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Mathematics 7 Quarter 4 Self-Learning Module 10 Measures of Central Tendency Module 10 of Ungrouped Data Mathematics – Grade 7 Quarter 4 – Module 10: Measures of Central Tendency of Ungrouped Data 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 royalties. 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 – Schools Division of Pasig City Development Team of the Self Learning Module Writers: Ma. Flor A. Pornela Editors: Ma. Victoria L. Peñalosa Reviewers: Ma. Cynthia P. Badana Illustrator: Name Layout Artist: Name Management Team: Ma. Evalou Concepcion A. Agustin OIC – Schools Division Superintendent Carolina T. Rivera, CESE OIC-Assistant Schools Division Superintendent Manuel A. Laguerta, EdD Chief, Curriculum Implementation Division Victor M. Javeña EdD Chief, School Governance and Operations Division Education Program Supervisors Librada L. Agon EdD (EPP/TLE/TVL/TVE) Liza A. Alvarez (Science/STEM/SSP) Bernard R. Balitao (AP/HUMSS) Joselito E. Calios (English/SPFL/GAS) Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports) Wilma Q. Del Rosario (LRMS/ADM) Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larangan) Perlita M. Ignacio PhD (EsP) Dulce O. Santos PhD (Kindergarten/MTB-MLE) Teresita P. Tagulao EdD (Mathematics/ABM) Printed in the Philippines by Department of Education – Schools Division of Pasig City Mathematics Quarter 4 Module 10 Measures of Central Tendency of Ungrouped Data 7 Introductory Message For the Facilitator: Welcome to the Mathematics Grade 7 Self – Learning Module on Measures of Central Tendency of Ungrouped data! This Self-learning module was collaboratively designed, developed and reviewed by educators both from the Schools Division Office of Pasig City headed by its Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A. Agustin, in partnership with the City Government of Pasig through its mayor, Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K to 12 Curriculum using the Most Essential Learning Competencies (MELC) in developing this instructional resource. This learning material hopes to engage the learners into guided and independent learning activities at their own pace and time. Further, this also aims to help learners acquire the needed 21st century skills especially the 5 Cs, namely: Communication, Collaboration, Creativity, Critical Thinking and Character while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module: Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners. As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Moreover, you are expected to encourage and assist the learners as they do the tasks included in the module. For the Learner: Welcome to the Mathematics Grade Level 7 Self – Leaning Module on Measures of central tendency of Ungrouped Data! This self-learning module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This self-learning module has the following parts and corresponding icons: Expectations - These are what you will be able to know after completing the lessons in the module Pretest - This will measure your prior knowledge and the concepts to be mastered throughout the lesson. Recap - This section will measure what learnings and skills that you understand from the previous lesson. Lesson- This section will discuss the topic for this module. Activities - This is a set of activities you will perform. Wrap Up- This section summarizes the concepts and applications of the lessons. Valuing-this part will check the integration of values in the learning competency. Posttest - This will measure how much you have learned from the entire module. EXPECTATION 1. To find the mean, median and mode of ungrouped data (M7SPIVf-g-1). PRETEST Directions: Read each item carefully. Choose the letter of the best answer. 1. What is the most stable and useful measure of central tendency? A. mean B. median C. mode D. range 2. When the data has extreme scores, what is the best measure of average? A. mean B. median C. mode D. range For numbers 3 – 5, The list gives the ages of students enrolled in swimming classes. 4 3 4 5 6 7 8 30 5 9 11 5 7 8 6 7 3. What is the median age of the students enrolled in swimming classes? A. 6.5 B. 7 C. 7.5 D. 8 4. Which of the following describes the distribution? A. unimodal B. bimodal C. trimodal D. no mode 5. What is the mean age of the students? A. 6.21 B. 7.34 C. 7.81 D. 8 RECAP A. Find at least 3 words associated with measures of central tendency. Words are arranged horizontally, vertically or diagonally. M E D I A N E F I S D S E W R R D I E A Y Q T E D O H D A T A Q L C A G O P L U E D S I S M P E A S R E V D U N W N F A N B I T B. Identify the measure of central tendency that is most appropriate to determine the following. 1. 2. 3. 4. The The The The leading presidential candidate. average annual salary of employees in a company. average grade in mathematics of the students in class. average rainfall for the month of August. LESSON A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode. Mean The mean is the most popular among the measures of central tendency for it is widely used. It is commonly called the average of a set of n numbers and it is the sum of all numbers divided by n. The formulas for the computation of the mean are as follows: Formula 𝑥̅ = ∑𝑥 𝑁 where ∑𝑥 = the summation of x (sum of the measures) N = number of values of x. Example 1 : The number of hours each student spends in studying and doing school projects each day is shown in the table. Find the mean amount of time the students spend in studying and doing school projects. Kristel 4 Roger 3 Roland 3 Liza 4 April 5 Gabby 4 Kim 3 Solution: 𝑥̅ = ∑𝑥 𝑁 4+3+3+4+5+4+3 7 26 𝑥̅ = 7 𝑥̅ = 3.7 Thus, the mean or average time the students spend in studying and doing school projects each day is 3.7 hours. 𝑥̅ = Example 2 : The set of data shows the age of 10 teachers of Pasig City Science High School. Calculate the mean. 39 24 25 26 32 25 23 50 26 27 Solution: 𝑥̅ = ∑𝑥 𝑁 39 + 24 + 25 + 26 + 32 + 25 + 23 + 50 + 26 + 27 10 297 𝑥̅ = 10 𝑥̅ = 29. 70 Thus, the mean or the average of the ages is 29.70 or 30 𝑥̅ = Weighted Mean 𝑥̅ = ∑𝑓𝑋 𝑁 where f = frequency X = score ∑𝑓𝑋 = sum of the product of frequency and score N = total frequency To find the mean of a set of data in a frequency table, 1. Multiply each “X” in the set of data by “f” to obtain “fX” 2. Find the sum of “fX” 3. Find the sum of f, (N) 4. The mean, 𝑥̅ = ∑𝑓𝑋 𝑁 Example 3 : Find the mean savings per student, given the frequency distribution table. Weekly Savings (X) Number of Students (f) Weekly Savings Number of Students (fX) 40 1 40 50 4 200 60 1 60 70 3 210 80 1 80 N = 10 ∑𝑓𝑋 = 590 Solution: 𝑥̅ = ∑𝑓𝑋 𝑁 590 𝑥̅ = 10 𝑥̅ = 59 Therefore, the mean weekly savings per student is Php 59.00. Median The median is the middle score for a set of data that has been arranged in order of magnitude (list after the scores are arranged in decreasing or 𝑁+1 increasing order). It is the value of the ( 2 )th item or position. Example 1 : The following are the number of students per section of Grade 7 level of Rizal High School in Pasig City. Find the median. 55, 60, 48, 50, 52, 50 and 51. Solution: Arrange the numbers in ascending order 48, 50, 50, 51, 52, 55, 60 N=7 𝑁 + 1 th ) 2 7 + 1 th 𝑥̃ = ( 𝑥̃ = ( 2 ) score score 8 𝑥̃ = (2)th score 𝑥̃ = 4th score 𝑥̃ = 51 Since 7 is an odd number, simply get the middle score, so the median is 51. Example 2 : The following set of data shows the height in centimeter of 10 students: 160, 154, 156, 160, 170, 159, 155, 145, 160 and 167. Solution: Arrange the numbers in ascending order 145, 154, 155, 156, 159, 160, 160, 160, 167, 170 N = 10 𝑁 + 1 th ) 2 10 + 1 th 𝑥̃ = ( 𝑥̃ = ( 2 11 th 𝑥̃ = ( 2 ) ) score score score 𝑥̃ = 5.5th score (the mean of the 5th and 6th scores) Since the number of measures is even, then the median is the mean of the two middle scores. 𝑥̃ = 159 + 160 2 𝑥̃ = 159.50 Hence, the median height of in centimeters of students is 159.50 Mode The mode is the measure or value which occurs most frequently in a set of data. It is the value with the greatest frequency. The word modal is often used when referring to the mode of a data set. If a data set has only one value that occurs most often, the set is called unimodal. A data set that has two values that occur with the same greatest frequency is referred to as bimodal. When a set of data has more than two values that occur with the same greatest frequency, the set is called multimodal. To find the mode for a set of data: 1. Select the measure that appears most often in the set. 2. If two or more measures appear the same number of times, then each of these values is a mode. 3. If every measure appears the same number of times, then the set of data has no mode. Example 1 : The sizes of 9 classes in a certain school are 58, 52, 58, 65, 68, 60, 57,64 and 60. Solution: 𝑥̂ = 58 and 60 (Bimodal) Example 2 : The following data shows the height in centimeter of 10 students: 160, 154, 156, 160, 170, 159, 155, 145, 160 and 167. Solution: 𝑥̂ = 160 (Unimodal) ACTIVITIES ACTIVITY 1: PRACTICE! Direction: Calculate the mean of ungrouped data. 1. 10, 12, 15, 25 2. 16, 19, 20, 30, 10 3. 21, 29, 27, 33, 28 4. 28, 30, 32, 45, 40, 50 5. 65, 45, 75, 88, 45, 91 ACTIVITY 2: KEEP ON PRACTICING Direction: Complete the table by calculating the median and mode. Median 1. 2. 3. 4. 5. Mode 3, 5, 2, 1, 2 5, 5, 2, 4, 6 8, 1, 4, 7, 7, 4 7, 6, 6, 8, 9, 10 11, 8, 9, 10, 10, 8, 9 ACTIVITY 3: TEST YOURSELF Direction: Calculate the mean, median and mode of the following. 1. Shoe sizes of selected students: 5, 7, 6, 6.5, 8, 5, 5.5, 7, 7.5 2. Tickets sold in a week: 45, 35, 56, 44, 63, 55, 60 3. Expenses of Maria in 10 days: 100, 90, 125, 140, 130, 100, 125, 125, 100, 90 4. Weight of a grade 7 students in pounds: 75.2, 81.5, 70.4, 88.1, 90.7, 85.9 5. Height of basketball students: 156, 125, 136, 155, 169, 140, 145, 170, 181 WRAP UP Remember… Mean is the most popular among the measures of central tendency for it is widely used. It is often called average. The formulas for the computation of the mean are as follows. Formula 𝑥̅ = ∑𝑥 Weighted Mean 𝑥̅ = 𝑁 ∑𝑓𝑋 𝑁 Median is the middle score for a set of data that has been arranged in order of magnitude (list after the scores are arranged in decreasing or increasing order). Mode is the number that occurs most frequently. If a data set has only one value that occurs most often, the set is called unimodal. A data set that has two values that occur with the same greatest frequency is referred to as bimodal. When a set of data has more than two values that occur with the same greatest frequency, the set is called multimodal. To find the mode for a set of data: 1. Select the measure that appears most often in the set. 2. If two or more measures appear the same number of times, then each of these values is a mode. 3. If every measure appears the same number of times, then the set of data has no mode. VALUING Reflections: (Journal Writing) As a student, why do you think your knowledge in finding the mean, median and mode is important? Identify a real–life situation/s wherein you can relate your knowledge about these measures of central tendency. Share your thoughts and insights in 3-5 sentences and write these in your notebook. POSTTEST A. Directions: Read each item carefully. Choose the letter of the best answer. 1. What is the measure of central tendency that divides an arranged (ascending or descending) distribution into two equal parts? A. mean B. Median C. Mode D. Range 2. Which set of data has a mean of 15, a median of 14, and a mode of 14? A. 3, 14, 14, 19, 25 B. 3, 7, 14, 15, 25 C. 4, 14, 14, 15, 22 D. 9, 14, 15, 15, 22 3. The mean score of a set of 40 scores is 81. Find the sum of the 40 test scores. A. 2, 250 B. 2, 280 C. 2, 820 D. 3, 240 B. Direction: Calculate for the mean, median and mode. 1. Mang Jose is a fish vendor. The following are his sales for a week. Sunday: Php 7, 800 Monday: Php 3, 200 Tuesday: Php 4, 500 Wednesday: Php 3, 900 Thursday: Php 5, 100 Friday: Php 6,300 Saturday: Php 8,200 RECAP A. Middle, Median, Data, Frequent B. 1. Mode 2. Mean 3. Mean 4. Mean PRETEST 1. 2. 3. 4. 5. A C A B C POSTTEST A. 1. B 2. A 3. D ACTIVITY 1 1. 15.5 2. 19 3. 27.6 4. 37.5 5. 68.17 ACTIVITY 2 Median 2 5 5.5 7.5 9 1. 2. 3. 4. 5. Mode 2 5 4 and 7 6 8, 9 and 10 ACTIVITY 3 Mean Median 6.5 6.39 51.14 55 112. 5 112.5 81.97 83.7 153 155 1. 2. 3. 4. 5. Mode 5, 7 no mode 100, 125 no mode no mode B. Mean – 5, 571. 43 Median – 5, 100 Mode – no mode KEY TO CORRECTION REFERENCES Oronce, Orlando, and Marilen Mendoza. E-MATH 7. Manila: Rex Book Store, Inc., 2015. Nivera, Gladys. Grade 7 Mathematics Patterns and Practicalities. Makati City: Salesiana Books by Don Bosco Inc., 2014 Mirabona, Isaac, and Custodio, Sergio. Interactive Mathematics Grade 7. Manila: Innovative Educational Materials Inc., 2013 http://www.riosalado.edu/web/oer/WRKDEV100 20011_INTER_0000_v1/lessons/Mod05_MeanMedianMode.shtml (accessed July 18, 2020) http://web.simmons.edu/~benoit/lis642/Hafner-Chapter5.pdf (accessed July 18, 2020) http://www.cimt.plymouth.ac.uk/mepjamaica/unit17/StudentText.pdf (accessed July 18, 2020)

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