Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

10

4th QUARTER – Module 2: CALCULATING SPECIFIED MEASURE OF

POSITION

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

1

Mathematics – Grade 10 Alternative Delivery Mode Quarter 4 - Module 2: Calculating Specified Measure 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: region9@deped.gov.ph

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.

This module focuses on measures of position. Specifically, this module will help

you understand how to calculate a specified measure of position in a given set of

ungrouped and grouped data following a step by step approach.

This module provides an opportunity for you to improve your critical thinking

and solving skills by accomplishing the series of practical tasks presented in each

lesson. As you go through this module, you may answer the question, “How do the

steps in solving measures of position for ungrouped data differ from that of the grouped

data?”

By the end of this module, you are expected to calculate a specified measures

of position (e.g. 90th percentile) of a set of data (M10SP-IVb-1):

• calculate for quartiles, deciles and percentiles for ungrouped data

• calculate quartiles for grouped data

• calculate deciles for grouped data, and

• calculate percentiles for grouped data

What I Need to Know

3

What I Know

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

a separate sheet.

For items 1-2, consider the score distribution of 15 students given below:

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

1. The lower quartile is ______________.

A. 72 B. 74 C. 79 D. 86

2. What is the 75th percentile?

A. 73 B. 77 C. 79 D. 82

3. All EXCEPT one is not equivalent to 50th percentile.

A. 5th Decile B. 𝑄2 C. Median D. lower quartile

For items 4-5:

The screen time (in minutes) spent per day by some senior citizens using their

smartphones.

Class Interval

(in minutes)

Frequency (f) Lower

Boundaries

<cf

51 - 60 5 50.5 31

41 - 50 6 40.5 26

31 - 40 7 30.5 20

21 - 30 10 20.5 13

11 - 20 3 10.5 3

4. What is the total frequency?

A. 50 B. 31 C. 60 D. 51

5. Which class interval contains the 2nd quartile?

A. 21 - 30 B. 31 - 40 C. 41 - 50 D. 51 – 60

What’s In

Directions: Perform as indicated. Write your answers on a separate sheet. A. Complete the Frequency Table

You Complete Me! ACTIVITY

4

Upper

Boundaries

<cf

51 - 60 5 50.5 31

6 40.5 50.5

31 - 40 7 40.5 20

10 20.5 30.5

11 - 20 3 10.5 3

I = N =

B. Answer the following:

1. What is the class size?

2. How many samples are in the given distribution?

3. Which class interval contains the 2nd quartile? 5th decile? 50th percentile?

4. What is the frequency of the median class?

5. What is the cumulative frequency below the 2nd quartile? 5th decile? 50th

percentile?

What’s New

Directions: Read and understand the situation carefully. Then, answer the question

that follows. Write your answer on a separate sheet.

Last December an elimination for on-the spot parol-making contest was held in

Barangay Pascua. This was joined by Junior High School students Malakas, Matalino

and Magaling. To qualify for the final round the participants need to have a score above

the 75th percentile. Upon the go signal of the organizer, the participants individually

crafted their parol.

With his strength, Malakas finished his parol and scored 50% higher than all

participants. Meanwhile, Matalino used his wit to complete his parol and obtained a

score between the 8th-9th decile. Magaling, on the other hand, showcased her creativity

and scored above the upper quartile.

Question: Who of the three (Malakas, Matalino and Magaling) qualified for the final

round?

PAROL ALL ACTIVITY

5

What is It

LESSON 1

Calculating quartiles, deciles and percentiles for Ungrouped Data

Position of:

EXAMPLE

Given the set of scores {21, 30, 10, 7, 24, 27, 19, 24, 11, 26 and 19}; n = 11 Find: 1.) 𝑸𝟏 2.) 𝑫𝟔 3.) 𝑷𝟖𝟓

STEPS ILLUSTRATIVE EXAMPLE

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

In ascending order: 7, 10, 11, 19, 19, 21, 24, 24, 26, 27, 30

n=11 STEP 2: Calculate the position of the following using the formula:

1. 𝑸𝟏

𝑄𝑘 =𝑘(𝑛 + 1)

4

1. 1st quartile

Q1 =1(11+1)

4

𝑄1 =12

4

𝑸𝟏 = 𝟑 This means that the value of 𝑄1

is located in the 3rd position

therefore, 𝑄1= 11

1. Arrange the n data entries from lowest to highest (ascending order). 2. Calculate the position of the kth quartile/decile/percentile using its

corresponding formula. Round off to the nearest integer if necessary.

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

STEPS IN FINDING THE VALUE OF QUARTILES, DECILES, PERCENTILES OF n DATA

FORMULAS FOR UNGROUPED DATA

Quartiles

𝑄𝑘 =𝑘(𝑛 + 1)

4

where: n - total number of data k - position of the quartile

Deciles

𝐷𝑘 =𝑘(𝑛 + 1)

10

where: n - total number of data k - position of the decile

Percentiles

𝑃𝑘 =𝑘(𝑛 + 1)

100

where: n - total number of data k - position of the percentile

6

2.) 𝑫𝟔

𝐷𝑘 =𝑘(𝑛 + 1)

4

3.) 𝑷𝟖𝟓

𝑃𝑘 =𝑘(𝑛 + 1)

4

2. 6th decile

𝐷6 =6(11+1)

10

𝐷6 =6(12)

10

𝐷6 =72

10

𝑫𝟔 = 𝟕. 𝟐 ≈ 𝟕 (round off to the nearest integer)

This means that the value of 𝐷6 is

located in the 7th position therefore,

𝐷6 = 24

3. 85th percentile

𝑃85 =85(11+1)

100

𝑃85 =85(12)

100

𝑃85 =1020

100

𝑷𝟖𝟓 = 𝟏𝟎. 𝟐 ≈ 𝟏𝟎 (round off to the nearest integer)

This mean that the value of 𝑃85 is located in the 10th position therefore, 𝑃85 = 27

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

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

LESSON 2

Calculating quartiles, deciles and

percentiles for Grouped Data

1. Find the position of the kth class for:

Quartiles: Qk =kN

4 Deciles: 𝐷𝑘 =

𝑘𝑁

10 Percentiles: 𝑃𝑘 =

𝑘𝑁

100

2. Find the class interval where the kth score is contained. 3. Compute for Qk or Dk or Pk, using the corresponding formula.

STEPS IN FINDING THE VALUE OF QUARTILES, DECILES, PERCENTILES OF N DATA (GROUPED DATA)

D6 Q1 P85

7

EXAMPLE

Calculate the 1st quartile, 7th decile and 95th percentile of the Mathematics test scores of 50 students.

Solutions:

The table above illustrates the scores of 50 students already grouped using a

class size/width of 5. The scores are tabulated in column 1 and the corresponding

frequencies in column 2.

Each lower boundary is obtained by subtracting 0.5 to each of the lower score

of the class interval (e.g. In the class interval 46 - 50, the lower score is 46 minus 0.5

so the lower boundary is 45.5)

Less than cumulative frequency can be obtained by adding the frequency

starting from the frequency of the lowest class interval up to the frequency of the

Class Interval/ Scores

Frequency (f)

46-50 4

41-45 8

36-40 11

31-35 9

26-30 12

21-25 6

𝑸𝒌 = 𝑳𝑩 + (

𝒌𝑵𝟒

− 𝒄𝒇𝒃

𝒇𝑸𝒌

) 𝒊

where; Qk = kth Quartile

LB = lower boundary of the

Qk class

N = total frequency

cfb = cumulative frequency

of the class before the

Qk class

𝑓𝑄𝑘 = frequency of the Qk

class

i = size of class interval

k = nth quartile, where

n = 1, 2, and 3

𝑫𝒌 = 𝑳𝑩 + (

𝒌𝑵𝟏𝟎 − 𝒄𝒇𝒃

𝒇𝑫𝒌

) 𝒊

where; Dk = kth Decile LB = lower boundary of the

Dk class N = total frequency cfb = cumulative frequency

of the class before the Dk class

𝑓𝐷𝑘 = frequency of the Dk

class i = size of class interval k = nth quartile, where n = 1, 2, 3, 4, 5, 6, 7, 8

and 9

𝑷𝒌 = 𝑳𝑩 + (

𝒌𝑵𝟏𝟎𝟎 − 𝒄𝒇𝒃

𝒇𝑷𝒌

) 𝒊

where: Pk = kth Percentile LB = lower boundary of the

Pk class N = total frequency cfb = cumulative frequency

of the class before the Pk class

𝑓𝑃𝑘 = frequency of the Pk

class i = size of class interval k = nth quartile, where

n = 1, 2, 3,…,97, 98 and 99

FORMULAS FOR GROUPED DATA

8

highest class interval. For example, the cumulative frequency up to and including the

class interval 41 - 45 in the table above, is 6 + 12 + 9 + 11 + 8 = 46. This signifies that

46 students have scores less than or equal to 45.

Class Interval/ Scores

Frequency (f)

Lower Boundaries

(LB)

Less than Cumulative Frequency

46 - 50 4 45.5 50

41 - 45 8 40.5 46

36 - 40 11 35.5 38

31 - 35 9 30.5 27

26 - 30 12 25.5 18

21 - 25 6 20.5 6

STEPS ILLUSTRATIVE EXAMPLE

GIVEN 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝒇𝒐𝒓 𝑸𝟏

class so, 𝒌 = 𝟏 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝒇𝒐𝒓 𝑫𝟕 class

so, 𝒌 = 𝟕 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝒇𝒐𝒓 𝑷𝟗𝟓 class

so, 𝒌 = 𝟗𝟓 STEP 1: Find the position of the kth class N=50

𝑄1 =𝑘𝑁

4

=1(50)

4

= 12.5

This means we need

to find a class interval

where the 12.5th score

is contained.

𝐷7 = 𝑘𝑁

10

=7(50)

10

= 35

This means we need to

find a class interval

where the 35th score is

contained.

𝑃95 =𝑘𝑁

100

=95(50)

100

= 47.5

This means we need to

find a class interval

where the 47.5th score is

contained.

STEP 2: Find the class interval where the kth score is contained.

Note that the 7th to 18th

scores belong to the 26 - 30 class interval, thus 12.5 is contained therein. (see the bottom arrows of the frequency table above) By inspection, we have the following:

Note that the 28th to 38th

scores belong to the 36-40 class interval, thus 35 is contained therein. (see the center arrows of the frequency table above) By inspection, we have the following:

Note that the 47th to 50th

scores belong to the 46-50 class interval, thus 16 is contained therein. (see the top arrows of the frequency table above) By inspection, we have the following:

STEP 3: Compute using the appropriate formula

Compute for 𝑄1:

Compute for 𝐷7:

Compute for 𝑃95:

(28th-38th score) D7 Class

(7th-18th score) Q1 Class

(47th-50th score) P95 Class

𝑄1 = 𝐿𝐵 + (

𝑁4

− 𝑐𝑓𝑏

𝑓𝑄1

) 𝑖

LB= 25.5 𝑐𝑓𝑏= 6 𝑓𝑄1

=12 i=5

𝐷7 = 𝐿𝐵 + (

7𝑁10

− 𝑐𝑓𝑏

𝑓𝐷7

) 𝑖

𝑃95 = 𝐿𝐵 + (

95𝑁100 − 𝑐𝑓𝑏

𝑓𝑃95

) 𝑖

LB= 35.5 𝑐𝑓𝑏= 27

𝑓𝐷7=11 i=5

LB= 45.5 𝑐𝑓𝑏= 46

𝑓𝑃95=4 i=5

9

𝑸𝟏 = 𝟐𝟖. 𝟐𝟏

𝑫𝟕 = 𝟑𝟗. 𝟏𝟒

𝑷𝟗𝟓 = 𝟒𝟕. 𝟑𝟖

What’s More

Directions: Arrange the given set of data below in ascending order and compute the

specified measure of position then write the corresponding letter of the answer on the

blank to decode the mystery word.

.

Consider the given data on the number of minutes per day spent by students

using their gadgets while at home:

84 120 10 30 43 25 110 15 38 55 90 60 50 45 75 100 S A R I M C O L P D Q G U T N E

____ ____ ____ ____ ____ ____ ____ ____ 1. P32 2. D9 3. D7 4. Q1 5. D4 6. P25 7. P88 8. P64

Directions: Complete the table then solve the following by showing your complete

solutions. Write your answers on a separate sheet.

The distribution shows the efficiency ratings of 45 teachers. Using the data

below, calculate the 1.) lower quartile 2. 85th percentile 3.) 4th decile.

Ratings Number of Teachers

LB <cf

96-100 11

91-95 6

86-90 9

81-85 7

76-80 8

71-75 4

TRACK MY ... ACTIVITY 1

= 25.5 + (12.5 − 6

12) 5

= 35.5 + (35 − 27

11) 5

= 45.5 + (47.5 − 46

4) 5

= 25.5 + (

504

− 6

12) 5

= 35.5 + (

7(50)10 − 27

11) 5

= 45.5 + (

95(50)100 − 46

4) 5

ACTIVITY 2

10

What I Have Learned

Directions: Write each step in finding the position/location in the given set of data using the step-up process. Write your answers on a separate sheet.

Finding the value of kth Quartile for Ungrouped Data

Directions: Write each step in finding the position/location in the given set of data

using the step-up process. Write your answers on a separate sheet.

Finding the value of kth Percentile for Grouped Data

STEP 1

STEP 2

STEP 3

_________________

_________________

_________________

_________________

_________________

_________________

Formulas:

STEP-UP 1 ACTIVITY

STEP-UP 2 ACTIVITY

STEP 1

STEP 2

STEP 3

_________________

_________________

_________________

_________________

_________________

_________________

Formulas:

11

What I Can Do

Directions: Perform as directed and write your answer on a separate sheet.

Class Interval

frequency Lower Boundaries

Less than Cumulative Frequency

61-65 5

56-60 7

51-55 9

46-50 5

41-45 4

A. Solve the following to complete the table above.

1. Lower boundaries

2. Less than Cumulative Frequency

B. Then, solve for the following:

3. Upper Quartile 4. 6th Decile 5. 20th Percentile

Assessment

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

For items 1 to 2, consider the score distribution of 15 students given below:

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

1. Calculate the value of the 2nd decile.

A. 72 B. 74 C. 85 D. 83

2. What is the position of the third quartile?

A. 10th B. 11th C. 12th D. 13th

3. A mutual fund achieved the following growth rates in 11-month period

(3%, 2%, 7%, 8%, 2%, 4%, 3%, 7.5%, 7.2%, 2.7%, 2.9%). What is the 5th decile

from the given data?

A. 4% B. 3% C. 2% D. 2.09%

MAKE ME COMPLETE! ACTIVITY

12

For items 4 to 5, refer to the table below.

Interval Frequency Cumulative Frequency

41-50 8 24

31-40 3 16

21-30 6 13

11-20 5 7

1-10 2 2

4. In solving for the 2nd quartile, which lower boundary should be used?

A. 0.5 B. 11.5 C. 20.5 D. 31.5

5. What is the value of the 8th decile? A. 39.5 B. 40.05 C.43.05 D.44.5

References: Orlando A. Oronce and Marilyn O. Mendoza, e-math Advanced Algebra and Trigonometry IV, Manila: Rex Bookstore, 2010, pp, 425-426, 431- 436 Melvin M. Callanta, et al, Mathematics Learner’s Module 10 DepEd Pasig: Rex Bookstore, Inc., 2015, pp 368-395 Ramon B. Cristobal, et al, Math World 10 Quezon City: C & E Publishing, Inc. 2015, pp 264 Olivia N. Buzon, et al, Advanced Algebra with Trigonometry and Statistics Explorations and Applications, Salesiana Publishers, Inc. 2003, pp. 298-300

13

Answer Key

What I Know:

1.B 2. D 3. D 4. B 5. B

What’s In:

A.

Class

Interval frequency

Lower

Boundaries

Upper

Boundaries <cf

51 - 60 5 50.5 60.5 31

41-50 6 40.5 50.5 26

31 - 40 7 30.5 40.5 20

21-30 10 20.5 30.5 13

11 - 20 3 10.5 20.5 3

I =10 N =31

B.

1. 10 2. 31 3. 31 – 40, 31 - 40, 31 - 40 4. 7 5. 13, 13, 13

What’s New:

MALAKAS

What’s More:

Activity 1: P O S I T I O N Activity 2:

Ratings Number of Teachers LB <cf

96-100 11 95.5 45

91-95 6 90.5 34

86-90 9 85.5 28

81-85 7 80.5 19

76-80 8 75.5 12

71-75 4 70.5 4

1.Lower Quartile = 80.03 2. P85 =97.43 5. D4 =84.79

14

What I Have Learned:

STEP-UP 1 1. Arrange the given data in ascending order.

2. Calculate the position of the kth quartile using the formula: 𝑄𝑘=𝑘(𝑛+1)

4

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

STEP-UP 2 1. Find the position of the kth class N.. 2. Find the class interval where the kth score is contained. 3. Compute using the appropriate formula

𝑷𝒌=𝑳𝑩+ ቆ𝒌𝑵

𝟏𝟎𝟎−𝒄𝒇𝒃

𝒇𝑷𝒌

ቇ𝒊

What I Can Do:

Class Interval

f LB

A.1

Less than cf

A.2

61-65 5 60.5 30

56-60 7 55.5 25

51-55 9 50.5 18

46-50 5 45.5 9

41-45 4 40.5 4

A B. 3. 𝑄3=58.7 4. 𝐷6=55.5 5. 𝑃20=47.5

Assessment:

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

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.”