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Republic of the Philippines
Department of Education Regional Office IX, Zamboanga Peninsula
Zest for Progress
Zeal of Partnership
7
4th QUARTER β Module 7: MEASURE OF VARIABILITY
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
1
Mathematics β Grade 7 Alternative Delivery Mode Quarter 4 - Module 7: Measure of Variability First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of
the Government of the Philippines. However, prior approval of the government agency or
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Every effort has been exerted to locate and seek permission to use these materials from their
respective copyright owners. The publisher and authors do not represent nor claim ownership
over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Printed in the Philippines
Department of Education β Region IX, Zamboanga Peninsula
Office Address: Tiguma, Airport Road, Pagadian City
Telefax: (062) β 215 β 3751; 991 β 5975
E-mail Address: [email protected]
Development Team of the Module
Writer: Asuncion B. Cruz
Editors: Anthony S. Ramillano
Francis Michael E. Francisco
Layout Artist: Abdurauf J. Baldomero
Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.
Principal Salvador C. Bucoy, LMD
Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI
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.
2
Introductory Message
This Self β 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
teacherβs 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:
β« illustrate the measures of variability (range, average deviation, variance, standard
deviation) of statistical data (M7SP-IVh-1); and
β« 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
Whatβs 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.
Whatβs 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 (πΉ) is the difference between the highest and the lowest values in a given
set of data.
a. FORMULA: πΉ = π―ππππππ π½ππππ β π³πππππ π½ππππ = π― β π³
b. EXAMPLE: Find the range of the given set of data {2, 4, 6. 8, 10}.
SOLUTION: πΉ = π» β πΏ = 10 β 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: π΄. π«. =β|πβοΏ½Μ οΏ½|
π΅
where, π΄. π«. is the Mean Deviation
π is the individual scores
π is the mean
π΅ is the number of scores
|π β π| 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 οΏ½Μ οΏ½.
οΏ½Μ οΏ½ =2 + 4 + 6 + 8 + 10
5=
30
5= 6
Step 2 and 3:
SCORES (π) π β π |π β π|
2 2β6 = β4 4
4 4β6 = β2 2
6 6β6 = 0 0
8 8β6 = 2 2
10 10β6 = 4 4
π΅ = 5 β|π β π| = ππ
π΄. π«. =β|π β π|
π΅=
π + π + π + π + π
π=
ππ
π= π. π
3. VARIANCE
Variance (π2) is the square of the mean deviation divided by the total number of
scores (π).
a. FORMULA: π2 =β(πβοΏ½Μ οΏ½)2
π
b. EXAMPLE: Find the variance of the given set of data {2, 4, 6, 8, 10}.
SOLUTION:
SCORES (π) π β π (π β π)2
2 β4 (βπ)π = 16
4 β2 (βπ)π = 4
6 0 (π)π = 0
8 2 (π)π = 4
10 4 (π)π = 16
π΅ = 5 β(π β π)2 = ππ
ππ =β(π β π)π
π΅=
ππ + π + π + π + ππ
π=
ππ
π= π
4. STANDARD DEVIATION
Standard Deviation (π) is simply the square root of the variance.
a. FORMULA: π = βππ = ββ(πβοΏ½Μ οΏ½)2
π
b. EXAMPLE: Find the standard deviation of the given set of data {2, 4, 6, 8, 10}.
SOLUTION:
π = βπ β π. ππ
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 (π₯) of each class interval.
3. Multiply frequency (π) and midpoint (π₯) of each class interval to get ππ₯.
4. Add ππ₯ of each interval to get β ππ₯.
5. Compute the mean using the formula οΏ½Μ οΏ½ =β ππ₯
β π
6. Calculate the deviation (π β π) by subtracting the mean from each midpoint.
7. Square the deviation of each interval to get (π β π)π.
8. Multiply frequency ( π ) and (π β π)π . Find the sum of each product to get
β π(π β π)π.
9. Calculate the variance (v) using the formula
ππ =β π(π β π)π
β π β π
Calculate the range, variance and standard deviation of scores the First Periodical Test
of Grade 7 β Justice in Mathematics.
πΉ = πππππ πππππ πππππ πππ ππ
πππ πππππππ ππππππππ β
πππππ πππππ πππππ πππ ππ πππ ππππππ ππππππππ
RANGE (πΉ)
ππ =β π(π β οΏ½Μ οΏ½)π
β π β π
Where, π = class frequency π₯ = class mark / midpoint οΏ½Μ οΏ½ = class mean β π = total number of frequency
VARIANCE (ππ)
π = ΰΆ₯ππ = ΰΆ¨β π(π β οΏ½Μ οΏ½)π
β π β π
Where, π = variance
STANDARD DEVIATION (π)
EXAMPLE
7
1. Range
πΉ = 25.5 β 0.5 = 25
2. Mean
π =β ππ
β π=
πππ
ππ= ππ
3. Variance
ππ =β π(πβοΏ½Μ οΏ½)π
β πβπ=
ππππ
ππβπ=
ππππ
ππβ ππ. ππ
4. Standard Deviation
π = ΰΆ₯ππ = ΰΆ¨ππππ
ππβ π. ππ
Whatβs More
Directions: Complete the table below then calculate the different measures of variability
(range, mean deviation, variance, and standard deviation). Use οΏ½Μ οΏ½ = 92.
SCORES (π) π β π (π β π)2
90
91
92
92
93
94
Range
Mean
Deviation
ππ
π
Scores Frequency π Class Mark
(π) ππ (π β π) (π β π)π π(π β π)π
21 - 25 3 23 69 10 100 300
16 β 20 8 18 144 5 25 200
11 β 15 10 13 130 0 0 0
6 β 10 4 8 32 -5 25 100
1 - 5 5 3 15 -10 100 500
β π = ππ β π π = πππ β π(π β οΏ½Μ οΏ½)π = ππππ
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 β π π?
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 π Class Mark (π) ππ (π β π) (π β π)π π(π β π)π
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 β 28 5 24 120
β π = β π π = β π(π β οΏ½Μ οΏ½)π =
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 {π, π, π, ππ, ππ, ππ }.
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
Whatβs New:
1. π₯1ΰ΄€ΰ΄€ΰ΄€=106.25 π₯2 ΰ΄€ΰ΄€ΰ΄€=112.5 2. π π΄=17 π π΅=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
Whatβs 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.π₯1 ΰ΄€ΰ΄€ΰ΄€=0.990 π₯2 ΰ΄€ΰ΄€ΰ΄€=0.998
c.The spread is closer/almost the same
d.Both e.Sample 1
Whatβs More:
SCORES (π) πβπ (πβπ)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
ππ 1.67
π 1.29
What I Have Learned:
Measures of Variability
Ungrouped Data
Grouped Data
1.Range π―βπ³ Upper boundary of the highest interval
- Lower boundary of the
lowest interval
2.Variance ππ
=(πβπ)π
π΅ π
π=
βπ(πβπ)π
βπβπ
3.Standard Deviation π=ΰΆ¨(πβπ)π
π΅ π=ΰΆ¨
βπ(πβπ)π
βπβπ
What I Can Do:
Ages Frequency π Class Mark (π) ππ (πβπ) (πβπ)π π(πβπ)π
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 β 28 5 24 120 27 729 3645
βπ=ππ βππ=ππππ βπ(πβπ)π=ππππ
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 β inheritor of a glorious past, hostage to the
uncertain future. As such, I must prove equal to a two-fold
task β 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 β 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 β hope in the free abundance of the new
land that was to be their home and their childrenβs 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 β 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 β 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
β 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 β 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 β 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:
βI am a Filipino born to freedom, and I shall not rest until
freedom shall have been added unto my inheritanceβfor
myself and my children and my childrenβs childrenβ
forever.β