MMW 2ND SHIFTING
​ B: 150 + 240 + 200
C:
D:
E:
→ B WINS
UNIT 5.1
THE MATHEMATICS OF VOTING
➢
the contribution of mathematics in
political science ​began during the
French Revolution in the 18th century
through Mary Jean Antoine Nicolas de
Caritat
Mary Jean Antoine Nicolas de Caritat
also known as M
​ arquis de Condorcet
a French political philosopher and
mathematician who raised important
ideas related to voting systems
VOTING
the action or process of indicating
choice, opinion, or will on a question,
such as the choosing of a candidate, by
or as if by some recognized means,
such as a ballot
the goal of voting is to have a general
agreement called c
​ onsensus
→ CONSENSUS HAS TO BE MADE
VOTING METHODS
1. Plurality Method
2. Plurality with Elimination Method
3. Instant Run-off Voting
4. Borda Count Method
5. Pairwise Comparison Method
➢
= 590
= 340
= 360
= 300
the problem with the plurality method is
that it may produce a winner, even if the
majority didn’t vote for that winner. this
is resolved by the plurality with
elimination method
PLURALITY WITH ELIMINATION METHOD
-
-
each person votes for his or her favorite
candidate (or choice)
if no candidate receives a majority, then
the candidate with the fewest votes is
eliminated and another election is held
this process continues until a candidate
receives a majority of the votes
example:
A company is planning a company outing next
summer. There are three possible locations for
the outing: Amanpulo in Palawan, Pandan Island
in Mindoro, and BellaRocca Island in
Marinduque. The 1000 employees including
managers and department heads have to decide
based on costs, amenities, and safety. The
results of the election are given in the following
table.
→ using the plurality method:
PLURALITY METHOD
-
the one with the most number of votes
or with the most first-preference votes
wins
example:
Consider the preference schedule below, in
which the voters in a barangay are voting on five
different candidates for Brgy. Captain. The
candidates are called A, B, C, D, and E here for
simplicity.
Summary of votes of 1000 employees
Amanpulo wins
Assuming the result of the new voting, with
Pandan Island being eliminated is given below.
Where will the company outing be held?
→ by plurality with elimination, ​Amanpulo wins
➢
Preference table of voters in a barangay election
the problem with this method, aside
from the fact that it requires more time
and effort, is that it may violate the
monotonicity criterion
a voting system is monotonic if it is
neither possible to
prevent the election of a candidate by
ranking them higher on some of the
ballots, nor possible to elect an
otherwise unelected candidate by
ranking them lower on some of the
ballots
→ eliminate P; all the candidates below
the eliminated candidate will be upped
by one rank higher
INSTANT RUN-OFF VOTING
-
-
-
or Ranked-choice method
each voter ​ranks all of the candidates;
that is, each voter selects his or her first
choice, second choice, third choice, and
so on
if no candidate receives a majority, then
the candidate with the fewest first
choice votes is eliminated and those
votes are given to the next preferred
candidate
if a candidate now has a majority of
first-choice votes, that candidate is
declared the winner. If no candidate
receives a majority, this process
continues until a candidate receives a
majority
example:
Consider the voting of the 1000 employees to
choose the island for their company outing. They
were asked to write their 1​st choice, 2​nd choice
and 3​rd choice. The results are shown in the
following table preference table.
a.
BORDA COUNT METHOD
-
-
-
named after ​Jean Charles de Borda​,
who developed the system in 1770
each voter ranks all of the candidates;
that is each voter selects his or her first
choice, second choice, third choice, and
so on
if there are ​k candidates, each
candidate receives
➔ k points for each first-choice
vote
➔ k ​– 1 points for each
second-choice vote
➔ k ​– 2 points for each
third-choice vote, and so on
the candidate with the most total points
is declared the winner
example:
Consider the voting of the 1000 employees to
choose the island for their company outing. They
were asked to write their 1​st choice, 2​nd choice
and 3​rd choice. The results are shown in the
following table preference table.
use the plurality method to determine
the winner
Inspecting the table for first-choice row:
A: 168 + 202 = 370
B: 215 + 105 = 320
P: 90 220 = 310
→ A wins
b.
Inspecting the table for first-choice row:
A: 168 + 202 + 90 = 460 votes
B: 215 + 105 + 220 = 540 votes
→B
​ wins
use the instant runoff to determine the
winner
Inspecting the table for first-choice row:
A: 168 + 202 = 370
B: 215 + 105 = 320
P: 90 220 = 310
→ but 370 is not yet the majority
a.
use Borda Count Method ​in coming up
with a winner
Sixth column: B: 220
A: 168 + 202 + 90 = 460
B: 215 + 105 + 220 =540
→ B gets 1 point
→ A wins
PAIRWISE COMPARISON METHOD
-
-
-
also known as the ​Copeland’s method
named after ​Arthur Herbert Copeland​,
who proposed the method in 1951
each voter ranks all of the candidates;
that is each voter selects his or her first
choice, second choice, third choice, and
so on
for each possible pairing of candidates
(head to head challenge), the candidate
with the most votes receives 1 point; if
there is a tie, each candidate receives ½
point
the candidate who receives the most
points is declared as the winner
example:
Let us consider again the decision of a company
to determine the island destination to have their
summer outing.
●
Because there are k=3 candidates, there
must be 3 pairwise comparisons; that is
3!
c23 = (3−2)!2!
=3
●
Specifically, we investigate A versus B,
A versus P, and B versus P
A versus B:
First column:
A: 168
Second column:
A: 202
Third column:
B: 215
Fourth column:
B: 105
Fifth column:
A: 90
A versus P:
First column:
Second column:
Third column:
Fourth column:
Fifth column:
Sixth column:
A: 168
A: 202
A: 215
P: 105
P: 90
P: 220
A: 168 + 202 + 215 = 585
P: 105 + 90 + 220 = 415
→ A gets 1 point
B versus P:
First column:
Second column:
Third column:
Fourth column:
Fifth column:
Sixth column:
B: 168
P: 202
B: 215
B: 105
P: 90
P: 220
B: 168 + 215 + 105 = 488
P: 105 + 90 + 220 = 512
→ P gets 1 point
➢
-
-
B > A, A > C, but C > B
Condorcet Paradox
collective preferences can be
cyclic, even if the preferences
of individual voters are not
cyclic
transitive property does not
apply
UNIT 5.2
●
APPORTIONMENT
-
the act of dividing items between
different groups according to some
plan, especially to make proportionate
distribution in a f​ air manner
STATES
the parties having stake in the
apportionment
may be cities, regions, towns or
municipalities
SEATS
the number of indivisible objects which
are assigned in different states
may also be the number of firemen,
number of teachers, or number of
nurses assigned to different cities
HAMILTON’S METHOD
named after Alexander Hamilton (1792)
first method used to apportion the
seats in the US Congress
adopted to apportion the US House of
Representatives every ten years
between 1852 and 1900
Hamilton’s method tends to favor larger
states
also called the ​largest remainder
method
Steps in Hamilton’s Method
1. Determine how many people each
representative should represent by
calculating the standard divisor​.
d=
2.
POPULATION
a positive integer ​n which refers to the
number of citizens or residents in a
locale
the basis of the seats to the states
3.
4.
APPORTIONMENT
the act of dividing people among places
METHODS OF APPORTIONMENT
●
●
●
●
●
Hamilton’s Method
Jefferson’s Method
Adam’s Method
Webster’s Method
Hill-Huntington’s Method
total population
no. of representatives
Divide each state’s population by the
divisor to determine how many
representatives it should have, aka
standard quota​.
q=
STANDARD DIVISOR (​d​)
ratio of the total population to the total
number of seats to be allocated
STANDARD QUOTA (​q​)
ratio of a state’s population to the
standard divisor
Lowndes’ Method
state′s population
d
Cut off all the decimal parts of all the
quotas. Add up the remaining whole
numbers. Get the ​lower quota ​of
each state, which is the rounded
down value of the ​standard quota
(add all the l​ ower quota​).
If the sum of the ​lower quota is less
than
the
total
number
of
representatives,
the
remaining
number of seats are assigned to the
state whose decimal value of the
standard quota i​ s the highest.
example/s:
● Suppose the Phil. Constitution was
amended and the 250 members of the
House of Representatives will be
apportioned among regions. Using the
data in the 2015 National Population
Census per Region as shown in the
table
below,
determine
the
apportionment
using
Hamilton’s
method.
4.
If that the total from Step 3 was less
than the total number of
representatives, assign the remaining
representatives, one each, to the states
whose decimal parts of the quota were
largest, until the desired total is
reached.
●
Suppose 100 doctors have to be
assigned to different parts of Metro
Manila during New Year’s eve due to
firecracker reported injuries that may
happen. Allocations will depend on the
number of reported injuries of the
previous years. Below is the report of
WHO of reported firecracker injuries
from 2010 to 2014.
Solution:
1. Compute for ​d
total population
d = no. of representatives
d=
100,981,437
250
d = 403, 925.75
2.
Compute for the ​standard quota ​of
each state
state′s population
q=
d
ex: q =
12,877,253
403,925.75
= 31.880
Solution:
1. Compute for d
​
2.
3.
Cut of all the decimal parts of all the
quotas. Add up the remaining whole
numbers.
3.
4.
Compute for the ​standard quota ​of
each state
Lower quota
Final quota
➢
The problem with this method is that
increasing the number of seats may
cause the party to lose a seat → called
Alabama Paradox (​happened in the
state of Alabama in 1980)
JEFFERSON’S METHOD
proposed by Thomas Jefferson
(1743-1826)
first used in US Congress in 1792 until
1840.
Jefferson’s method tends to favor
larger states
first steps are the same with Hamilton’s
method
example/s:
● The House of Representatives of a
certain country needs ​41 new members
to divide among three states A, B, and
C. The population data of the 3 states
are as follows:
Solution:
1. Compute for d
​
total population
d = no. of representatives
d=
2.
897,934
41
d = 21, 900.83
Compute for the ​standard quota ​of
each state
q = state′s population
d
Steps in Jefferson’s Method
1. Determine how many people each
representative should represent by
calculating the standard divisor​.
d=
2.
total population
no. of representatives
Divide each state’s population by the
divisor to determine how many
representatives it should have, aka
standard quota​.
q=
3.
4.
3.
Cut of all the decimal parts of all the
quotas. Add up the remaining whole
numbers.
4.
Reduce the standard divisor (21,900.83)
until the sum of lower quota is 41.
state′s population
d
Cut off all the decimal parts of all the
quotas. Add up the remaining whole
numbers. Get the ​lower quota ​of
each state, which is the rounded
down value of the ​standard quota
(add all the ​lower quota​).
If the total from Step 3 was less than
the total number of representatives,
reduce the divisor and recalculate
the quota and allocation. ​Continue
doing this until the total in Step 3 is
equal to the total number of
representatives.
➔ The divisor we end up using
is called the ​modified
divisor or adjusted divisor​.
Trying modified divisor as 21,000
●
Use the Jefferson method to determine
how many doctors have to be assigned
at different parts of Metro Manila.
➔
Using Hamilton’s method, with standard
divisor of 28.10
Reduce ​the divisor. Trying divisor as 27.
➔
ADAM’S METHOD
Proposed by John Quincy Adams
(1767-1848)
Adams proposed using a modified
divisor greater than the standard
divisor, and he rounded his modified
quotas quota up to the upper quota.
It tends to favor smaller states.
Steps in Adam’s Method
➢ Opposite of Jefferson’s method
➢ Instead of rounding down to get the
standard quota, r​ ound up​.
➢ If the sum of the upper quotas does
not equal the total number of seats
to be apportioned, choose a
modified divisor ​greater ​than the
standard divisor and calculate the
modified quotas and upper modified
quotas.
example/s:
● The House of Representatives of a
certain country needs 41 new members
to divide among three states A, B, and
C. The population data of the 3 states
are as follows:
MODULO ARITHMETIC
-
-
-
-
-
Related to that of the remainder in
division
Operation of finding the remainder is
sometimes referred to as modulo
operation
Ex: clock → always on 1 to 12
Dividing an integer Z by 5 will have the
remainders {0, 1, 2, 3, 4)
25/5 = 5 + 0/5 → remainder
24/5 = 4 + ⅘ → remainder
Thus we define Zn as the set of integers
from 0, 1, 2, …, n-1 modulo n, i.e.
Zn = {0, 1, 2, …,n-1}
Z3 = {0, 1, 2} modulo 3
Z5 = {0, 1, 2, 3, 4, 5} modulo 5
Z8 = {0, 1, 2, 3, 4, 5, 6, 7, 8}
modulo 8
Zn has exactly n non-negative
integers
In Zn, modulo is simply the remainder r
when an integer a ​∑ Z is simply divided
by n
a/n has a remainder r < n
PERFORM THE FOLLOWING OPERATIONS
(REMAINDER)
1. In Z8 what is 4+9
Answer is 5
5 is the remainder when 13 is divided by
8
4+9 = 13
13/8 = 5 → remainder
2. In Z8, what is 15 + 21
15 + 21 = 36
36/8 = 32 +4/8 = 4
3. In Z8 what is 106+102
106 + 102 = 208
Divisible by 8 → 0 is the remainder
4. In Z8, what is 22-3
22-3 = 19
19/8 = 3
5. In Z8, what is 3-20
3 - 20 = -17
-17/8 = -16 - 1 / 8 = 7
8 was added to -1
6. In Z8, what is 11 x 5
48 + 7 = 55
Answer is 7
THE MODULO TABLE
Zn is closed under the binary operations
of addition and multiplication of
integers modulo n
Any 2 numbers in the set, the answer
are also in the set
Examples for addition
a.
b.
In Z3 = {0, 1, 2} → 3 integers only starting from 0
to 2 (limitations)
2 + 1 = 0 remainder
2 + 2 = 1 remainder
Example for multiplication
2 x 2 = 4 → remainder 1
THE MODULO CONGRUENCE
2 integers a and b are said to be
congruent modulo n, where n is a
natural number if a-b/n is an integer
Example
a = 13
b=4
n=9
-
a is congruent to b modulo n
Additive Inverse
In Zn , two numbers a and b are additive
inverse of each other if a + b ≡ 0 mod n
-
TRUE OR FALSE
1. 29 ​≡ 8 mod 3
3 │29-8
3 │21
TRUE
2. 15 ​≡ 4 mod 6
6 │15-4
6 │11
FALSE
3. 37 ≡ 9 mod 7
7 ​│37-9
7 │28
TRUE
4. 115 ≡ 10 mod 3
3 │115-10
3 │105
TRUE
5. 84 ​≡ 3 mod 9
9 │84-3
9 │81
TRUE
EQUATIONS INVOLVING MODULO
CONGRUENCE
1.
Example: Find the additive inverse of 7 in mod 16
Answer: 9 because 7 + (9) = 0 mod 16
Find the additive inverse of 2 in mod 12
Answer: 10
Find the additive inverse of 5 in Z7
5+ (2) ≡ 0 mod 7
Answer: 2
Multiplicative Inverse
In Zn , two numbers a and b are
multiplicative inverse of each other if ab
≡ 1 mod n.
Example: Find the multiplicative inverse of 5 in
Z7
Answer: 3 because (5)(3) = 1 mod 7
Find the multiplicative inverse of 3 in Z10
Answer: 7 because (3)(7) = 1 mod 10
Find the multiplicative inverse of 4 in mod 6
Answer: none
No multiple of 6 can be divided by 4 to
have a remainder of 1
Every number has an additive inverse but not
necessarily a multiplicative inverse.
2.
APPLICATIONS OF MODULO ARITHMETIC
Zeller’s congruence
Julius Christian Johannes Zeller (June
24 1822 - May 31, 1899)
X - any real number
N - whole number lang
EX. Floor Function of ½ = 0.5 → floor func is 0
-
3/2= 1.5 → floor func is 1
PI → 3
-
Thus when we divide a certain number
by another number, the floor function is
the quotient excluding the remainder
Kumbaga parating round down
Cautions:
● [x] is the rounded down value of x.
● Jan and Feb are the 13th and 14th
months of the previous year.
● January 20, 2011 is like the 13th month
of 2010
-
EXAMPLE - Using Gregorian Calendar
On what day is University of Santo Tomas
founded? → April 28, 1611
On what day is January 15, 2019
Modulo Function
Two integers a and b are said to be
congruent modulo n, where n is natural
number a - b is divisible by n. In case we
write:
DETERMINE THE DAY YOU WERE BORN
VIDEO ON ZELLER'S CONGRUENCE
Floor Function​ - For any real number x,
-
No need for the decimal point for the
floor function of x → only care about
the quotient not the remainder
-
The number n is called the modulus.
The statement a=b mod n is called the
congruence
If b mod n only means the remainder
when b is divided by n. Sometimes
written as
Rem (b, n) = b mod n
5 mod 2 = 1
Floor Function =​ quotient only
Modulo Function = r​ emainder only
Julian Calendar
365.25 days in a year (leap year is every
4 years
Proposed by Julius Cesar
Gregorian Calendar
calendar we use today
365.2422 in a year
Leap year occurs if the year is divisible
by 4 except for years divisible by 100
(i.e. 1700, 1800). But if the year is
divisible by 400 (like year 2000) its a
leap year
The calendar repeats itself every 400
years
Difference of gregorian calendar and
julian calendar is just 0.0078 → but has
a very big effect if we use diff ones
Introduced by Pope Gregory XII (June 7
1502- April 10 1585)
Accepted in 1752 onwards
Example:
If today is June 29, 2020 in Gregorian Calendar,
in Julian Calendar it is stil June 16, 2020 → 13
days
The discrepancy of both both
calendars, 0.0078 → is actually more
than 10 mins
Kaya Ganyan cause
Leap Year
John Harton Conway
December 26, 1937 - April 11, 2020
Doomsday Method
INTRODUCTORY CONCEPTS IN STATISTICS
STATISTICS
- comes from the Latin word “​statisticum
collegium​”, meaning “council of state”; Italian
word “​statista​”, meaning “statesman” or
“politician” or “political state”
- a discipline concerned with the analysis of data
and decision-making based upon data
- involves collecting, organizing, summarizing,
interpreting and presenting data
- why is it being learned:
● in medicine, it is used to determine the
efficacy of a drug
● in business and economics, it is used to
forecast sales, stocks, etc.
● in everyday life, it is used to see
weather condition forecasts
● in marketing, it is used to study
consumer behavior
● in psychology, it is used to study human
behavior
● in sports, it is used to summarize the
performance of athletes
- two categories of statistics:
● Descriptive Statistics
● Inferential Statistics
DESCRIPTIVE STATISTICS
- ​involves the method of organizing,
summarizing and presenting the data in an
informative form
- computing for the average or presenting data
through graphs
- totality of methods and treatments employed in
the collection, description and analysis of
numerical data
- its purpose is to tell something about a
particular group of observation
- limited to data set
INFERENTIAL STATISTICS
- also called statistical inference or inductive
statistics
- from the data, we go beyond the presentation;
from the analysis and presentation of the
available data, we predict or come up with an
intelligent guess on what’s next upon careful
investigation
- it is the logical process from sample analysis to
a generalization or a conclusion about a
population
- involves using information from a sample to
draw conclusions about the population
➔ Population ​- all the members of the
subject of interest
→ result: p
​ arameter
➔ Sample ​- selected members of the
subject of interest; not all possible
respondents under a certain study are
always available for a certain procedure
of statistical treatment
→ result: s
​ tatistics
➔ statistics ​is an estimate of the
parameter
➔ Variables ​- the one being measured; a
characteristic of objects, people or
events that can take of different values,
can vary in quantity
➢ Qualitative variables
➢ Quantitative variables
➔ Constants ​- held fixed; a characteristic
of objects or groups, people or events
that does not vary
Sources of Data
● Primary Data
comes from one’s personal encounter;
data collected personally
● Secondary Data
comes from other resources, may come
from journals, books, etc.
Given the following scenarios, identify:
● population and sample
● parameter and statistics
● variable and constant
Scenario 1:
● When all University of Sleep freshmen
students were asked, it was found that,
on the average, they spend 3.7 hours of
sleep per day during exam week.
● But from a 30 randomly selected
University of Sleep freshmen students,
it was found to be 3.6 hours per day.
➢ population​: all University of Sleep
freshmen students
➢ samples​:
30
randomly selected
University of Sleep freshmen students
➢
➢
➢
➢
parameter​: 3.7 hours
statistics​: 3.6 hours
variable​: number of hours of sleep
constant​: year level of University of
Sleep students
Scenario 2:
● From 100 randomly selected residents
of northern Alkovia, it was found that
13% of them had Dengue fever in 2019.
But according to the Institute of
Health’s Epidemiology Center (IHEC)
11.9% of all residents of northern
Alkovia had Dengue fever in 2019.
➢ population​: all residents of northern
Alkovia
➢ samples​: 100 randomly selected
residents of northern Alkovia
➢ parameter​: 11.9%
➢ statistics​: 13%
➢ variable​: occurrence of Dengue fever
➢ constant​: disease (Dengue), year
Scenario 3:
● 5% of Asian men suffer from red-green
color blindness. From 250 randomly
selected men in Southeast Asia, it was
found that 3% suffer from this type of
color blindness.
➢ population​: all Asian men
➢ samples​: 250 randomly selected men in
Southeast Asia
➢ parameter​: 5%
➢ statistics​: 3%
➢ variable​: occurrence of color blindness
➢ constant​: type of color blindness, race
DATA PRESENTATION
- results of data may be presented through:
● TEXTUAL
results are expressed in ​declarative
form
this type of presentation is advised
when you are trying to give an overview
of a single group, like reporting their
age, sex, and others
examples:
→ The mean age of the respondents is
28.7 years.
→ Twenty (40%) of the customers are
male.
●
-
TABULAR
results are displayed in rows and
columns
this method of presentation is advised
when giving a summary or repetition of
categories
example:
Male (n=20)
Female (n=30)
21.1
20.8
No. of smokers
3 (15.0%)
1 (3.3%)
Monthly income
21.2
21.7
Age (years)
* values expressed as mean, or counts (%)
●
-
GRAPHICAL
results are presented in diagrams
Types of Graphs:
Bar graphs
used to compare average,
counts, or percentage of
different groups
Line graphs
used to observe trends and to
compare trends or observe
gaps between groups across
time; best used when the
horizontal axis is an element
of time like data expressed in
daily, weekly, monthly, or
yearly time intervals
Pie graphs
used to express parts of a
whole; recommended to have
a maximum of 5 to 7 sectors
only
Scatterplots
used
to
describe
relationship
of
quantitative variables
Statistical
maps
used to present information
with respect to geographical
location
Pictogram
series of repeated resembling
icons to visualize simple data
Population
Pyramid
also known as the age-gender
pyramid;
shows
the
the
two
distribution of various age
groups in a population
according to gender
Boxplots
shows
the
five-number
summary (quartiles) of a given
set of data
Violin plots
similar to box plot, with the
addition of putting densities
on each side
- also called the o
​ utcome variable
LEVELS OF DATA MEASUREMENT
-
In statistics, it is important that you can
classify
the
variables
correctly.
Statistical treatments depend on the
classification of variables​.
➔
1.
example/s:
Are you a smoker?
❏ Yes
❏ No
= nominal
How frequently do you smoke?
❏ Always (>19 sticks/day)
❏ Oftentimes (10-19 sticks/day)
❏ Sometimes (1-9 sticks/day)
❏ Rarely (at most 1 stick/week)
❏ Never
= ordinal
How many sticks of cigarette do you
smoke in a day?
Specify: _______ sticks
= ratio
VARIABLES
● Quantitative variables
● Qualitative variables
● Independent variable
● Dependent variable
Quantitative variables
- expressed in numerical form
- reveals the pattern of a certain data
- continuous or discrete variable
➔ Continuous variable
every fraction, decimal is considered
(ex: height, temperature)
➔ Discrete variable
whole numbers are the only ones to be
considered (ex: no. of people)
- may be categorized into two levels:
➔ Interval
➔ Ratio
Qualitative variables
- also known as the categorical variables
- non-numeric data; expressed in textual form
- provides the answer to the question “why?”
- may be further classified into two levels:
➔ Nominal
➔ Ordinal
Independent variable
- variable controlled by the experimenter or the
researcher and expected to have an effect on the
behavior of the subject
- also called the ​explanatory variable
Dependent variable
- some measure of the behavior of subject and
expected to be influenced by the independent
variable
2.
3.
NOMINAL LEVEL OF MEASUREMENT
categorical
variables
in
which
responses have no order
all responses are in the same level
example/s: race, color, sex, disease
occurrence, subscription
Keyword: Category
It is used to differentiate classes or
category for purely classification or
identification purposes
SSS number
Ex. Survey in Tapsilogan about the taste sa isang
menu → pakaplog
Commonly in survey questionnaires
there is a demographic profile
Mutually exclusive and exhaustive →
for every category there should only one
answer
Nominal data is ​qualitative variable
ORDINAL LEVEL OF MEASUREMENT
categorical
variables
in
which
responses have order
responses may be ranked or arranged
ascendingly or descendingly
example/s: position in organization,
BMI interpretation, smoking frequency,
Likert scale
Keyword: RANK, qualitative
Somewhat stronger than the nominal
level, because in an original data, aside
from categorization, one data can be
identified as greater or lesser than the
other data
However there is no numerical
specification of the difference between
two data. We only know that one data is
lesser or higher than the other data
Example: a teacher asks a question to the
students “what is nominal data”
Student ans: the one you discussed a
while ago (GOOD)
Another question was asked, what is
ordinal data? - student answered: the
one that we are discussing now (VERY
GOOD)
Teacher Asked again “what is
statistics” student answered It is a
branch of mathematics that examines
and investigates ways and processes to
-
analyze the data gathered…
(EXCELLENT)
We only know the ranking however we
don't know the difference between the
two (ex the diff of good and very good
→ numerically)
INTERVAL LEVEL OF MEASUREMENT
has r​ elative zero
→ zero response does not indicate
absence
→ example/s: temperature in Celsius or
Fahrenheit (The temperature in A (40​°C)
is 30°C more as compared to B (​10​°C))
responses are compared through their
difference
Higher than nominal and ordinal
Used to classify order and differentiate
between classes or categories in terms
of degrees of differences
Either discrete or continuous, In other
words we are dealing with numerical
values
Calculation
Example: Temp reading (C and F = numerically
diff), calendar, scores of students, clock
RATIO LEVEL OF MEASUREMENT
has a
​ bsolute zero
→ zero response indicates absence
→ example/s: temperature in Kelvin, no.
of books, liters of alcohol, height
can be expressed as factor of one
response over the other
→ example/s: amount of money (Php
100 is 5 times the value of Php 20)
responses are compared through their
quotient
Pinakamataas na level of measurement
Only one diff with interval which is
absolute zero or it has a true zero point
Example: Weight, height → you can not have a
measurement that is below zero, unlike interval
data pwede like -10 C, However if Kelvin, ratio na
siya, cause in Kelvin no zero → zero reading is
considered a null point
Almost all that you can use in ratio can
be used in interval
Sometimes if the data is numeric it is
called ​scale data
-
SAMPLING TECHNIQUE
RANDOM / PROBABILITY
Simple
Systematic
Stratified
Cluster
NON RANDOM / NON PROBABILITY
Purposive/ Judgement
Convenience
Quota
Snowball
Voluntary
RANDOM SAMPLING
1. Simple -​ process of selecting a certain
number of sample size in a population
via random numbers or lottery
Example: Lotto (in 6/42) - the machine will pick
six numbers out of 42
6 samples in 42 population
Classroom Setting
Names in the jar for recitation
2. Systematic ​- process of selecting an
nth element in the population until the
desired number is obtained
Example: Need ng 5 students out of 40 students
in the class
POPULATION / SAMPLE
40 / 5 = 8
From 1 - 8 you randomly select a
number (lets say nabunot ay 2 → so
lahat ng number 2 selected)
What if may remainder → then you can
roundoff nalang
Stratified​ Sampling - is the process if
subdividing the population into
subgroups or strata and drawing
members at random from each
subgroup or stratum
In accordance with the
proportion of each stratum
Strata- g
​ roup with
commonality
Example: Surveying 200 reviewees out of 50
reviewees in review center
Out of 50k the breakdown of reviewees
are
Sample Size
3.
50% of 200
16% of 200
And so on
Progra
m
# of
reviewee
%
Sample
Size
LET
25 000
50%
100
CSE
8 000
16%
32
NLE
7 000
14 %
28
CLE
10 000
20%
40
Total
50 000
100%
200
Cluster- s
​ omewhat related with
stratified sampling, in a sense where
both are subdividing the population
Process of selecting clusters
from a population which is
very large or widely spread out
over a wide geographical area
Subgroups = “cluster”
Paghati ng cluster in
accordance with
geographical area
Ex: Philippines- widely dispersed so subdivided
into regions
Classroom - by row
*In reality usually combination of techniques
ginagamit
4.
NON RANDOM SAMPLING
Is called nonrandom if the participant
do not have equal chance in becoming
part of the study
Biased
Sampling procedure or technique
whose sample are selected in a
deliberate manner with little or no
attention to randomization
1. Purposive/ Judgemental ​sampling or
subjective sampling - process of
selecting based on judgement to select
as a sample which the researcher
believed, based on prior information,
will provide the data they need. →
subjective
Ex: interview
Convenience- t​ he researcher chooses
members merely based on proximity,
and does not consider whether they
represent the entire population or not →
willing & available
Ex: research- where you ask your friends, parents
and stuff
3. Quota- ​applied when an investigator
survey or collect information from an
assign number or quota of individual
from one of several sample units
fulfilling a certain prescribed criteria or
belonging to one stratum
Ex: you will conduct a survey to people with diff
gender → to make it easier you get a sample that
is based on a certain criteria
Example: honor students lang interview
4. Snowball- t​ echnique in which one or
more members of a population are
located and used to lead the researcher
to other members of the population
Aka referral method, referral
sampling, chain referral
sampling
Ask referrelas from others
with the same interest → like a
snowball na lumalaki habang
gumugulong
5. Voluntary- ​technique when samples are
composed of respondents who are self
select into the survey or study. Most of
the time the respondents have a strong
interest in the study
Ex: national election
2.
Cochran sample sizehttp://www.raosoft.com/samplesize.html