LP in Stat & Proba.pdf

i
PROTOTYPE
CONTEXTUALIZED
DAILY LESSON
PLANS IN
GRADE 11/12
STATISTICS AND
PROBABILITY

iii
LIST OF DEVELOPMENT TEAM MEMBERS
PROTOTYPE AND CONTEXTUALIZED DAILY LESSON PLANS IN GRADE
11/12 (STATISTICS AND PROBABILITY)
WRITERS
MILADEN DESPABILADERAS JASMIN A. JAO
ERLYN M. LACSA PINKY D. DESTACAMENTO
ANALYN M. VELOSO MARKSON B. MEJIA
ADELFA C. DITAN AMORY R. BORINGOT
RICHELLE D. DIONEDA KATHLEEN DUCAY
MA. JECCA L. AZAS ARWIN D. BONTIGAO
RUEL G. FRAGO NELVIN EBIO
MA. CIELO BERMUNDO CATHERINE GERONIMO
MICHAEL DOMANAIS
DEMONSTRATION TACHERS
TITO GUATNO PINKY O. DESTACAMENTO
ERLYN M. LACSA JASMIN A. JAO
RUEL G. FRAGO ANNALYN M. VELOSO
AMORY R. BORINGOT MARKSON B. MEJIA
MARILYN HULAR JECCA AZAS
RICHELLE DIONEDA MILADEN DESPABILDERAS
KATHLEEN DUCAY ARWIN BONTIGO
EDITORS
MA. THERESA DUAZO ROWENA H. BORJA
ELENA D. HUBILLA
Education Program Supervisor-1
Mathematics
MONSERAT D. GUEMO, Ph. D.
CID Chief Supervisor
MARIVIC P. DIAZ, Ed. D.
OIC, Assistant Schools Division Superintendent
Dr. NYMPHA D. GUEMO
Schools Division Superintendent

1
I.OBJECTIVES
A. Content
Standards
The learner demonstrates understanding of key concepts of
random variables and probability distributions.
B. Performance
Objective
The learner is able to apply an appropriate random variable for
a given real-life problem (such as in decision making and
games of chance).
C. Learning
Competencies/
Objectives
( Write the LC
code for each)
 The learner illustrates a random variable (discrete
and continuous). (M11/ 12SP- IIIa-1)
 The learner distinguishes between a discrete and
continuous random variable. (M11/ 12SP – IIIa-2)
II.CONTENT EXPLORING RANDOM VARIABLES
III.LEARNING
RESOURCES
A. References
1.Teachers Guide
pages
2.Learners
Material Pages
3. Textbook
Pages
Statistics and probability
Rene R. Belecina, Elisa S. Bacacay, Efren B. Mateo,pp.2 –8
B. Other Learning
Resources
Worksheets
IV. PROCEDURE
A. Reviewing past
lesson or
Presenting the
new lesson
COUNTABLE or MEASURABLE?
Identify whether the given situation is countable or measurable.
The students will raise their right hand if the situation is
countable, left if it is measurable.
1. Number of students inside the classroom
2. Amount of salt needed to cook chicken tinola
3. Number of likes your recent post received
4. Capacity of an auditorium
5. Length of the chalkboard
School Grade Level 11
Teacher Learning Area Statistics and
Probability
Time & Date Quarter 3rd

2
B. Establishing a
purpose of the
new lesson
The teacher should have summarized the learners’ answers in
the previous activity as follows:
COUNTABLE MEASURABLE
Number of students inside
the classroom
Amount of salt needed to
cook tinola
Number of likes your recent
post received
Capacity of an auditorium
Length of chalkboard
Let the learners identify the key words identifying countable
and measurable variables.
C. Presenting
Examples/
instances of the
new lesson
1. The teacher will discuss what variable is.
A variable is a characteristic that is observable or
measurable in every unit of the universe. Variables can be
broadly classified as either qualitative or quantitative. And
quantitative can be classified into discrete and
continuous.
2. The students will be asked to determine the variables in the
activity they performed.
3. The teacher will explain quantitative and qualitative
variables, as well as discrete and continuous variables.
4. Let the students classify discrete and continuous variables
from the given situations in the activity.
D. Discussing new
concepts and
practicing new
skills no.1.
CREATE YOUR GROUP PROFILE
To create a group profile in statistics class, the members
of each team will fill up the following data:
 NAME OF THE STUDENT
 GENDER
 AGE
 NUMBER OF SIBLINGS
 DAILY ALLOWANCE
 RELIGION
 HEIGHT IN CM
 WEIGHT IN KG
 FINAL GRADE IN GENERAL MATH SUBJECT
After gathering the data, each team will make a creative group
profile on a cartolina.
RUBRICS:
ORGANIZATION OF THE DATA – 15
CREATIVITY – 10
TOTAL: 25
The students will make a summary on the classifications of the
data gathered in their group profile through a table.
QUANTITATIVE QUALITATIVE
Age Name of student
Number of siblings Gender
Daily allowance Religion
Height in cm
Weight in kg
Final grade in General Math

3
Guide Questions:
1. When do you say that the variable is qualitative?
2. When do you say that the variable is quantitative?
3. Among the quantitative variables, which are discrete? Why?
4. Among the quantitative variables, which are continuous?
Why?
E. Discussing new
concepts and
practicing new
skills no.2
The students will classify the listed quantitative variables in the
activity CREATE YOUR GROUP PROFILE as discrete or
continuous by putting the data in the correct column.
DISCRETE CONTINUOUS
Age Height in cm
Number of siblings Weight in kg
Daily allowance Final grade in Gen. Math
The teacher will discuss what discrete and continuous
variables are.
A random variable is a discrete random variable if
its set of possible outcomes is countable.
A random variable is a continuous random variable
if it takes on values of a continuous scale. Often,
continuous random variables represent measured
data, such as heights, weights and temperatures.
F. Developing
Mastery
(Leads to
Formative
Assessment 3.)
WHAT AM I?
The students will classify the listed quantitative variables below
as discrete or continuous by putting the data in the correct
column.
1.the number of patients attributed to dengue
2. the average amount of electricity consumed per household
per month
3. the number of patient arrivals per hour at a hospital
4. the number of voters who reported for registration
5. the amount of sugar in a cup of coffee
DISCRETE CONTINUOUS
The number of patients
attributed to dengue
The average amount of
electricity consumed per
household per month
the number of patient arrivals
per hour at a hospital
the amount of sugar in a cup
of coffee
the number of voters who
reported for registration
G. Finding
practical
application of
concepts and
skills in daily living
Make a survey regarding the use of cellphone of 5 of your
classmates using the following variables. For each of them,
classify the qualitative and the quantitative. Distinguish a
quantitative variable as to discrete or continuous.
1. Number of family members with cellphone
2. Type of ownership
3. Length (in minutes) of longest call made on each cellphone

4
4. Amount paid for cellphone load per month
H. Making
Generalization
and abstraction
about the lesson
1. How do you classify quantitative and qualitative variable?
2. How do you distinguish discrete and continuous variable?
I. Evaluating
learning
Classify whether the variable is qualitative or quantitative. If
quantitative, distinguish if discrete or continuous.
ADVANCED LEARNERS AVERAGE LEARNERS
1. the number of dropouts in
a school for a period of 10
years
1. gender of athletes for
Palarong Bikol
2. the number of defective
computers produced by a
manufacturer per year
2. speed of a car
3. the number of points
scored in a basketball game
3. number of school days per
semester
4. the heights of a varsity
players in a school in meters
4. the number of accidents
per year in an intersection
5. the length of time spent in
playing video games in
minutes
5. the number of deaths
attributed to lung cancer
J. Additional
activities for
application and
remediation
Reflection:
In life, what are countable treasures? What are measurable
treasures? If you are to choose, which do you prefer to keep,
countable or measurable treasures? Why?
V- REMARKS
VI-REFLECTION
VII-OTHERS
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the
remedial
lessons work?
No. of learners
who have caught
up with the
lesson
D. No. of learners
who continue to

5
require
remediation
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What
difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What
innovation
or localized
materials did I
use/discover
which I wish to
share with other
teachers?

6
I.OBJECTIVES
A. Content
Standards
random variables and probability distributions.
B. Performance
Objective
The learner is able to apply an appropriate random variable for
a given real-life problem (such as in decision making and games
of chance).
C. Learning
Competencies/
Objectives
( Write the LC
code for each)
 The learner finds the possible values of a random
variable.
( M11/12SP – IIIa-3)
II.CONTENT EXPLORING RANDOM VARIABLES
III.LEARNING
RESOURCES
A. References
1.Teachers
Guide pages
2.Learners
Material Pages
3. Textbook
Pages
B. Other Learning
Resources
Statistics and probability
Rene R. Belecina, Elisa S. Baccay, Efren B. Mateo, pp.2 8
IV.PROCEDURE
A. Reviewing
past lesson or
Presenting the
new lesson
Each team will perform an experiment using coins and dice to
answer the following questions. An answer board is provided for
each team. Every correct answer is equivalent to 5 points. The
three teams with the highest score will be declared winners.
1. In how many ways can a coin fall?2
2. In how many ways can a die fall?6
3. In how many ways can two coins fall?4
4. In how many ways can two dice fall?36
B. Establishing a
purpose of the
new lesson
GIVE ME MY SAMPLE SPACE
Each team will complete the table by identifying the sample
space for the given event.
EVENT SAMPLE SPACE
1. Tossing a coin H,T
2. Rolling a die 1,2,3,4,5,6
3. Tossing two coins (H,T), (H,H), (T,H), (T,T)
4. Rolling two dice (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
Probability

7
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)
Guide Questions:
1. How many outcomes did you have in tossing a coin?
2. How many outcomes did you have in rolling a die?
3. How many outcomes did you have in tossing two coins?
4. How many outcomes did you have in rolling two dice?
5. How did you represent the outcomes of each event?
This activity leads you to the understanding of Sample Space
and Finding the Value of the Random Variable.
C. Presenting
Examples/
instances of the
new lesson
Two dice are rolled. Let X be the random variable representing
the 6 spots/dots that occur. Find the value of random variable
X.
SAMPLE SPACE VALUE OF THERANDOM
VARIABLE X
1,1 0
1,2 0
1,3 0
1,4 0
1,5 0
1,6 1
2,1 0
2,2 0
2,3 0
2,4 0
2,5 0
2,6 1
3,1 0
3,2 0
3,3 0
3,4 0
3,5 0
3,6 1
4,1 0
4,2 0
4,3 0
4,4 0
4,5 0
4,6 1
5,1 0
5,2 0
5,3 0
5,4 0
5,5 0
5,6 1
6,1 1
6,2 1
6,3 1

8
6,4 1
6,5 1
6,6 2
The value of random variable X are 0,1 and 2
D. Discussing
new concepts
and practicing
new skills no.1
The teacher will discuss how to find the Value of Random
Variable.
E. Discussing
new concepts
and practicing
new skills no.2
Suppose an experiment is conducted to determine the distance
that a certain type of car will travel using 10 L of gasoline over a
prescribed test course. If distance is a random variable, can you
determine the value of random variable? Why? Why not?
Lead the students to understanding on continuous random
variable.
F. Developing
Mastery (Leads
to Formative
Assessment 3.)
Three coins are tossed. Let Z be the random variable
representing the number of heads that occur. Find the values
of the random variable Z.
SAMPLE SPACE VALUE OF THERANDOM
VARIABLE Z
H,H,H 3
H,H,T 2
H,T,H 2
H,T,T 1
T,H,H 2
T,H,T 1
T,T,H 1
T,T,T 0
The value of the random variable Z are 0,1,2 and 3
G. Finding
practical
application of
concepts and
skills in daily
living
Suppose a cellphone buyer wants to buy four units of
cellphones. Randomly, how would he know that the cellphone
he chose is defective or not?
Let D represent the defective cellphone and N represents the
non-defective cellphone. If we let X be the random variable
representing the number of defective cellphones, show the
values of the random variable x. Complete the table below to
show the values of the random variable.
SAMPLE
SPACE
VALUE OF THE RANDOM VARIABLE X
(number of defective cellphones)
D,D,D,D 4
D,D,D,N 3
D,D,N,D 3
D,D,N,N 2
D,N,D,D 3
D,N,D,N 2

9
D,N,N,D 2
D,N,N,N 1
N,D,D,D 3
N,D,D,N 2
N,D,N,D 2
N,D,N,N 1
N,N,D,D 2
N,N,D,N 1
N,N,N,D 1
N,N,N,N 0
The value of the random variable X are 0,1,2,3and 4.
H.Making
Generalization
and abstraction
about the lesson
How do you find the values of a random variable?
I. Evaluating
learning
Find the possible values of the random variable.
ADVANCED LEARNERS
From a box containing 4 black balls and 2 green balls, 3 balls
are drawn in succession. Each ball is placed back in the box
before the next draw is made. Let G be a random variable
representing the number of green balls that occur. Find the
values of the random variable G.
POSSIBLE
OUTCOMES
VALUE OF RANDOM VARIABLE G
AVERAGE LEARNERS
A shipment of five computers contains two that are slightly
defective. If a retailer receives three of these computers at
random, list the elements of the possible outcomes using D
for defective and N for non- defective computers. To each
sample point assign a value x of random variable x of the
random variable X representing the number of computers
purchased by the retailer which are slightly defective. Find the
values of the random variable X.
POSSIBLE
OUTCOMES
VALUE OF RANDOM VARIABLE G
J. Additional
activities for
application and
remediation

10
V- REMARKS
VI-REFLECTION
VII-OTHERS
A. No. of
learners
who earned 80%
in the evaluation
B. No. of
learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the
remedial
lessons work?
No. of learners
who have caught
up with the
lesson
D. No. of
learners
who continue to
require
remediation
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What
difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What
innovation
or localized
materials did I
use/discover
which I wish to
share with other
teachers?

11
Probability
I. OBJECTIVES
A. Content Standard The learner demonstrates understanding of key
concepts of random variables and probability
distributions.
B. Performance
Standard
The learner is able to apply an appropriate random
variable for a given real-life problem (such as in
decision making and games of chance).
C. Learning
Competencies/
Objectives
The learner is able to:
 Illustrate a probability distribution for a discrete
random variable and its properties. M11/12SP-
IIIa-4
 Construct the probability mass function of a
discrete random variable and its corresponding
histogram. M11/12SP-IIIa-5
 Compute probabilities corresponding to a given
random variable. M11/12SP-IIIa-6
II. CONTENT CONSTRUCTING PROBABILITY DISTRIBUTION
III. LEARNING
RESOURCES
A. References
1. Teacher’s guide
pages
117-127
2. Learner’s material
pages
NONE
3. Textbook Pages
4. Additional
materials from
learning resource
(LR) portal:
NONE
B. Other Learning
Resources
Statistics and Probability by Rene R. Belecina, Elisa S.
Bacacay, and Efren B. Mateo
IV. PROCEDURES
A. Reviewing previous
lesson or presenting new
lesson
Ask the learners to provide information on how many
siblings they have by asking them to raise their hands
as the teacher calls the no. of siblings they have by
starting with 0,1,2,..
No. of
Siblings
Frequency Relative
Frequency
0
1
2
3
4
5
6
7
8

12
9
10
Total:
Draw a histogram to represent relative frequency.
Emphasize that the values on the y- axis represent
these relative frequencies (in percent). Have them add
the areas, and show that the sum is 100%. Ask them if
this is a coincidence or this is expected?
B. Establishing a purpose
for the lesson
 Present the objectives of the lesson:
 Illustrates a probability distribution for a discrete
random variable and its properties.
 Constructs the probability mass function of a
discrete random variable and its corresponding
histogram.
 Computes probabilities corresponding to a given
random variable.
C. Presenting
examples/instances of the
new lesson
 Introduce the properties of the probability.
Properties of Probability Distributions of Discrete
Random Variable
1. The probability of each value of the random
variable must be between or equal to 0 and 1. In
symbol, we write it as 0 ≤ P(x) ≤ 1.
2. The sum of the probabilities of all values of the
random variable must be equal to 1. In symbol,
we write it as ∑ P(x) = 1.
 Present the example.
Suppose three coins are tossed. Let Y be the
random variable representing the number of tails
that occur. Find the probability of each values of the
random variable Y.
Solution:
STEPS SOLUTION
1. Determine the
sample space.
Let H
represent head
and T
represent Tail.
The sample space for this experiment
is:
S=
{TTT, TTH,THT,HTT, HHT,HTH, THH,HHH)}
2. Count the
number of tails
in each
outcome in the
sample space
and assign a
number to this
outcome.
Possible
Outcomes
Value of the
Random Variable
Y (Number of tails)
TTT 3
TTH 2
THT 2
HTT 2
HHT 1

13
HTH 1
THH 1
HHH 0
3. Write the
possible values
of the random
variable Y
representing
number of tails.
Assign
probability
Values P(Y) to
each value of
the random
variable.
Number
of tails
(Y)
Probability
P(Y)
0 1/8
1 3/8
2 3/8
3 1/8
The probability Distribution or the Probability Mass
function of Discrete Random Variable Y
Number of tails
Y
0 1 2 3
Probability P(Y) 1/8 3/8 3/8 1/8
D. Discussing new
concepts and practicing
new skills #1
 The students will look for a partner and distribute a
worksheet for each pair.
Let T be a random variable giving the number of heads
in three tosses of a coin. List the elements of the
sample space S for the three tosses of the coin and
find the probability of each of the values of the random
variable T. (10 mins)

14
STEPS SOLUTION
1. Determine the
sample space.
S=
{TTT, TTH,THT,HTT, HHT,HTH, THH,HHH)}
2. Count the number
of heads in each
outcome in the
sample space
and assign this
number to this
outcome.
Possible
Outcome
s
Value of the
Random
Variable
T(Number of
heads)
TTT 0
TTH 1
THT 1
HTT 1
HHT 2
HTH 2
THH 2
HHH 3
3. Write the number
of possible values
and assign
probability values
to each random
variable.
Number
of
heads
(T)
Probability of
P(T)
0 1/8
1 3/8
2 3/8
3 1/8
 Make a probability Mass Function of Discrete
Random Variable T.
Number of
Heads (T)
0 1 2 3
Probability
P(T)
1/8 3/8 3/8 1/8
 Construct a Histogram.

15
 Call two volunteer pairs to share their output to the
class.
E. Discussing new
concepts and practicing
new skills #2
 The students will form 5 groups and the teacher
will provide the worksheet to each group. The
group activity is good for 15 mins.
Two balls are drawn in succession without replacement
from an urn containing 5 red balls and 6 blue balls. Let
Z be the random variable representing the number of
blue balls. Construct the probability distribution of the
random variable Z.
STEPS SOLUTION
1. Determine the
sample space.
S= { RR, RB, BR, BB }
2. Count the number of
blue balls in each
outcome in the
sample space and
assign this number to
this outcome.
Possibl
e
Outco
mes
Value of
the
Random
Variable Z
(Number
of Blue
Balls)
RR 0
RB 1
BR 1
BB 2
3. Write the number of
possible values and
assign probability
values to each
random variable.
Number
of Blue
Balls (Z)
Probabili
ty P(Z)
0 ¼
1 ½
2 ¼
 Make a probability Mass Function of Discrete
Random Variable Z.
Number of blue balls (Z) 0 1 2
Probability P(Z) 1/4 1/2 1/4
 Construct a Histogram.
F. Developing mastery

16
 Let them post their output on the board and each
group will critic the output of the other group.
G. Finding practical
applications of concepts,
and skills in daily living
Practical Application
 The daily demand for copies of a newspaper at a
variety store has the probability distribution as
follows:
Number of copies X Probability P(X)
0 0.06
1 0.14
2 0.16
3 0.14
4 0.12
5 0.10
6 0.08
7 0.07
8 0.06
9 0.04
10 0.03
What is the probability that three or more copies will be
demanded in a particular day?
0.64 or 64 %
What is the probability that the demand will be at least
two but not more than six?
0.6 or 60%
H. Making generalization  How do you construct probability distribution?
 How do you make the histogram for a probability
distribution? Give the steps in constructing the
histogram for a probability distribution.
I. Evaluate learning The following data show the probabilities for the
number of Banana Chipssold in SHS Canteen:
Number of Banana
Chips
Probability P(x)
0 0.100
1 0.150
2 0.250
3 0.140
4 0.090
5 0.080
6 0.060
7 0.050
8 0.040
9 0.025
10 0.015
a. Find P(X≤ 2) =0.5
b. Find P(X≥ 7)=0.13
c. Find P(1≤ X ≤ 5)= 0.81

17
d. Construct a histogram
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional activities
for remediation
who scored below
80%
C. Did the remedial
lessons work?
No. of learners
who have caught
up with the lesson
D. No. of learners
who continue to
require
remediation
E. Which of my
teaching strategies
worked well? Why
did this works?
F. What difficulties
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation or
localized materials
did I use/discover
which I wish to
share with other
teachers?

18
School Grade Level Eleven
Teacher
Learning
Area
Statistics and
Probability
Time & Date Quarter Third Quarter
I – OBJECTIVES
A. Content Standards The learner demonstrates understanding of
key concepts of random variables and
probability distributions.
B. Performance
Standard
The learner is able to apply an appropriate
random variable for a given real-life problem
(such as in decision making and game of
chance).
C. Learning
Competencies/
Objectives3
 The learner illustrates and calculates the
variance of a discrete random variable.
M11/12SP-IIIb-1, M11/12SP-IIIb-2
 The learner interprets the variance of a
discrete random variable. M11/12SP-
IIIb-3
 The learner solves problems involving
the variance of a discrete random
variable. M11/12SP-IIIb-4
II – CONTENT Computing the Variance of a Discrete Random
Variable
III – LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learners’ Materials
3. Textbook pages 31 - 40
4. Additional Materials
from Learning
Resource (LR)
portal
5. Other Learning
Resources
Statistics and Probability (Rex Book Store)
by: Rene R. Belecina
IV – PROCEDURE
A. Reviewing past
lesson or
presenting the new
lesson
1. Start the lesson by “The Longer the
Better Game”
Mechanics:
a. Group the class into 4 or 5 groups
b. Give each group a printed pictures of
bananas
c. Each group will measure the sizes of the
bananas
d. They will compute the mean, variance
and standard deviation of the data
gathered

19
e. The group that finishes first will be the
winner.
Ask: How do you get the mean, the
variance and the standard deviation?
 (Recall that the average of a given set of
data is a measure of central tendency.
Inform them that the expected value –
being an average – measures the center
of the distribution of the possible values
of X.)
 The variance and standard deviation
describe the amount of spread,
dispersion, or variability of the items in a
distribution. So using the standard
deviation we have a “standard way of
knowing what is normal, what is extra
large or extra small.
B. Establishing a
purpose for the
new lesson
Ask:
How do you describe the spread or
dispersion in a probability distribution?
Our lesson for today will teach us how to
compute the variance and standard
deviation of a discrete probability
distribution.
C. Presenting
examples/instances
of the new lesson
Present a contextualized problem to the class.
 The number of pentel pens sold per day
at the canteen, along with its
probabilities, is shown in the table
posted on the board. Compute the
variance and the standard deviation of
the probability distribution by following
the given steps.
Number of Pentel
Pens Sold (X)
Probability
P(X)
0 1
10
1 2
10
2 3
10
3 2
10
4 2
10

20
D. Discussing new
concepts and
practicing new skill
# 1
(After filling in the table, add another column on
the right and let the class subtract the mean
from the value of the random variable X)
No. of
Pentel
Pens
Sold
(X)
Probability
P(X)
𝑋
∙ 𝑃(𝑋) 𝑋 − 𝜇
0
1
10
0
0 – 2.2 =
- 2.2
1
2
10
2
10
1 – 2.2 =
1.2
2
3
10
6
10
2 – 2.2 =
-0.2
3
2
10
6
10
3 – 2.2 =
0.8
4
2
10
8
10
4 – 2.2 =
1.8
(Let them square the result an write it on the
column added in the right side)
No. of
Pentel
Pens
Sold
(X)
P(X)
𝑋 ∙ 𝑃(𝑋)
𝑋
− 𝜇
(𝑋
− 𝜇)2
0
1
10
0 - 2.2 4.84
1
2
10
2
10
1.2 1.44
2
3
10
6
10
-0.2 0.04
3
2
10
6
10
0.8 0.64
4
2
10
8
10
1.8 3.24
(Let the students multiply the result in the 5th
column by the corresponding probability P(X))
No. of
Pentel
Pens
Sold
(X)
P(X) 𝑋
∙ 𝑃(𝑋)
𝑋
− 𝜇
(𝑋
− 𝜇)2
(𝑋
− 𝜇)2
∙ 𝑃(𝑋)
0
1
10
0 - 2.2 4.84 0.484

21
1
2
10
2
10
1.2 1.44 0.288
2
3
10
6
10
-0.2 0.04 0.012
3
2
10
6
10
0.8 0.64 0.128
4
2
10
8
10
1.8 3.24 0.648
(Tell the students to get the sum in the 6the
column)
𝜎2
= ∑(𝑥 − 𝜇)2
∙ 𝑃(𝑋) = 1.56
 This is now the variance of the
probability distribution
Ask:
- How do we get the standard deviation?
 To get the standard deviation, simply
get the square of the variance.
E. Discussing new
concepts and
practicing new skill
# 2
(Present the alternative Procedure in Finding
the Variance and Standard Deviation of a
Probability Distribution found in page 35 of the
textbook.)
F. Developing
Mastery
Ask:
What does the variance tell us?
How about the standard deviation?
applications of concepts
and skills in daily living
Present a sample word problem to the class.
(The teacher will decide if the activity will be
done by pair or by group)
 When three coins are tossed, the
probability distribution for the random
variable X representing the number of
heads that occur is given below.
Compute the variance and the standard
deviation of the probability distribution.

22
Number of
Heads (X)
Probability P(X)
0
1
8
1
3
8
2
3
8
3
1
8
(Checking of output)
H. Making
Generalization
Ask:
-What does the variance of the probability
distribution tell us?
-How do you interpret the variance of a
probability distribution?
-How do you get the variance of discrete
random variable?
-How do you get the standard deviation of
discrete random variable?
(Solicit ideas/answers from the class and post
it on the board)
(Present the formula for the variance of the
discrete random variable)
Formula for the Variance and Standard
Deviation of a Discrete Probability Distribution
The variance of a discrete random variable with
a discrete probability distribution is given by the
formula:
𝜎2
= ∑(𝑥 − 𝜇)2
∙ 𝑃(𝑋)
The standard deviation of a discrete random
variable with a discrete probability distribution
is given by the formula:
𝜎2
= √(𝑋 − 𝜇)2 ∙ 𝑃(𝑋)
where:
X = value of the random variable
P(X) = probability of the random variable X
𝜇 = mean of the probability distribution

23
I. Evaluate learning
(The teacher will distribute an activity sheet for
the evaluation)
Solve.
Find the variance and standard deviation of the
probability distribution of the random variable
X, which can take only the values 1, 2 and 3,
given the P(1) =
10
33
,
P(2) =
1
3
, and P(3) =
12
33
.
J. Additional Activities
V. REMARKS
VI. OTHERS
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson worked? No.
of learners who
have caught up
with the lesson
D. No. of learners who
continue to require
remediation
E. Which of the
teaching strategies
worked well? Why
did these work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
localized materials
did I used/discover
which I wish to
share with other
teachers?

24
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
A. Content Standards
The learner demonstrates understanding of key
concepts of normal probability distribution.
B. Performance
Standards
The learner is able to accurately formulate and solve
real-life problems in different disciplines involving
normal distribution.
C. Learning
Competencies /
Objectives
(Write the LC code for
each)
The learner illustrates a normal random variable and its
characteristics.( M11/12SP-IIIc-1)
The learner construct normal curve (M11/12SP-IIIc-2)
Specific Objectives:
1. Illustrate a normal random variable
2. Determine / enumerate the characteristics of a
normal random variable / probability distribution.
3. Cite real- life examples involving normal distribution.
4. Sketch / Construct a normal curve which represents
a normal distribution.
II. CONTENT
Normal Probability Distribution and its
Characteristics
III.
LEARNINGRESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
for Learning
B. Other Learning
Resources
 Next Century Mathematics (Statistics and
Probability) Senior High School by Jesus P.
Mercado pages 308-321
 Statistics and Probability by Rene R. Belecina et.
Al. First Edition
IV. PROCEDURES
A. Reviewing
previous lesson
or presenting the
new lesson
I. Ask the leader of the day to do the routinely activities:
a. lead the prayer
b. do the head counting
c. recapitulation of the previous lesson
(the teacher thank the leader of the day’s effort)
II. Ask the students to do the activity “Let’s be United”
(refer to sheet no. 1)

25
Group Activity
Students are to form a figure out of the pieces of the
puzzle
Group1: Graph skewed to the right
Group 2:Graph skewed to the left
Group 3:Graph of a normal distribution
Group4: Sketch of a negatively skewed
Group 5: Sketch of a positively skewed
Group 6: Sketch of a normal curve
III. Class discussion:
The teacher facilitates the discussion on the different
aspects or characteristics of each graph/ sketch/ figure
through the following questions:
1. What have you formed? Say something about
the figure.
2. Is there similar graphs? In what sense?
3. If we are to group the graphs / figures you
formed, which should be together?
4. How do these grouped figures differ from the
other groups?
(for a bigger class the teacher can select
representatives to do the activity especially those who
were identified as good performers in class, then the
rest of the class observes)
B. Establishing a
purpose for the
lesson
The teacher would say: “Today we are guided by
the following objectives … “
A. The teacher presents a power point (any visual) of
the objectives of the lesson.
B. (The presentation of today’s rule during class
discussion is encouraged if any)
C. Presenting
examples/
instances of the
new lesson
A. Based on the observations from the previous
activity the teacher discusses the difference among
the positively and negatively skewed and the graph
of normal distribution.
(Mention that there are so many continuous random
variables, such as IQ scores, heights of people, or
weights have histograms that have bell-shaped
distributions.)
B. Show them the picture to let them see the real- life
application of the normal curve.

26
1. What have you noticed with the picture shown?
2. If we are to locate the middle part, what can you say
on the left or right part of the figure?
3. Is the given figure best describes a normal probability
distribution? Why?
C. Discussing
new concepts
and practicing
new skills #1
1. Let the students watch the video on normal
distribution and its properties
(the students has to take down notes on the
properties)
2. Discussion of the properties (this can be done
through the video or after watching the video)
3. (include) Draw a picture of the normal (bell-
shaped) curve
Emphasize the following statements about the normal
curve:
• The total area under the normal curve is equal to 1.
• The probability that a normal random variable X
equals any particular value a, P(X=a) is zero (0) (since
it is a continuous random variable).
• Since the normal curve is symmetric about the mean,
the area under the curve to the right of m equals the
area under the curve to the left of m which equals ½,
i.e. the mean m is the median.
 Emphasize also to learners that every normal
curve (regardless of its mean or standard
deviation) conforms to the following "empirical
rule" (also called the 68-95-99.7 rule):
• About 68% of the area under the curve falls within 1
standard deviation of the mean.
• About 95% of the area under the curve falls within 2
standard deviations of the mean.
• Nearly the entire distribution (About 99.7% of the
area under the curve) falls within 3 standard
deviations of the mean.
4. Explain that the graph of the normal distribution
depends on two factors: the mean m and the
standard deviation σ.
5.
D. Discussing
new concepts
and practicing
new skills #2
Sketching Normal Curve
The teacher shows the normal curve to the class and
the process on how to sketch the curve. (the teacher

27
should give emphasis on the properties of the normal
curve)
E. Developing
mastery
(Leads to
Formative
Assessment 3)
A. Present the properties through PPT then ask
the students to perform the activity with the
group of 10 students (any desired group size of
the teacher):
Group 1: Sketch a normal curve then label the
parts of the curve showing the properties of the
curve.(puzzle like or anything related to arts)
Group 2: Create a convo (conversation about
the properties of the normal distribution )
Group 3: Make a Jingle of about the properties
of the normal distribution.
*The teacher can add other skills/ talents of the
students ass observed by the teacher.
F. Finding
practical
applications
of concepts
and skills in
daily living
Let students cite some example in real- life where
they can see the normal curve or distribution.
G. Making
generalization
s and
abstractions
about the
lesson
Let them answer the question;
“What are the properties of the normal distribution?”
H. Evaluating
Learning
Distribute sheet 1 to the students.
V. REMARKS
VI. REFLECTION
earned 80% in the
evaluation.
require additional activities
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.

28
continue to require
remediation
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
localized materials did I
use/discover which I
wish to share with other
teachers?
Evaluation
Test I. TRUE OR FALSE:
1. The standard normal distribution is also called normal curve.
2. The area under a normal curve is 100.
3. The mean of a standard normal curve is 3.
4. The curve of a normal distribution extends indefinitely at the tails.
5. The shape of the normal probability distribution is symmetric about the mean.
Test II
Give and label each normal curve below with the correct characteristics / properties.

29
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
B. Performance
Standards
C. Learning
Competencies /
Objectives
each)
The learner identifies regions under the normal
curve corresponding to different standard
normal values. (M11/12SP-IIIc-3)
Specific Objectives: The learner will be able to:
Read and utilize the z- table correctly.
Draw a sketch of a normal curve
Identify regions under the normal curve
corresponding to different standard normal
values.
II. CONTENT Regions under the Normal Curve
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
pages
3. Textbook pages Next Century Mathematics (Statistics and
Statistics and Probability by Rene R. Belecina
et. Al. First Edition
4. Additional Materials for
Learning
B. Other Learning
Resources
https://int.search.myway.com/search/AJimage.j
html?&n=7858e4f9&p2=%5E0D%5Exdm495%
5ETTAB03%5Eph&pg=AJimage&pn=4&ptb=2E
73C0C4-6C67-4509-B7E8-
BCE1002E511C&qs=&searchfor=normal+distri
bution+curve&si=6127991364903-
6127991365303&ss=sub&st=tab&tpr=jrel2&trs=
wtt&ots=1570210308912&imgs=1p&filter=on&i
mgDetail=true
IV. PROCEDURES
Reviewing previous
lesson or
presenting the new
lesson
lead the prayer
do the head counting
recapitulation of the previous lesson

30
II. Ask the students to recall the definition of the
standard normal curve
(A standard normal curve is a normal probability
distribution that has a mean (μ) equals 0 and a
standard deviation (σ) equals 1)
*the teacher may present the concept on the
board or any visual aid he/ she may have then
show them a picture or an example of a normal
curve with the properties of a normal
distribution / curve.
Establishing a
purpose for the
lesson
A. The teacher would say: “Today we are
guided by the following objectives … “
The teacher presents a power point (any visual)
of the objectives of the lesson.
B. (The presentation of today’s rule during class
discussion is encouraged if any)
Presenting
examples/
instances of the
new lesson
A. Let them recall the role of the standard deviation in
the normal curve.
(the distance or units at the bottom part of the
curve is the standard deviation σ)
B. The teacher presents the normal curve
divided in desired portions.
*through this, the teacher can give preview of
the lesson about the regions under the normal
curve.
I. Discussing
new concepts
and practicing
new skills #1
The teacher present the video on “Normal
Distribution Table - Z-table Introduction”
Ask the students to perform the activity. (to
check their skill on the utilization of the z- table)
By triad: Give the corresponding area between z= 0
and each of the following: (Check the answer after 5
minutes)
P(z=0.23) answer : 0.0910
P(z=1.09) answer: 0.3621
P(z=2.01) answer: 0.4778
P(z=-0.98) answer: 0.3365
P(z=-0.03) answer: 0. 0120
The teacher shall do the corrections/ fixing of
the mistakes committed by the students in the
utilization of the z- table. The teacher has to
inform the students of the other possible z-

31
table. (the teacher can present the different z-
table trough a power point presentation)
Discussing new
concepts and
practicing new skills
#2
A. The teacher shall discuss the proportions of
areas under the normal curve through a video
presentation on “ Normal Distribution -
Explained Simply (part 1)” and “b Normal
Distribution - Explained Simply (part 2)”
B. GROUP ACTIVITY
In a group of 5 students, the teacher asks the
students to give their ideas of the following
statement ask them to support their answers
illustrating each situation in a normal curve:
(answer can be written in a manila paper or
through a PPT. The students have to identify
the correct statement base on the equivalent
proportions of areas under the normal curve).
1. Z – score -2 and 2 covers 95.44%
2. The area from z- score 1 to 2 is 15%.
3. The total area between z= -1 and z= +1 is
0.6826
Answers:
1. The statement is correct or true by adding
the values from the table P(Z=2)=0.4772 or
47.72% and P(Z=-2)=0.4772 or 47.72% the
sum is 0.9544 or 95.44%.
2. The area covered from z= 0 to z=+1 is
0.3413 or 34.13% and from z=0 to z= +2 is
0.4772 0r 47.72% then the difference of the
values is 0.1359 or 13.59% not 15%. Therefore,
the statement is incorrect or false.
3. The sum of the values from z= 0 to z= -1
which is 0.3413 and from z= 0 to z=+1 which is
0.3413, is 0.6826. So, the statement is true or
correct.

32
*the teacher should give the correct illustration
of each statement as she/ he checks and
explains the answers (the teacher can use a ppt
or an IM for normal curve)
**ICT INTEGRATION
If computers are available, show learners that
we could alternatively use Excel to obtain (a)
and (b). Merely enter the command =
NORMSINV(0.5832)
and generate the value of z as 0.210086 for
(a). While for (b), we enter the command
= NORMSINV(1-0.8508)
and thus find z as –1.03987.
* the teacher can explore some z- scores for further
drills on the ICT integration.
A. Developing
mastery
(Leads to Formative
Assessment 3)
Let the students perform the activity on the areas
under the normal curve. (see attached sheet 1) ;
(different colors can be use if desired)
B. Finding
practical
applications of
concepts and
skills in daily
living
Ask the students to give their real- life examples of
having regions or areas or a figure parallel or related
to the lesson. (example: covered area in cleaning the
floor/ applying floor wax in an specific area / region of
the floor) * Creativity and imaginative skill of the
teacher is highly encouraged.
C. Making
generalizations
and
abstractions
about the
lesson
Present the normal curve with the common / usual
proportions under the normal curve. Let the students
give at least 1 visible proportions of the areas under
the normal curve.
D. Evaluating
Learning
Let the students perform attached sheet 2
V. REMARKS
VI. REFLECTION
earned 80% in the
evaluation.
for remediation who scored
below 80%.

33
C. Did the remedial lessons
work? No. of learners
who have caught up with
the lesson.
continue to require
remediation
Why did these work?
encounter which my
can help me solve?
use/discover which I wish
to share with other
teachers?

34
Sheet 1
Shade the normal curves with its given corresponding z- score
then identify the proportions of areas under the normal curve.
Greater than z= 1.065
Between z= 0. 12 and
z= 1.96
Between z= -2. 08 and
z= 0.78.
P(-1.53 < 0.45)
From mean (μ) to z= 2.05 Between z=-1.12 and z= 1.12

35
Sheet 2
EVALUATION
A. Determine the proportions of the areas under the following normal
curves.
B. Illustrate and give the proportions of the regions under the normal
curve with the following z- scores.
5. 𝑧 = 0.38 ; 𝑓𝑟𝑜𝑚 𝑧 = 0 6. 𝑧 = −1.29 ; 𝑓𝑟𝑜𝑚 𝑧 = 0
7. 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧 = 2.01 𝑎𝑛𝑑 𝑧 = 2.93 8. 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧 =
−0.67
9. 𝑏𝑒𝑙𝑜𝑤 𝑧 = 1.37 10. 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑜𝑓 𝑧 = 2.03
2.
1.
3. 4.
-0.64
-2
-2 1 -1 1 2
2 2.5 2
-1
-2

36
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
B. Performance
Standards
C. Learning
Competencies /
Objectives
each)
The learner converts a normal random variable
to a standard normal variable and vice versa.
(M11/12SP-IIIc-4)
1. Find the z- value that corresponds to a score X
2. Utilize/ use z- table independently and correctly.
3. Convert a normal random variable to a standard
normal variable and vice versa
4. Sketch the normal curve with convert a normal
random variable to a standard normal variable
and vice versa
II. CONTENT
Conversion of a normal random variable to a
standard normal variable and vice versa
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
pages
3. Textbook pages Next Century Mathematics (Statistics and Probability)
Senior High School by Jesus P. Mercado pages 308-
321
Statistics and Probability by Rene R. Belecina et. Al.
First Edition
for Learning
B. Other Learning
Resources
IV. PROCEDURES
Reviewing
previous lesson or
presenting the new
lesson
lead the prayer

37
II. Ask the students to recall the properties of a normal
curve/ distribution through the activity:
A. Sketch the normal curve with the following
properties
1. μ= 0, σ=1
2. μ= 35, σ=3
3. μ=98, σ= 2.5
4. μ=105, σ= 4
5. μ= 100, σ= 20
*the teacher shall focus on the baseline of the
normal curve and the standard deviation. The
distance of each unit should be reviewed.
Establishing a
purpose for the
lesson
A. The teacher presents the objectives of the
lesson through a power point presentation.
B. The teacher presents a normal curve with the
converted raw scores. Let the students
determine the μ and σ.
Presenting
examples/
instances of the
new lesson
A. Let them recall the role of the standard deviation in
the normal curve.
(the distance or units at the bottom part of the curve is
the standard deviation σ)
J. Discussing
new concepts
and practicing
new skills #1
The teacher presents the video on the derivation of
the formula
The areas under the normal curve are given in terms
of z- values or scores. Either the z- score locates X
within a sample or within a population.
The formula for calculating z is :
Where :
X- given measurement
μ- population mean
σ- population standard deviation
For population data
𝑧 =
𝑋 − 𝜇
𝜎
For sample data
𝑧 =
𝑋 − 𝑋
̅
𝑠

38
s- sample standard deviation
X- sample mean
*raw scores may be composed of large values, but
large values cannot be accommodated at the base
line of the normal curve. So, they need to be
converted into scores for convenience without
sacrificing the meaning associated to the raw score.
Discussing new
concepts and
practicing new
skills #2
A. Group Activity: Solve for the equivalent z- score
of the problem assigned to your group, then sketch
the normal curve showing the calculated z- score that
corresponds to the raw score X.
Group 1- Given the mean μ= 60 and the standard
deviation σ= 5 of a population, find the z- value that
corresponds to score X= 54.
deviation σ= 2.6 of a population, find the z- value that
deviation σ= 6.5 of a population, find the z- value that
corresponds to score X= 147.5.
B. Let them present their output through a
manipulative normal curve made – up of
cardboard.
*The teacher can make her/ his own rubrics
according to the ability of the students.
*the teacher should give the correct illustration of
each statement as she/ he checks and explains
the answers (the teacher can use a ppt or an IM
for normal curve)
E. Developing
mastery
(Leads to
Formative
Assessment 3)
A. By triad. : Give the missing value;
1. X= 23, μ=32, σ=8, z=?
2. μ=231, σ= 120, X= 250, z=?
3. μ=127, σ= 5, X= 98, z=?
4. μ=450, σ= 15, z=-1.5, X=?
5. σ= 5, X= 98, z=2.21, μ=?
B. Checking of the answer may be done through a
quick check where the teacher will give the
answers or if the students seem to be slow in
understanding the concept, the solution of each
problem shall be presented.

39
F. Finding
practical
applications
of concepts
and skills in
daily living
Ask the students to give their real- life examples of
having small or large things which need to be converted
just to fit in an actual scenario
* Creativity and imaginative skill of the teacher is highly
encouraged.
G. Making
generalization
s and
abstractions
about the
lesson
Ask the students to give the summary of the
lesson.
The teacher shall present the formulae to the students
through a PPT.
The formula for calculating z is :
Where :
X- given measurement
μ- population mean
σ- population standard deviation
s- sample standard deviation
X- sample mean
H. Evaluating
Learning
V. REMARKS
VI. REFLECTION
earned 80% in the
evaluation.
for remediation who
scored below 80%.
C. Did the remedial
learners who have
caught up with the
lesson.
continue to require
remediation
Why did these work?
encounter which my
can help me solve?
For population data
𝑧 =
𝑋 − 𝜇
𝜎
For sample data
𝑧 =
𝑋 − 𝑋
̅
𝑠

40
wish to share with other
teachers?
EVALUATION
Shade the normal curves with its corresponding z- score after
converting the raw score to its standard normal variable.
Solve for the missing value.
X= 219
μ=200
σ=21
z=?
X= 950
μ=1000
σ=25
z=?
X= 69
μ=75
σ= 14
z=?
X= 12
μ=20
σ=6.5
z=?
X= 250
σ=15.5
z=1.65
μ=?
X= 100
z=-0.98
μ= 112
σ=?

41
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
B. Performance Standards
C. Learning
Competencies /
Objectives
each)
The learner computes probabilities and
percentiles using the standard normal table.
(M11/12SP-IIIc-d1)
5. Recall the concept on the reading of
probabilities on the z- table.
6. Find the z- scores when probabilities are
given.
7. Computes the probabilities and percentiles
using the standard normal table.
II. CONTENT Locating Percentiles Under the Normal Curve
III.
LEARNINGRESOURCES
A. References
Teacher’s Guide
pages
pages
3. Textbook pages Next Century Mathematics (Statistics and
Statistics and Probability by Rene R. Belecina
et. Al. First Edition
4. Additional Materials for
Learning
B. Other Learning
Resources a. https://www.google.com/search?q=percentile
&oq=percentile&aqs=chrome..69i57j0l5.2828j
0j9&sourceid=chrome&ie=UTF-8
b.
IV. PROCEDURES
lesson or presenting the
new lesson
I. Ask the leader of the day to do the routinely
activities:
lead the prayer

42
II. Ask the students to recall the process on how to
read values from the z- table by asking the students to
give the equivalent probability of the following; (this
can be a quiz bee with the help of the power point)
1. z= 0.12
2. z=-2.13
3. z=1.28
4. z=2.48
5. z=-0.87
6. z=-1.24
7. z= -2.09
8. z= 2.01
9. z= 1.72
10. z= 0.04
B. Establishing a purpose
for the lesson
C. The teacher presents the objectives of the
lesson through a power point presentation.
D. The teacher ask: “Which of the following are
familiar to you?”
a. First Honor
b. Top five
c. Eliminated candidates are the below 10%
d. Scholars are the top two
e. Remediation session is for students at the
bottom 5.
*the teacher shall ask the students to give the
meaning of each situation above.
C. Presenting examples/
instances of the new lesson A. (optional) the teacher can make a huge
normal curve and ask the students to stand on
the position of the following:(this can be done
by group)
1. Above z= 2.00
2. Below z = 0.08
3. More than z= 1.54
4. Less than or equal to z=-1.34
5. To the right of z= 0.49
D. Discussing new concepts
and practicing new skills #1 The teacher presents and asks the opinion of
the class about the picture.

43
*the idea of the percentile shall be given
emphasis and be defined.
Percentile - each of the 100 equal groups into
which a population can be divided according to
the distribution of values of a particular
variable.
A percentile is a measure used in statistics
indicating the value below which a given
percentage of observations in a group of
observations fall
E. Discussing new concepts
and practicing new skills #2
C. The teacher shall present the following
considerations or important things to
remember when we are given probabilities
and we know their corresponding z- scores.
1. A probability value corresponds to an area
under the normal curve.
2. In the Table of Areas Under the Normal Curve,
the numbers in the extreme left and across the
top are z- scores, which are the distances
along the horizontal scale. The numbers in the
body of the table are areas or probabilities.
3. The z- scores to the left of the mean are
negative values.
D. Group Activity:
Ask the students to sketch the following:
Group 1: P25
Group 2: P65
Group 3: P88
Group 4: P90
Group 5: P98
E. Let them give the meaning of the assigned
percentile to their group.
F. Ask them to present the illustrations(for the
wrong sketch the teacher should check or
correct the illustration)
G. Discussion of how to determine the z- score of
every percentile.
The 95th
percentile is z= 1.645
 .95/2 = 0.45 where there is no exact
0.45 in the table so therefore we get
the nearest values z=1.65 (0.4505)
and the z= 1.64 (0.4495) by
interpolation the value now is z=
1.645.
H. Ask the students to give the z – score of their
assigned percentile as stated above (B)
F. Developing mastery
(Leads to Formative
Assessment 3)
A. Let them perform the activity by pair:
1. Find the upper 10% of the normal curve. Illustrate
the normal curve.

44
2. The results of a nationwide aptitude test in
Mathematics are normally distributed with m=80
and s= 15. What is the percentile rank of a score
84?
I. Check their answer and resolve the
misconceptions committed by the students.
applications of concepts and
skills in daily living
Ask them to give their own example of the percentile
rank (students can mention their rank after taking the
quiz or any test they had)
H. Making generalizations
and abstractions about the
lesson
Is a normal curve useful in visualizing the positions of
the scores or the rank? Why do you think so? Write
your thoughts in a piece of paper.
I. Evaluating Learning
V. REMARKS
VI. REFLECTION
earned 80% in the
evaluation.
for remediation who scored
below 80%.
C. Did the remedial lessons
work? No. of learners who
have caught up with the
lesson.
continue to require
remediation
Why did these work?
encounter which my
principal or supervisor can
help me solve?
use/discover which I wish
to share with other
teachers?

45
EVALUATION
1. Sketch the 85th
percentile.
2. Present the procedure in calculating the P99 of the normal curve then draw.
3. What is the percentile rank of a score of 56 from the normally distributed NAT
results with mean of 75 and σ= 20. Draw.

46
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
A. Content
Standard
sampling ad sampling distributions of the sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and sampling
distributions of the sample mean to solve real-life problems
in different disciplines.
C. Learning
Competency/Obj
ectives
The learner illustrates random sampling.
M11/12SP-IIId-2.
II. CONTENT Random Sampling
III. LEARNING RESOURCE
References 1. Jose Dilao S., Orines F and Bernabe J. (2009).
Advanced Algebra, Trigonometry and Statistics. SD
Publications, Inc. pp 234-236.
2. Ocampo J. & Marquez W. (2016). Senior High
Conceptual Math and Beyond Statistics and Probability.
Brilliant Creations Publishing, Inc. pp.86-93.
Other Learning
Resource
https://www.youtube.com/watch?v=xh4zxC1OpiA
IV. PROCEDURES
A. Reviewing
previous lessons
or presenting the
new lesson
Recall from our study of probability that the number of
combinations of n objects taken r at a time is obtained by
using the formula.
C (n, r) =
𝑛!
(𝑛−𝑟)!𝑟!
𝑤ℎ𝑒𝑟𝑒 𝑛 ≥ 𝑟
Evaluate the following:
1.C (5, 3)
2. C (10, 4)
3. C (9, 6)
4. C (8, 2)
5. C (7, 6)
The students will explain their solutions.
B. Establishing a
purpose for the
lesson
To prepare the students in the lesson, activities are as
follows:
A.A sample of investment experts was asked to give their
opinion as to where they would invest their money. The
following are their responses.
Stocks Real estate Real estate
Precious metals Art Precious metals
Real state Precious metals Commodities
Art Precious metals Foreign money

47
Precious metals Commodities Commodities
Stocks Foreign money Stocks
Stocks Stocks Real estate
Real estate Stocks Real estate
Commodities Stocks Real estate
Stocks Precious metals Real estate
Real estate Real estate Foreign money
Construct a table to show the frequency distribution of the
given responses.
Types of Investment Frequency
C. Presenting
Examples/Instan
ces of the
Lesson
Norma wants to know the common number of children her
classmates’ families have. Which of the following samples is
a good representation of the class? Why?
1.A sample consisting of Norma’s friends
2.A sample consisting of students belonging to rich families.
3. A sample consisting of students whose names were drawn
from a box all the names of students in Norma’s class.
Wrong conclusion may be inferred from samples given in
numbers 1 and 2. This sample will not represent the correct
number of children the families of Norma’s classmates have.
The sample in a number 3 in the best representation of the
class.
This is idea of representativeness leads to the importance of
random sampling, a method of drawing out a sample from a
population without a definite plan, purpose, or pattern.
D. Discussing New
concepts and
Practicing New
Skills # 1
Let students analyze the video in the link-
After watching the video presentation, the students will
define random sampling and state its uses.
E. Developing
Mastery
Group activity for 10 minutes. The students are task
to:
1. Create problem that involves random sampling.
2. Construct a table that show frequency distribution of
the samples.
3. What learning discovered in doing such activity?
Would you be able to use this in your life? How and
why?

48
The rubrics will be used in scoring the performance of the
group.
Categor
ies
4
Excellen
t
3
Satisfact
ory
2
Developi
ng
1
Beginning
Mathe
matical
Concep
t
Demons
trates a
thoroug
h
underst
anding
of the
topic
and
uses it
accurate
ly to
solve
the
problem
Demons
trates a
satisfact
ory
underst
anding
of the
uses it
to
simplify
the
problem
.
Demonstr
ates
incomplet
e
understa
nding
and has
some
misconce
ptions.
Shows lack
of
understandin
g and have
severe
misconceptio
ns.
Accura
cy of
comput
ation.
All
computa
tion are
correct
an are
logically
present
ed
The
computa
tion are
correct.
Generally
, most of
the
computati
ons are
not
correct.
Errors in
computations
are severe.
Organiz
ation of
the
report
Highly
organize
d, flows
smoothl
y, and
observe
s logical
connecti
ons of
points.
Satisfact
orily
organize
d.
Sentenc
e flow is
generall
y
smooth
and
logical.
Somewh
at
cluttered.
Flow is
not
consisten
tly
smooth,
appears
disjointed
.
Illogical and
obscure. No
logical
connections
of ideas.
Difficult to
determine
the meaning.
Particip
ation of
the
membe
rs
All
member
s take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
Almost
90-99%
take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
Almost
80-89%
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
Almost 70-
79% take
part in the
activity,
support and
encourage
others in the
group. There
isa conflict
relationship
in doing the
activity.

49
s do not
find fault
in one
another,
open to
commen
ts and
criticism
.
s do not
find fault
in one
another,
open to
commen
ts and
criticism
.
in one
another,
open to
comment
s and
criticism.
F. Making
generalization
and abstraction
about the lesson
What is random sampling?
Random sampling is a method by which every
element of a population has a chance of being
included in a sample. That is, the elements that
compose the sample are taken without purpose. The
more elements in the sample, the better the chances
of getting a true picture of the whole population.
G. Evaluating
Learning
Determine whether the following is a random sample or not.
Explain your answer.
1.To select the students to attend the summer workshop in
Sorsogon, the teacher told her class to count off, and then
selected those even-numbered students for the workshop.
2. To study the average number of years a family has stayed
in Barangay Guinlajon, the barangay captain chose to
interview the families around his residence.
3. To find the average number of dengue victims in hospitals
per day, a researcher made a list of all hospitals in Sorsogon
Province, and then selected every fifth in the list.
4. A survey of the prevailing cost of rice was undertaken in
the seven key cities of the country.
5. To select students for MTAP competition, the school
math coordinator decided to screen competitive students
from junior high school.
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80% in
the evaluation.
B. No. of learners
who require
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners
who continue to
require remediation

50
E. Which of my
teaching strategies
worked well? Why
did these work?
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation
or localized
materials did I
wish to share with
other teachers?

51
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
distributions of the sample mean to solve real-life problems in
different disciplines.
C. Learning
Competency/
Objectives
The learner distinguishes between parameter and statistic.
M11/12SP-IIId-3.
II. CONTENT Parameter and Statistic
References
3. Ocampo J. & Marquez W. (2016). Senior High Conceptual
Math and Beyond Statistics and Probability. Brilliant
Creations Publishing, Inc. pp.86-93.
2. Supplementary Statistics Topics. Retrieved from
https://www2.southeastern.edu/Academics/Faculty/dgurne
y/Math241/StatTopics.html
5. Surbhi (2017). Difference Between Statistic and Parameter
Retrieved from https://keydifferences.com/difference-
between-statistic-and-parameter.html
Other Learning
Resource
https://www.youtube.com/watch?v=M-L8C2aOf7E
IV. PROCEDURES
A. Reviewing
previous
lessons or
presenting the
new lesson
Jumble the letters that corresponds to the given definition.
1. AATD- facts and statistics collected together for reference or
analysis.
2. NIOTALUPOP- an aggregate observation of subjects
grouped together by a common feature
3. ELPMSA- a small part or quantity intended to show what the
whole is like.
4.UAIESMMRZ- give a brief statement of the main points of
(something).
5. PRMTRSAAEE- a numerical or other measurable factor
forming one of a set that defines a system or sets the conditions
of its operation.
B. Establishing a
purpose for
the lesson
Let students analyze the given definition and comparison
chart of statistic and parameter
In statistics vocabulary, we often deal with the terms parameter
and statistic, which play a vital role in the determination of the
sample size. Parameter implies a summary description of the
characteristics of the target population. On the other extreme,

52
the statistic is a summary value of a small group of population
i.e. sample.
-Definition of Statistic
A statistic is defined as a numerical value, which is obtained
from a sample of data. It is a descriptive statistical measure and
function of sample observation. A sample is described as a
fraction of the population, which represents the entire
population in all its characteristics. The common use of statistic
is to estimate a particular population parameter.
From the given population, it is possible to draw multiple
samples, and the result (statistic) obtained from different
samples will vary, which depends on the samples.
-Definition of Parameter
A fixed characteristic of population based on all the elements of
the population is termed as the parameter. Here population
refers to an aggregate of all units under consideration, which
share common characteristics. It is a numerical value that
remains unchanged, as every member of the population is
surveyed to know the parameter. It indicates true value, which
is obtained after the census is conducted
C. Presenting
Examples/Inst
ances of the
Lesson
The students will distinguish the parameter and statistic in the
given statements.
1.A researcher wants to know the average weight of females
aged 22 years or older in Sorsogon. The researcher obtains
the average weight of 54 kg, from a random sample of 40
females.
-Solution: In the given situation, the statistics are the average
weight of 54 kg, calculated from a simple random sample of
40 females, in Sorsogon while the parameter is the mean
weight of all females aged 22 years or older.
2.A researcher wants to estimate the average amount of water
consumed by male teenagers in a day. From a simple random
sample of 55 male teens the researcher obtains an average of
1.5 litres of water.

53
-Solution: In this question, the parameter is the average
amount of water consumed by all male teenagers, in a day
whereas the statistic is the average 1.5 litres of water
consumed in a day by male teens, obtained from a simple
random sample of 55 male teens
D. Discussing
New concepts
and Practicing
New Skills # 1
Let the students analyze the video in the link-
https://www.youtube.com/watch?v=M-L8C2aOf7E
After watching the video presentation, the students will reflect
to the difference between parameter and statistic and connect
it to real life.
E. Developing
Mastery
Group activity for 10 minutes. The students are task to:
4. Create statements that involves parameter and
statistic.
5. What learning discovery did you found useful in your
daily life activities?
group.
Categories 4
Excellent
3
Satisfactory
2
Developing
1
Beginning
Mathem
atical
Concept
Demonstr
ates a
thorough
understa
nding of
the topic
and uses
it
accuratel
y to solve
the
problem
Demonstr
ates a
satisfacto
ry
understa
nding of
the uses
it to
simplify
the
problem.
Demonstra
tes
incomplete
understan
ding and
has some
misconcep
tions.
Shows
lack of
understan
ding and
have
severe
misconcep
tions.
Accurac
y of
computa
tion.
All
computati
on are
correct
an are
logically
presente
d
The
computati
on are
correct.
Generally,
most of
the
computatio
ns are not
correct.
Errors in
computatio
ns are
severe.
Organiza
tion of
the
report
Highly
organize
d, flows
smoothly,
and
observes
logical
connectio
ns of
points.
Satisfact
orily
organize
d.
Sentence
flow is
generally
smooth
and
logical.
Somewhat
cluttered.
Flow is not
consistentl
y smooth,
appears
disjointed.
Illogical
and
obscure.
No logical
connection
s of ideas.
Difficult to
determine
the
meaning.

54
Participa
tion of
the
member
s
All
members
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
in one
another,
open to
comment
s and
criticism.
Almost
90-99%
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
in one
another,
open to
comment
s and
criticism.
Almost 80-
89% take
part in the
activity,
support
and
encourage
others in
the group
members
do not find
fault in one
another,
open to
comments
and
criticism.
Almost 70-
79% take
part in the
activity,
support
and
encourage
others in
the group.
There isa
conflict
relationshi
p in doing
the
activity.
F. Making
generalization
and
abstraction
about the
lesson
Differentiate parameter to statistic.
-Parameters are numbers that summarize data for an entire
population. Statistics are numbers that summarize data from a
sample, i.e. some subset of the entire population
G. Evaluating
Learning
Problems (1) through (6) below each present a statistical
study*. For each study, identify both the parameter and the
statistic in the study.
1) A researcher wants to estimate the average height of women
aged 20 years or older. From a simple random sample of 45
women, the researcher obtains a sample mean height of 63.9
inches.
2) A nutritionist wants to estimate the mean amount of sodium
consumed by children under the age of 10. From a random
sample of 75 children under the age of 10, the nutritionist
obtains a sample mean of 2993 milligrams of sodium
consumed.
3) Nexium is a drug that can be used to reduce the acid
produced by the body and heal damage to the esophagus. A
researcher wants to estimate the proportion of patients taking
Nexium that are healed within 8 weeks. A random sample of
224 patients suffering from acid reflux disease is obtained, and
213 of those patients were healed after 8 weeks.
4) A researcher wants to estimate the average farm size in
Kansas. From a simple random sample of 40 farms, the
researcher obtains a sample mean farm size of 731 acres.

55
5) An energy official wants to estimate the average oil output
per well in the United States. From a random sample of 50 wells
throughout the United States, the official obtains a sample
mean of 10.7 barrels per day.
6) An education official wants to estimate the proportion of
adults aged 18 or older who had read at least one book during
the previous year. A random sample of 1006 adults aged 18 or
older is obtained, and 835 of those adults had read at least one
book during the previous year.
J. Additional
activities for
application or
remediation
V. REMARKS
VI. REFLECTION
A. No. of
learners who
earned 80% in
the evaluation.
B. No. of
learners who
require
additional
activities for
remediation
who scored
below 80%.
C. Did the
remedial
lessons work?
No. of learners
who have
caught up with
the lesson.
D. No. of
learners who
continue to
require
remediation
E. Which of my
teaching
strategies
worked well?
Why did these
work?
F. What
difficulties did I
encounter
which my
principal or

56
supervisor can
help me solve?
G. What
innovation or
localized
materials did I
use/discover
which I wish to
share with
other
teachers?

57
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competency/O
bjectives
M11/12SP-IIId-4. The learner identifies sampling
distributions of statistics (sample mean)
II. CONTENT Identifying Sampling Distributions of statistics (sample
mean)
References Ocampo J. & Marquez W. (2016). Senior High
Conceptual Math and Beyond Statistics and Probability.
Brilliant Creations Publishing, Inc. pp.86-93.
Other Learning
Resource
IV. PROCEDURES
A. Reviewing
previous
lessons or
presenting the
new lesson
Find the mean of the following sets of data.
Set of data Mean
1.18, 19, 20, 21, 22,21, 20, 19, 17, 17, 16, 16, 16
2.5,3,6,9, 7,2,10,8
3.18,16,19,22,20, 15,23,21,21
4.76,69,63,82,29,83,64,71,76
5.36,37,37,38,23,30,35
B. Establishing a
purpose for the
lesson
Suppose we have a population of size N with a mean 𝜇, and
we draw or select all possible samples of size n from this
population. Naturally, we expect to get different values of the
means for each sample. The sample means may be less
than, greater than, or equal to the population mean 𝜇.
The sample means obtained will from a frequency and the
corresponding probability distribution can be constructed.
This distribution is called the sampling distribution of the
sample means.
C. Presenting
Examples/Insta
nces of the
Lesson
How do we construct the sampling distribution of the sample
means? Study the given example.
A population consists of five values (Php2, Php 3, Php 4, Php
5, Php6). A sample of size 2 is to be taken from this
population.
a. How many samples are possible? List them
and compute the mean of each sample.

58
b. Construct the histogram of the sampling
distribution of the sample means.
D. Discussing
New concepts
and Practicing
New Skills # 1
The following table gives the monthly salaries
Officer Salary
A 8
B 12
C 16
D 20
E 24
F 28
1.How many samples are possible? List them and compute
the mean of each sample?
2. Construct the sampling distribution of the sample means.
3. Construct the histogram of the sampling distribution of the
sample means.
E. Developing
Mastery
Group activity for 10 minutes. The students are task
to:
1. Create problem that involves sampling distributions
of statistics (sample mean).
2. Construct sampling distribution and histogram of the
sample means
3. What learning discovered in doing such activity?
Would you be able to use this in your life? How and
why?
group.
Categories 4
Excellent
3
Satisfactory
2
Developing
1
Beginning
Mathe
matical
Concep
t
Demons
trates a
thoroug
h
underst
anding
of the
topic
and
uses it
accurate
ly to
solve
the
problem
Demons
trates a
satisfact
ory
underst
anding
of the
uses it
to
simplify
the
problem
.
Demonstr
ates
incomplet
e
understa
nding
and has
some
misconce
ptions.
Shows lack
of
understandin
g and have
severe
misconceptio
ns.
Accura
cy of
comput
ation.
All
computa
tion are
correct
an are
logically
present
ed
The
computa
tion are
correct.
Generally
, most of
the
computati
ons are
not
correct.
Errors in
computations
are severe.

59
Organiz
ation of
the
report
Highly
organize
d, flows
smoothl
y, and
observe
s logical
connecti
ons of
points.
Satisfact
orily
organize
d.
Sentenc
e flow is
generall
y
smooth
and
logical.
Somewh
at
cluttered.
Flow is
not
consisten
tly
smooth,
appears
disjointed
.
Illogical and
obscure. No
logical
connections
of ideas.
Difficult to
determine
the meaning.
Particip
ation of
the
membe
rs
All
member
s take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
s do not
find fault
in one
another,
open to
commen
ts and
criticism
.
Almost
90-99%
take
part in
the
activity,
support
and
encoura
ge
others in
the
group
member
s do not
find fault
in one
another,
open to
commen
ts and
criticism
.
Almost
80-89%
take part
in the
activity,
support
and
encourag
e others
in the
group
members
do not
find fault
in one
another,
open to
comment
s and
criticism.
Almost 70-
79% take
part in the
activity,
support and
encourage
others in the
group. There
isa conflict
relationship
in doing the
activity.
F. Making
generalization
and abstraction
about the
lesson
What is sampling distribution of sample means?
-It is the frequency distribution of the sample means taken
from a population.
G. Evaluating
Learning
A. Determine the number of different samples of the given
size n that can be drawn from the given population of size
N.
N N Number of Possible Samples
7 3
15 5
50 4
10 3
25 4
B. Random samples of size n=2 are drawn from a finite
population consisting of numbers 5, 6,7,8,and 9.
a. How many possible samples are there?

60
b .List all the possible samples and the corresponding
mean for each sample.
c. Construct the sampling distribution of the sample means.
d. Construct the histogram for the sampling distribution of
the sample means. Describe the shape of the histogram.
V. REMARKS
VI. REFLECTION
A. No. of learners
who earned 80% in
the evaluation.
B. No. of learners
who require
for remediation who
scored below 80%.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners
who continue to
require remediation
E. Which of my
teaching strategies
worked well? Why
did these work?
did I encounter
which my principal
or supervisor can
help me solve?
G. What innovation
or localized
materials did I
wish to share with
other teachers?

61
Teacher
Learning
Area
Statistics and
Probability
I. OBJECTIVES
A. Content Standard The learner demonstrates understanding of key
concepts of sampling and sampling distributions of the
sample mean.
B. Performance
Standard
The learner is able to apply suitable sampling and
sampling distributions of the sample mean to solve
real-life problems in different disciplines.
C. Learning
Competency/Objectives
Write the LC code for
each
The learners shall be able to finds the mean, variance
and the standard deviation of the sampling distribution
of the sample mean.
M11/12SP-IIId-5
I.CONTENT Sampling and Sampling Distributions
II.LEARNING
RESOURCES
A. Reference K-12 Curriculum Guide
Statistics and Probability by Belencina, Baccay &
Mateo
1.Teacher’s Guide pages
2.Learner’s Material
pages
3.Textbook pages pp. 110-119
4.Additional Materials
from Learning
Resource(LR) Portal
B. Other Learning
Resources
Calculator, manila paper, pentel pen, projector and
laptop
IV. PROCEDURES
lesson or
presenting the new
lesson
Tell the class that the
sampling distribution of
the sample means is
actually the probability
distribution of the
sample mean
Start the lesson with a
review on how to
construct the sampling
distribution of the
sample mean.
Consider a population
consisting of 1,2,3,4
and 5. Suppose
sample size 2 are
drawn from this
population
Construct the sampling
distribution of the
sample mean.

62
Step 1:Determine the
number of possible
samples that can be
drawn from the
population using the
combination formula
n𝐶𝑟 =
𝑛!
(𝑛−𝑟)!𝑟!
Step 2:List all possible
samples and compute
the mean of each sample
Sample Mean
Step 3: Construct a
frequency distribution of
the sample means
obtained in step 2
Sam
ple
Mea
n
𝑥̅
Frequ
ency
Proba
bility
P(x)
Tot
al
5𝐶2 =
5!
(5−2)!2!
=
5𝑥4𝑥3!
3!2!
= 20/2
5𝐶2= 10
Or through the use of scientific
Calculator
Keystroke:
5 n𝐶𝑟 2 = Display
Sample Mean
1,2 1.50
1,3 2.00
1,4 2.50
1,5 3.00
2,3 2.50
2,4 3.00
2,5 3.50
3,4 3.50
3,5 4.00
4,5 4.50
Sample
Mean
𝑥̅
Frequency Probability
P(x)
1.50 1 1/10
2.00 1 1/10
2.50 2 2/10 or
1/5
2 2/10 or
1/5
3.50 2 2/10 or
1/5
4.00 1 1/10
4.50 1 1/10
Total 10 1.00

63
B. Establishing a
purpose for the lesson
Telltheclass thaton this lesson weshallcontinue tocomputethe
meanandvarianceofthesampling distributionofthesample mean
C. Presenting
examples/instances of
the
new lesson
consisting of 1,2,3,4 and
5. Suppose sample size
2 are drawn from this
population
Compute the population
mean from the given
example
Challenge the students
to compare this mean to
the mean of the sampling
distribution of the sample
mean after the next
activity had been done.
𝜇 =
Σ𝑥
𝑛
𝜇 =
1+2+3+4+5
5
𝜇=3.00
D. Discussing new
concepts and
practicing new
skills # 1
Discuss the steps on
how to find the mean and
variance of the given
sampling distribution
(PPT)
ICT Integration
Activity 2
consisting of 1,2,3,4 and
5. Suppose sample size
2 are drawn from this
population
Find the mean and
variance of the sampling
mean?
Remind the students to
follow the s
1. Construct the
sampling distribution of
the sample mean.
2. Compute the mean of
the sampling distribution
of the sample mean (𝜇𝑥̅
Answers:
𝑋
̅ P(𝑋
̅) 𝑋
̅ ∙ P(𝑋
̅)
1.50 1/10 0.15
2.00 1/10 0.20
2.50 1/5 0.50
3.00 1/5 0.60
3.50 1/5 0.70
4.00 1/10 0.40
4.50 1/10 0.45
Total 1.00 𝚺𝑿
̅ ∙ P(𝑿
̅)=
3.00
𝜇𝑥̅ = 𝚺𝑿
̅ ∙ P(𝑿
̅)
𝜇𝑥̅ = 3.00 mean of the
sampling distribution of the
means

64
by multiplying the sample
mean by the
corresponding probability
and add the results.
3. Compute the variance
(𝜎2
𝑥̅) of the sampling
Mean using the
formula
𝜎2
𝑥̅ = Σ P(𝑋
̅) ∙ (𝑋
̅
– 𝜇)2
4. Compute the standard
deviation by finding the
square root of the
variance
𝜎𝑥̅ = √Σ P(𝑋
̅) ∙ (𝑋
̅ – 𝜇)2
Complete the table below
𝑋
̅ P
(
𝑋
̅
)
𝑋
̅
–
𝜇
(𝑋
̅
–
𝜇)
2
P(𝑋
̅)
∙ (𝑋
̅
– 𝜇)2
T
o
t
a
l
Σ
P
(
𝑋
̅
)
=
Σ
P(𝑋
̅)
∙ (𝑋
̅
– 𝜇)2
=
What is now the mean
and the variance of the
given sampling
distribution?
𝑋
̅ P
(
𝑋
̅
)
𝑋
̅
–
𝜇
(𝑋
̅ –
𝜇)2
P(𝑋
̅) ∙ (𝑋
̅ –
𝜇)2
1.
5
0
1/
1
0
-
1.
50
2.2
5
0.225
2.
0
0
1/
1
0
-
1.
00
1.0
0
0.100
2.
5
0
1/
5
-
0.
50
0.2
5
0.050
3.
0
0
1/
5
0.
00
0.0
0
0.000
3.
5
0
1/
5
0.
50
0.2
5
0.050
4.
0
0
1/
1
0
1.
00
1.0
0
0.100
4.
5
0
1/
1
0
1.
50
2.2
5
0.225
T
ot
al
1.
0
0
ΣP(𝑋
̅)∙(𝑋
̅–
𝜇)2
=0.750
𝜎2
𝑥̅ = Σ P(𝑋
̅) ∙ (𝑋
̅ – 𝜇)2
𝜎2
𝑥̅ = 0.750 Variance of the
sampling distribution of the
sample mean

65
E. Discussing new
concepts and
practicing new
skill #2
From the activity ask the
students to compute the
standard deviation by
finding the square root of
the variance.
̅) ∙ (𝑋
̅ – 𝜇)2
= √0.750
𝜎𝑥̅ = 0.87
So, the standard deviation of
the sampling distribution of the
sample mean is.87
F. Developing
mastery leads
to Formative
Assessment
After the discussion, divide the class into 4 groups and
distribute worksheets and materials.
Group 1. Construct Me
Group 2- Meant to be
Group 3.Difference and its Square
Group 4:Your Square Root, My Standard
G. Making
generalization
and abstraction
about the
lesson
Give the summary through question and answer.
1. What are the steps in computing the mean, variance
and standard Deviation of the sampling distribution of the
sample mean?
2. How do you compare mean of the sample means and
the mean of population?
H. Evaluating
Learning
Evaluate the students base on the results of their output.
J. Additional
activities for
application or
remediation
From a group of eight students in your class. Determine
the general weighted average of the members of the group
and list all possible samples of size 2 and their
corresponding mean. Construct the sampling distribution
and solve the mean , variance and standard deviation of
the sampling distribution of the sample mean.
V. REMARKS
VI. REFLECTION
A. No of learners who
earned 80% in the
evaluation
B. No of learners who
require additional
activities for
remediation who
scored
below 80%
C. Did the remedial
learners who have
caught up with the
lesson
D. No of learners who
continue to require
remediation

66
E. Which of my
teaching strategies
worked well? Why did
these work?
encounter which
my principal or
supervisor can help me
solve?
wish to share with
other teachers?

67
Worksheets
Group1. Construct Me
Given the population 1,3,4,6 and 8.
Suppose the sample size of 3 are drawn
from this population.
Construct the sampling distribution of the
Sample Mean
Step1. List all possible samples of size 3
and their corresponding mean
Step 2 Construct the sampling distribution
of the sample means
Sample Mean
2.67
3.33
3.67
4.00
4.33
5.00
5.67
6.00
Sample
Mean
𝑥̅
P(𝑥̅)
2.67 1 1/10
3.33 1 1/10
3.67 1 1/10
4.00 1 1/10
4.33 2 2/10 or 1/5
5.00 2 2/10 or 1/5
5.67 1 1/10
6.00 1 1/10
Total 10 1.00
Sample
Mean
𝑥̅
P(𝑥̅)
Total
Sample Mean
1,3,4 2.67
1,3,6 3.33
1,3,8 4.00
1,4,6 3.67
1,4,8 4.33
1,6,8 5.00
3,4,6 4.33
3,4,8 5.00
3,6,8 5.67
4,6,8 6.00

68
Group 2: Meant to Be.
Solve the mean of the sampling distribution
of the mean.
Group 3: Difference and its square
If the mean 𝜇 of the population is 5.
Sample
Mean
𝑥̅
Probability
P(𝑥̅)
𝑥̅ ∙ P(𝑥̅)
2.67 1/10
3.33 1/10
3.67 1/10
4.00 1/10
4.33 2/10 or 1/5
5.00 2/10 or 1/5
5.67 1/10
6.00 1/10
Total 1.00
Sample
Mean
𝑥̅
Probability
P(𝑥̅) 𝑥̅ ∙ P(𝑥̅)
2.67 1/10 0.267
3.33 1/10 0.333
3.67 1/10 0.367
4.00 1/10 0.400
4.33 2/10 or 1/5 0.866
5.00 2/10 or 1/5 1.00
5.67 1/10 0.567
6.00 1/10 0.600
total 1.00 Σ𝑥̅ ∙ P(𝑥̅) =4.40
What is now the mean of the sampling
distribution of the sample mean?
Step 1. Subtract the population mean (𝜇)
from each sample (𝑥̅).
Sampl
e
Mean
𝑥̅
Probabilit
y
P(𝑥̅)
𝑥̅ - 𝜇
2.67 1/10
3.33 1/10
3.67 1/10
4.00 1/10
4.33 2/10 or
1/5
5.00 2/10 or
1/5
5.67 1/10
6.00 1/10
2.67 1/10
total 1.00
𝜇𝑥̅ = 𝚺𝑿
̅ ∙ P(𝑿
̅)
𝜇𝑥̅ = 4.40 mean of the sampling
distribution of the means
Sampl
e
Mean
𝑥̅
Probability
P(𝑥̅)
𝑥̅ - 𝜇
2.67 1/10 -1.73
3.33 1/10 -1.07
3.67 1/10 -0.73
4.00 1/10 -0.40
4.33 2/10 or 1/5 -0.07
5.00 2/10 or 1/5 0.60
5.67 1/10 1.27
6.00 1/10 1.60
total 1.00

69
Group 4: Your Square Root, My Standard
Compute the Variance and Standard Deviation of the sampling distribution of the
Means if the mean 𝜇 of the population is 5.
Step 1: Multiply (𝑥̅ − 𝜇)2 by its corresponding Probability P(𝑥̅) and add the results
Sample
Mean
𝑥̅
Probability
P(𝑥̅) 𝑥̅ - 𝜇 (𝑥̅ − 𝜇)2
2.67 1/10 -1.73 2.993
3.33 1/10 -1.07 1.145
3.67 1/10 -0.73 0.533
4.00 1/10 -0.40 0.160
4.33 2/10 or
1/5
-0.07 0.005
5.00 2/10 or
1/5
0.60 0.360
5.67 1/10 1.27 1.613
6.00 1/10 1.60 2.560
total 1.00
Sample
Mean
𝑥̅
Probabili
ty
P(𝑥̅) 𝑥̅ - 𝜇
(𝑥̅
− 𝜇)2
P(𝑥̅)∙ (𝑥̅ −
𝜇)2
2.67 1/10 -1.73 2.993
3.33 1/10 -1.07 1.145
3.67 1/10 -0.73 0.533
4.00 1/10 -0.40 0.160
4.33 2/10 or
1/5
-0.07 0.005
5.00 2/10 or
1/5
0.60 0.360
5.67 1/10 1.27 1.613
6.00 1/10 1.60 2.560
total 1.00
Step 2:Square the difference 𝑥̅ - 𝜇
Sampl
e
Mean
𝑥̅
Probabilit
y
P(𝑥̅)
𝑥̅ - 𝜇 (𝑥̅ − 𝜇)2
2.67 1/10 -1.73
3.33 1/10 -1.07
3.67 1/10 -0.73
4.00 1/10 -0.40
4.33 2/10 or
1/5
-0.07
5.00 2/10 or
1/5
0.60
5.67 1/10 1.27
6.00 1/10 1.60
total 1.00
Samp
le
Mean
𝑥̅
Probab
ility
P(𝑥̅)
𝑥̅ - 𝜇 (𝑥̅ − 𝜇)2
P(𝑥̅)∙ (𝑥̅ − 𝜇)2
2.67 1/10 -1.73 2.993 0.300
3.33 1/10 -1.07 1.145 0.115
3.67 1/10 -0.73 0.533 0.053
4.00 1/10 -0.40 0.160 0.016
4.33 2/10
or 1/5
-0.07 0.005 0.001
5.00 2/10
or 1/5
0.60 0.360 0.072
5.67 1/10 1.27 1.613 0.161
6.00 1/10 1.60 2.560 0.256
total 1.00 Σ P(𝑥̅)∙ (𝑥̅ −
𝜇)2
=0.974
What is now the variance of the sampling
distribution of the sample mean?
𝜎2
𝑥̅ = Σ P(𝑋
̅) ∙ (𝑋
̅ – 𝜇)2
𝜎2
𝑥̅ = 0.974 Variance of the sampling distribution
of the sample mean
̅) ∙ (𝑋
̅ – 𝜇)2
= √0.974
𝜎𝑥̅ = 0.990 standard deviation of the sampling
distribution of the sample mean

70
Daily Lesson Plan in Statistics and Probability
Grade 11/12
Quarter 3 Week 5
I. OBJECTIVES
A. Content
Standard
The learner demonstrates understanding of key concepts
of sampling ad sampling distributions of the sample mean.
B. Performance
Standard
C. Learning
Competency/Obj
ectives
M11/12SP-III-e-2. The learner illustrates the Central Limit
Theorem.
II. CONTENT Central Limit Theorem
References 1. Commission on Higher Education & Philippine Normal
University (2016). Teaching Guide for Senior High
School: Statistics and Probability. pp.242-261
2. Woodward, E. (2019). Ed's Intro to Prob and Stats.
Retrieved from
https://legacy.cnx.org/content/col12133/1.1/ pp. 364.
3. Holmes, A., Illowsky, B., & Dean, S. (2019). Retrieved
from
https://opentxtbc.ca/introbusinessstatopenstax/chapter
/usng-the-central-limit-theorem.
Other Learning
Resource
1. Calculator, manila paper, permanent markers,
ruler/meter stick, Diagram of the different shapes of
distributions retrieved from
http://mathcenter.oxford.emory.edu/site/math117/shap
eCenterAndSpread/
IV. PROCEDURES
A. Reviewing
previous
lessons or
presenting the
new lesson
1. Describe the shape of the following distribution.
Image downloaded from
http://mathcenter.oxford.emory.edu/site/math117/shapeCenterAndSpread/

71
Options
A. Symmetric, unimodal, bell-shaped
B. Uniform
C. Skewed right
D. Skewed left
E. Symmetric, bimodal
F. Non-symmetric, bimodal
B. Establishing a
purpose for the
lesson
What will be the effect of increasing the sample size on the
shape of the sampling distribution of the sample mean
given that the samples are selected at random?
The learners will be asked to write their hypothesis on their
notebook. The teacher will inform the learners that in order
to test their hypotheses, they will be asked to perform an
activity. At this point, the learners will be divided into six
groups. Each group will be given a copy of the worksheet
to be used and other materials needed to accomplish the
task.
C. Presenting
Examples/Instan
ces of the
Lesson
The class will be divided into 4 groups. Provide each group
with the materials needed in accomplishing their tasks such
as dice, Hand-outs, permanent markers, calculator, manila
paper and coloring materials. (See attached Hand-outs.)
Tasks:
Group 1: Construct a probability distribution of the random
variable X defined by the outcomes of rolling a die. Draw its
corresponding histogram. What is the shape of the
distribution?
Group 2. Ask one member of the group to roll 2 dice.
Consider this as trial 1. Compute the sample mean of the
faces showing. For example, 1 and 2, thus the mean is
1+2
2
= 1.50. Record the outcomes and the mean of the
samples on the hand-out provided to your group (Hand-out
1.A) The same person will continue rolling the dice until 20
trials. After completing all the required trials, construct a
probability distribution of the sample means and construct
its corresponding histogram. Describe the shape of the
distribution.
faces showing. For example, 1, 3,4,5 and 2, thus the mean
is
1+3+4+5+2
5
= 3.00. Record the outcomes and the mean of
the samples on the hand-out provided to your group (Hand-
out 1.B) The same person will continue rolling the dice until
20 trials. After completing all the required trials, construct a
probability distribution of the sample means and construct
its corresponding histogram. Describe the shape of the
distribution.

72
faces showing. For example, 1,1,1,4,4,5,6,2,3 and 2, thus
the mean is
1+1+1+4+4+5+6+2+3+2
10
= 2.90. Record the
outcomes and the mean of the samples on the hand-out
provided to your group (Hand-out 1.C) The same person
will continue rolling the dice until 20 trials. After completing
all the required trials, construct a probability distribution of
the sample means and construct its corresponding
histogram. Describe the shape of the distribution.
D. Discussing New
concepts and
Practicing New
Skills # 1
The learners will be given at most 2 minutes to present their
group outputs. The teacher then checks the histogram
constructed by each group. Once all of the groups’ outputs
are checked, ask the learners to compare the histograms of
the sampling distributions of the sample mean when n=2,
n=5 and n=10 and compare this to the original population
distribution constructed by Group 1. What happens to the
shape of the sampling of the sample means when the
sample size increases?
E. Developing
Mastery
Show the following sets of diagrams to the learners. Let
them answer the guide questions afterwards.
(A) (B) (C)
Images retrieved from https://opentxtbc.ca/introbusinessstatopenstax/chapter/usng-the-
central-limit-theorem
Guide Questions
1. What is the shape of the population distribution in Set
A? in Set B? in Set C?

73
2. What happens to the shape of the sampling of the
sample means when the sample size increases?
3. Complete the statement below about central limit
theorem.
The central limit theorem for sample means says that
as the sample size_________, the sampling distribution
of the sample mean grows closer to a ________,
regardless of the shape of the original population
distribution. (increases, normal distribution)
F. Making
generalization
and abstraction
about the lesson
The central limit theorem for sample means says that
as the sample size increases, the sampling distribution
of the sample mean grows closer to a normal
distribution, regardless of the shape of the original
population distribution.
When the variable has a distribution that is not a Normal
distribution, the sample means are not normally
distributed unless the sample size is large enough.
(Generally, a good rule of thumb is to use a sample size
of at least 30, to ensure a sampling distribution that will
be approximately normal. Unless of course the original
population is known to be normal, in which case the
sampling distribution of the sample mean will be
guaranteed to normal.)
G. Evaluating
Learning
Choose only 1 of the suggested tasks. (See attached
worksheets 1-A to 1-D)

74
WORKSHEET 1-A
I. Write O if the statement is TRUE and X if otherwise.
1. The Central Limit Theorem tells us that as sample sizes get larger, the sampling
distribution of the sample means will become normally distributed, even if the
data within each sample are not normally distributed. (TRUE)
2. The shape of the sampling distribution of the means becomes left skewed if
random samples of size n becomes larger. (FALSE)
3. The central limit theorem states that as the sample size increases, the shape
of the distribution of the sample values look more and more normal. (FALSE)
II. Read and analyze the situations below. Write a short explanation for your answer.
4. A certain study involving senior high school students’ number of hours spent in
social media in a day shows a strongly skewed distribution with a mean of 5.2
hours and a standard deviation of 2.4 hours. What is the shape of the sampling
distribution of the sample means of 55 randomly selected senior high school
students if 55 is considered to be a large sample? Justify your answer.

75
WORKSHEET 1-B. What’s Your Muddiest Point?
I’m most confused about _____________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
_________________________________________________________________

76
WORKSHEET 1-C. 3-2-1
Accomplish the table below by writing 3 things that you have learners today, 2 things
that you found interesting and 1 question that you still have in mind.
3
Things I learned today
2
Things I found interesting
1
Questions I still have

77
WORKSHEET 1-D. T-L-R
Accomplish the table below by writing your initial hypothesis in the first column. In the
second column, write all the things that you have learned throughout the session and
in the third column, write a short reflection about your learnings. Is your hypothesis
correct? Can you cite real life situations or phenomena wherein the concept of central
limit theorem can be applied?
What I think
(Write your initial
hypothesis before the
conduct of the activity)
What I learned?
(Write the things that
you learned today.)
Reflection

78
HAND-OUT 1.A
SAMPLING DISTRIBUTION OF SAMPLE MEANS (n=2)
Name: _______________________ _______________________
_______________________ _______________________
_______________________ _______________________
_______________________ _______________________
Tabulation of Results.
Trials Samples (X)
Sample Means
(round off to the
nearest hundredths)
(𝑿
̅)
Example 1,2 1.50
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Complete the probability distribution of the sample means below. You may add columns if
needed.
(𝑿
̅)
P (𝑿
̅)
Draw the histogram of the sampling distribution of the sample means (n=2).
Describe the shape of the distribution.

79
HAND-OUT 1.B
SAMPLING DISTRIBUTION OF SAMPLE MEANS (n=5)
Name: _______________________ _______________________
_______________________ _______________________
_______________________ _______________________
_______________________ _______________________
Tabulation of Results.
Trials Samples (X)
Sample Means
(round off to the
nearest hundredths)
(𝑿
̅)
Example 1, 3,4,5,2 3.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Complete the probability distribution of the sample means below. You may add columns if
needed.
(𝑿
̅)
P (𝑿
̅)
Draw the histogram of the sampling distribution of the sample means (n=2).
Describe the shape of the distribution.

LP in Stat & Proba.pdf

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