Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

7

4th QUARTER – Module 7: MEASURE OF VARIABILITY

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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Mathematics – Grade 7 Alternative Delivery Mode Quarter 4 - Module 7: Measure of Variability First Edition, 2020

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

Principal Salvador C. Bucoy, LMD

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ASDS Jay S. Montealto, CESO VI

ASDS Norma T. Francisco, DM, CESE

EPS Mathematics Vilma A. Brown, Ed. D.

EPS LRMS Aida F. Coyme, 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.

This module is an aid in the basic statistics lesson of the fourth quarter grade 7

mathematics. The module follows an approach to computational statistics supported by

examples and exercises. It covers the key concept of measures of variability for grouped and

ungrouped data.

This module was designed to cater the academic needs of diverse learners in

achieving and improving the twin goals of mathematics in basic education levels which are

enhancing the critical thinking and problem solving skills. The language used recognizes the

vocabulary level of grade 7 students. The lessons followed a developmentally sequenced

teaching and learning processes to meet the curriculum requirements.

After going through the module, the learners are expected to:

⚫ illustrate the measures of variability (range, average deviation, variance, standard

deviation) of statistical data (M7SP-IVh-1); and

⚫ calculate the measures of variability of ungrouped and grouped data. (M7SP-IVh-I-1)

What I Need to Know

3

What I Know

Directions: Choose the letter that corresponds to your answer.

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

A. range B. mode C. variance D. standard deviation.

2. Which measures of variability is considered most reliable?

A. Variance B. standard deviation C. range D. average deviation.

3. For the set of data consisting of 5, 4, 3, and 2, what is the range?

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

4. If the range of a set of data is 16, and the highest score is 25, what is the lowest score?

A. 8 B. 9 C. 10 D.12

5. What is the standard deviation of the given sets of data {1.3, 5, 7}?

A. 2.0 B. 2.12 C. 2.21 D. 2.23

What’s In

Directions: Analyze the given situation and answer the given procedure below.

A housewife surveyed canned pork ham for a special family gathering. She picked 6

cans each from the two boxes packed by company 1 and company 2. Both boxes have the

same weight. Consider the following weights in kilograms of the canned pork ham packed by

the two companies (1 and 2).

Company 1 0.98 1.01 0.95 0.88 1.02 1.10

Company 2 1.05 1.02 0.89 1.00 0.91 1.12

Help the housewife choose the best sample by doing the following procedure.

A. Arrange the weights of the sample in increasing order.

B. Find the mean weight of each sample.

C. Analyze the spread of the weights of each sample from the mean.

D. Which sample has weights closer to the mean?

E. If you were to choose from these two samples which would you prefer? Why?

Analysis of data can lead us to a right decision in life.

What’s New

The IQs of 4 members of 2 Families A and B are presented below.

Family A 98 102 115 110

Family B 110 108 120 112

WHO IS SMARTER? ACTIVITY

4

A. Find the mean IQ of the two families.

B. Find the range of the IQ of both families.

QUESTIONS:

1.What have you observed about the range of the IQ of the two families?

2. What does it tell you?

LESSON

1 Measure of Variability

What is It

Smaller dispersion of scores arising from the comparison often indicates more

consistency and more reliability.

For ungrouped data, the commonly used measures of variability are the Range, Average

or Mean Deviation, Variance and Standard Deviation.

1. RANGE

Range (𝑹) is the difference between the highest and the lowest values in a given

set of data.

a. FORMULA: 𝑹 = 𝑯𝒊𝒈𝒉𝒆𝒔𝒕 𝑽𝒂𝒍𝒖𝒆 − 𝑳𝒐𝒘𝒆𝒔𝒕 𝑽𝒂𝒍𝒖𝒆 = 𝑯 − 𝑳

b. EXAMPLE: Find the range of the given set of data {2, 4, 6. 8, 10}.

SOLUTION: 𝑹 = 𝐻 − 𝐿 = 10 − 2 = 8

2. AVERAGE / MEAN DEVIATION (M.D.)

Average/ Mean Deviation (M.D.) is the dispersion of data about the average of these

data.

a. FORMULA: 𝑴. 𝑫. =∑|𝒙−�̅�|

𝑵

where, 𝑴. 𝑫. is the Mean Deviation

𝒙 is the individual scores

𝒙 is the mean

𝑵 is the number of scores

|𝒙 − 𝒙| is the absolute value of

the deviation from the

mean

b. STEPS IN COMPUTING THE MEAN DEVIATION

1. Find the mean of all cases

2. Find the absolute value difference between each score and the mean.

3. Find the sum of the difference and divide it by the number of scores N.

Measures of Variability refer to the spread of values about the Mean.

MEASURE OF VARIABILITY

Measures of Variability for Ungrouped Data A

5

c. EXAMPLE: Find the mean deviation of the given set of data {2, 4, 6, 8, 10}.

SOLUTION:

Step 1: Find the mean �̅�.

�̅� =2 + 4 + 6 + 8 + 10

5=

30

5= 6

Step 2 and 3:

SCORES (𝒙) 𝒙 − 𝒙 |𝒙 − 𝒙|

2 2−6 = −4 4

4 4−6 = −2 2

6 6−6 = 0 0

8 8−6 = 2 2

10 10−6 = 4 4

𝑵 = 5 ∑|𝒙 − 𝒙| = 𝟏𝟐

𝑴. 𝑫. =∑|𝒙 − 𝒙|

𝑵=

𝟒 + 𝟐 + 𝟎 + 𝟐 + 𝟒

𝟓=

𝟏𝟐

𝟓= 𝟐. 𝟒

3. VARIANCE

Variance (𝜎2) is the square of the mean deviation divided by the total number of

scores (𝑁).

a. FORMULA: 𝜎2 =∑(𝒙−�̅�)2

𝑁

b. EXAMPLE: Find the variance of the given set of data {2, 4, 6, 8, 10}.

SOLUTION:

SCORES (𝒙) 𝒙 − 𝒙 (𝒙 − 𝒙)2

2 −4 (−𝟒)𝟐 = 16

4 −2 (−𝟐)𝟐 = 4

6 0 (𝟎)𝟐 = 0

8 2 (𝟐)𝟐 = 4

10 4 (𝟒)𝟐 = 16

𝑵 = 5 ∑(𝒙 − 𝒙)2 = 𝟒𝟎

𝝈𝟐 =∑(𝒙 − 𝒙)𝟐

𝑵=

𝟏𝟔 + 𝟒 + 𝟎 + 𝟒 + 𝟏𝟔

𝟓=

𝟒𝟎

𝟓= 𝟖

4. STANDARD DEVIATION

Standard Deviation (𝜎) is simply the square root of the variance.

a. FORMULA: 𝜎 = √𝝈𝟐 = √∑(𝒙−�̅�)2

𝑁

b. EXAMPLE: Find the standard deviation of the given set of data {2, 4, 6, 8, 10}.

SOLUTION:

𝜎 = √𝟖 ≈ 𝟐. 𝟖𝟑

Now you will study the different measures of variability for grouped data. The

different measures of variability for grouped data are the Range, Variance and Standard

Deviation.

Measures of Variability for Grouped Data B

6

To find the range, variance, and standard deviation of grouped data, we need to take

note of the following formula:

In calculating the variance, do the following steps:

1. Prepare a frequency distribution with appropriate class interval and write the

corresponding frequency.

2. Get the midpoint (𝑥) of each class interval.

3. Multiply frequency (𝑓) and midpoint (𝑥) of each class interval to get 𝑓𝑥.

4. Add 𝑓𝑥 of each interval to get ∑ 𝑓𝑥.

5. Compute the mean using the formula �̅� =∑ 𝑓𝑥

∑ 𝑓

6. Calculate the deviation (𝒙 − 𝒙) by subtracting the mean from each midpoint.

7. Square the deviation of each interval to get (𝒙 − 𝒙)𝟐.

8. Multiply frequency ( 𝑓 ) and (𝒙 − 𝒙)𝟐 . Find the sum of each product to get

∑ 𝒇(𝒙 − 𝒙)𝟐.

9. Calculate the variance (v) using the formula

𝝈𝟐 =∑ 𝒇(𝒙 − 𝒙)𝟐

∑ 𝒇 − 𝟏

Calculate the range, variance and standard deviation of scores the First Periodical Test

of Grade 7 – Justice in Mathematics.

𝑹 = 𝒖𝒑𝒑𝒆𝒓 𝒄𝒍𝒂𝒔𝒔 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒐𝒇

𝒕𝒉𝒆 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 −

𝒍𝒐𝒘𝒆𝒓 𝒄𝒍𝒂𝒔𝒔 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒐𝒇 𝒕𝒉𝒆 𝒍𝒐𝒘𝒆𝒔𝒕 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍

RANGE (𝑹)

𝝈𝟐 =∑ 𝒇(𝒙 − �̅�)𝟐

∑ 𝒇 − 𝟏

Where, 𝑓 = class frequency 𝑥 = class mark / midpoint �̅� = class mean ∑ 𝑓 = total number of frequency

VARIANCE (𝝈𝟐)

𝝈 = ඥ𝝈𝟐 = ඨ∑ 𝒇(𝒙 − �̅�)𝟐

∑ 𝒇 − 𝟏

Where, 𝜎 = variance

STANDARD DEVIATION (𝝈)

EXAMPLE

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1. Range

𝑹 = 25.5 – 0.5 = 25

2. Mean

𝒙 =∑ 𝒇𝒙

∑ 𝒇=

𝟑𝟗𝟎

𝟑𝟎= 𝟏𝟑

3. Variance

𝝈𝟐 =∑ 𝒇(𝒙−�̅�)𝟐

∑ 𝒇−𝟏=

𝟏𝟏𝟎𝟎

𝟑𝟎−𝟏=

𝟏𝟏𝟎𝟎

𝟐𝟗≈ 𝟑𝟕. 𝟗𝟑

4. Standard Deviation

𝝈 = ඥ𝝈𝟐 = ඨ𝟏𝟏𝟎𝟎

𝟐𝟗≈ 𝟔. 𝟏𝟔

What’s More

Directions: Complete the table below then calculate the different measures of variability

(range, mean deviation, variance, and standard deviation). Use �̅� = 92.

SCORES (𝒙) 𝒙 − 𝒙 (𝒙 − 𝒙)2

90

91

92

92

93

94

Range

Mean

Deviation

𝝈𝟐

𝝈

Scores Frequency 𝒇 Class Mark

(𝒙) 𝒇𝒙 (𝒙 − 𝒙) (𝒙 − 𝒙)𝟐 𝒇(𝒙 − 𝒙)𝟐

21 - 25 3 23 69 10 100 300

16 – 20 8 18 144 5 25 200

11 – 15 10 13 130 0 0 0

6 – 10 4 8 32 -5 25 100

1 - 5 5 3 15 -10 100 500

∑ 𝒇 = 𝟑𝟎 ∑ 𝒇 𝒙 = 𝟑𝟗𝟎 ∑ 𝒇(𝒙 − �̅�)𝟐 = 𝟏𝟏𝟎𝟎

SOLUTION

COMPLETE ME! ACTIVITY

8

What I Have Learned

Directions: Complete the table by stating the difference in their formula among the measures

of variability for ungrouped and grouped data.

MEASURES OF VARIABILITY UNGROUPED DATA GROUPED DATA

1. Range

2. Variance

3. Standard Deviation

What I Can Do

Directions: Complete the table and answer the questions that follow.

Ages of Covid-19 positive in Zamboanga City for the first three months with a mean

QUESTIONS:

1. What is the ∑ 𝒇 𝒙?

2. What is the value of the mean in the given distribution table?

3. What is the upper-class boundary of the top interval?

4. What is the lower-class boundary of the bottom interval?

5. What is the range?

6. What is the variance of the given distribution table?

7. What is the standard deviation?

Assessment

Directions: Choose the letter that corresponds to your answer.

1. Which of the following is the dispersion of data about its mean?

A. range C. variance

B. average/mean deviation D. standard deviation

Ages Frequency 𝒇 Class Mark (𝒙) 𝒇𝒙 (𝒙 − 𝒙) (𝒙 − 𝒙)𝟐 𝒇(𝒙 − 𝒙)𝟐

65 - 73 6 69 414

56 - 64 13 60 780

47 - 55 19 51 969

38 - 46 4 42 168

29 - 37 3 33 99

20 – 28 5 24 120

∑ 𝒇 = ∑ 𝒇 𝒙 = ∑ 𝒇(𝒙 − �̅�)𝟐 =

DIFFERENTIATE ME!

ACTIVITY

9

2. What measures of variability simply refers to the difference of the highest value from the

lowest value of a given data set?

A. range C. variance

B. average/mean deviation D. standard deviation

3. If one of the given intervals of a given frequency distribution table is 12 - 20, what is the

class mark of these data?

A. 8 B. 16 C. 20 D. 32

4. What data are needed to compute for standard deviation?

A. range C. variance

B. average/mean deviation D. all of the above

5. Which measures of variability is considered most reliable?

A. range C. variance

B. average/mean deviation D. standard deviation

For items 6 to 7 refer to the data {𝟑, 𝟔, 𝟗, 𝟏𝟐, 𝟏𝟓, 𝟏𝟖 }.

6. What is the range of the given data?

A. 9 B. 12 C. 15 D. 18

7. What is the mean of the data?

A. 10 B. 10.5 C. 11 D. 11.5

8. How do you compute for standard deviation for ungrouped and grouped data?

A. Get the square of the mean

B. Get the class interval of the data

C. Get the square root of the variance

D. Divide the sum of the data by the total number of data.

9. If the range of the data is 5 and the highest data is 16, what is the lowest data?

A. 21 B. 16 C. 11 D. 9

10. What does measures of variability tell us?

A. The collection of data C. the average of the data

B. The spread of the information D. the spread of the values about the mean.

References:

Pogoso C. Montana, Introductory Statistics; Advance Algebra, Trigonometry and Statistics,

270-273.

Ann Xavier Gantent, Integrated Algebra 1 United States of America: Amsco School

Publication, Inc., 2007, 690.

Canonigo, A. (2012) Statistics, K - 12 Facilitators Guide.

10

Answer Key

What’s New:

1. 𝑥1തതത=106.25 𝑥2 തതത=112.5 2. 𝑅𝐴=17 𝑅𝐵=12

These range tell us that the IQ of family

A are more scattered than the IQ of

family B.

What I Know:

1.B222. B 3. C 4. B 5. D

What’s In:

a.Sample 1: 0.88, 0.95, 0.98, 1.01, 1.02, 1.10

Sample 2: 0.89, 0.91, 1.00, 1.02, 1.05, 1.12

b.𝑥1 തതത=0.990 𝑥2 തതത=0.998

c.The spread is closer/almost the same

d.Both e.Sample 1

What’s More:

SCORES (𝒙) 𝒙−𝒙 (𝒙−𝒙)2

90 -2 4

91 -1 1

92 0 0

92 0 0

93 1 1

94 2 4

10

Range 4

Mean

Deviation 1

𝝈𝟐 1.67

𝝈 1.29

What I Have Learned:

Measures of Variability

Ungrouped Data

Grouped Data

1.Range 𝑯−𝑳 Upper boundary of the highest interval

- Lower boundary of the

lowest interval

2.Variance 𝝈𝟐

=(𝒙−𝒙)𝟐

𝑵 𝝈

𝟐=

∑𝒇(𝒙−𝒙)𝟐

∑𝒇−𝟏

3.Standard Deviation 𝝈=ඨ(𝒙−𝒙)𝟐

𝑵 𝝈=ඨ

∑𝒇(𝒙−𝒙)𝟐

∑𝒇−𝟏

What I Can Do:

Ages Frequency 𝒇 Class Mark (𝒙) 𝒇𝒙 (𝒙−𝒙) (𝒙−𝒙)𝟐 𝒇(𝒙−𝒙)𝟐

65 - 73 6 69 414 18 324 1944

56 - 64 13 60 780 9 81 1053

47 - 55 19 51 969 0 0 0

38 - 46 4 42 168 9 81 324

29 - 37 3 33 99 18 324 972

20 – 28 5 24 120 27 729 3645

∑𝒇=𝟓𝟎 ∑𝒇𝒙=𝟐𝟓𝟓𝟎 ∑𝒇(𝒙−𝒙)𝟐=𝟕𝟗𝟑𝟖

1. 2550 2. 51 3. 73.5 4. 19.5 5. 53 6. 162 7. 12.73

Assessment:

1.B 3.B 5. D 7. B 9. C

2.A 4. C 6. C 8. C 10. D

11

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.”