7 MEASURE OF CENTRAL TENDENCY - 4th QUARTER - Module 6: Department of Education - ZNNHS

Republic of the Philippines
 Department of Education
 Regional Office IX, Zamboanga Peninsula

 7 Zest for Progress
 Z Peal of artnership

 4th QUARTER – Module 6:
 MEASURE OF CENTRAL TENDENCY

Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________

Mathematics – Grade 7
Alternative Delivery Mode
Quarter 4 - Module 6: Measures of Central Tendency
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

 Development Team of the Module
 Writer: Abdurauf J. Baldomero
 Editors: Divine T. De Guzman
 Haifa D. Ternida
 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.

Printed in the Philippines
Department of Education – Region IX, Zamboanga Peninsula
Office Address: Tiguma, Airport Road, Pagadian City
Telefax:
E-mail Address:

 1

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
discussion 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 need to 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. And read the instruction
carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answer the tasks in this module,
do not hesitate to consult your teacher or facilitator.
Thank you.

 What I Need to Know
 This module was written as an aid in the basic statistics lesson of the fourth quarter of
Grade 7-Mathematics. The module follows a step – by – step approach to computational
statistics supported by examples and exercises. It covers the key concepts of measures of
central tendency for grouped and ungrouped data.
 After going through the module, the learner is expected to:
 ⚫ illustrate the measures of central tendency (mean, median, and mode) of a
 statistical data. (M7SP–IVf–1)
 ⚫ calculate the measures of central tendency of ungrouped and grouped data.
 (M7SP– IVf–g–1)

 What I Know
Directions: Read and understand each statement carefully. Write the letter of the correct
answer on a separate sheet.
1. Which of the following is NOT a measure of central tendency?
 A. Mean B. Median C. Mode D. Range
2. What is the median of 9, 5, and 7?
 A. 6 B. 7 C. 8 D. 11

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3. Which of the measures of central tendency will you use if you want to know the common
 age of grade 7 students?
 A. Mean B. Median C. Mode D. Range
4. What is the mean of the grouped data below?
 CLASS INTERVAL ∙ 
 51 – 55 3 53 159 A. 19
 56 – 60 8 58 464 B. 25
 61 – 65 10 63 630 C. 61
 66 – 70 4 68 272 D. 73
 25 1525
5. Using the grouped data given in item number 4, which is the modal class?
 A. 51 – 55 B. 56 – 60 C. 61 – 65 D. 66 – 70

 LESSON MEASURES OF CENTRAL
 TENDENCY FOR UNGROUPED
 1
 DATA

 What’s In
 ACTIVITY GUESS WHAT
Directions: Organize the data below in the frequency distribution table given and answer the
question that follows.
 The data show the age of 15 grade 7 students at Don Pablo Lorenzo Memorial High School
 11 12 12 11 12 13 12 12 12 12 12 12 14 12 11
 AGE TALLY FREQUENCY QUESTION:
 11 III 3 If Mario is an incoming grade 7 student
 12 and belongs to the most common age, what
 13 could be his age?
 14
 TOTAL
 Organizing data through tables and graphs can help in describing the situations, drawing
conclusions, and even make inferences about events. But summarizing the data can help in
predicting outcomes in the future and make sound decisions and judgments.

 What’s New
 ACTIVITY WHO IS WHO
Directions: Compare the two set of data below.
 The data show the Mathematics grades of sections A and B.
 Section A 82 83 84 84 84 85 85 85
 Section B 70 70 80 81 85 90 93 95

QUESTION: Which of the two sections performed better? Justify your answer.

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What is It
 A Data
 Data in statistics are a collection of facts, such as numbers, words, measurements,
observation or just description of things.
 Data can be classified into two forms namely: grouped and ungrouped data.
 Data
 Ungrouped Grouped
 Classification
 Ungrouped or raw data are Grouped data are data that are sorted or
 data that are not yet sorted, grouped in different classes called class
 Definition classified, or grouped. They intervals.
 are data that are first gathered
 in an experiment or study.
 21 23 19 17 12
 CLASS INTERVAL TALLY 
 15 15 17 17 19
 Example 1 – 10 0
 23 23 21 23 25
 Marks 11 – 20 |||| - |||| - | 11
 25 21 19 19 19
 obtained by 20 21 - 30 |||| - |||| 9
 students in a
 These data are ungrouped The data are grouped using a frequency
 Mathematics
 since they are just gathered distribution table to serve as a
 Examination
 data without classification or convenient way to present the data since
 even arrangement. it is easy to read and understand.

 B Measures of Central Location or Tendency
 Measures of Central Location or Tendency is one way of summarizing data by
representing a data set using a single value. It is commonly done by finding the central value
of a data set or the typical value. The most common measures of central tendency are the
mean, median, and mode.
 Measure of
 Central MEAN MEDIAN MODE
 tendency
 Definition The Mean is commonly Median value of a The Mode or the modal
 known as the
 data set is simply value of a data is the
 “average” in your the middle value most frequent value or
 grade school
 of the set arranged the value that appears
 mathematics. It made in ascending or most often. A set with
 use of all the data to descending order. two modes is called
 describe the set of data.
 In case there are bimodal set. If no data
 two middle values, value or values occur
 add the two values most frequently, the data
 and divide it by 2. have no mode.
 Illustrative STEP 1: Add all the STEP 1: Arrange Find the value that
 Example 1 data. the set in appears most often or the
 increasing order value that has the highest
 Data Set: 9 + 7 + 4 + 7 + 3 = 30 (also called Data number of appearances.
 9, 7, 4, 7, 3 Array).
 9, , 4, , 3

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Number of STEP 2: Divide the sum 3, 4, , 7, 9
 Data: 5 by the number of data 7 appeared twice while
 to get the mean. STEP 2: Get the the rest of the data
 middle value. appeared once
 = ̅= = 
 Middle value: 7 = ̂= 
 = ̃= 
 Illustrative Sum of data: Data Array: Note:
 Example 2 80 + 92 + 83 + 80 + 83 + 81 80, 80, , , 83, 92 80 and 83 appeared most
 = often
 Data Set: Note: There are 2
 80, 92, 83, 80, 83, 81 middle values. ̂ = { , }
 
 ̅=
 = . 
 
 Number of + 
 ̃=
 =
 Data: 6 
 ̃ = 
 
 What’s More
Directions: Find the measures of central tendency (mean, median, and mode) of the
ungrouped data below.
 The data show the age of fifteen
 randomly selected grade 7 MEAN MODE
 students at Don Pablo Lorenzo Total Number of Data
 Memorial High School. ̅)
 MEAN ( 
 11 12 12 13 12 MEDIAN
 16 13 13 18 13 DATA ARRAY
 11 13 12 14 12 ̃)
 MEDIAN ( 

 LESSON MEASURE OF CENTRAL
 2
 TENDENCY FOR GROUPED DATA

 What is It

 Mean of Grouped Data

 To compute for the mean of grouped data, first, we need to extend the basic column of
frequency distribution table by adding two additional columns. For this example, we will
remove the tally column.
 CLASS INTERVAL FREQUENCY ( ) CLASS MIDPOINT ( ) ∙ 
 16 – 20 6

 a. CLASS MIDPOINT ( )
 In computing for the mean, we will use class midpoint instead of the entire class interval
 to represent each class interval by a single number. We can compute for the class midpoint

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by getting the average of the lower limit and the upper limit of the class interval or
 simply adding the lower and upper limit divided by 2.
 = ( + ) ÷ 
 b. FREQUENCY × CLASS MIDPOINT ( ∙ )
 To solve this column, we need to multiply the frequency and the class midpoint.
 ( ) ∙ ( )
 c. TABLE COMPLETION
 CLASS
 ∙ 
 INTERVAL
 16 – 20 6 ( + ) ÷ = ÷ = ∙ = 
 21 – 25 14 ( + ) ÷ = ÷ = ∙ = 
 26 – 30 8 ( + ) ÷ = ÷ = ∙ = 
 31 – 35 2 ( + ) ÷ = ÷ = ∙ = 

 d. APPLYING THE FORMULA
 After extending the frequency distribution table, we can now compute for the mean.
 FORMULA SYMBOL DESCRIPTION EXAMPLE

 ∑( ∙ ) ∑( ∙ ) Sum of 4th column ( ∙ ) 720
 ̃ =
 ∑ Sum of 2nd column
 ∑ or 30
 
 To solve for the mean, we just need to substitute the necessary value from the table
 to the formula.
 ∑( ∙ ) 108 + 322 + 224 + 66 720
 ̃ = = = = 24
 ∑ 30 30
 Therefore, the mean of the grouped data above is 24.

 Median of Grouped Data

 Similar to the mean, before we can solve for the median, we need to extend the basic
columns of the frequency distribution table by one column called cumulative frequency.
 CLASS INTERVAL FREQUENCY ( ) CUMULATIVE FREQUENCY ( )
 16 – 20 6

 a. CUMULATIVE FREQUENCY ( )
 Cumulative frequency is the continuous addition of the frequencies in a frequency
 distribution table. To solve for the cumulative frequency, we need to get the first frequency
 and add the frequency of the next row to get the cumulative frequency of the next row.
 ROW NO. CLASS INTERVAL SOLUTION 
 1 16 – 20 6 Start by getting the first frequency 6 6
 To get the next , add the next frequency 14
 2 21 – 25 14 to the previous 6 20
 6 + 14 = 20
 3 26 – 30 8 20 + 8 = 28 28
 4 31 – 35 2 28 + 2 = 30 30
 We need to make sure that the last is equal to the sum of the frequencies ∑ . In
 this case, ∑ = + + + = which is equal to the last 30.

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b. MEDIAN CLASS
 Before we can compute for the median, first, we need to identify which among the rows
 ∑ 
 is the median class. To find the median class, we need to compute for the value of and
 
 look for the row which has a cumulative frequency ( ) equal or greater than the value of
 ∑ 
 .
 
 ∑ 
 In the example above, our ∑ is 30. So, our is 15. Looking at the , 15 belongs to
 
 row number 2 with a of 20 since it is the first row with greater than 15. Therefore,
 our median class is row number 2.
 c. APPLYING THE FORMULA
 After extending the frequency distribution table FORMULA
 and finding the median class, we can now compute
 ∑ 
 for the median by identifying the value of each − 
 ̃ = + ( 
 ) 
 
 symbol in the formula.
 SYMBOL DESCRIPTION EXAMPLE
 LOWER BOUNDARY of the
 median class. (Subtract 0.5 to The lower limit of the median class is 21. So,
 
 the lower limit of the median = − . = . 
 class)
 ∑ Sum of 2nd column (frequency) ∑ 
 divided by 2 = = 
 
 CUMULATIVE FREQUENCY
 We need the of row number 1 which is 6
 BEFORE the median class
 FREQUENCY of the median We need to find the of row number 2 which
 
 class is 14
 Using the class interval of row number 2 which
 CLASS WIDTH
 21-25, we can count that the class width is 5
 To solve for the median, we just need to substitute the necessary value from the table
 to the formula given above.
 ∑ 
 − − 
 ̃ = + ( ) = . + ( ) = . + ( ) = . + ≈ . 
 
 Therefore, the median of the grouped data above is approximately 23.71.

 Mode of Grouped Data

 Unlike the mean and median of grouped data, we don’t need to extend the basic columns
of a frequency distribution table. We only need the class interval and frequency.
 a. MODAL CLASS
 Before we can compute for the mode, we need to identify which among the rows is the
 modal class or the row with the highest frequency.
 CLASS INTERVAL FREQUENCY ( )
 16 – 20 6
 21 – 25 14 Modal Class
 26 – 30 8
 31 – 35 2
 b. APPLYING THE FORMULA
 FORMULA
 To compute for the mode of a grouped data, we
 
 need to identify the value of each symbol in the ̂ = + (
 ) 
 + 
 formula.

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SYMBOL DESCRIPTION EXAMPLE
 The lower limit of the
 LOWER BOUNDARY of the modal class modal class is 21. So,
 = − . = . 
 Difference in the frequency of the modal class ( ) and = 14 and = 6
 the frequency of the class interval before it ( ) 1 = − 
 NOTE: If there is no row before the modal class, is zero. = − = 
 Difference in the frequency of the modal class ( ) and = 14 and = 8
 the frequency of the class interval after it ( ) 2 = − 
 NOTE: If there is no row after the modal class, is zero. = − = 
 CLASS WIDTH The class width is 5
 To solve for the mode, we just need to substitute the necessary value from the table
 to the formula given above.
 
 ̃ = + (
 ) = . + ( ) = . + ( ) = . + ≈ . 
 + + 
 Therefore, the mode of the grouped data above is approximately 23.36.

 What’s More
Directions: Find the mean, median, and mode of the grouped data below. Show your solution.

 ROW NO. CLASS INTERVAL ∙ 
 1 90 - 98 6
 2 99 -107 22
 3 108 - 116 43
 4 117 - 125 28
 5 126 - 134 9

 What I Have Learned
 ACTIVITY FILL ME UP
Directions: Differentiate the process in solving for the different measures of central tendency
of ungrouped and grouped data.

 Measure of Central
 Ungrouped Data Grouped Data
 Tendency
 Mean
 Median
 Mode

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What I Can Do
 ACTIVITY WHICH IS WHICH
Directions: Identify which measure of central tendency (mean, median, or mode) is applicable
for each given situation.
1. Favorite subject of students 2. General average of a student in a quarter
3. Most common age of grade 7 students 4. Middle salary of a country or city
5. Average income of a country

 Assessment
Directions: Read and understand each statement carefully.
1. Which of the following is NOT a measure of central tendency?
 A. Mode B. Outlier C. Median D. Mean
2. What measure of central tendency can be applied to get the average salary of government
 employees?
 A. Mode B. Outlier C. Median D. Mean
For items 3 to 5, refer to the data below showing the age of 9 newly hired public school teachers
 21, 25, 19, 19, 24, 21, 19, 25, 25
3. What is the mean score of the data set?
 A. 21 B. 22 C. 25 D. 27
4. What is the median score of the data set?
 A. 21 B. 22 C. 25 D. 27
5. Which of the following is a modal score of the data set?
 A. 21 B. 22 C. 25 D. 27
For items 6 to 9, refer to the grouped data below showing the scores of students in a
20-item quiz.
 CLASS INTERVAL ∙ 
 1–5 10
 6 – 10 12
 11 – 15 6
 16 – 20 2
6. How many students were surveyed?
 A. 10 B. 12 C. 22 D. 30
7. What is the mean of the grouped data above?
 A. 8 B. 10 C. 30 D. 240
8. In which row will the median class fall?
 A. 1 - 5 B. 6 - 10 C. 11 - 15 D. 16 - 20
9. What is the frequency of the modal class?
 A. 2 B. 6 C. 10 D. 12
10. What does the measure of central tendency tell us?
 A. The central value of the data set C. The collection of information
 B. The position of a value relative to other D. The spread of the data set
 value

 9

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data/central-measures.html.
"Finding a Central Value", Math Is Fun, Accessed June 2, 2020, http://www.mathsisfun.com/
com/data/data.html.
"What is Data?," Math Is Fun, accessed June 1, 2020, http://www.mathsisfun.
 References:
 Assessment: What I Can Do:
 1. b 2. d 3. b 4. a 5. c 1. Mode 2. Mean 3. Mode
 6. d 7. a. 8. b 9. d 10. a 4. Median 5. Mean
 What’s More: Lesson 2
 ROW NO. CLASS INTERVAL ∙ 
 1 90 - 98 6 94 564 6
 2 99 -107 22 103 2, 266 28
 3 108 - 116 43 112 4, 816 71
 4 117 - 125 28 121 3, 388 99
 5 126 - 134 9 130 1, 170 108
 108 12, 204
 Mean Median (The Median Class is row number 3)
 − 
 
 ̅= = 
 ̃ = . + ( ) ≈ . 
 
 Mode (The Modal Class is row number 3)
 
 ̃ = . + ( ) = . 
 + 
 What’s More: Lesson 1 What I Have Learned:
 Total: 195 Number of Data: 15 Answer will vary
 Mean ( ̅): 13 Mode: 12 and 13
 Data Array:
 11,11,12,12,12,12,12,13,13,13,13,13,14,16,18
 ̃): 13
 Median ( What’s New:
 Section A performed better since its
 average (84) is greater than the
 What’s In:
 average of section B (83).
 Age Mario could be 12
 11 3 years old since it is
 12 10
 13 1 the common age of What I Know:
 14 1 grade 7 students 1. d 2. b 3. c
 TOTAL 15 4. c 5. c
 Answer Key

I AM A FILIPINO
 by Carlos P. Romulo
I am a Filipino – inheritor of a glorious past, hostage to the It is the mark of my manhood, the symbol of my dignity as
uncertain future. As such, I must prove equal to a two-fold a human being. Like the seeds that were once buried in the
task – the task of meeting my responsibility to the past, and tomb of Tutankhamen many thousands of years ago, it shall
the task of performing my obligation to the future. grow and flower and bear fruit again. It is the insigne of my
I am sprung from a hardy race – child many generations race, and my generation is but a stage in the unending
removed of ancient Malayan pioneers. Across the centuries, search of my people for freedom and happiness.
the memory comes rushing back to me: of brown-skinned I am a Filipino, child of the marriage of the East and the
men putting out to sea in ships that were as frail as their hearts West. The East, with its languor and mysticism, its passivity
were stout. Over the sea I see them come, borne upon the and endurance, was my mother, and my sire was the West
billowing wave and the whistling wind, carried upon the that came thundering across the seas with the Cross and
mighty swell of hope – hope in the free abundance of the new Sword and the Machine. I am of the East, an eager
land that was to be their home and their children’s forever. participant in its struggles for liberation from the imperialist
This is the land they sought and found. Every inch of shore yoke. But I know also that the East must awake from its
that their eyes first set upon, every hill and mountain that centuried sleep, shake off the lethargy that has bound its
beckoned to them with a green and purple invitation, every limbs, and start moving where destiny awaits.
mile of rolling plain that their view encompassed, every river For I, too, am of the West, and the vigorous peoples of the
and lake that promised a plentiful living and the fruitfulness West have destroyed forever the peace and quiet that once
of commerce, is a hollowed spot to me. were ours. I can no longer live, a being apart from those
By the strength of their hearts and hands, by every right of whose world now trembles to the roar of bomb and cannon
law, human and divine, this land and all the appurtenances shot. For no man and no nation is an island, but a part of the
thereof – the black and fertile soil, the seas and lakes and main, and there is no longer any East and West – only
rivers teeming with fish, the forests with their inexhaustible individuals and nations making those momentous choices
wealth in wild and timber, the mountains with their bowels that are the hinges upon which history revolves. At the
swollen with minerals – the whole of this rich and happy land vanguard of progress in this part of the world I stand – a
has been for centuries without number, the land of my forlorn figure in the eyes of some, but not one defeated and
fathers. This land I received in trust from them, and in trust lost. For through the thick, interlacing branches of habit and
will pass it to my children, and so on until the world is no custom above me I have seen the light of the sun, and I
more. know that it is good. I have seen the light of justice and
I am a Filipino. In my blood runs the immortal seed of heroes equality and freedom, my heart has been lifted by the vision
– seed that flowered down the centuries in deeds of courage of democracy, and I shall not rest until my land and my
and defiance. In my veins yet pulses the same hot blood that people shall have been blessed by these, beyond the power
sent Lapulapu to battle against the alien foe, that drove Diego of any man or nation to subvert or destroy.
Silang and Dagohoy into rebellion against the foreign I am a Filipino, and this is my inheritance. What pledge
oppressor. shall I give that I may prove worthy of my inheritance? I
That seed is immortal. It is the self-same seed that flowered shall give the pledge that has come ringing down the
in the heart of Jose Rizal that morning in Bagumbayan when corridors of the centuries, and it shall be compounded of the
a volley of shots put an end to all that was mortal of him and joyous cries of my Malayan forebears when first they saw
made his spirit deathless forever; the same that flowered in the contours of this land loom before their eyes, of the battle
the hearts of Bonifacio in Balintawak, of Gregorio del Pilar cries that have resounded in every field of combat from
at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in Mactan to Tirad Pass, of the voices of my people when they
flowers of frustration in the sad heart of Emilio Aguinaldo at sing:
Palanan, and yet burst forth royally again in the proud heart “I am a Filipino born to freedom, and I shall not rest until
of Manuel L. Quezon when he stood at last on the threshold freedom shall have been added unto my inheritance—for
of ancient Malacanang Palace, in the symbolic act of myself and my children and my children’s children—
possession and racial vindication. The seed I bear within me forever.”
is an immortal seed.

 11