class)
- Taken with equal number of significant figures as the
given data.
Class boundaries (true class limits)
- More precise expression of the class interval
- It is usually one significant digit more than the class
limit.
- Acquired as the midpoint of the upper limit of the
lower class and the lower limit of the upper class
Frequency Distribution Table
Frequency
- The number of observations falling within a particular
class.
- Counting and tallying
Class width (class size)
- Numerical difference between the upper and lower class
boundaries of a class interval.
Class mark (class midpoint)
- Middle element of the class
- It represents the entire class and it is usually
symbolized by x.
Frequency Distribution Table
Cumulative Frequency Distribution
- can be derived from the frequency distribution and can
be also obtained by simply adding the class frequencies
- Partial sums

Types of Cumulative Frequency Distribution

- Less than cumulative frequency (<cf) refers to the
distribution whose frequencies are less than or below
the upper class boundary they correspond to.
- Greater than cumulative frequency (>cf) refers to the
distribution whose frequencies are greater than or
above the lower class boundary the correspond to.
Frequency Distribution Table
Relative Frequency
- Percentage frequency of the class with respect to the
total population
- For presenting pie charts

Relative Frequency (%rf) Distribution

- The proportion in percent the frequency of each class
to the total frequency
- Obtained by dividing the class frequency by the total
frequency, and multiplying the answer by 100
Frequency Distribution Table
Class Interval Frequency x LCB UCB <cf >cf %rf
Frequency Distribution Table
Steps in Constructing a Frequency Distribution Table (FDT)
1. Get the lowest and the highest value in the
distribution. We shall mark the highest and lowest
value in the distribution.

2. Get the value of the range. The range denoted by R,

refers to the difference between the highest and the
lowest value in the distribution. Thus,

R = H ─ L.
Frequency Distribution Table
3. Determine the number of classes. In the
determination of the number of classes, it should be
noted that there is no standard method to follow.
Generally, the number of classes must not be less than
5 and should not be more than 15. In some instances,
however, the number of classes can be approximated
by using the relation
𝑘 = 1 + 3.322 log 𝑛 (Sturges’ Formula),
where k = number of classes and n = sample size. is
the ceiling operator (meaning take the closest integer
above the calculated value).
Square root principle: 𝑘 = 𝑛
Frequency Distribution Table
4. Determine the size of the class interval. The value of C
can be obtained by dividing the range by the desired
number of classes. Hence, 𝐶 = 𝑅Τ𝑘.

5. Construct the classes. In constructing the classes, we

first determine the lower limit of the distribution. The
value of this lower limit can be chosen arbitrarily as
long as the lowest value shall be on the first interval
and the highest value to the last interval.
Frequency Distribution Table
6. Determine the frequency of each class. The
determination of the number of frequencies is done
by counting the number of items that shall fall in each
interval.
Frequency Distribution Table
2. Using the steps discussed, construct the frequency
distribution of the following results of a test in statistics of
50 students given below.

88 62 63 88 65
85 83 76 72 63
60 46 85 71 67
75 78 87 70 43
63 90 63 60 73
55 62 62 83 79
78 43 51 56 80
90 47 48 54 77
86 55 76 52 76
43 52 72 43 60
Frequency Distribution Table
2. Using the steps discussed, construct the frequency
distribution of the following results of a test in statistics of
50 students given below.
Answer:
88 62 63 88 65
Class Interval Frequency 85 83 76 72 63
43-49 7 60 46 85 71 67
75 78 87 70 43
50-56 7
63 90 63 60 73
57-63 10
55 62 62 83 79
64-70 3 78 43 51 56 80
71-77 9 90 47 48 54 77
78-84 6 86 55 76 52 76
43 52 72 43 60
85-91 8
Frequency Distribution Table
3. The following are the scores of 40 students in a Math
quiz. Prepare a frequency distribution for these scores
using a class size of 10.
22 31 55 76 48 49 50 85 17 38
92 62 94 88 72 65 63 25 88 88
86 75 37 41 76 64 66 58 66 76
52 40 42 76 29 72 59 42 54 62
 Frequency Distribution Table
3. The following are the scores of 40 students in a Math
quiz. Prepare a frequency distribution for these scores
using a class size of 10.
Answer: 22 31 55 76 48 49 50 85 17 38
Class Interval Frequency 92 62 94 88 72 65 63 25 88 88
17-26 3 86 75 37 41 76 64 66 58 66 76
27-36 2 52 40 42 76 29 72 59 42 54 62
37-46 6
47-56 6
57-66 9
67-76 7
77-86 2
87-96 5
Frequency Distribution Table
4. The thickness of a particular metal of an optical
instrument was measured on 121 successive items as they
came off a production line under what was believed to be
normal conditions. The results are shown in Table 4.5.
Frequency Distribution Table
4. Answer
 Data Presentation
Graphical Form of Frequency Distribution

Frequency Polygon
- Line graph
- The points are plotted at the midpoint of the classes.

Histogram (Frequency Histogram or Relative Frequency

Histogram)
- Bar graph
- Plotted at the exact lower limits of the classes
 Data Presentation
Graphical Form of Frequency Distribution

Ogive
- Line graph
- Graphical representation of the cumulative frequency
distribution
- The < ogive represents the <cf while the > ogive
represents the >cf.
 Data Presentation
5. Construct a frequency polygon, histogram, and ogives
of the given distribution.
Class Interval Frequency
25-29 1
30-34 1
35-39 5
40-44 8
45-49 15
50-54 4
55-59 4
60-64 3
65-69 4
70-74 3
75-79 2
 Data Presentation
In the preparation of a polygon, the frequency values are
always plotted on the y-axis (vertical) while the classes are
plotted on the x-axis (horizontal). Here we use the class
midpoints.
Frequency Polygon
16
15
14
13
12
11
10
Frequency (f)

9
8
7
6
5
4
3
2
1
0
17 22 27 32 37 42 47 52 57 62 67 72 77 82 87
Class Midpoint (x)
 Data Presentation
The preparation of the histogram is similar to the construction
of the frequency polygon. While the frequency polygon is
plotted using the frequencies against the class midpoints, the
histogram is plotted using the frequencies against the exact limit
of the classes. Frequency Histogram
16
15
14
13
12
11
10
Frequency (f)

9
8
7
6
5
4
3
2
1
0
19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5 74.5 79.5
Exact Class Limit
 Data Presentation
Frequency Histogram
Frequency Histogram
16
15
14
13
12
11
10
Frequency (f)

9
8
7
6
5
4
3
2
1
0
22 27 32 37 42 47 52 57 62 67 72 77 82
Class Midpoint (x)
 Data Presentation
Ogives
Ogives
55

50

45 < ogive
> ogive
40
Cumulative Frequency (CF)

35

30

25

20

15

10

0
19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5 74.5 79.5 84.5
Class Boundary (CB)
 Data Presentation
6. Construct a frequency polygon, histogram, and ogives
of the frequency distribution from problem #2.

Class Interval Frequency

43-49 7
50-56 7
57-63 10
64-70 3
71-77 9
78-84 6
85-91 8
 Summary
• Stem-and-leaf diagram is one way of data presentation
tabular form.
• Frequency distribution can be depicted in two ways:
tabular and graphical (frequency polygon, histogram,
and ogives) forms.
 References
• DeCoursey. Statistics and Probability for Engineering
Applications with Microsoft® Excel © 2003
• Montgomery and Runger. Applied Statistics and
Probability for Engineers, 5th Ed. © 2011
Normal Distribution Checking and Normalizing Data
Activity
https://www.statskingdom.com/kolmogorov-smirnov-test-calculator.html
• Check Kolmogorov-Smirnov Test for normality.
Conclusion
By applying normalization techniques, you can reduce noise in the dataset and potentially improve the
normality of the distribution. Visual tools like histograms and Q-Q plots are essential to assess the
effectiveness of these methods. Confirming it with other methods such as KS test will prove normality.