4 th QUARTER Module 5: PROBLEMS ON PERCENTILE AND PERCENTILE RANK FOR UNGROUPED DATA

Republic of the Philippines

Department of Education

Regional Office IX, Zamboanga Peninsula

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QUARTER – Module 5:

PROBLEMS ON PERCENTILE AND

PERCENTILE RANK FOR UNGROUPED DATA

Name of Learner:

___________________________

Grade & Section:

___________________________

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Mathematics – Grade 10 Alternative Delivery Mode

Quarter 4 - Module 5: Problems on Percentile and Percentile Rank for Ungrouped Data First Edition, 2020

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Development Team of the Module

Writer: Ciara Mae P. Jamorol

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.

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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.

The Department of Education yearly administers the National Career Assessment Examination (NCAE) to Grade 9 students. Now that you are in Grade 10, do you still remember your score? What was your rank? Have you ever thought of comparing your grade with that of your classmates? Whenever your teacher asks your class to form a line according to your height, what is your position in relation to your classmates? What General Weighted Average (GWA) would get you to qualify for honors?

Why do you think certain students in any national examination get higher rank than other students? Some state colleges and universities are offering scholarship programs for graduating students who belong to the upper 5%, 10%, or even 25%. What does this mean to you?

After going through the module, you are expected to:

• Solve Problems involving Measures of Position (M10SP-Ivd-e-1)

a. solve word problems involving real - life situations on percentiles for ungrouped data

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b. solve word problems involving real - life situations on percentile ranks for ungrouped data

What I Know

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

1. Which of the following is NOT a measure of position?

A. Decile B. Mean C. Percentile D. Quartile

2. Katarina’s average score in Mathematics test was 85 and is equivalent to the 86th percentile rank. Which of the following is NOT TRUE about Katarina’s score? A. 86% of the students are below Katarina’s score.

B. 14% of the students got a score higher than Katarina. C. If the passing score is the median, Katarina passed. D. 14% of the students are below Katarina’s score. 3. The 50th _{percentile is also called ______________.}

A. 1st _{Quartile B. Median C. 3}rd _{Quartile D. 9}th _Decile

4. The scores of the top students in grade 10 class are listed below, what is the 25th

Percentile?

80, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 94

A. 80 B. 81 C. 82 D. 83

5. Refer to item #4. If Marie obtained a score of 86, what is her percentile rank? A. 25th _{Percentile C. 75}th _Percentile

B. 50th _{Percentile D. 99}th _Percentile

What’s In

Directions: Read the given problem below carefully and answer the questions that follow:

The set of data below shows the daily sales (in pesos) of Pedro’s Lugawan arranged in ascending order for the month of November.

680 699 720 750 750 775 790 800 800 800 810 820 830 845 855 855 870 875 880 895 900 905 915 925 925 930 940 950 955 965 a. What is the n total number of data?

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c. If the 7th _{Decile is equal to 900, what is its equivalent if the given data set is}

divided to 100 equal parts?

What’s New

Directions: Study the following figures

Figure 1:

Figure 2:

Figure 3:

Questions:

a. Observe figure 1, how many parts does it have? b. Observe figure 2, how many parts does it have?

c. What measures of positions do figures 1 and 2 illustrate?

d. If figure 3 is divided into 100 equal parts, what measure of position will it illustrate?

e. Which among the three measures of positions should be used to find the position of a certain value in a very large data set?

What is It

LESSON

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Problems on Percentile and

Percentile Rank for Ungrouped Data

PERCENTILES

The percentiles are the ninety-nine score points which divide a distribution into one hundred equal parts, so that each part represents the data set. It is used to characterize values according to the percentage below them. For instance, the first percentile (P1) separates the lowest 1% from the other

99%, the second percentile (P2) separates the lowest 2% from the other 98%,

and so on.

5 P₇₅ =k(N + 1) 100 P₇₅ =75(10 + 1) 100 P₇₅ =825 100 P75 = 8.25 P₇₅ ≈ 8 (8𝑡ℎ 𝑣𝑎𝑙𝑢𝑒)

NOTE: Follow the steps in finding the value of percentiles of n data

STEPS ILLUSTRATIVE EXAMPLE

STEP 1: Arrange the n data values from lowest to highest.

In ascending order:

6900, 7000, 7100, 7500, 7800, 8000, 8200, 8700, 9200, 9800

Since there 10 data values then n = 10 STEP 2: Find the position of

the 75th percentile (P75) using

the formula:

𝐏_𝐤= 𝐤(𝐧 + 𝟏) 𝟏𝟎𝟎

(Round off to the nearest integer if necessary.)

PERCENTILE RANK

Percentile ranks are particularly useful in relating individual scores to

their positions in the entire group. A percentile rank is typically defined as the proportion of scores in a distribution that a specific score is greater than or equal to. For instance, if you obtained a score of 95 in a math test and it is greater than or equal to the scores of 88% of the students who took the test then the percentile rank is 88.

What is Percentile Rank?

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A. To solve for the position of the kth percentile score, follow the given formula below.

P

_k

=

k(n+1)

100

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

Percentile =

# of values below X + 0.5

total # of values

× 100%

FORMULA: Ungrouped Data

EXAMPLE 1: PERCENTILE

The ten Fast Food stores located at Paseo del Mar in Zamboanga City have the following daily sales in Philippine Peso; 7100, 7500, 6900, 8200, 7000, 9200, 7800, 8700, 9800, and 8000. What is P₇₅ and what does it imply?

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STEP 3: Locate the value of the specified measures of position in the given set of data.

6900, 7000, 7100, 7500, 7800, 8000, 8200, 8700, 9200, 9800

Since P75 is the 8th value then P75 = 8700.

Therefore, this implies that 75% of the data distribution fall below 8700.

STEPS ILLUSTRATIVE EXAMPLE

STEP 1: Arrange the n data values from lowest to highest.

In ascending order:

52 64 71 75 77 82 88

90 93 98

Note that 𝑋 = 77 and hence there are 4 scores below 77

STEP 2: Find the percentage of scores below 77.

Percentile

= # of values below X+0.5

total # of values × 100%

Therefore, a score of 77 has a percentile rank of 45. This implies that 45% of the data distribution fall below 77.

Remember that Percentiles are expressed by ordinal numbers (1st, 2nd, 3rd...99th) that refer to actual values or scores in a data set while as percentile ranks are expressed by percentages. Hence, we use the formula for percentile to characterize values according to the percentage below a certain data value; on the other hand, percentile rank is the percentage of the data value.

EXAMPLE 2: PERCENTILE RANK

Suppose you scored 77 on a test in a class of 10 students. With the 10 scores listed below, what percent of the students got a score lower than you?

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What’s More

Directions: Refer to the given problem below. Compute for the percentile scores and interpret the results. Show a proper solution and BOX your final answer on a separate sheet of paper.

The crisis brought about by the COVID-19 has posted a lot of challenges for everyone. One of which is the implementation of the Enhanced Community Quarantine (ECQ) which caused many to lose their jobs. With no definite end in sight yet for the COVID-19 crisis, many Filipinos are counting on the government for financial help. This is where its Social Amelioration Program (SAP) comes in. This program aims to administer emergency cash assistance to families with minimum wage who are greatly affected by the ECQ. Below are the daily minimum wages of forty (40) SAP beneficiaries of a certain barangay in Zamboanga City.

When we are dividing data into equal parts we are commonly talking about quartiles, deciles and percentiles. Quartiles divide the data into four parts; deciles divide the data into 10 parts; and percentiles divide the data into 100 parts.

Quartiles are often used as a measure of spread of the data in what is called the interquartile range (IQR). The IQR is simply the difference between the third quartile and first quartile.

Deciles and percentiles are usually applied to large data sets. Deciles divide a data set into ten equal parts. One example of the use of deciles is in school awards or rankings such as students in the top 10%.

Percentiles divide the data set into groupings of 1%. Standardized tests often report percentile scores. These scores help compare students’ performances to that of their peers. Growth charts are another common example of an application of percentiles. To help doctors and parents determine if a child is developing normally, his or her measurements are compared with others in the same sex and age groups.

When do we use Quartiles, Deciles and Percentiles?

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300 300 300 315 316 320 320 325 330 332 332 335 335 335 340 342 345 348 350 352 355 355 355 358 360 362 364 364 368 370 371 372 373 375 375 380 382 383 385 385 Solve and interpret the following percentile:

a. 25th _Percentile

b. 70th _Percentile

c. 95th _Percentile

Directions: Refer to the given problem below. Compute for the percentile ranks and interpret the results. (Use separate sheet to answer this activity)

Phillip Magsaysay is a front liner working as security guard in a local convenience store in Zamboanga City. After the Enhanced Community Quarantine (ECQ) was lifted, he noticed that the number of customers aged 18 and above has increased. One day, he listed down the age of 20 customers who went inside from 8am to 11am. Below are the data gathered by Phillip in ascending order.

18 18 19 19 20 21 22 23 23 25

26 26 27 28 29 30 31 32 33 33

a. A customer named Maria Alfonso went to buy essential goods at the convenience store. She is 29 years old. What is her percentile ranking?

b. If only customers aged 22 and above have quarantine pass, what percentage of the customers does not have quarantine pass??

Directions: Refer to the given statement below then answer the questions that follow. (Use separate sheet to answer this activity)

In the class of 55, Ross got a percentile rank of 81 in the Math Quiz. What does this percentile rank imply?

Answer: _________________________________________________ Is Ross’ score bad or good?

Answer: _________________________________________________

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How many students got scores lower than Ross’ score?

Answer: _________________________________________________

What I Have Learned

Directions: How well did you learn? Test your knowledge by filling out the graphic organizer below: (Use another sheet for your answer)

What I Can Do

Directions: Study the given problem below and answer the questions that follow: (Write your answer and solution on a separate sheet)

Questions:

1. Will the presentation of the data help you solve the percentile rank? If yes, why? If no, how should it be arranged?

Twenty (20) cellular data users were asked by a Tech.Mobile representative to determine how much money they spent monthly on mobile load for cellular data. The data recorded were as follows:

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2. What is the 80th _{percentile? What does it imply?}

3. Carla spent 460Php on cellular data, what is her percentile rank? Show your solution.

4. Justin spent 435Php on cellular data, what is her percentile rank? Show your solution.

5. What implications can you draw from Carla and Justin’s percentile rank?

Assessment

Directions: Read each item carefully and write only the letter which corresponds to the correct answer on a separate sheet of paper.

1. In a 40-item test, she got a score of 30 which is the 75th _{percentile. This means}

that:

A. her score is higher than 25% of her classmates. B. she surpassed 75% of her classmates.

C. she got the lowest score.

D. 75% of the class did not pass the test.

2. What is the value of the 60th _percentile?

A. 92 B. 100 C. 91 D. 74

3. What is the value of the median?

A. 86 B. 85 C. 78 D. 83

4. What does it imply if the diastolic pressure is 70% above?

A. It is in the 30th _{percentile. C. It is in the 40}th _percentile.

B. It is the median. D. It is in the 100th _percentile.

5. What does it imply if the diastolic pressure is 100 mmHg? A. It is the 100th _percentile.

B. 50% of the diastolic pressure is less than 100 mmHg. C. It is the median.

D. It is the 10th _percentile.

For items 2 to 5, refer to the problem below:

The blood pressure of a person varies periodically in every heartbeat. The lowest pressure measured in a cycle is called the diastolic pressure. Listed below are the diastolic pressures (mmHg) of 15 males aged 30 to 45:

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6. Given the scores of 9 students in their Statistics exam; 48, 39, 57, 32, 28, 63, 51, 54, and 36. If Via got a score of 54. What does it mean?

A. 25% of the scores has a value below the 25th _percentile.

B. 50% of the scores has a value below the 50th _percentile.

C. 72% of the scores has a value below the 72nd _percentile.

D. There are three scores that belong to the 25th _{percentile which is the 25% of}

the scores with values less than 7.

7. A group of students got the following scores in their biology quiz; 44, 40, 40, 36, 35, 33, 33, 33, and 30. What is the percentile rank of Sevi who got a score of 36?

A. 58 B. 61 C. 65 D. 70

8. Fernando got a score of 90 in a test given to the class of 50 students. Forty (40) of his classmates got scores lower than 90. This implies that ________.

A. 9% who took the test did better than Fernando. B. 81% who took the test did better than Fernando.

C. Fernando achieved a lower score than 81% of the students who took the test. D. Fernando achieved a higher score than 81% of the students who took the test. 9. A 50-item test was conducted in a class of 55 students. Sucre obtained a score of

38 which is equivalent to the 75th _{percentile, what does it imply?}

A. 50% of his classmates got a score higher than 38.

B. He belong to the top 25% of students with the best scores. C. He belong to the top 25% of students with the lowest scores. D. His score is below the 3rd _decile.

10. The scores of Miss World candidates from seven judges were recorded as follows; 8.45, 9.20, 8.56, 9.13, 8.67, 8.85, and 9.17. If Miss Philippines got a score of 9.17, what is her percentile rank?

A. 90 B. 82 C. 80 D. 79

Additional Activities

Directions: Solve and interpret for the following percentile scores in the given problem below. Write the answer on a separate sheet.

Skin care is a range of practices that supports skin integrity, enhances its appearance, and relieves skin conditions. Victoria, a beauty and health vlogger asked her 30 randomly chosen female followers to share how many minutes do they spend doing their daily skin care routines. The following are the data recorded:

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Arranged Data: ___________________________________________ ___________________________________________ Solve and interpret for the following percentile scores:

1. P12: ________ Interpretation: __________________________________ 2. P32: ________ Interpretation: __________________________________ 3. P58: ________ Interpretation: __________________________________ 4. PR = 24: ________ Interpretation: __________________________________ 5. PR = 29: ________ Interpretation: __________________________________

References:

Melvin M. Callanta, et. al, Mathematics Learner’s Module 10 Pasig City: Department of Education 2015, pp. 376-384

Percentiles and Percentile Ranks, last modified 2016,

https://sites.google.com/site/bphsstatistics/percentiles-percentile-ranks

Segmenting Data: Quartiles, Deciles, Percentiles, last modified 2019,

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Answer Key

Wha t’s M ore: AC TIV IT Y 1 a. P 25 =10 .25 ≈ 10 ( 10𝑡ℎ 𝑑 𝑎𝑡𝑎 𝑣 𝑎𝑙𝑢 𝑒); Si nce P 25 is th e 1 0 th d ata val ue th en P 25 = 33 2. Th ere fo re, th is imp lie s th at 25 % o f t he d ata d ist rib uti on fa ll be lo w 332. b. P 70 =28 .70 ≈ 29 ( 29𝑡ℎ 𝑑 𝑎𝑡𝑎 𝑣 𝑎𝑙𝑢 𝑒); Si nce P 70 is th e 2 9 th d ata va lu e t he n P 70 = 36 8. Th ere fo re, th is imp lie s th at 70 % o f t he d ata d ist rib uti on fa ll be lo w 368. c. P 95 =38 .95 ≈ 39 ( 39𝑡ℎ 𝑑 𝑎𝑡𝑎 𝑣 𝑎𝑙𝑢 𝑒); Si nce P 95 is th e 3 9 th d ata val ue th en P 95 = 38 5. Th ere fo re, th is imp lie s th at 95 % o f t he d ata d ist rib uti on fa ll be lo w 385. AC TIV IT Y 2 Y 2 : a. Perce nti le = 72.5 ≈ 73; Th ere fo re M ari a ha s a p erce nti le ra nk of 73 . Th is imp lie s th at 73 % o f t he d ata va lu es are b elo w 2 9. b. Perce nti le = 32.5 ≈ 33; Th ere fo re th e p erc en tile ra nk o f c us to me rs a ge d 2 2 is 33 . Th is imp lie s th at 33 % o f t he cu sto me rs a re n ot allo we d i ns id e t he co nve nie nce sto re. ACTI VITY 3 do es th is pe rcen tile ra nk imp ly? a. It imp lie s th at Ro ss g ot a sco re h ig he r th an 8 1% o f t he cl ass. b. Ross’ sco re is be tte r be caus e h is sco re is abo ve 81 % o f t he class a nd on

ly im. h ove ab is ass cl he f t % o 19

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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: