G10MATHQ4M3.pdf - ZNNHS

Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

10

4th QUARTER – Module 3: INTERPRETING MEASURES OF POSITION

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

1

Mathematics – Grade 10 Alternative Delivery Mode Quarter 4 - Module 3: Interpreting Measures of Position First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines

Department of Education – Region IX, Zamboanga Peninsula

Office Address: Tiguma, Airport Road, Pagadian City

Telefax: (062) – 215 – 3751; 991 – 5975

E-mail Address: [email protected]

Development Team of the Module

Writer: Mildred P. Caceres

Editors: Rosemary B. Garcia

Elizabeth M. Raciles, Ed.D.

Illustrator: Mildred P. Caceres

Layout Artist: Zandro G. Sepe, M.S.

Reviewers: EPS, Mathematics Vilma A. Brown, Ed.D.

Principal Zandro G. Sepe, M.S.

Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI

ASDS Jay S. Montealto, CESO VI

ASDS Norma T. Francisco, Ed.D.,CESE

EPS Mathematics Vilma A. Brown, Ed.D.

EPS LRMS Aida F. Coyme, Ed.D.

2

Introductory Message

This Self – Learning Module (SLM) is prepared so that you, our dear learners, can continue

your studies and learn while at home. Activities, questions, directions, exercises, and

discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you

discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell

you if you can proceed on completing this module or if you need to ask your facilitator or your

teacher’s assistance for better understanding of the lesson. At the end of each module, you

need to answer the post-test to self-check your learning. Answer keys are provided for each

activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our

facilitators and parents for strategies and reminders on how they can best help you on your

home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use

a separate sheet of paper in answering the exercises and tests. Read the instructions carefully

before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this

module, do not hesitate to consult your teacher or facilitator.

Thank you.

In our country, all Grade 9 students are required to take the National Career

Assessment Examination (NCAE) which is intended to predict which profession

matches their skills and interests. The graduating students may use these results as

their guide on what track to take when they proceed to Senior High School.

If you have seen your result on this test, it does not only show the student’s

percentage score but also your individual percentile rank. Do you have any idea on

what your rank means? The percentile rank relates a student’s score to those of the

other examinees.

In this module, you will know more about solving percentile rank, and

interpreting the computed measures of position. As you go through this module, you

will be able to answer questions like: How can the position of data be described and

used in solving real-life problems? What is the importance of interpreting measures of

position?

What I Need to Know

3

At the end of this module, you are expected to interpret measures of position (M10SP-IVc-1):

• Interpret the computed upper and lower quartile of ungrouped data

• Interpret the computed deciles of ungrouped data

• Interpret the computed percentile of ungrouped data

• Interpret the computed percentile rank of a specific value in ungrouped set

of data

What I Know

Directions: Choose the letter that corresponds to your answer. Write your answer on

a separate sheet.

1. Agapito Flores ranks 10th in a class of 40. What is Agapito Flores percentile

rank?

A. 90 B. 75 C. 30 D.10

2. Consider the score distribution of 15 students below:

83 72 87 79 82 77 80 73 86 81 79 82 79 74 74

How can median be interpreted?

A. Seven students scored below and above 79.

B. Seven students scored below 79.

C. Fourteen students scored below 79.

D. Seven students scored above 79.

3. The third quartile of the ages of 1600 Grade 10-students is 18 years. Which of

the following statements is TRUE?

A. One thousand four hundred students are older than 18 years old.

B. Majority of the students are above 18 years old.

C. Seventy-five percent of the students are 18 years old and below.

D. Seven Hundred Fifty students are 18 years old.

4. Francis’ score in a 75-item test was the median score. What is his percentile

rank?

A. 35th B. 50th C. 75th D. 25th

5. Fifty employees of a certain private company were rated and ranked. Those

who are above the 90th percentile rank received an additional incentive. How

many of the employees received an additional incentive?

A. 45 B. 10 C. 5 D. 2

4

What’s In

Direction: Answer the activity as fast as you can. Write your answers on a separate sheet.

The following are scores of 10 students in their 40-item quiz:

34 23 15 27 36 21 20 13 33 25

1. What are the scores of the students which are less than or equal to 25% of the data? ___________________________________________________________________



What’s New

Directions: Read the situation carefully and answer the questions that follow. Write

your answer on a separate sheet.

A non-government organization grants scholarship to students based on their

academic performance, in comparison with other applicants. There is a total of 1500

applicants.

Full scholarship

with Allowance

Full Scholarship

without

allowance

Partial

Scholarship with

Allowance

Partial

Scholarship

without

allowance

95th to 99th

Percentile

90th to 94th

Percentile

85th to 89th

percentile

80th to 84th

percentile

QUESTIONS:

1. How many students were not granted any scholarship?

2. How many students were granted full scholarship with allowance?

3. How many students were granted partial scholarship without allowance?

Do you have a sharp memory? ACTIVITY

ACTIVITY

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What is It

LESSON 1

Interpreting Measures of Position

EXAMPLE

A group of students obtained the following scores in their latest 30-item quiz

in Math 10, find the lower quartile and upper quartile, and interpret the results.

7 10 11 19 19 21 24 24 26 27 30

ANSWER:

By computation, the first quartile and third quartile is 11 and 26,

respectively.

7, 10, 11, 19, 19, 21, 24, 24, 26, 27, 30

Thus,

GIVEN INTERPRETATION/S

1st Quartile 11

It means that 25% of the students have scores

lower than or equal to 11 while 75% have scores

higher than 11.

3rd Quartile 26

It means that 75% of the students have scores

lower than or equal to 26 while 25% have scores

higher than 26.

Recall that quartiles divide a distribution into four equal parts, and that the

Lower or first quartile (𝑄1) means there are 25% of the data fall below

or equal to 𝑄1 and 75% of the data is above 𝑄1.

Upper and third quartile (𝑄3) means there are 75% of the data fall below

or equal to 𝑄3and 25% of the data is above 𝑄3.

LOWER AND UPPER QUARTILES

Q3 Q1

How to interpret computed lower and upper quartiles of ungrouped data?

A

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EXAMPLES

1. Mrs. Magalang gave a test to her students in Statistics. The students finished their

test in 35 minutes. This time is the 2.5th decile of the allotted time. What does it

mean?


2.5th Decile 35 minutes

The students finished the test in less than or equal

to 35 minutes which is 25% lower than the allotted

time. (Giving a low quartile or percentile would be

considered good because the students finished

the test earlier than the given time frame.)

2. Florence works as a Covid-19 nurse in a private hospital in Zamboanga City. Her

salary is the 7th decile of the salaries of all employees. Should Florence be glad

about her salary or not? Explain your answer.

Deciles show the 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80% and 90% of the scores, such that 10% of the data fall below or equal to D1 and 90% of the distribution is above D1, 20% of the distribution fall below or equal to D2, and so on.

DECILES FOR UNGROUPED DATA

How to interpret the computed deciles of ungrouped data?

B

D2.5

35 minutes

𝐷7

Florence’s ₱ salary

7


7th Decile Florence’s salary

Yes, Florence should be pleased with her salary

considering that 70% of the employees received a

salary lower than hers while only 30% have salary

higher than hers.

EXAMPLE

Consider the test scores in Geometry. The 75th percentile score is 15. If the 75th

percentile is the passing score, what does it mean? What is its measure of position in

relation to the other data?


75th Percentile 15 passing score

Students should answer correctly at least 15

items to pass the test.

It also means that 75% of all the scores

obtained is less than or equal to the score of

15.

How to interpret the computed percentiles of ungrouped data?

C

Percentiles are denoted by 𝑃1 , 𝑃2 , …, 𝑃99. Read as “P sub 1 or first percentile”, “P sub 2 or “second percentile,” and so on.

The 1st percentile means 1% of the data fall below or equal to 𝑃1, the 2nd

percentile means 2% of the data fall below or equal to 𝑃2, and so on.

PERCENTILES FOR UNGROUPED DATA

𝑃75

Score 15

8

+

EXAMPLES

1. Consider the scores of 12 students of Grade 10-Hope in a Math quiz: 15, 13, 20,

16, 14, 19, 11, 15, 19, 10, 18, and 9. What is the percentile rank of 16 and what

does the result mean?

Solution:

2. If you obtained a score of 95 in a mathematics test and this score is greater than

or equal to the scores of 88% of the students who took the test, then your percentile

rank would be 88.

STEPS Illustrative Example

1

Arrange the scores in ascending

order.

9 10 11 13 14 15 15 16 18 19 19 20

2

Count the number of values or data

lower than given score.

9, 10, 11, 13, 14, 15, 15, 16, 18, 19, 19, 20

*Note that there are 7 values below 16.

3

Apply the formula for percentile rank.

Remember to round off to the nearest

whole number.

Percentile = # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑋 + 0.5

𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠× 100

= 7 + 0.5

12× 100

= 62.5 ≈ 63

4

Interpret the result.

Therefore, 16 is the 63rd percentile and

thus, 63% of the distribution fall below or

equal to 16.

Percentile ranks refer to the percentage of the total scores that are less than or equal to a given score. These are particularly useful in relating individual scores to their positions in the entire group or distribution.

Finding the Percentile Rank Corresponding to a Value

The percentile corresponding to a given value x is computed by using

the following formula:

Percentile = ሺ# 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑋ሻ+0.5

𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠× 100

PERCENTILE RANK

How to interpret the computed percentile rank in an ungrouped set of data?

D

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What’s More

Directions: Study the following situations. Then, answer the questions that follow.

1. Emilio, who is a 14-year old grade 10 student, believes that he is generally young

for his grade level. Upon gathering of data, it has been found out that the first

quartile is 15 years old. Would you agree with Emilio? Why?

Ans._____________________________________________________________

____________________________________________________________

2. Hidilyn Diaz received a score of 90 from the judges in an international weightlifting

competition joined by 50 participants. Of that, 40 participants had scores lower than

90. What does this imply?

Ans._____________________________________________________________

____________________________________________________________

3. The class of 10-Mathpagmahal is composed of 20 female students. Josefa is said

to be the 8th tallest among her female classmates. What does it mean?

Ans._____________________________________________________________

____________________________________________________________

Consider the situation below, then answer items 4-5.

The scores of 22 students in a Math quiz are the following: 22, 23, 21,

22, 25, 22, 18, 24, 24, 24, 24, 20, 22, 22, 22, 15, 26, 22, 22, 24, 28, 24

Compute and interpret the percentile rank of Ralph who got a score of 25.

4. Solution:

5. Interpretation:

_________________________________________________________________

_________________________________________________________________

ACTIVITY

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What I Have Learned

Directions: Write the steps in finding the Percentile Rank using the clouds below.

Write your answers on a separate sheet.

1. 2. 3. 4.

Directions: Fill in each blank to make a true statement. Write your answers on a

separate sheet.

1. The first quartile means there are _______ of the data fall below or equal to Q1 .

2. _____________ corresponds to a position where 70% of the data are below or

equal to its given value.

3. The 42nd percentile of the distribution is the score 15, this implies that 58% of the

distribution have scores ________ than 15.

4. 3rd Decile or ____________means that 30% of the distribution is less than or equal

to the value of 𝐷3.

5. The upper quartile corresponds to a position where ________ of the data are

above the given value.

What I Can Do

Directions: Read each situation carefully, then answer the questions that follow.

1. Your older sister tells you that you seem to be short for your age. In school, your

teacher measured your height and that of your classmates to compute for your

BMI. Your teacher tells you that your height belongs to the 9th decile in the class.

JUSTIFY ME! ACTIVITY

Cloud Process ACTIVITY 1

ACTIVITY 2

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Assuming you and your classmates have similar ages, would you agree with what

your sister says? Why?

Ans._____________________________________________________________

____________________________________________________________

2. Many professional schools require applicants to take a standardized test. Suppose

that 1000 students took the test, and you found out that your score of 63 (out of

100) was the 73rd percentile. How would you interpret your position?

Ans._____________________________________________________________

_____________________________________________________________

3. Among the class, your score is at the 65th percentile in the final exam, what

conclusion can you draw?

Ans._____________________________________________________________

_____________________________________________________________

4. A year ago, Kaye Masigasig began working at a computer store. Her supervisor

asks her to keep a record of the number of sales she made each month. The

following data set is a list of her sales for the last 12 months: 34, 47, 1, 15, 57, 24,

20, 11, 19, 50, 28, 37. Using Kaye’s sales: 𝑄1=17 and 𝑄3=42. Where does 34 fall

and what percent of the sales are below 34?

Ans._____________________________________________________________

_____________________________________________________________

5. Marikit ranked 84th percentile in an examination. What does this mean?

Ans._____________________________________________________________

_____________________________________________________________

Assessment

Directions: Read carefully each item and choose the letter that best answers the

question.

1. Given the set of scores {48, 39, 57, 32, 28, 63, 51, 55, 36), if a student received a

score of 55, what does it mean?

A. 25% of the scores has a value ≤ 𝑄1

B. 50% of the scores has a value ≤ 𝑄2

C. 75% of the scores has a value ≤ 𝑄3

D. There are three scores below 𝑄1, which is the 25% of the scores with values ≤ 55

2. In a 60-item test, the passing mark is the 3rd quartile. What does it imply?

A. The students should answer at least 45 items correctly to pass the test.

B. The students should answer at least 50 items correctly to pass the test.

C. The students should answer at most 45 items correctly to pass the test.

D. The students should answer at most 50 items correctly to pass the test.

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3. Josefina’s score in an 85-item test was the median score. What is her percentile

rank?

A. 45 B. 50 C.75 D. 85

4. Jian and her brother are both 62 inches tall. Jian is the 85th percentile for her age

and her little brother is in the 90th percentile. Who is taller for their age?

A. They are the same, both are 62 inches tall.

B. Jian, she’s taller for more people of her age than her brother.

C. The brother of Jian, which is taller for more people of his age than to Jian.

D. You can’t tell if you don’t know the mean and standard deviation.

5. Cyclists also want to finish the race in less time than everyone else. Which of the

following measures of position would mean a cyclist finished the race in less time?

A. 5th percentile B. 3rd quartile C. 2nd decile D. 1st quartile

Additional Activities

Directions: Read the given situation carefully, then answer the questions that

follow.

Forty Grade 10-Narra students compared their NCAE results with each

other in the Numerical Skills Area. Their standard scores are shown below.

QUESTIONS:

1. Jose scored 713. What is his percentile rank within the group?

2. What percentage of the students is higher than Jose?

3. If Antonio’s score is the 3rd quartile, who scored higher between Jose and

Antonio?

4. Students who scored higher than P90 will be granted a scholarship. How

many students will be given scholarships?

5. Given the scores of Jose and Antonio will they receive a scholarship

grant? Why?

HOW’S YOUR RANK? ACTIVITY

455 462 479 481 482 508 508 526 534 538 548 559 580 583 585 595 598 605 606 611 616 628 640 642 655 655 667 677 681 700 712 713 719 727 729 746 752 770 774 779

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Answer Key

What I Know:

1.B 2. A 3. C 4. B 5. C

What’s In:

1.13 and 15 2. 13, 15, 20, 21, 23, and 25 3. NONE

What’s New:

1.1200 students 2. 75 students 3. 75 students

What’s More:

1. Yes, because only 25% of the students have ages 15 years and below while

75% of the students have ages 15 years and above.

2.It means that Heidelyn Diaz achieved a higher score compared to 80% of her

competitors.

3.Since Josefa is the 8th tallest, it means that there are 12 female students who are

shorter than her. Furthermore, it also implies that 60% of her female classmates

are shorter than her while the remaining 40% of her female classmates are taller

than Josefa.

4.Percentile = # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑋+0.5

𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠×100% =

19 +0.5

22×100% = 88.64≈89

5.Therefore, 25 is the 89 percentile and thus, 89% of the distribution falls below 25.

What I Have Learned: A.Cloud Process

1.25% 2. 7th

Decile or 70th Percentile 3. Greater than/above 4. 30

th Percentile 5. 25%

What I Can Do:

1.No, I am taller than 90% of my classmates, therefore 90% of them are shorter than me.

2.This means that 270 students got 63 or better score. 3.65% of the class scored not as good than I did or 65% of the class scored less

than mine. 4.It means 34 falls in the third quartile which means 75% of the sales are less than

or equal to 42. 5.84% obtained a score below than Marikit’s score in the exam.

Assessment:

1. C 2. A 3. B 4. C 5. A

Additional Activities: (Optional)

1.PR=79 2. 21% 3. Jose 4. 4 students 5. None, as they both scores below P90

Arrange the data from lowest to highest

Count the number of

values below the given

score

Apply the Formula for Percentile

Rank

Interpret the result.

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References: Orlando A. Oronce and Marilyn O. Mendoza, e-math Advanced Algebra and Trigonometry IV, Manila: Rex Bookstore, 2010, pp 431- 435 Melvin M. Callanta, et al, Mathematics Learner’s Module 10 DepEd Pasig: Rex Bookstore, Inc., 2015, pp 283, 364-377, 392-394 Ramon B. Cristobal, et al, Math World 10 Quezon City: C & E Publishing, Inc. 2015, pp 245-252, 264-268 Math DLL

15

I AM A FILIPINO

by Carlos P. Romulo

I am a Filipino – inheritor of a glorious past, hostage to the

uncertain future. As such, I must prove equal to a two-fold

task – the task of meeting my responsibility to the past, and

the task of performing my obligation to the future.

I am sprung from a hardy race – child many generations

removed of ancient Malayan pioneers. Across the centuries,

the memory comes rushing back to me: of brown-skinned

men putting out to sea in ships that were as frail as their hearts

were stout. Over the sea I see them come, borne upon the

billowing wave and the whistling wind, carried upon the

mighty swell of hope – hope in the free abundance of the new

land that was to be their home and their children’s forever.

This is the land they sought and found. Every inch of shore

that their eyes first set upon, every hill and mountain that

beckoned to them with a green and purple invitation, every

mile of rolling plain that their view encompassed, every river

and lake that promised a plentiful living and the fruitfulness

of commerce, is a hollowed spot to me.

By the strength of their hearts and hands, by every right of

law, human and divine, this land and all the appurtenances

thereof – the black and fertile soil, the seas and lakes and

rivers teeming with fish, the forests with their inexhaustible

wealth in wild and timber, the mountains with their bowels

swollen with minerals – the whole of this rich and happy land

has been for centuries without number, the land of my

fathers. This land I received in trust from them, and in trust

will pass it to my children, and so on until the world is no

more.

I am a Filipino. In my blood runs the immortal seed of heroes

– seed that flowered down the centuries in deeds of courage

and defiance. In my veins yet pulses the same hot blood that

sent Lapulapu to battle against the alien foe, that drove Diego

Silang and Dagohoy into rebellion against the foreign

oppressor.

That seed is immortal. It is the self-same seed that flowered

in the heart of Jose Rizal that morning in Bagumbayan when

a volley of shots put an end to all that was mortal of him and

made his spirit deathless forever; the same that flowered in

the hearts of Bonifacio in Balintawak, of Gregorio del Pilar

at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in

flowers of frustration in the sad heart of Emilio Aguinaldo at

Palanan, and yet burst forth royally again in the proud heart

of Manuel L. Quezon when he stood at last on the threshold

of ancient Malacanang Palace, in the symbolic act of

possession and racial vindication. The seed I bear within me

is an immortal seed.

It is the mark of my manhood, the symbol of my dignity as

a human being. Like the seeds that were once buried in the

tomb of Tutankhamen many thousands of years ago, it shall

grow and flower and bear fruit again. It is the insigne of my

race, and my generation is but a stage in the unending

search of my people for freedom and happiness.

I am a Filipino, child of the marriage of the East and the

West. The East, with its languor and mysticism, its passivity

and endurance, was my mother, and my sire was the West

that came thundering across the seas with the Cross and

Sword and the Machine. I am of the East, an eager

participant in its struggles for liberation from the imperialist

yoke. But I know also that the East must awake from its

centuried sleep, shake off the lethargy that has bound its

limbs, and start moving where destiny awaits.

For I, too, am of the West, and the vigorous peoples of the

West have destroyed forever the peace and quiet that once

were ours. I can no longer live, a being apart from those

whose world now trembles to the roar of bomb and cannon

shot. For no man and no nation is an island, but a part of the

main, and there is no longer any East and West – only

individuals and nations making those momentous choices

that are the hinges upon which history revolves. At the

vanguard of progress in this part of the world I stand – a

forlorn figure in the eyes of some, but not one defeated and

lost. For through the thick, interlacing branches of habit and

custom above me I have seen the light of the sun, and I

know that it is good. I have seen the light of justice and

equality and freedom, my heart has been lifted by the vision

of democracy, and I shall not rest until my land and my

people shall have been blessed by these, beyond the power

of any man or nation to subvert or destroy.

I am a Filipino, and this is my inheritance. What pledge

shall I give that I may prove worthy of my inheritance? I

shall give the pledge that has come ringing down the

corridors of the centuries, and it shall be compounded of the

joyous cries of my Malayan forebears when first they saw

the contours of this land loom before their eyes, of the battle

cries that have resounded in every field of combat from

Mactan to Tirad Pass, of the voices of my people when they

sing:

“I am a Filipino born to freedom, and I shall not rest until

freedom shall have been added unto my inheritance—for

myself and my children and my children’s children—

forever.”

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