7 MEASURE OF CENTRAL TENDENCY - 4th QUARTER - Module 6: Department of Education - ZNNHS
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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 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 office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalty. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. 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 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 2
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. 3
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 4
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 5
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. 6
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. 7
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 8
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
10 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
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