Table of Contents
Introduction to Logic
Statement
Truth value and truth table
Negation
Compound Statements and Grouping Symbols
Alternative Method for Truth Table
Conditional Statement
STRUCTURES OF MATHEM ATICS
Basic Concepts on Sets
PATTERNS IN NATURE
Symmetry
a)
b)
Bilateral Symmetry
Radial Symmetry
Set
Elements
Methods of Defining Set
Types of Set Theory Symbol
Venn Diagrams Set Operation
Properties of the Union Operation
Properties of the Intersect Operation
Symbols to Word Expressions
Shapes
Fractals
Parallel Lines
Fibonacci Spiral
FIBONACCI SEQUENCE AND THE
Fibonacci Spiral G O L D E N R A T I O
Golden Ratio
Golden Rectangles
INDUCTIVE REASONING
MATHEMATICS FOR OUR DAILY LIFE
Counterexamples
DEDUCTIVE REASONING
Mathematics for Organization
Mathematics for Prediction
Mathematics for Control
TREE DIAGRAM “LISTING METHOD”
POLYA’S PROBLEM-SOLVING STRATEGY
LANGUAGE OF MATHEMATICS
Convention Letters
Mathematical Symbols
STATISTICS
CLASSIFICATION OF DATA
TRANSLATION FROM ENGLISH TO
M AVersus
T H ESentences
MATICAL STATEME NTS
Expression
Differentiation of Mathematical Expression and Equation/Sentence
STRUCTURES OF MΑΤΗΕΜΑΤΙCS
Qualitative Data
Quantitative Data
Types of Quantitative Variable
Types of Statistics
Descriptive Statistics
Inferential Statistics
LEVELS OF MEASUREMENT
Basic Concepts on Sets
Set
Elements
Methods of Defining Set
Types of Set Theory Symbol (THIS IS OPTIONAL)
Venn Diagrams Set Operation
Properties of the Union Operation
Properties of the Intersect Operation
Symbols to Word Expressions (THIS IS OPTIONAL)
LOGIC
Nominal Data
Ordinal Data
Interval Data
Ratio Data
MEASURES OF CENTRAL TENDENCY, AND VARIANCE OR
DISPERSION
Measure of Central Tendency
Mean
Median
Mode
Measure of Variance or Dispersion
Range
Variance
Standard Deviation
Definition of symbols and Variables
Formula of Measures for Ungrouped an Grouped Data
Types of Frieze Patterns
1.
2.
3.
4.
5.
6.
7.
Hop Pattern
Jump
Step
Sidle
Spin hop
Spin sidle
Spin jump
CORRELATION & REGRESSION ANALYSYS
WALLPAPER PATTERN
Correlation Analysis
TESSELLATION
Interpretation: When the value of “r” is
Pearson’s Product Moment
Regression Method
Least Square Regression Equation
Regression Method Formula:
Correlation Between Ordinal Variable
Spearman Rank Order Correlation Coefficient
KINDS OF DATA DISTRIBUTION
Symmetrical or Normal Distribution
Positively Skewed Distribution
Negatively Skewed Distribution
BASIC CONCEPTS OF GRAPHS
Terminologies Of Concepts Of Graphs
Complete Graph
Equivalent Graphs
HYPOTHESIS TESTING
TYPES OF HYPOTHESIS TESTING
EULER CIRCUITS
Eulerian Graph Theorem
Parametric tests
t-test for Dependent Samples (paired)
t-test for Independent Samples (unpaired)
z-test
F-test
Non-parametric tests
Hypothesis
EULER PATH
Euler Path Theorem
HAMILTONIAN GRAPHS
Dirac’s Theorem
Null Hypothesis (Ho)
Alternative Hypothesis (Ha)
WEIGHTED GRAPH
The Greedy Algorithm
The Edge-Picking Algorithm
Applications of Weighted Graphs
GRAPH COLORING
EUCLIDEAN TRANSFORMATION
TYPES OF ISOMETRY
1.
2.
3.
4.
Translation
Reflections
Rotations
Glide Reflections
SYMMETRIC PATTERNS
TYPES OF SYMMETRIC PATTERNS
Rosette patterns (finite designs)
1.
2.
Cyclic Symmetry
Dihedral Symmetry
Frieze patterns
Four-Color Theorem
2-Colorable Graph Theorem
Mathematical
Systems
MODULO N
Arithmetic Operations Modulo n
Examples: Arithmetic Operations Modulo n
Examples: Arithmetic Operations Modulo n
Examples: Arithmetic Operations Modulo n
Solving Congruence Equations
Problems: Solving Congruence Equation
Additive and Multiplicative Inverses
th e Natu reo f
Mathematics
H I S T O R Y
O F
M A T H E M A T I C S
Shapes
o
Fractals
o
o is sometimes called “Science of Pattern”
o there are specific pattern rules and it can be
applied to different things
o Math has been studied ancient times and some
formulas are still used today. (e.g. Pythagorean
Theorem)
o “The Laws of Nature are but the
Mathematical thoughts of God” –Euclid
o It is anywhere and everywhere. We rarely
notice its existence; we don’t always see math
in everything but it’s simply there from the
tiniest human cell to the power of the sun, there
is math.
o It begs the question, “Was Math discovered or
was invented?”
o No matter what the answer is, Math has been a
fundamental part of our history, It has a special
link, even with nature as everything can be
governed or at least related to a mathematical
concept. And man is trying to understand or go
with its rhythm, its very pattern.
P A T T E R N S
I N
N A T U R E
two sides that are mirror images of one another
o
there is special line called the “line of
symmetry” that separates the two objects
equally
o
o
o
Leonardo Fibonacci discovered the sequence
o
Sequence begin with 0 and 1
o
Each Subsequent number is the sum of the
two-proceeding number.
o
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
o
tool to create the Fibonacci spiral
Golden Ratio
o
a special number found by dividing a line into
2 parts so that the longer part divided by the
smaller part is also equal to the whole length
divided by the longer part
o
symbolized using “phi”
o
used in mathematics and arts
o
the ratio between the sum of those quantities
and the larger one is the same as the ratio
between the larger one and the smaller.
aa +
+ bb aa
=
= =
=φ
φ
aa
bb
R
DET
I ARLYS Y M M E T R Y
BA
IL
there is a center point and numerous lines that
can be branched out
o
unlimited number of lines, so long as they are
of the same size and shape and connected to
the center point as reference
A series of squares with lengths equal to the
Fibonacci numbers would end up creating a
spiral by following the edges
FIBONACCI SEQUENCE AND THE GOLDEN RATIO
an object has two sides that are mirror images
of one other
o
lines that do not intersect or touch at all but
runs at the same direction/course
Fibonacci Spiral
B
B II LL A
AT
TE
ER
RA
A LL SSY
YM
MM
ME
ET
TR
RY
Y
o
same patterns/shapes but different sizes
Parallel Lines
Symmetry
o
Geometry is the branch of mathematics that
describes Shapes (e.g. rectangle, triangles,
circle, diamond, etc.)
o
constant approximation of Golden Ratio =
1.6180339887….
o
Golden Ratio can be derived from the
Fibonacci numbers when you divide a number
from the sequence from a number before that
number (ex: 377/233)
Golden Rectangles
Mathematics for our Daily Life
o
Ratio of the length is longer and the width is
shorter
o
expresses itself everywhere, in almost every
facet of life
o
A rectangle that can be cut up into a square
and a rectangle similar to the original one.
o
it is considered as the language of science and
engineering
T hae longer
+ b siad e = φ
= =φ
sh or
a ter si dbe
Mathematics for Organization
o
used as a tool to help us make sound analysis
and better decision
o
you can probably think of different situations
with mathematical tools being used
o
it also develop strategies of problem-solving
Mathematics for Prediction
o
prediction through the analysis and
interpretation of existing data
o
probability and patterns
o
usually predictions is used in weather
forecasting, and also as a basis for predicting
patterns based on your observation
Mathematics for Control
o
Influence the behavior of a system and has the
control in order to achieve a desired goal
o
Money – mathematics of manipulation
o
Man is able to exert control over himself and
effects of nature through math
o
Human Behavioral pattern can change the
society and the natural world
Mathematical Symbols
Mathematical
an
Language nd Symbols
L A N G U A N G E
O F
SYMBOL
SYMBOL NAME
MEANING/
DEFINITION
=
equals sign
equality
≠
not equal sign
inequality
≈
similar/
approximately
equal
approximation
M A T H E M A T I C S
o
facilitates communication and clarifies
meaning for many things
o
system of communication which consists of sets
of sounds and written symbols which are used
by the people.
>
strictly inequality
greater than
<
strictly inequality
less than
o
allows people to express themselves and
maintain their identity
≥
inequality
greater than
or equal to
o
it consists of words, letters, and symbols that
are known as natural language. Non-verbal
signs or images that can be used to
communicate can be called as mathematical
langauge
≤
inequality
less than or equal to
o
Math is a universal language
o
o
o
TRANSLATION FROM ENGLISH TO MATHEMATICAL STATEMENTS
Nouns, Verbs, Sentences
o
the principles and foundations of math
are the same everywhere around the
world
Nouns could be your integers, numbers, or
expressions
Ex. 5, 2(5-1/2)
o
improves our mental ability as it
teaches us logical ways of thinking
Verb may be your equals sign “=”, or an
inequality (>,<)
o
Pronouns are your variables x and y
Ex. 5x-2, xy
make it easier to express their thoughts
because it is:
o
Sentence would be formed
Ex. 3x + 7 = 22
▪
Precise – able to make very fine
distinctions
▪
Concise – able to say things
briefly
▪
Powerful – able to express
complex thoughts with relative
ease
Convention Letters
o
not rules, but often used that way
o
some letters have special uses
EXAMPLES
WHAT THEY USUALLY MEAN
a, b, c, …
constants (fixed values)
i, j, k, l,m, n
positive integers
(for counting)
… x, y, z
variables (unknown)
Expression vs. Sentences
ENGLISH
MATHEMATICS
name given
to an object
of interest
Noun
(person,
place, thing)
Ex. Carol,
Idaho
Expression
Ex: 5, 2+3, 1/2
a complete
thought
Sentence
Ex: The capital
of Idaho is
Boise.
Sentence
Ex: 3+4=7
Differentiation of Mathematical Expression & Equation
S T R U C T U R E S O F M A T H E M A T I C S
BASIS FOR
COMPARISON
EXPRESSION
Meaning
Expression is a
mathematical
phrase which
combines
numbers,
variables and
operators to
show the value
of something.
An equation is a
mathematical
statement wherein
two expressions are
set equal to each
other.
A sentence
fragment, that
stands for a
single
numerical
value.
A sentence that
shows equality
between two
expressions.
Result
Simplification
Solution
Relation symbol
No relation
symbol
Yes, equal sign (=)
Sides
One sided
Two sided, left and
right
Answer
Numerical
value or
simplified
expression
Assertion, i.e. true
or false.
Example
7x - 2(3x + 14)
7x - 5 = 19
What is it?
EQUATION
Basic Concepts on Sets
SE
SE
TT
o
a well-defined collection of objects (elements)
similar/related or dissimilar/non-related in
some ways.
o
Set theory – created by Georg Ferdinand
Ludwig Philip cantor
o
elements are enclosed in braces “{ }”
E
E LL E
EM
ME
EN
NT
T SS
o
objects/ random symbols/numbers/names that
belong to a set, denoted by “U”
o
elements in a set do not repeat. I repeat, they
DO NOT repeat.
o
order of elements in a set is not important, but
used for organizing purposes
Methods of Defining Set
R O SE
TL
EE
RMM
EE
NTTH
SOD
o
A = {1, 2, 3, 4, 5}
o
defined by enumerating the elements of the
set.
Five increased by ten
15
Twenty subtracted from twelve
-8
Two more than the sum of six and four
12
o
A = {x|x is a counting number less than 6}
Thirty decreased by eight
22
o
The quotient of twenty- four by three, plus
ten
18
Is by using descriptive phrases in the form of
x”such that x”
8 times the sum of 5 and 3
64
1/2 the difference of 20 and 8
6
3 less than the product of 4 and 6
21
7 more than the quotient of 16 by 8
9
3 increased by the sum of 7 and 13
23
The sum of two consecutive integers is 15.
Five years ago, John’s age was half of the
age he will be in 8 years. How old is he
now?
7 and 8
x-5=1/2(x
+8)
Six subtracted from thrice a number
3x-6
Five taken from a number
x-5
The sum of your crush and you
NA
R UE
LL
E EM
MEETNHTO
SD
Types of Set Theory Symbol
Note: Some examples tend to change the elements in
Set A or/and B to illustrate how the symbol functions
in a statement properly.
SYMBOL
SYMBOL NAME
MEANING/DEFINITION
EXAMPLE
{}
set
a collection of
elements
A=
{3,7,9,14},
B = {9,14,28}
|
such that
so that
A = {x | x∈
R, x<0}
A⋂B
intersection
objects that belong
to set A and set B
A⋂B=
{9,14}
A⋃B
union
objects that belong
to set A or set B
A⋃B=
{3,7,9,14,28}
A⊆B
subset
A is a subset of B.
set A is included in
set B.
{9,14,28} ⊆
{9,14,28}
A⊂B
proper subset /
strict subset
A is a subset of B,
but A is not equal to
B.
{9,14} ⊂
{9,14,28}
not subset
set A is not a subset
of set B
{9,66} ⊄
{9,14,28}
superset
A is a superset of B.
set A includes set B
{9,14,28} ⊇
{9,14,28}
proper superset /
strict superset
A is a superset of B,
but B is not equal to
A.
{9,14,28} ⊃
{9,14}
not superset
set A is not a
superset of set B
{9,14,28} ⊅
{9,66}
power set
Number of subset in
a set; n= # of
elements
A⊄B
A⊇B
A⊃B
A⊅B
2n
P(A)
cardinality of
countable ordinal
numbers set
Ø
empty set
Ø = {}
u
universal set
set of all possible
values
N0
natural numbers /
whole numbers
set (with zero)
N0= {0,1,2,3,4,...}
0 ∈ N0
n1
natural numbers /
whole numbers
set (without zero)
n1= {1,2,3,4,5,...}
6 ∈ n1
Z
integer numbers
set
Z=
{...-3,-2,-1,0,1,2,3,...}
-6 ∈ Z
Q
rational numbers
set
Q = {x | x=a/b, a,b∈
Z and b≠0}
2/6 ∈ Q
R
real numbers set
R = {x | -∞ < x <∞}
6.343434
∈R
C
complex numbers
set
C = {z | z=a+bi, ∞<a<∞,
-∞<b<∞}
6+2i ∈ C
# of subsets
in B = 8
power set
all subsets of A
equality
both sets have the
same members
Ac
complement
all the objects that
do not belong to set
A
A'
complement
all the objects that
do not belong to set
A
A’= {28}
A=B
aleph-one
A={3,9,14},
B={3,9,14},
A=B
A=Ø
Ex:
P(B) = {Ø, {9}, {14}, {28}, {9,14}, {9,28}, {14,28},
{9,14,28}}
A\B
relative
complement
objects that belong
to A and not to B
A = {3,9,14},
B = {1,2,3},
A\B=
{9,14}
A-B
relative
complement
objects that belong
to A and not to B
A = {3,9,14},
B = {1,2,3},
A-B=
{9,14}
A∆B
symmetric
difference
objects that belong
to A or B but not to
their intersection
A = {3,9,14},
B = {1,2,3},
A∆B=
{1,2,9,14}
A⊖B
symmetric
difference
objects that belong
to A or B but not to
their intersection
A = {3,9,14},
B = {1,2,3},
A⊖B=
{1,2,9,14}
a∈A
element of,
belongs to
set membership
A={3,9,14},
3∈A
x∉A
not element of
no set membership
A={3,9,14},
1∉A
(a,b)
ordered pair
collection of 2
elements
A×B
Cartesian product
set of all ordered
pairs from A and B
Hindi kasya
|A|
cardinality
the number of
elements of set A
A={3,9,14},
|A|=3
#A
cardinality
the number of
elements of set A
A={3,9,14},
#A=3
aleph-null
infinite cardinality of
natural numbers set
B = {9,14,28}
Ex
A = {1,2,3}
B = {1,3,5}
A x B = {(1,1), (1,3), (1,5), (2,1), (2,3), (2,5), (3,1),
(3,3), (3,5)}
Formal definition of a Cartesian product:
AxB = {(a,b) | a∈A and b ∈B}
Venn Diagrams Set Operation
o
o
L O G I C
Introduction to Logic
o
o
Symmetric Difference (extra)
Properties of the Union Operation
AU∅=A
Identity law
AUU=U
Domination law
AUA=A
Idempotent law
AUB=BUA
Commutative law
A U (B U C) = (A U B) U C
Associative law
Properties of the Intersect Operation
A∩U=A
Identity law
A∩∅=∅
Domination law
A∩A=A
Idempotent law
A∩B=B∩A
Commutative law
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Associative law
o
Logic is the science of correct reasoning.
Logic allows us to determine the validity of
arguments in and out of mathematics.
Illustrates the importance of precision and
conciseness of the language of mathematics.
Statement
o a declarative sentence that is either true (T) or
false (F) but NOT both.
SIMPL
TA
EE
L ESM
ET
NE
TM
S ENT
o a statement that conveys a single
idea
C O M P OEULNEDM SETNATTSE M E N T
o a statement that conveys two or
more ideas.
Use words/phrases such as “and”, “or”, “if-then”
and “if and only if” to create a compound
statements.
Let A and B be simple statements
Type
Statement
A
Negation
Not A
A‚B
Conjunction
A and B
AƒB
Disjunction
A or B
ADB
Condition
If A, then B
AGB
Bicondition
A if and only if B
Symbols to Word Expressions
Symbols
A⋂B
A and B
A⋃B
A or B
A-B
A only and not B
∆ or ⊖
A or B, but not A and B
A’
Not A
A’ ⋃ B’ = (A ⋂ B)’
Not A and B
A’ ⋂ B’ = (A ⋃ B)’
Not A or B
A’ ⋃B
Not A, or B
A’ ⋂ B
Not A, and B
Truth Value & Truth Table
o
The truth value of a simple
statement is either true (T) or false
(F).
o The truth value of a compound
statement depends on the truth
values of its simple statements and
connectives.
A truth table is a table that shows the truth value of a
compound statement for all possible truth values of its
simple statements.
Negation
o
o
o
o
o
Say P is a statement.
The negation of P means not
𝑃 and is denoted by ~𝑷.
If the statement is false, its
negation is true.
If the statement is true, its
negation is false.
The negation of the negation
of a statement is the original
statement
Compound Statements & Grouping Symbols
C
CO
OM
MPPO
OUUN
ND
D SSTTA
ATTEEM
MEEN
NTTSS &
&G
GR
RO
OUUPPIIN
NG
G SSY
YM
MB
BO
OLLSS
➔ Given a compound statement in symbolic form,
parentheses are used to indicate which simple
statements are grouped together.
Ex:
P‚ (Q ƒ~R)
(P‚~Q) ƒ R
(P‚~Q)!(P ƒ Q)
➔ Given a compound statement written in English,
comma is used to indicate which simple
statements are grouped together.
Difference when using commas
P, and Q or R P‚(QƒR)
P and Q, or R (P‚Q)ƒR
Say P and Q be statements.
The conjuction denoted by 𝑷 ∧ 𝑸
is TRUE if and only if BOTH P and
Q are true.
The disjuction denoted by 𝑷 ∨ 𝑸 is
TRUE if and only if P is TRUE, Q is
TRUE, or BOTH P and Q are true.
Alternative Method for Table
•
•
•
•
2n = # of rows in a truth table; n = # of Statement
Use the truth values for each simple
statement and their negations to enter
the truth values under each connective
within a pair of grouping symbols—
parentheses ( ), brackets [ ], braces { }.
If some grouping symbols are nested
inside other grouping symbols, then
work from the inside out.
In any situation in which grouping
symbols have not been used, then we
use the following order of precedence
agreement. First assign truth values to
negations from left to right, followed by
conjunctions from left to right, followed
by disjunctions from left to right,
followed by conditionals from left to
right, and finally by biconditionals from
left to right.
Tautology – Statement that is always true
Self – Contradiction – Statement that is always false
Conditional Statement
Problem
Solving Reasoning
and
T R E E
o
Problem Solving
o
o
Problem - an inquiry starting from given
conditions to investigate or demonstrate a fact,
result or law.
Problem Solving is a fundamental means of
developing mathematical knowledge at any
level.
Inductive Reasoning
o
It is an approach to logical thinking that
involves making generalizations based on
specific details.
Conjecture
o
o
a mathematical statement that has not yet been
rigorously proved.
One counterexample means that the statement
is false.
Counterexamples
o
An example that opposes or contradicts an
idea or theory.
Deductive Reasoning
o
It is a type of logic where general statements,
or premises, are used to form a specific
conclusion.
o
It moves from generalities to specific
conclusions.
D I A G R A M
“ L I S T I N G
M E T H O D “
a tool in the field of general mathematics,
probability, and statistics that helps calculate
the number of possible outcomes of an event
or problem, and to cite those potential
outcomes in an organized way.
P O L Y A ‘ S
P R O B L E M - S O L V I N G
S T R A G E T E G Y
o
George Polya was a mathematician in the 1940s
o
He devised a systematic process for solving
problems that is now referred to by his name:
the Polya 4-Step Problem-Solving Process
P FE Q
S UO
F TSI T
A TTI V
I E
S T
I R
C ISA B L E
TY P T
E SY O
AN
TA
VA
Data
Management
➡ Descriptive Statistics
Collecting, organizing, presenting, and
analyzing numerical data
Focuses on quantitatively describing the
collection of data
Summary of the samples with corresponding
measures is stated
This is the organizing and summarizing data
using numbers and graphs
S T A T I S T I C S
Branch of science that deals with the
collection, presentation, organization, analysis,
and interpretation of data
➡ Inferential Statistics
๏
Population - collection of all elements
under consideration in a statistical
inquiry
๏ Sample - subset of a population
๏ Variable - characteristics & attributes of
the elements in a collection that
can
assume different values for the different
elements
C L A S S I F I C A T I O N
O F
D A T A
Qualitative Data
Represents differences in quantity, character
or kind but not amount
Measure of “types” and may be represented
by names or symbols
Describes individuals or objects by their
categories or groups
Descriptive data based on observations and
usually involves 5 senses (see, feel, taste,
hear, smell)
Quantitative Data
Numerical in nature and can be ordered/
ranked
Measure of “values” or “counts” and
expressed in numbers
T
TY
YP
PE
E SS O
O FF Q
QU
UA
AN
NT
T IIT
TA
AT
T IIV
VE
EV
VA
AR
R II A
AB
B LL E
E
Discrete Quantitative Variable (Countable)
Ex: 8 cats (base in whole numbers and can be
counted)
➡ Continuous Variable Quantitative Values
(integers, rational, irrational values)
Ex: 5.5m (anything can be measured such as
distance, speed, weight)
➡
๏
Analyzing the organized data
Leading to prediction
Assume from the sample data of the
population might probably be
Using sample Data to make an interference
or conclusion of the population
Measurement – Process of determining the value
or label of the variable based on what has been
observed
E EVSEOLF SQO
F NM
S IUV R
E AMR E
TL
YP
UA
T IET A
AT
E V
I ANBTL E
➡ Nominal Data
Labeling variables, without quantitative value
Nominal scales -"labels"
Cannot be arranged in an ordering scheme
➡ Ordinal Data
Measures of non-numeric concepts
Difference between the values of the data
cannot be determined
Interval is meaningless
The order of the values is important and
significant
➡ Interval Data
Quantitative measurements used to identify
and rank
Has negative values
Can’t compare two values
Has no true zero point
➡ Ratio Data
Similar to interval scale but has a true zero
point
No negative values
Multiples are significant
MEASURES OF CENTRAL TENDENCY AND VARIANCE/DISPERSION
MEANINGFUL
ORDER
MEASURABLE
DIFFERENCES
DATA
LABELED
Nominal
Yes
Ordinal
Yes
Yes
Interval
Yes
Yes
Yes
Ratio
Yes
Yes
Yes
TRUE ZERO
STARTING
POINT
Yes
๏
Measure of Central Tendency – a typical value of
a set of data or observation where they tend to
cluster. Moreover, it depends on the values
(mean, median, mode) of what the shape of the
graph and skewness looks like.
Mean
Represents the sum of all values in a dataset
divided by the total number of the values.
doesn’t always locate the center of the data
accurately
in a skewed distribution, the mean can miss the
mark.
Extreme values in an extended tail pull the
mean away from the center
best to use the mean as a measure of the central
tendency when you have a symmetric
distribution
Best used for Interval and Ratio Data
Median
the middle value that splits the dataset in half.
Outliers and skewed data have a
smaller effect on the median
When you have a skewed distribution, the
median is a better measure of central tendency
than the mean
Best used for Ordinal and Interval Data and ratio
Mode
the value that occurs the most frequently in
your data set.
If the data have multiple values that are tied
for occurring the most frequently, you have a
multimodal distribution.
no value repeats, the data do not have a
mode.
Best used for Nominal Data
๏
Measure of Variance or Dispersion - Indicates the
degree or extent to which numerical values are
dispersed or spread out above the average value
in a distribution.
Range
how varied the data set is
how spread out numbers in the data set really
are
estimate the measure of spread of standard
deviation
a very crude measurement of the spread of
data because it is extremely sensitive to
outliers
single data value can greatly affect the value
of the range
Variance
a measurement of the spread between
numbers in a data set.
it measures how far each number in the set is
from the mean and therefore from every
other number in the set
A large variance indicates that numbers in
the set are far from the mean and from each
other, while a small variance indicates the
opposite
Variance can be negative. A zero value
means that all of the values within a data set
are identical.
Standard Deviation
Standard deviation looks at how spread out a
group of numbers is from the mean
If the data points are further from the mean,
there is a higher deviation within the data set;
thus, the more spread out the data, the higher
the standard deviation
Definition of Symbols & Variables
➡ Weighted Mean
Xm
Midpoint of each class
f
Frequency or Class Frequency
n
Number of population/ sum of class
frequency
a +Σx
b −w
a
=
= =φ
a Σw b
Lower class boundary of median/modal
class
Note: Values for modal and median class
may not be the same
LCBme/
mo
Respective weight of the data
Example: Grouped Data
CLASS
1-10
f
cf
xm
fxm
f(x-x̄)2
71
71
5.5
390.5
10874.2
0
i
Class size or Class width
11-20
32
103
15.5
496
180.60
fm
Frequency of median class
21-30
50
153
25.5
1275
2906.50
n/2
Location of median class
31-40
20
173
35.5
710
6212.33
41-50
8
181
45.5
364
6104.82
fm1
f of modal class – f of the previous class
from modal class
x̄= 17.88
fm2
f of modal class – f of the next class from
modal class
N = 181
ª f(x-x̄)2 = 6569.61
Formula of Measures for Ungrouped Data & Grouped Data
UNGROUPED DATA
Median Class = (n/2)th = 90.5th
Modal Class = Class 1-10
cfb = 71
fm = 32
i = 10
LCBme = 11 - .5 = 10.5
LCBmo = 1 -.5 = .5
fm1 = 71 - 0 = 71
fm2 = 71 – 32 = 39
GROUPED DATA
Mean
x̄ =
Σx
n
x̄ =
Σ f xm
n
Median
If “n” is odd
~x = x n + 1
2
~x = LCB + i
me
If “n” is even
~x =
xn + x
2
n
( 2 )+1
(
n
2
+ cf b
fm
)
2
Mode
^x = Most high est f r equ en c y valu e
If there are 2 equal modes – Bimodal
If 3 – Trimodal
^x = LCBme + i
f m1
( f m1 + f m 2 )
Range
= UCBhci − UCBlci
= Hi gh est va l ue − L owest va l ue High class interval +.5
Low class interval -.5
Variance
s2 =
Σ(x − x̄)2
n−1
s2 =
Σf(x − x̄)2
n−1
Standard Deviation
s=
s2
s=
s2
Class 11 -20
CORRELATION AND REGRESSION ANALYSIS
Correlation Analysis
r × y=
a statistical method used to determine
whether a relationship between two
variables exists
12(724) − 76(119)
2
2
[12(678) − (76) ][12(1561 ) − (119) ]
2
Note : m a d e a cor r ect ion on on e valu e;
"Σx 2" f r om 670 to 678
r = − 0.11
Therefore, there is a very low correlation between
the results of the scores obtained by the reading
comprehension and vocabulary test
Regression Method
A linear regression is used to make
predictions about a single value.
TINTERPRETATION:
Y P E S O F Q U A NWHEN
T I T ATHE
T I VVALUE
E V A OF
R I “r”
A BIS
LE
VALUES ( r )
INTERPRETATION
0.0
no correlation
±1.00
perfect correlation
±0.01-±0.25
very low
±0.26-±0.50
moderately low
±0.51-±0.75
high
±0.75-±0.99
very high
PRODUCT
MOMENT
T Y P EPEARSON’S
S OF QUA
NTITAT
I V E V A (RrI) A B L E
REGRESSION
EQUATION
T Y PLEAST
E S O SQUARE
F QUAN
T IT AT IV E
VARIABLE
+ ba +a bX
Ya =
= =φ
a
WHERE:
Y= DEPENDENT
X= INDEPENDENT
a= Y-int (x=0)
b= SLOPE
T Y P E REGRESSION
S O F Q U A NMETHOD
T I T A T IFORMULA
VE VARIABLE
Measure of the linear association between 2
variables that are measured on interval or ratio
scales.
2
Σx
(Σya)(
) −a (Σx)(Σxy)
+
b
a=
= = φ2
an Σx 2 −b (Σx)
It was developed by Karl Pearson that is why the
correlation coefficient is sometimes called
"Pearson's r“. The formula is defined by:
r =
b
n Σx y − (Σx)(Σy)
b = a + b = a = φ2
na Σx 2 −b(Σx)
N ΣX Y − N X ΣY
a+b
a
=
=φ
2
2
2
b N ΣY 2 − (ΣY ) ]
[N ΣX −a(ΣX ) ][
From the previous example:
a =
119(678) - 76(724)
12(678) - (76)²
Y = 10.87 − 0.15X
= 10.87
b=
12(724) - 76(119)
12(678) - (76)²
= -0.15
Correlation Between Ordinal Variable
SPEARMAN
RANK
TY P E S O
F QORDER
U A N TCORRELATION:
I T A T I V E VCOEFFICIENT
ARIABLE
Used to calculate the correlation of ordinal data
w/c are classified according to order or rank.
KINDS OF DATA DISTRIBUTION
Symmetrical/Normal Distribution
the mean, median, and mode all fall at the same point
or equal
6(Σd )
rs =a 1+−b = a = φ
a n(nb2 − 1)
2
WHERE:
d = difference between ranks
n = number of paired observations
Positively Skewed Distribution
the extreme scores are larger, thus the mean is larger
than the median
rs = 1 −
6(Σd²)
n(n 2 − 1)
rs = 1 −
6(14.5)
= 0.83
8(82 − 1)
CONCLUSION:
•
The rs = 0.83 indicates that there is a very
high positive correlation between the two
judges.
Negatively Skewed Distribution
The order of the measures of central tendency would
be the opposite of the positively skewed distribution,
with the mean being smaller than the median, which
is smaller than the mode
HYPOTHESIS TESTING
Method of using simple data to decide between
two competing claims(hypothesis) about a
population characteristic.
Concerns itself with the decision-making rules
for choosing alternatives while controlling and
minimizing the risks of wrong decisions.
TYPES OF HYPOTHESIS TESTING
t-test for Dependent Samples (paired)
A parametric test applied to one group of samples
It can be used in evaluation of a certain program or treatment
It is applied when the mean before and the mean after are being
compared
t-test for Independent Samples (unpaired)
Used when we compare the means of two independent groups
Used when the sample is less than 30
Parametric Test
The parametric tests are tests applied to data
that are normally distributed.
A statistical test, in which specific
assumptions are made about the population
parameter is known as parametric test.
It is assumed that the measurement of
variables of interest is done on interval or
ratio level.
The measure of central tendency in the
parametric test is mean.
There is complete information about the
population.
Non-Parametric Test
tests that do not require a normal distribution
a statistical test used in the case of non-metric
independent variables, is called nonparametric test
the variable of interest is measured on
nominal or ordinal scale
the measure of central tendency in the
parametric test is median
there is no information about the population
Z-test
It is used to compare two means: the sample means and the
perceived population mean.
It is also used to compare the two sample means taken from the
same population. When samples are equal to or greater than 30.
It can be applied in two ways: the One-sample mean test and the
two sample mean test.
F-test
It is another parametric test used to compare the means of two or
more independent groups.
It is also known as the analysis of variance (ANOVA)
Kinds of ANOVA: One-way, two-way, three-way
We used ANOVA to find out if there is a significant difference
between and among the means of two or more independent
groups.
Hypothesis
Example:
claim/statement either about the value of a
single population characteristics or about the
values of several population characteristics
NULL
TY P E S O F Q
U AHYPOTHESIS
N T I T A T I (Ho)
VE VARIABLE
a claim about a population characteristic that is
initially assumed to be true
researcher tries to reject or disprove
have the “equal sign”
ALTERNATIVE
TY P E S O
F Q U A N T HYPOTHESIS
I T A T I V E (Ha)
VARIABLE
the competing claim
what we are attempting to demonstrate in an
indirect way
➡ One-Tailed Test
o Ha is directional (<,>) e.g. Ho:μ=21
Ha:μ>21
➡ Two-Tailed Test
o
Ha is nondirectional (≠) e.g. Ho:μ=21
Ha:μ≠21
➡ Significance Level (α)-related to the degree of
certainty we require in order to reject ho in favor
of ha)
➡ Test Statistic
o a quantity calculated from the sample
data
o
its value is used to decide whether or
n o t h o s h o u l d b e re j e c t e d i n a
hypothesis test (z-test, t-test, anova)
➡ Critical/Tabular Value - threshold to which the
value of the test statistic in a sample is
compared to determine whether or not ho is
rejected
➡ Critical/Rejection Region- set of values of the
test statistic for which Ho is rejected in a
hypothesis test (graph on the board)
➡ z-test & t-test - used to compare or study 2
groups of data through the value of their means
➡ z-test- used when s is known and n≥30
➡ t-test- used when s is known and n<30
The above figure shows a 99 % confidence interval,
the remaining 1 percent is the percentage of error
denoted by an alpha(a). Hence, the critical value is
± .005, the .01 (1%) is divided into two since it is a two
tailed graph.
Reflection
Geometric Design
o
o
EUCLIDEAN TRANSFORMATION
Isometry or isometric transformations is a type of transformation in
which the angles, size and side measurements of the figure remains the
same.
A reflection flips a shape/figure/object over to create a
mirror image.
The mirror is a line called the axis of reflection.
The triangle on the right has
been reflected over the red
dotted line, thus, creating a
mirror image.
Whenever a figure is
reflected, each of its points
must be of the same distance
from the line of reflection.
TYPES OF ISOMETRY
Translation
o
Examples of reflection in patterns:
A translation in the plane moves or slides a
shape/figure/object.
The figure has been
moved. The new figure is
now on the upper right.
o
Real-life examples of reflection:
A translation is not allowed to turn, flip or change its
size. Each point is moved to the same exact distance
and the same direction.
Reflection of the mountains in
the water
Human body is a reflection
when split in half
Rotation
The figure did not
change its size, only
the position in which
where it stands.
o
o
o
Examples of translation in patterns:
A rotation turns a shape/figure/object.
You have to know the pivot point, called the center of the
rotation, as well as the angle of the rotation.
In every rotation, the figure should always be the same size
and shape.
Sometimes figures are
rotated just a few
degrees and other
times they may be
rotated in a very
obvious manner.
Real-life examples of translation:
The rotation
happened but the
figure remains the
same size and shape.
Design statues in a park
G-clef note in a music sheet
Real-life examples of glide rotation:
If a figure is rotated all
the way around back
to where it started, it is
a full rotation with an
angle of 360°.
Man’s footsteps
Leaves in branches
SYMMETRIC PATTERNS
A plane pattern has a symmetry if there is an isometry present in the
plane.
If a figure is rotated
only half of the full
rotation, it has an
angle of 180°.
A transformation of a pattern is a symmetry of the pattern if the pattern
stays the same.
Examples of rotation in patterns:
TYPES OF SYMMETRIC PATTERNS
Rosette patterns (finite designs)
CYCLIC SYMMETRY:
Real-life examples of rotation:
o
there is a rotation symmetry around a center point
but no mirror lines; it only goes continuously in
circles.
Examples of cyclic symmetry in patterns:
Ferris wheel
Bicycle wheels
Glide Reflections
Real-life examples of cyclic symmetry:
→ A glide reflection is a combination of a translation and a
reflection.
→ The axis of reflection must be parallel to the direction of the
translation.
→ You can reflect and then translate or vice versa.
Dart board
London eye (Ferris wheel)
DIHEDRAL SYMMETRY
o
Examples of glide rotation in patterns:
rotation symmetry around a center point with mirror lines
through the center point. You can distinguish the different equal
parts.
Example of dihedral symmetry in patterns:
Real-life examples of dihedral symmetry:
Snowflake
Starfish
SIDLE
Frieze patterns
•
•
- involves translations and vertical reflection lines with a
180° rotation.
Frieze patterns are patterns that have translational symmetry in
one direction
they go on to infinity directions, both left and right
SPIN HOP
Example of cyclic symmetry in patterns:
Types of Frieze Patterns
HOP PATTERN
JUMP
- involves translations and a 180° rotation.
SPIN SIDLE
– translation then either vertical reflection or 180°
rotation.
- just a translation done in a repetitive manner
- involves horizontal reflections followed by translations
SPIN JUMP
– horizontal and vertical reflections and
translations
STEP
- all are glide reflections
WALLPAPER PATTERN
Wallpaper pattern is a pattern with translation in two directions, done in
a repetitive manner.
It is an arrangement of frieze patterns stacked upon one another to fill
the entire plane.
A wallpaper pattern can be made up of a combination of rotation,
reflection, and glide reflection.
TESSELLATION
Covering any flat surface with a pattern of multiple shapes and styles
such that no part remains uncovered or overlaps
Terminologies Of Concepts Of Graphs ....................2
Complete Graph .............................................2
Equivalent Graphs ...........................................2
Eulerian Graph Theorem ...................................2
Euler Path Theorem .........................................2
The Greedy Algorithm ......................................3
The Edge-Picking Algorithm ................................3
Four-Color Theorem .........................................4
2-Colorable Graph Theorem ...............................4
Mathematicsof Graphs
Basic Concepts of Graphs
A graph is a diagram or a set of points and lines that
are connected to each point.
Euler Circuits
•
Terminologies Of Concepts Of Graphs
• In general, graphs can contain vertices that are
not connected to any edges.
•
If two or more edges connect the same vertices,
they are called multiple edges.
•
If an edge begins and ends at the same vertex, it
is called a loop.
•
A graph is connected if any vertex can be
reached from any other vertex by tracing along
edges.
•
A connected graph in which every possible edges
is drawn between vertices is called complete
graph.
Complete Graph
•
If each pair of the graph vertices is
connected by an edge.
•
If there is almost all one edge between any two
vertices in the graph but there is no loop it is
called SIMPLE.
The definition of a Euler Circuit is a circuit
that uses every edge, but never uses the
same edge twice. So basically, when using a
Euler circuit, the only thing you need to look
forward to is that you start and end with the
same vertex. You could pass through the
same points more than once, the most
important thing is just end with the same
vertex. A graph than contains a Euler Circuit
is a Eulerian Graph.
A graph that contains an Euler circuit is a Eulerian
graph.
Eulerian Graph Theorem
• A connected graph is Eulerian ( has an Eulerian
circuit) if and only if each vertex of the graph is of
even degree.
Euler Path
•
So basically, Euler Path is way different from a
Euler Circuit, where in Euler Path, it doesn’t allow
the passing through an edge more than once. It
also differ when it comes to the vertex, where it
ends up with a different vertex.
Euler Path Theorem
•
A connected graph contains an Euler path if
and only if the graph has two vertices of odd
degree with all other vertices of even
degree.
•
Furthermore, every Euler path must start at
one of the vertices of odd degree and end at
the other
Equivalent Graphs
• Two or more graphs are equivalent if they have
the same vertex and run or pass through the same
edges.
Hamiltonian Graphs
•
•
Hamiltonian circuit is a circuit that visits
vertex once with no repeats. Being a
circuit, it must start and end at the same
vertex.
Hamiltonian path also visits every vertex
once with no repeats, but does not have to
start and end at the same vertex.
The Edge-Picking Algorithm
• Another method of finding a Hamiltonian circuit in
a complete weighted graph is given by the
following edge-picking algorithm.
•
Mark the edge of smallest weight in the graph.
•
Mark the edge of the next smallest weight in the
graph, as long as it does not complete a circuit
and does not add a third marked edge to a single
vertex.
•
Continue the process until you can no longer mark
any edges. Then mark the final edge that
completes the Hamiltonian circuit.
Planarity and Graph Coloring
Weighted Graph
•
A weighted graph is a graph in which each edge
is associated with a value, called a weight.
The Greedy Algorithm
•
greedy algorithm is basically choosing the
smallest value option at every chance we
get.
•
it focuses on picking the vertex with the
minimum amount all throughout until it
travels along all the vertices.
•
finding efficient Hamiltonian circuits in
complete weighted graphs
•
choose a vertex as a starting point, and travel
along the edge that has the smallest weight.
•
A planar graph is a graph that can be drawn so
that no edges intersect each other (except at
vertices).
Graph Coloring
•
If the map is divided into regions in some manner,
what is the minimum number of colors required if
the neighboring regions are to be colored
differently?
•
There is a connection between map coloring and
graph theory. Maps can be modeled by graphs
using the countries as the vertices and two
vertices (countries) are adjacent if they share a
common boundary.
•
In graph coloring, each vertex of a graph will be
assigned one color in such away that no two
adjacent vertices have the same color. The
interesting idea here is to determine the minimum
number of (distinct) colors to be used so that we
can color each vertex of a graph with no two
adjacent vertices have the same color.
Four-Color Theorem
Every planar graph is 4-colorable.
The Chromatic Number of a Graph
The minimum number of colors needed to color a
graph so that no edge connects vertices of the same
color is called the chromatic number.
2-Colorable Graph Theorem
• A graph is 2-colourable if and only if it has no
circuits that consist of an odd number of vertices.
The time can be determined by (5 – 71) 𝑚𝑜𝑑 12. Observe that (5 – 71)
𝑚𝑜𝑑 12 = (−66) 𝑚𝑜𝑑 12.
Mathematical
Find a whole number x less than 12 such that −66 = 𝑥 𝑚𝑜𝑑 12 .
−66 = 𝑥 𝑚𝑜𝑑 12 ⇔
−66 − x
12
So that, −66 𝑚𝑜𝑑 12 = 6.
Systems
MODULO N
To say that two integers are considered as congruent modulo n, wherein n is
a natural number, if
𝑎−𝑏
is equivalent to an integer.
𝑛
Therefore, if it’s 5 o’clock now, 71 hours ago is 6 o’clock.
EXAMPLES: ARITHMETIC OPERATIONS MODULO N
3. In 2005, April 15 fell on a Friday. On what day of the week will April 15
fall in 2013?
There are 8 years between two dates.
Each year has 365 days (except for 2008 and 2012).
we write it as: a ≡ b modulo n
So the total number of days in between two dates are 8(365) + 2 = 2,922.
Thus, we want 2,922 𝑚𝑜𝑑 7 But 2,922 ÷ 7 = 417 remainder 3.
Moreover, the integer value of modulo n is equal to the remainder left of the
number when it is divided by the n.
We want a day that is same as the day 3 days after April 15, 2005.
The statement 𝑎 ≡ 𝑏 𝑚𝑜𝑑 𝑛 is called a congruence.
Modulo n example:
29 ≡ 8 mod 7
60 ≡ 0 mod 15
17 ≡ 2 mod 5
Arithmetic Operations Modulo n
perform the indicated operations first, afterwards divide your
answer by modulus. The number you will get is the remainder and should be
a whole number and less than the modulus.
Note: The result of an arithmetic operation mod n is always a whole
number less than n.
Perform the modular arithmetic.
12+9 ≡ 1 mod 5
12+3 ≡ 0 mod 5
EXAMPLES: ARITHMETIC OPERATIONS MODULO N
Therefore, April 15, 2013 is Monday.
Solving Congruence Equations
It is defined as finding of all the values (which is a whole number
less than the modulus) of the variables for which the congruent is true. It
is not your ordinary mathematical equation that needs certain solution to
be solved. It is only satisfied once the value of the variable is found. It does
not require any specific solution rather once a single solution was found
additional solutions can also be found by just repeatedly adding the
modulus to the original equation. For instance, just like for common
problems, we look for solutions that would be best to solve our problems
without the need to base it to any mathematical formulas.
Remarks:
A congruence equation can have more than one solution among the whole
numbers less than the modulus.
1. Disregarding AM or PM, if it is now 5 o’clock, what time will be 45 hours
from now?
A congruence equation can have no solution.
The time can be determined by (5 + 45) 𝑚𝑜𝑑 12.
Find all whole number solutions of the congruence equation.
Observe that (5 + 45) 𝑚𝑜𝑑 12 = 50 𝑚𝑜𝑑 12 .
But 50 𝑚𝑜𝑑 12 = 2.
Therefore, if it’s 5 o’clock now, 45 hours from now is 2 o’clock.
EXAMPLES: ARITHMETIC OPERATIONS MODULO N
2. Disregarding AM or PM, if it is now 5 o’clock, what time was it 71 hours
ago?
PROBLEMS: SOLVING CONGRUENCE EQUATION
1.
2.
3.
4.
𝑥 ≡ 7 𝑚𝑜𝑑 4
2𝑥 ≡ 5 𝑚𝑜𝑑 9
2𝑥 + 1≡ 6 𝑚𝑜𝑑 5
5𝑥 + 1≡ 3 𝑚𝑜𝑑 5
Additive and Multiplicative
Inverses
If the sum of two numbers is 0 (mod m), then the numbers are additive
inverses of each other (mod m).
If the product of two numbers is 1 (mod m), then the numbers are
multiplicative inverses of each other (mod m)
It is necessary to check only the whole numbers less than the modulus.
Examples: Find the additive inverse and the multiplicative inverse, if it
exists, of the given number.
5 in mod 7 arithmetic
7 in mod 12 arithmetic