Republic of the Philippines

UNIVERSITY OF NORTHERN PHILIPPINES Tamag, Vigan City

2700 Ilocos Sur

SENIOR HIGH SCHOOL AFFAIRS

UNIVERSITY OF NORTHERN PHILIPPINES

SEA TEACHER PROJECT

VIGAN CITY, PHILIPPINES

FEBRUARY, 2018

MEAN, VARIANCE, AND STANDARD DEVIATION OF

DICRETE RANDOM VARIABLE

A DETAILED LESSON PLAN

Prepared by:

RIKA AULIA NANDA

1406103020062

HELEN R. SIEMBRE

Cooperating Teacher

A LESSON PLAN FOR FINAL DEMONSTRATION – FEBRUARY 13, 2018

A. Component Identity

1. School : Senior High School Affairs

2. Subject : Statistics and Probability

3. Class/Semester: 11 – 2nd Semester

4. Time : 12:30 – 01:30

5. Topic : Random Variables

6. Sub-Topic : Mean, Variance and Standard Deviation

B. Objectives

Through a lesson and group collaboration students will be able to :

1. Illustrate the mean, variance, and standard deviation of discrete random variable

2. Calculate the mean, variance, and standard deviation of discrete random variable

3. Solve problems involving mean, variance, and standard deviation of probability

distribution

C. Subject Matter

1. Topic : Random Variables

2. References : Mercado, J.P., and Fernando B. Orines. 2016. Next Century

Mathematics Statistics and Probability. Philippines: Phoenix Publishing House,

Inc.

3. Materials : Power Point, Marker, Board, Activity Sheets, Coins

D. Methodology

1. Elicit ( Review/ recall )

The teacher will ask students about the lesson that they have learned in the

previous meeting regarding random variables

2. Engage (Motivation)

The teacher will elicit students responses about the probability of tossing coins

3. Development of the lesson

a. The teacher will show the topic about mean, variance and standard

deviation using power point.

b. The teacher will discuss how to find mean, variance and standard

deviation and give some examples.

c. The students will be divided into 5 groups. Then, each group will be

giventhe group activity in finding the mean, variance and standard

deviation. They will be given 15 minutes to discuss about their responses

regarding the group activity.

d. The students will present the result of the discussion in front of the class.

E. Assesment / Evaluation

After the discussion, the teacher will administer a short quiz.

F. Assignment / Agreement

The teacher will make a conclusion and tell the students what will be the next topic.

TEACHER’S ACTIVITY

STUDENTS’ ACTIVITY

I. PREPARATION

Prayer

Good Morning everyone!

There’s no better way of doing it than

acknowledging the goodness of our

Creator. Let’s pray by our own faith

before we start the lesson.

Checking of Attendance

Where are our classroom monitors?

Is there anyone who is absent? Check

your classmates and please report.

Statement of Objectives

By the end of the lesson, hopefully you

are able to

• Illustrate the mean, variance, and

standard deviation of discrete

random variable

• Calculate the mean, variance, and standard deviation of discrete

random variable

• Solve problems involving mean, variance, and standard deviation

of probability distribution

Recall

Can someone give me a recap of our

lesson last time?

In tossing 3 coins, how many probability

in getting 3 heads?

That was wonderful! Yes, that is 1/8.

Thank you for that wonderful responses.

Motivation

Now, look at what is in my hand ?

Yes, you are correct. This is the dices. We

have 2 dices here. What do you usually

the dices ?

A student will be called to lead the prayer

The class monitor will report if someone

is absent.

Ma’am all boys are present.

All girls are present except….

Last meeting, we discussed about the

random variables. I learned that the

random variable is the variable which

assigning values determined by chance.

Its 1/8 Ma’am !

It is dices ma’am!

We role the dices ma’am.

Yes, it’s true dear. What for we role the dices?

Yes, and then what we got after playing

dices?

Yes, very good. Then, let’s role the dices.

In indonesia, we called it by dadu. Okay

guys, let’s role the dices until 5 times.

Please someone write the outcomes on

board.

We finished role the dices, what we got

guys ?

Compute the value guys! Add all of the

numbers and divided by how many trial

experiment that we had done?

So the result is ?

Okay guys, according to the computed

value that we got, we had ... as the result.

We called it mean right ?

The mean is the average, where you add

up all the numbers and then divide by the

number of numbers.

Do you understand guys ?

Okay, Today, you will learn how to find

the mean, variance and standard deviation

of discrete random variable.

For playing ma’am! hahaha

We got numbers and we add the numbers

of the dices ma’am!

Okay ma’am. Oo I see ma’am.

Yes, ma’am.

We got, ....

.../.... = ... ma’am!

It is....

Yes, ma’am!

Oo I see ma’am ..

Yes, sure ma’am.

Alright, ma’am.

II. PRESENTATION

Now, I will be going to present to you

about the first topic.

Showing a Power Point

Slide 1

Mean of Discrete Random Variable

The mean of a discrete random variable X

is also called the expected value of X. The

discrete random variable X assumes

values or outcomes in every trial of an

experiment with their corresponding

probabilities.

Alright, Ma’am.

The students listen the discussion of the

teacher.

The expected value of X is denoted by E(X) / µ

Do you understand the topic guys ?

What is mean ?

Brilliant! Can you read the next slide?

Yes, that’s true!

Okay. May I show you some example!

Example 1.

A researcher surveyed the households in a

small town. The random variable X

represents the number of college

graduates in the households. The

probability distribution of X is shown

below:

x 0 1 2

P(x) 0.25 0.50 0.25

Find the mean or expected value of X.

I answer the question by using the formula

of mean.

Are you understand guys ?

You just need to multiply x and P(x) then

get the sum of it. That’s it guys! The table

will be like this :

X P(x) x.P(x)

0 0.25 0

1 0.50 0.50

2 0.25 0.50

∑[𝑥𝑃(𝑥)]

= 1.00

Can you tell me what is the result of the

mean?

Yes, correct! The expected value is 1. So

the average number of college graduates

in the household of the small town is one.

Yes Ma’am!

Mean is the Expected Value of X Ma’am!

The mean or expected value of a discrete

random variable x is computed using the

following formula:

E (X) = ∑[𝑥 𝑃 (𝑥)]

Sure, Ma’am!

The students will pay attention while the

teacher solve the example.

Okay, Ma’am!

It is 1.00 Ma’am!

Yes, Ma’am.

Is it clear guys ?

Okay. Can we proceed to the next slide

and read it for me guys ?

Very good guys!

Do you understand what is that ?

Okay, I will explain it to you.

Guys, you have to know the symbol of the

variance is “small letter sigma”

I will show you the formula of the

variance for you to understand well.

𝑉𝑎𝑟(𝑋) = ∑[ (𝑥 − 𝜇)2𝑃(𝑥)] OR

σ2 = ∑[ (𝑥 − 𝜇)2𝑃(𝑥)]

I will again discuss what variance is all

about.

This means that you are going to subtract

the values of x to mean, then you are

going to squared it, after that you are

going to multiply it from the probability

of x, then get the sum of it.

Understand class?

I will give you the same example, but we

are going to find the variance of the X, so

lets add another 3 columns from the table

above.

Aah, yaa sure! You may.

While the student copy the topic, I add

the another 3 columns to find the

variance.

Then I ask to student. Any body who can

still remember what i’ve told you to a

while ago in finding variance?

Can you tell once again ?

That’s clear Ma’am!

Of course Ma’am !

The variance of a random variable X is

denoted by σ 2

The variance of a random variabe is the

expected value of the square of the

difference between the assumed value of

random variable and the mean.

Not really Ma’am.

Yes Ma’am!

Yes Ma’am!

Ooo, i see Ma’am!

Yes Ma’am!

Wait Ma’am! We will copy first!

The students are copying.

Yes Ma’am!

Yes Ma’am. We are going to subtract the

values of x to mean, then we are going to

squared it, after that we are going to

Great guys. Very good !

Look at the table. So the result of the

variance is .....

Is it clear with variance guys?

Okay, that’s good!

Now, lets talk about the standard

deviation!

Kindly read the slide guys ?

Yes students, and then ?

Yes, that’s the formula of the standard

deviation. That’s is so easy, we just need

to square root the result of the variance.

Guys, lets see the example of standard

deviation.

Yes, dear. Sure, we can, because we just

have to square root of the variance.

So, what is the result guys ?

Yeah, very good student. So, is that clear

for mean, variance and standard deviation

?

So good.

You have to know that all of this formulas

are related to each others, so if we want to

find the variance, we have to know first

the mean, same goes with the standard

deviation. If we want to find the standard

deviation, we have to know first the

variance. That’s the point.

Do you understand guys?

So glad to hear that guys!

multiply it from the probability of x, then get the sum of it.

..... Ma’am !

Yes, Ma’am!

The standard deviation of a discrete

random variable X is written as σ.

It is square root of the variance.

The standard deviation is computed as :

σ = √∑[ (𝑥 − 𝜇)2𝑃(𝑥)]

Yes ma’am. We understand ma’am.

Ma’am, can we just add the row in same

table above ?

The result is ... ma’am

Yes ma’am!

Yes ma’am!

Thank You, ma’am!

Now, I will write another example on the board. The example is :

Determine the variance and standard

deviation of the following probability

mass function:

x 0 1 2 3 4

P(x) 0.1 0.2 0.3 0.3 0.1

Guys, do you know how to answer the

question?

Okay guys, please fill in the blank , write

your answer on the board.

Who wants to fill the answer ? please

come in front of the class.

Yes, sure dear!

Yes, please everyone fill the blank until

the table is complete!

Very good students! Thank you!

Yes ma’am!

Yes, Ma’am!

Me ma’am!

Me ma’am!

Yes ma’am!

Ma’am, it is done!

You’re welcome ma’am!

III. PRACTICE

Group Discussion

Now, after you have understood about

mean, variance and standard deviation, I

want to divide you into five groups.

Then, each group will discuss about the

questions in the group activity sheets. You

have 10 minutes to discuss about your

response. Afterwards, you will match

your answer with the heart shape that I

had prepare. So you have to paste in the

kartunila on the board until the kartunila

completed! The group that posted the

most answers will get the highest points.

One patch, you’ll get one point.

Any question about the instruction?

Class, class. Have you done with the

discussion?

Okay, now, who wants to take the first

chance?

Guys, please go forward to paste your

answer !

Yes/No, Ma’am.

Yes ma’am !

Me, Ma’am!

Yes ma’am!!

IV. PURPOSEFUL CLOSURE

Conclusion

Before we end the class, is there anyone

wants to conclude what we have learned

today?

Yes, that’s correct guys, very well.

So, the mean of a discrete random

variable x is the expected value of x. The

variance of a random variable x is the

expected value of the square of the

difference between the assumed value of

random variable and the mean. And the

last, the standard deviation of a random

variable X is the square root of the

variance.

Short Quiz

Find the mean, variance and standard

deviation of the following situation. An

officer at a prison questioned each inmate

to find out how many times the inmate has

been convicted. The officer came up with

the following table that shows the relative

frequencies of x:

x 0 1 2 3 4

P(x) 0.16 0.53 0.20 0.00

8

0.03

Next Topic

Next meeting, we will learn about

sampling and sampling distribution.

Closing Prayer

Let’s pray together by our own faith..

Goodbye class.

We have learned about mean, variance

and standard deviation Ma’am.

It has its own formulas. The formulas

realted to each other.

Good bye Ma’am.

Group Name : …………………………………………………..

Member’s Name : …………………………………………………..

& Presense Number …………………………………………………..

…………………………………………………..

…………………………………………………..

………………………………………………….

Class :

Learning Content : Mean, Variance, and Standars deviation of Discrete Random

Variable

Indicators :

1. Student will be able to illustrate the mean, variance, and standard deviation of discrete

random variable

2. Student will be able to calculate the mean, variance, and standard deviation of discrete

random variable

Direction :

1. Fill the Group’s Identity

2. Work in your group.

3. Answer the question by fill in the blank using pen

4. If any question, raise your hand and tell to teacher

1. When four coins are tossed, the probability distribution for the random variable X

representating the number of heads. Compute the mean, variance and standard deviation

of the probability distribution.

x P(x)

0 .....

1 .....

2 .....

3 .....

4 .....

Answer :

x P(x) x P(x) x-μ (𝐱 − 𝛍)2 (𝐱 − 𝛍)2 P(x)

0 ..... ..... ..... ..... .....

1 ..... ..... ..... ..... .....

2 ..... ..... ..... ..... .....

3 ..... ..... ..... ..... .....

4 ..... ..... ..... ..... .....

∑(𝐱 𝐏(𝐱)) = ..... ∑[(𝐱 − 𝛍)𝟐 𝐏(𝐱)] = .....

GROUP ACTIVITY

TASK

μ = ........................ ( write the formula)

= ........................

So, the mean of X is ................................., the variance of X is ......................................,

and the standard deviation of X is ..................................... .

2. Find the mean, variance and standard deviation of the probability distribution of the

random variable X

x 11 12 13 14 15

P(X) 0.1 0.3 0.3 0.1 0.2

Answer :

x P(x) x P(x) x-μ (𝐱 − 𝛍)2 (𝐱 − 𝛍)2 P(x)

..... ..... ..... ..... ..... .....

..... ..... ..... ..... ..... .....

..... ..... ..... ..... ..... .....

..... ..... ..... ..... ..... .....

..... ..... ..... ..... ..... .....

∑(𝐱 𝐏(𝐱)) = ..... ∑[(𝐱 − 𝛍)𝟐 𝐏(𝐱)] = .....

μ = ........................ ( write the formula)

= ........................

σ2 = ......................... ( write the formula)

= .........................

σ 2 = ......................... ( write the formula)

= .........................

σ = √. . . . . . . . . . . . . . . .( write the formula)

= √. . . . . . . . . . . . … . = .........................

So, the mean of X is ................................., the variance of X is ......................................,

and the standard deviation of X is ..................................... .